WIN Special Session: Work from Women in Numbers
Organizers: Katherine Stange, Beth Malmskog "Obstructions to the Hasse princple on Enriques surfaces" Jennifer Berg, Rice University Abstract: In 1970, Manin showed that the Brauer group of a variety can obstruct the
existence of rational points, even when there exist points everywhere locally. Later, Skorobogatov
defined a refinement of this BrauerManin obstruction, called the ´etaleBrauer
obstruction. We show that this refined obstruction is necessary to understand failures of
the Hasse principle on Enriques surfaces. This completes the case of Kodaira dimension 0
surfaces. "A polynomial sieve in a geometric setting" Alina Bucur, University of California, San Diego Abstract: We consider an application of a polynomial sieve to counting points
of bounded height on a cyclic cover of P
n over the rational function field. This is
joint work with A.C. Cojocaru, M. Lal´ın and L. Pierce and it was started at WIN3. "Shadow Lines in the Arithmetic of Elliptic Curves" Mirela Ciperiani, Institute for Advanced Study/ Univ of Texas at Austin Abstract: Let E be an elliptic curve of analytic rank 2 over Q, and p a prime of ordinary
reduction such that the ppart of the TateShafarevich group of E/Q is finite. This
implies that E(Q)⊗Qp ' Q2
p
. Consider imaginary quadratic fields K satisfying the
Heegner hypothesis, such that the corresponding twisted elliptic curve has analytic
rank 1 over Q. Each such field K gives rise to a copy of Qp in E(K)⊗Qp called the
shadow line. We will describe work initiated in WIN3 which allows us to compute
these shadow lines, and verify a conjecture which states that they in fact lie in
E(Q) ⊗ Qp. We will also discuss the use of this work in analyzing the distribution
of shadow lines in E(Q) ⊗ Qp as K varies.
The WIN3 work described in this talk is joint with J. S. Balakrishnan, J. Lang,
B. Mirza, and R. Newton. "Galois action on homology of Fermat curves" Rachel Davis, University of WisconsinMadison Abstract: We prove a result about the Galois module structure of the Fermat curve
using commutative algebra, number theory, and algebraic topology. Specifically,
we extend work of Anderson about the action of the absolute Galois
group of a cyclotomic field on a relative homology group of the Fermat curve.
By finding explicit formulae for this action, we determine the maps between
several Galois cohomology groups which arise in connection with obstructions
for rational points on the generalized Jacobian. Heisenberg extensions
play a key role in the result. This is joint work with R. Pries, V. Stojanoska,
and K. Wickelgren. "Families of padic automorphic forms on unitary groups" Jessica Fintzen, University of Michigan Abstract: We will start with an introduction to padic automorphic forms and then discuss a
variant of the qexpansion principle (called the SerreTate expansion principle) for padic
automorphic forms on unitary groups of arbitrary signature. We outline how this can be used to
produce padic families of automorphic forms on unitary groups, which has applications to the
construction of padic Lfunctions. This is done via an explicit description of the action of certain
differential operators on the SerreTate expansion. The talk is based on joint work with Ana Caraiani, Ellen Eischen, Elena Mantovan and Ila
Varma. "KneserHeckeOperators for Codes over Finite Chain Rings" Jingbo Liu, The University of Hong Kong Abstract: In this talk we will extend results on KneserHeckeoperators for codes over finite
fields, to the setting of codes over finite chain rings. In particular, we consider chain
rings of the form Z/p2Z for p prime. This is a joint work with Amy Feaver, Anna Haensch and Gabriele Nebe. "Zeta functions of curves with many automorphisms" Padmavathi Srinivasan, Georgia Institute of Technology Abstract: We will describe specific families of ArtinSchreier curves with many automorphisms in
odd characteristic. These curves admit the action of a large extraspecial group, which
helps us compute the zeta functions of these curves over the field of definition of these
automorphisms. This generalizes results of Van der Geer and Van der Vlugt from characteristic
two, and gives rise to new examples of maximal curves. This is joint work with Irene Bouw,
Wei Ho, Beth Malmskog, Renate Scheidler and Christelle Vincent. "Hypergeometric varieties and hypergeometric series" Holly Swisher, Oregon State University Abstract: We study generalized Legendre curves y
N = x
i
(1 − x)
j
(1 − λx)
k
,
using periods to determine for certain N a condition for when the endomorphism
algebra of the primitive part of the associated Jacobian variety contains
a quaternion algebra over Q. In most cases this involves computing Galois
representations attached to the Jacobian varieties using Greene’s finite field hypergeometric
functions. From here it is natural to explore higher dimensional
hypergeometric algebraic varieties from this perspective, including relationships
to classical hypergeometric functions and hypergeometric functions over finite
fields.
All of this work is joint with Alyson Deines, Jenny Fuselier, Ling Long, and
FangTing Tu. Part of this work is also joint with Ravi Ramakrishna.
(Organizers: Radmila Sazdanovic, Shirley Yap, Emilie Purvine)
"Topological Complexity in Protein Structures"
Erica Flapan, Pomona College
Abstract: For DNA molecules, topological complexity occurs exclusively as the result
of knotting or linking of the polynucleotide backbone. By contrast,
while knots and links have been found within the polypeptide backbones of
some protein structures, nonplanarity can also result from the connectivity
between a polypeptide chain and inter and intrachain linking via cofactors
and disulfide bonds. In this talk, we survey the knots, links, and nonplanar
graphs that have been identified in protein structures and present models
explaining how protein knots might occur and why certain nonplanar con
figurations are more likely to occur than others. "Data Science for
Topologists" Jenn Gamble, Noodle Analytics, Inc. Abstract: In this talk, we will begin with the question “What is Data
Science?” and outline some common statistical and machinelearningbased
approaches. We will next describe how a geometric/topological approach is also useful,
and why this makes topologists/mathematicians uniquely situated to contribute
to this growing field.
Concepts will be illustrated with a number of applications, including examples from network analysis (iterative simplicial collapse to identify coreperiphery structure and community groups), and eventseries analysis of the perioperative period to identify surgical best practices. “Identification of Copy Number Aberrations in
Breast Cancer Subtypes
Using Persistence Topology” Georgina Gonzalez, University of California, Davis Abstract: Chromosome
aberrations are a hallmark of cancer initiation and progression. DNA copy
number aberrations (CNAs), such as copy number gains and losses, are of
particular interest because they may harbor oncogenes or tumor suppressor genes
(driver aberrations). Genome wide experimental detection of copy number
aberrations across the genome is achieved through microarray and DNA sequencing
technologies. However, the identification of driver CNAs remains a challenge.
Supervised methods address this problem by detecting CNAs that are common and
specific to a given category (such as cancer subtype) or a cancer with specific
clinical characteristics. We will present Topological Analysis of aCGH
(TAaCGH), our complementary supervised method that identifies CNAs based on the
topological properties of the CGH profile. TAaCGH focuses on the relationships
between multiple genomic regions by mapping overlapping fragments of aCGH
profiles into a 2D point cloud using a sliding window method. We then use the
theory of computational algebraic homology to find patterns and associations
within the data with β0, the number of connected components of a simplicial. As
a result, β0 provides us with a measure of genomic instability that help us to
identify aberrant regions. "The Convergence of Mapper" Elizabeth Munch, University at Albany  SUNY Abstract: Mapper, a powerful tool for topological data analysis, gives a summary of the structure of data with respect to a filter function and a cover of the function range. Assuming that this data comes from a true, underlying (but possibly not accessible) topological space, a related construction, the Reeb graph, can be thought of as the ground truth and Mapper, its approximation. In particular, working with a better data sample and/or a more refined cover intuitively results in a Mapper graph which is more similar to the Reeb graph. In this talk, we will discuss a method for defining a distance on these objects via the interleaving distance idea from persistent homology. We can look at various ways to rigorously quantify the idea that Mapper converges to the Reeb graph, as well as ideas for approximation methods of the distance itself. "Towards Spectral Sparsification of Simplicial Complexes based on Generalized Effective Resistance" Bei Wang, University of Utah Abstract: As a generalization of the use of graphs to describe pairwise interactions, simplicial complexes have recently emerged as a useful tool for modeling higherorder interactions between three or more objects in complex systems. To apply spectral methods in learning to massive datasets modeled as simplicial complexes, we work towards the sparsification of simplicial complexes based on preserving the spectrum of the associated Laplacian operators. In particular, we introduce a generalized effective resistance for simplexes; provide an algorithm for sparsifying simplicial complexes at a fixed dimension; and verify a specific version of the generalized Cheeger inequalities for weighted simplicial complexes. This is a joint work with Braxton Osting and Sourabh Palande. "Parameterfree topology inference and sparsification for data on manifolds" Yusu Wang, The Ohio State University Abstract: In recent years, a considerable progress has been made in analyzing data for inferring the topology of a space from which the data is sampled. Current popular approaches often face two major problems. One concerns with the size of the complex that needs to be built on top of the data points for topological analysis; the other involves selecting the correct parameter to build them. In this talk, I will describe some recent progress we made to address these two issues in the context of inferring homology from sample points of a smooth manifold sitting in an Euclidean space. I will describe how we sparsify the input point set and to build a complex for homology inference on top of the sparsified data, without requiring any user supplied parameter. Our sparsification algorithm guarantees that the data is sparsified at least to the level as specified by the socalled local feature size; and at the same time, the sparsified data is adaptive as well as locally uniform. This is joint work with Tamal K. Dey and Dong Zhe. "On minimumarea homotopies of curves in the plane" Carola Wenk, Tulane University Abstract: We study the problem of computing a homotopy from a planar curve C to a point that minimizes
the area swept. The existence of such a minimum homotopy is a direct result of the solution
of Plateau..s problem. We provide structural properties of minimum homotopies that lead
to an algorithm. In particular, we prove that for any normal curve there exists a minimum
homotopy that consists entirely of contractions of selfoverlapping subcurves (i.e.,
boundaries of immersed disks). "Local Maxima of the Distance Function for Delaunay Triangulations on the Plane" Shirley Yap, California State University East Bay Abstract: Given a set of points P, its Voronoi diagram can be defined as the set of singularities
(nonsmooth points) of the distance function d. The minima of d are of the points of P
themselves. The critical points of d coincide with points at which a face of the Voronoi
diagram intersects its dual Delaunay triangulation. Such triangles are called anchored
triangles are useful for reconstructing two dimensional surfaces.
In this talk, I will discuss recent results about the density of the set of anchored triangles
in the Delaunay triangulation of a set P of points in the plane under certain sampling,
boundary, and packing conditions.
(Organizers: Carrie Manore, Erica Graham) "Models for Vector Transmitted Viral Disease of Crops with
Different Replanting Strategies" Vrushali Bokil, Oregon State University Abstract: Vectortransmitted diseases of plants have had devastating effects on agricultural production
worldwide, resulting in drastic reductions in yield for crops such as cotton, soybean, tomato and
cassava. In this investigation, we formulate a new plantvectorvirus model with continuous replanting
from densitydependent replanting of healthy and some infected plants. The new model
is an extension of a model formulated by Holt et al., An epidemiological model incorporating
vector population dynamics applied to African cassava mosaic virus disease, Journal of Applied
Ecology, pages 793806, 1997. Both models are analyzed and thresholds for disease elimination
are defined in terms of the model parameters. Parameter values for cassava, whiteflies, and the
virus, in African cassava mosaic virus serve as a case study. A numerical investigation illustrates
how the equilibrium densities of healthy and infected plants for both models vary with changes
in parameter values. Applications of insecticide and roguing to reduce plant disease and to
increase the number of plants harvested are studied using optimal control theory. "Mathematical modeling of hepatic insulin resistance in adolescent girls" Cecilia Diniz Behn, Colorado School of Mines Abstract: Insulin resistance (IR) is a crucial element of the pathology of the metabolic syndrome,
which now affects more than a third of the population in the United States.
Understanding the contribution to hyperglycemia of abnormal hepatic glucose release
following a meal is crucial for the assessment of potential new medications. Using an
oral glucose tolerance test (OGTT) protocol with two stable isotope tracers, both the
rate of appearance of exogenous glucose coming from the drink and the suppression of
endogenous glucose in response to the drink may be computed. We adapt a
mathematical model of glucoseinsulin dynamics during a labeled OGTT to describe
hepatic IR in adolescent girls. We investigate the structural identifiability of the model,
and this analysis informs the implementation of appropriate numerical approaches for
subjectspecific parameter estimation. Improved understanding of interactions
between exogenous and endogenous hepatic glucose dynamics will facilitate the
characterization of IR in individual patients and different disease conditions and may
support the development of targeted therapeutic approaches. "Identifiability and Parameter Estimation in Modeling Disease Dynamics" Marisa Eisenberg, University of Michigan, Ann Arbor Abstract: Connecting dynamic models with data to yield predictive results often requires a variety
of parameter estimation, identifiability, and uncertainty quantification techniques. These
approaches can help to determine what is possible to estimate from a given model and
data set, and help guide new data collection. Here, we will discuss differential algebraic
and simulationbased approaches to identifiability analysis, and examine how parameter
estimation and disease forecasting are affected when examining disease transmission
via multiple types or pathways of transmission. Using examples taken from cholera
outbreaks in several settings, as well as the West Africa Ebola epidemic, we illustrate
some of the potential difficulties in estimating the relative contributions of different
transmission pathways, and show how alternative data collection may help resolve this
unidentifiability. We also illustrate how even in the presence of large uncertainties in the
data and model parameters, it may still be possible to successfully forecast disease
dynamics. "Toward
a computational model of hemostasis" Karin Leiderman, Colorado School of Mines Abstract: Hemostasis
is the process by which a blood clot forms to prevent bleeding at a site of
injury. The formation time, size and structure of a clot depends on the local
hemodynamics and the nature of the injury. Our group has previously
developed computational models to study intravascular clot formation, a process
confined to the interior of a single vessel. Here we present the first stage of
an experimentallyvalidated, computational model of extravascular clot formation
(hemostasis) in which blood through a single vessel initially escapes through a
hole in the vessel wall and out a separate injury channel. This stage of the model
consists of a system of partial differential equations that describe platelet
aggregation and hemodynamics, solved via the finite element method. We also
present results from the analogous, in vitro, microfluidic model. In both
models, formation of a blood clot occludes the injury channel and stops flow
from escaping while blood in the main vessel retains its fluidity. We discuss
the different biochemical and hemodynamic effects on clot formation using
distinct geometries representing intra and extravascular injuries. "Simulating WithinVector Generation of the Malaria Parasite Diversity" Olivia Prosper, University of Kentucky Abstract: Plasmodium falciparum, the malaria parasite causing the most severe disease in humans, undergoes
an asexual stage within the human host, and a sexual stage within the vector host, Anopheles
mosquitoes. Because mosquitoes may be superinfected with parasites of different genotypes, this
sexual stage of the parasite lifecycle presents the opportunity to create genetically novel parasites.
To investigate the role that mosquitoes’ biology plays on the generation of parasite diversity, which
introduces bottlenecks in the parasites’ development, we first constructed a stochastic model of
parasite development withinmosquito, generating a distribution of parasite densities at five parasite
lifecycle stages: gamete, zygote, ookinete, oocyst, and sporozoite, over the lifespan of a mosquito.
We then coupled a model of sequence diversity generation via recombination between genotypes to
the stochastic parasite population model. Our model framework shows that bottlenecks entering the
oocyst stage decrease diversity from the initial gametocyte population in a mosquito’s blood meal,
but diversity increases with the possibility for recombination and proliferation in the formation of
sporozoites. Furthermore, when we begin with only two distinct parasite genotypes in the initial
gametocyte population, the probability of transmitting more than two unique genotypes from
mosquito to human is over 50% for a wide range of initial gametocyte densities. "Viruses like it hot: modeling the effects of temperature variation on dengue
transmission" Helen Wearing, The University of New Mexico Abstract: The recent emergence of Zika and chikungunya viruses has shed a spotlight on
the potential of mosquitoborne viruses to cause disease outbreaks beyond
tropical climes. Dengue virus, which is transmitted by the same principal
mosquito, has also been causing minor outbreaks in more temperate zones during
the past decade, following introductions from tropical regions where it is
endemic. In this talk, we discuss mathematical models of dengue transmission
that explicitly account for temperature effects on mosquito life history traits
and on viral dissemination within the mosquito. We use these models to examine
the impact of seasonal and diurnal temperature fluctuations on the potential
for dengue outbreaks in six U.S. cities with differing temperature profiles. We
demonstrate that the timing of viral introduction and the temperature profile
of the city interact to determine the potential for, and magnitude of, a
subsequent outbreak. In addition, we highlight how different assumptions about
the relationship between temperature, mosquito mortality, and viral
dissemination within the mosquito affect our results. We discuss our findings
in the context of a current collaboration to integrate mathematical model development
and experimental data collection, which aims to improve our understanding of
how temperature impacts the transmission dynamics of other mosquitoborne
viruses. Nakeya Williams, United States Military Academy Abstract: This study considers pulsatile and nonpulsatile models for
the prediction of shortterm cardiovascular responses during headup tilt
(HUT). HUT refers to tilting a patient from supine position to an upright
position. To explore potential deficits within the autonomic control system,
which maintains the cardiovascular system at homeostasis, many people suffering
from chronic fainting or lightheadedness are often exposed to the headup tilt
test. This system is complex and difficult to study in vivo. As a result, we
show how mathematical modeling can be used to extract features of the cardiovascular
system that cannot be measured experimentally. More specifically, we show that
it is possible to develop a mathematical model that can predict changes in
cardiac contractility and vascular resistance, quantities that cannot be
measured directly, but which are useful to assess the state of the system.
The cardiovascular system is pulsatile, yet predicting the control in response to headup tilt for the complete system is computationally challenging, and limits the applicability of the model. In this work we show how to develop a simpler nonpulsatile model that can be interchanged with the pulsatile model, which is significantly easier to compute, yet it is still able to predict internal variables. The models are validated using headup tilt data from healthy young adults. "Determining NearOptimal Treatment Protocols via Nonlinear Cancer Models"Shelby Wilson, Morehouse College Abstract: This work aims to develop evidencebased treatment protocols designed to optimize the
effectiveness of combined cancer therapies. Here, we study chemotherapy in a context where it is
combined with antiangiogenic drugs (drugs that prevent blood vessel growth). Model
parameters corresponding to tumor growth and monotherapy are estimated in a mixedeffect
manner using Monolix (Lixoft) while parameters corresponding to drug synergism are estimated
in a fixedeffect manner using a NelderMead Simplex Method. We then evaluate the hypothesis
that our two drugs interact synergistically when administered together. A direct consequence of
this interaction is the creation of a therapeutic window in which the relative timing between drug
administrations is most effective. Finally, we use our model, in combination with heuristic
algorithms, to propose drug treatment schedules that are designed to maximize treatment
outcomes.
(Organizers: Gizem Karaali, Hélène Barcelo)
"Combinatorial models for Schubert polynomials" Sami Hayes Assaf, University of Southern California Abstract: Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this talk, I will introduce several new combinatorial models for Schubert polynomials that relate them to other known bases including key polynomials and fundamental slide polynomials. I will present a generalization of the insertion algorithm of Edelman and Greene to give a bijection between reduced expressions and pairs of tableaux of the same key diagram shape and use this to give a simple formula, directly in terms of reduced expressions, for the key polynomial expansion of a Schubert polynomial. Finally, I will show how this model can be used to give a simple proof of Kohnert's algorithm for computing Schubert polynomials. "Uniqueness of BerlineVergne's valuation" Fu Liu, University of California, Davis Abstract: BerlineVergne constructs a valuation that assigns values to faces
of polytope, and it satisfies what we call "the McMullen's formula". There are
different solutions to the McMullen's formula. Any solution provides a way to
write the coefficients of the Ehrhart polynomial of a polytope as positive sums of these
values. We study the BerlineVergne's valuation on generalized permutohedra, and show that their
construction is the unique solution to the McMullen's formula that is symmetric about
the coordinates. This is joint work with Federico Castillo. "The Remarkable Ubiquity of Standard Young Tableaux of Bounded Height" Marni Mishna, Simon Fraser University Abstract: Standard Young tableaux are a classic family of combinatorial objects that appeared in algebra early in the previous century. Their utility is widely appreciated. The subfamily of tableaux of bounded height also appears in many guises in bijective and enumerative combinatorics. The generating functions are particularly lovely for their algebraic and analytic properties. We will explore these combinatorial classes, focusing on recent bijections that illustrate new, nontrivial connections between some very classic objects. We conclude by tracing the shadows of these results in representation theory. Work in collaboration with Julien Courtiel, Eric Fusy and Mathias Lepoutre."Discrete affairs with Macdonald and GromovWitten" Jennifer Morse, University of Virginia Abstract: After discussing the nature of problems in Schubert calculus, we will see how our lasting affair with Macdonald symmetric functions has revealed that the LascouxSch\"utzenberger charge on tableaux can be used as a tool in quantum, affine and equivariant Schubert calculus. We will also give a new formula for the monomial expansion of Macdonald polynomials using the charge statistic."Face numbers and the fundamental group" Isabella Novik, University of Washington Abstract: We will discuss a proof of Kalai's conjecture positing a lower bound on the number of edges of a $(d1)$dimensional triangulated manifold $\Delta$ in terms of $d$, the minimum number of generators of the fundamental group of $\Delta$, and the number of vertices of $\Delta$. Our proofs rely on the $\mu$numbers introduced by Bagchi and Datta and on their algebraic and topological interpretations. "The partition algebra, symmetric functions and
Kronecker coefficients" Rosa Orellana, Dartmouth College Abstract: The SchurWeyl duality between the symmetric group and the general
linear group allows us to connect the representation theory of these two
groups. A consequence of this duality is the Frobenius formula which
connects the irreducible characters of the general linear group and the
symmetric group via symmetric functions. The symmetric group is also in Schur Weyl duality with the partition
algebra. This duality allows us to introduce a new Frobenius type formula
that connects the characters of the symmetric group and those of
the partition algebra. Due to this connection we have introduced a new
basis of the ring of symmetric functions which specialize to the characters
of the symmetric group when evaluated at roots of unity. Furthermore,
the structure coefficients for this new basis of symmetric functions are the
stable (or reduced) Kronecker coefficients. In this talk we will discuss how
this new basis allows us to use symmetric functions to study the representation
theory of the partition algebra and the Kronecker coefficients. This is joint work with Mike Zabrocki. "Schur expansion of parabolic HallLittlewood polynomials"
Anne Schilling, University of California at Davis
Abstract: In 2000, Shimozono and Weyman carried out a combinatorial study of the
Poincar\'e polynomials of isotypic components of a natural family of
graded $\mathrm{GL}(n)$modules supported in the closure of a nilpotent
conjugacy class. These polynomials, $H_{\lambda^\bullet}[X;t]$, defined
for any sequence of partitions $\lambda^{\bullet} = (\lambda^{(1)},\ldots,
\lambda^{(d)})$, generalize the KostkaFoulkes polynomials, are $q$analogues
of LittlewoodRichardson coefficients, and relate to MacdonaldSchur
transition matrices in special cases. Shimozono and Weyman conjectured that
the Schur expansion of parabolic HallLittlewood polynomials can be elegantly
described as the generating functions of katabolizable semistandard
tableaux with the charge statistic of Lascoux and Sch\"utzenberger.
We outline how to attack this conjecture. "Chromatic bases for symmetric functions" Stephanie van Willigenburg, University of British Columbia Abstract: The chromatic polynomial was generalized to the chromatic symmetric function by
Stanley in 1995. This function has recently experienced a renaissance, such as
Shareshian and Wachs introducing a quasisymmetric refinement to study the positivity
of chromatic symmetric functions into the basis of elementary symmetric functions, that
is epositivity. In this talk we approach the question of epositivity from a different angle through
resolving which of the classical symmetric functions can be realized as the chromatic
symmetric function of some graph, and identifying families of graphs whose chromatic
symmetric functions give rise to new epositive bases of the algebra of symmetric
functions. This is joint work with Soojin Cho, Samantha Dahlberg and Angele Hamel.
WINASC Special Session: Recent Research Development on Numerical Partial Differential Equations and Scientific Computing
(Organizers: ChiuYen Kao, Yekaterina Epshteyn) "Minimization of the Principal Eigenvalue of a Mixed Dispersal Model" Baasansuren Jadamba, Rochester Institute of Technology Abstract: In this work, we study a mixed dispersal model of population dynamics and its
corresponding linear eigenvalue problem. The model describes the evolution of a
population which disperses both locally and nonlocally. We investigate how longterm
dynamics depend on parameter values. We also study the minimization of the positive
principal eigenvalue; the problem that is motivated by the determination of optimal
spatial arrangement of favorable and unfavorable regions for the species to die out more
slowly or survive more easily. Numerical results are presented to show various scenarios
for the case of Dirichlet and Neumann boundary conditions. This work is in collaboration
with Marina Chugunova, ChiuYen Kao, Christine Klymko, Evelyn Thomas and Bingyu
Zhao. " Decoupling algorithms for a fluidporoelastic structure interaction problem" Aycil Cesmelioglu, Oakland University Abstract: In this work, we propose decoupling algorithms for the finite element solution of the
interaction of a free fluid and a poroelastic structure described as a coupled StokesBiot system. The
decoupling of the problem is done by casting it as a constrained optimization problem which enforces
the continuity of the normal stress on the interface through a Neumann type control. The objective
functional is designed to minimize the violation of the other interface conditions. Numerical
algorithms based on a residual updating technique will be presented together with some numerical
results. This is joint work with H. Lee, A. Quaini, K. Wang, S.Y. Yi. " An efficient and high order accurate solution technique for partial differential
equations" Adrianna Gillman, Rice University Abstract: Solving linear boundary value problems often limits what practitioners (scientist
and engineers) simulate numerically. Having efficient and high accuracy solution
techniques for these boundary values problems will increase the range of problems
that can be modeled numerically. This talk presents a high order discretization
technique that comes with a fast direct solver. The high order discretization
technique is robust even for problems with highly oscillatory solutions. The
computational cost of the direct solver scales linearly (or nearly
linearly) with respect to the number of unknowns. The tiny constant prefactor
makes this technique ideal for problems involving many solves. Numerical
results will illustrate the performance of the method and its use in applications. "The effect of the sensitivity parameter in
weighted essentially nonoscillatory methods" Yan Jiang, Michigan State University Abstract: Weighted essentially nonoscillatory methods (WENO) were
developed to capture shocks in the solution of hyperbolic conservation laws
while maintaining stability and without smearing the shock profile. WENO
methods accomplish this by assigning weights to a number of candidate stencils,
according to the smoothness of the solution on the stencil. These weights
favor smoother stencils when there is a significant difference while combining
all the stencils to attain higher order when the stencils are all smooth. When
WENO methods were initially introduced, a small parameter . was defined
to avoid division by zero. Over time, it has become apparent that . plays the
role of the sensitivity parameter in stencil selection. WENO methods allow
some oscillations, and it is wellknown that these oscillations depend on the
size of .. In this work we show that the value of . must be below a certain
critical threshold .c, and that this threshold depends on the function used
and on the size of the jump discontinuity captured. Next, we analytically
and numerically show the size of the oscillations for one timestep and over
long time integration when . < .c and their dependence on the size of ., the
function used, and the size of the jump discontinuity.
"Computational model of biofilm evolution with a variational inequality" Malgorzata Peszynska, Oregon State University Abstract: In the talk we present analysis and computational results for a recently developed numerical
model of flow and transport in which the geometry of the (porescale) domain is changing in
time. This research is inspired by available experimental microimaging data showing
biofilm growing at porescale. Biofilm is a collection of microbial cells which adhere to each other
and to fluid and fluidsolid interfaces. We propose a new model for this process using a
parabolic
variational inequality. In our model, a system of advectiondiffusionreactions for biomass and
nutrient evolution is coupled to viscous (NavierStokes) fluid model, and the fluidbiofilm
interface is described similarly to that in onephase Stefan problem. The most interesting and
challenging part is how to account for the constraint on the maximum density of biofilm that can
be present, and for the associated growth through interfaces. This is joint work with
Anna Trykozko, Interdisciplinary Centre for Modeling, University of Warsaw, and Azhar
Alhammali from Oregon State. "A C0 finite element method for elliptic distributed optimal control problems with pointwise state constraints" Sara Pollock, Wright State University Abstract: We consider a class of nonconforming finite element methods for elliptic distributed optimal control problems with pointwise state constraints, in threedimensional convex polyhedral domains. The optimal control problem can be reformulated as a fourthorder variational inequality, to which a quadratic C0 interior penalty method may be applied. Taking this approach, we obtain numerical results for the threedimensional problem, demonstrating predicted convergence rates. "Numerical methods for solving linear poroelasticity equations" Beatrice Riviere, Rice University Abstract: The modeling of poroelastic deformation arises in many fields including biomechanics,
energy and environmental engineering. We propose and analyze discontinuous Galerkin
methods for solving the linear poroelasticity equations. In a first approach, the flow
and
mechanics equations are solved fully implicitely. In a second approach, the equations
are
decoupled and solved sequentially at each time step. Theoretical error estimates are derived.
Applications to reservoir engineering and biomedicine are shown. "On Metrics for Computation of Strength of Coupling in Multiphysics Simulations" Anastasia Wilson, Augusta University Abstract: Many multiphysics applications arise in the world of mathematical modeling and simulation.
Much of the time in scientific computation these multiphysics applications are solved by decoupling the
physics, giving no heed to how this affects the numerical results. However, a fully coupled approach is often
not computationally cost effective. Consequently, having a metric for determining the strength of coupling
could give insight into whether a simulation should be decoupled in the computation. If the fully coupled
approach is not available, then a metric that measures the strength of coupling dynamically in time could
help determine when smaller time steps are required to better incorporate coupling into the split solution.
In this paper, we report on an Institute for Mathematics and Its Applications student project where we
explored metrics for dynamically measuring the strength of coupling between two physical components in
a model multiphysics simulation. Four metrics were considered: two based on measured components of the
Jacobian matrix, one on error estimates, and the last on time scales of the system components. The metrics
are all developed based on previous work found in the literature and tested on a diffusionreaction problem.
WINART Special Session: Representatives of Algebras
(Organizers: Susan Montgomery, Maria Vega)
"Mutation of friezes" Karin Baur, University of Graz Abstract: A frieze pattern is a lattice of shifted rows positive integers satisfying the diamond
rule:
for any four entries
b
a d
c
we have adbc=1.
These patterns were first studied in the 70s by Coxeter and Conway who
proved that frieze patterns with finitely many rows are in correspondence
with triangulations of polygons.
Over the last years, they gained fresh interest because of connections between triangulations
of surfaces and cluster algebras.
Mutation is a key notion in cluster theory. We introduce mutation as an operation on frieze
patterns and show how it is compatible with mutation in the cluster
theory setup.
This is joint work with E. Faber, S. Gratz, K. Serhiyenko and G. Todorov. "Noncommutative Resolutions of Toric Rings" Eleonore Faber, University of Michigan Abstract: We consider endomorphism rings A = EndR(M) of Cohen–Macaulay modules
M over commutative rings R. If A has finite global dimension, then
it is called a noncommutative resolution of singularities (NCR) of R (or
Spec(R)). When R is a domain of characteristic p > 0, one possible M to
consider is the module of p
e
th roots. In this talk, we consider toric rings R, where the module of p
e
th roots gives
a noncommutative resolution, and show how the precise module structure
of the endomorphism ring can be described combinatorially. In particular,
we are interested in the case when the global dimension of A is equal to the
Krulldimension of R. This is joint work with Greg Muller and Karen E.
Smith. "Separating Ore sets for Prime Ideals of Quantum Algebras" Sian Fryer, University of California Santa Barbara Abstract: The prime ideals of various families of quantized coordinate rings can be studied via a finite set of primes known as the Hprimes, which stratify the prime spectrum. This allows us to phrase questions about the Zariski topology of the prime spectrum in terms of quotients and localizations of the algebra with respect to the Hprimes. Of course, localization in noncommutative algebras isn't necessarily easy: we can only invert sets of elements which satisfy the Ore conditions. I will talk about what these Ore sets should look like in general, and then relate them to a combinatorial construction called the Grassmann necklace which allows us to easily compute examples. (Joint work with M Yakimov and K Casteels.) "Representations of signed affine Brauer algebras"
Mee Seong Im, United States Military Academy Abstract: I will explain the construction of unital associative algebras called signed
affine Brauer algebras, which are an extension of Brauer algebras constructed
by D. Moon. Our algebras could also be realized as the periplectic version
s ⩔ of the affine NazarovWenzl algebras. I will introduce s ⩔ algebraically
and diagrammatically, and I will discuss the representation theory of these
algebras.
This is joint with M. Balagovic, Z. Daugherty, I. EntovaAizenbud, I.
Halacheva, J. Hennig, G. Letzter, E. Norton, V. Serganova, and C. Stroppel.
"Auslander’s Theorem for Permutation Actions on A = k−1[x1, . . . , xn]" Ellen Kirkman,Wake Forest University Abstract: Let k be an algebraically closed field of characteristic zero. Maurice Auslander
proved that when a finite subgroup G of GLn(k), containing no reflections, acts on
A = k[x1, . . . , xn] naturally, with fixed subring AG, then the skew group algebra
A#G is isomorphic to EndAG (A) as algebras. We prove that Auslander’s Theorem
holds for A = k−1[x1, . . . , xn] under the action of any group of permutations
of {x1, . . . , xn}. In some cases AG is a graded isolated singularity in the sense of
MoriUeyama (work wtih J. Gaddis, W. F. Moore, and R. Won). "Clustertilted and quasitilted algebras" Khrystyna Serhiyenko, University of California, Berkeley Abstract: It is known that relationextensions of tilted algebras are clustertilted algebras,
and various resulting relationships between them have been studied with great interest. In
this work, we investigate a wider class of algebras of global dimension at most two, namely the
quasitilted algebras. We show that relationextensions of quasitilted algebras are 2CalabiYau
tilted. To study the module category of clustertilted algebras of euclidean type, we
generalize the notion of reflections of local slices and develop an algorithm for constructing
the tubes. Finally, we characterize all quasitilted algebras whose relationextensions are
clustertilted of euclidean type. This is joint work with Ibrahim Assem and Ralf Schiffler. "Dominant Dimension and Tilting Modules" Gordana Todorov,Northeastern University Abstract: We study which algebras have a tilting module which is both generated and
cogenerated by projectiveinjective modules. Auslander algebras have such a
tilting module and for algebras of global dimension 2, Auslander algebras are
classified by the existence of such a tilting module. In this paper we show that, independently of global dimension, the existence
of such a tilting module is equivalent to the algebra having dominant dimension
at least 2. Furthermore, as special cases, we show that algebras obtained
from Auslander algebras by extensions on certain injective modules, have such
a tilting module. We also give a description of which Nakayama algebras have
such a tilting module. WiSh Special Session: Shape Modeling and Applications
(Organizers: Kathryn Leonard, Asli Genctav)
"3D Shape Representations for Interactive Images and Videos" Duygu Ceylan, Adobe Research Abstract: Images and videos are becoming one of the most popular medium for capturing the 3D world
due to the increasing availability of mobile and lowcosthighresolution devices. This
popularity is creating a desire for interacting with this content similar to realworld
interactions, e.g. rotating an object in an image, viewing a video from a different viewpoint.
Such interactions pave the road to a much more immersive user experience. One way to accomplish
such interactions is to compute an intermediate 3D shape representation from the available
image and video data and use this to guide an interactive editing process. Even though
accurate 3D shape modeling from images and videos is still a challenging task, often an
intermediate 3D proxy representation is sufficient for various editing applications. In this talk, I will present our recent work that converts monocular 360videos to sterescopic
videos that can be viewed on a headset with 6 degreeoffreedom (DOF). We first compute
an intermediate 3D representation of the scene from the monocular video. We then playback
the video in a VR headset where we track the 6DOF motion of the headset. We synthesize
novel views for each eye in real time by a warping method that is guided by the intermediate
3D representation. This results in a full 3D VR experience that responds both to translational
and rotational motion of the headset significantly eliminating motion sickness and increasing
immersiveness. "Discriminating
Placentas of Increased Risk for Autism with Chorionic Surface Vascular Network
Features"
JenMei Chang, Colorado State University
Abstract: It was hypothesized
that variations in the placental chorionic surface vascular network (PCSVN)
structure may reflect both the overall effects of genetic and environmentally
regulated variations in branching morphogenesis within the conceptus and the
fetus' vital organs. Significant differences in certain PCSVN features in
children at increased risk for autism have been identified; however, a
comprehensive understanding of how these, and possibly a lot more, features
work together to provide discriminating power is lacking. 28 vesselbased and 8
shapebased PCSVN attributes from a highrisk ASD cohort of 89 placentas and a
populationbased cohort of 201 placentas were examined for ranked relevance
using the Boruta algorithm. Principal component analysis (PCA) was applied to
isolate principal effects of vessel growth on the fetal surface. Linear
discriminant analysis with a 10fold cross validation was performed to establish
classification statistics. Boruta algorithm selected 15 vesselonly attributes
as relevant, implying the difference in high and low ASD risk is better
explained by the vascular features alone. The five principal features, which
accounted for about 88% of the data variability in PCA, indicated that PCSVNs
associated with placentas of highrisk ASD pregnancies generally had fewer
branch points, thicker and less tortuous vessels, better extension to the
surface boundary, and smaller branch angles than their populationbased
counterparts. "Medial Fragments for Segmentation of
Articulating Objects in Images" Ellen Gasparovic, Union College Abstract: We propose a method for extracting objects from natural images by combining fragments
of the Blum medial axis, generated from the Voronoi diagram of an edge map
of a natural image, into a coherent whole. Using techniques from persistent homology
and graph theory, we combine image cues with geometric cues from the medial fragments
in order to aggregate parts of the same object. We demonstrate our method on
images containing articulating objects, with an eye to future work applying articulationinvariant
measures on the medial axis for shape matching between images. This is joint
work with Erin Chambers and Kathryn Leonard. "A Joint Segmentation
and Nonlinear Elasticity Registration algorithm using FFT" Weihong Guo, Case Western Reserve University Abstract: We present a Fourier transform based solution of joint image
registration and segmentation. The combined registration and segmentation
framework is optimized in a way that the displacement (solution to
registration) and the segmenting curve of the deforming template converge at
the same time. The images are modeled as hyperelastic (specifically St. Venant
Kirchhoff) materials allowing for nonlinear straindisplacement relationship
and consequently larger deformation. We iteratively solve the segmentation and registration
subproblems by first solving for the displacement using the fast fourier
transform. The second task, which is the segmentation of the template image is
based on the dual formulation of the piecewise constant MumfordShah in the
framework of active contour model with a weighted TV regularity. Numerical
experiments show the advantages of the proposed method. "Shapes and Other Things" Terry Knight, Massachusetts Institute of Technology Abstract: Shape grammars
have offered a unique computational theory of design over the past forty or so
years. Shape grammars are comprised of visual, shape rules that specify seeing and doing actions (see this ® do that). Shape rules
apply in computations to generate, or
compute, designs made of shapes. Underpinning
shape grammar computations are formal definitions of shapes based on their
visual properties. Recently, shape grammars have been
adapted to define making grammars comprised
of rules that apply to compute material, realworld objects or things, as opposed to abstract shapes.
Underpinning making grammars and their computations are formal definitions of
things based on their physical, sensory properties. In this talk, I will overview (1)
shape computing with shape grammars, (2) different ways that shapes in shape
grammars have been augmented with material properties to describe physical things,
and (3) new work with making grammars for computing physical things and their
properties directly. I will highlight some merits, drawbacks, and peculiarities
of computing with shapes, computing with augmented shapes, and computing with things
in the realm of design. "Measures on the Blum medial axis: Toward automated shape understanding" Kathryn Leonard, California State University Channel Islands Abstract: The Blum medial axis offers a skeletal shape representation whose desirability
has long been hampered by the perceived instability of its branching structure. Instead
of the usual approach of trying to prune spurious medial points, we define measures on
the medial axis that capture salient qualities of the associated shape regions. These
measures provide a framework for tasks beyond pruning, including decomposing a shape into
parts, determining the similarity of parts, and evaluating the shape’s inherent complexity.
We present an overview of these measures and apply the resulting framework to a wide range
of shapes. "Conveying and Analyzing Shapes: From Art to Science" Alla Sheffer, University of British Columbia Abstract: Humans have developed multiple ways to communicate about both tangible and
abstract
shape properties. Artists and designers can quickly and effectively convey complex shapes
to a broad
audience using traditional mediums such as paper, while both experts and the general public
can
analyze and agree on intangible shape properties such as style or aesthetics. While perception
research
provides some clues as to the mental processes involved, concrete and quantifiable explanations
of this
process are still lacking. Our recent line of research aims to quantify the geometric properties and tools involved
in shape
communication and analysis, and to develop algorithms that successfully replicate human
abilities in this
domain. In my talk I will survey our efforts in this domain  describing methods for creation
of 3D looking
shaded production drawings from concept sketches; sketch based modeling algorithms that
automatically create complex 3D shapes from artistgenerated line drawings in a range
of domains,
including industrial design, character modeling, and garment design; and methods for
style analysis and
transfer for a range of manmade shapes. The common thread in these approaches is the
use of insights
derived from perception and design literature combined with subsequent perceptual validation
via a
range of user studies.
"The Intelligent Search and Mapping of Shipwrecks in the Coastal
Waters of Malta" Zoe Wood, Cal Poly  San Luis Obispo Abstract: With its rich maritime history, the coastal waters of Malta contain numerous
shipwrecks of archeological importance. For marine archeologists searching for
undiscovered wrecks, the sheer magnitude of the search space is a major
challenge. Where should the archeologist begin? Towed side scan sonar
has been used to detect potential wreck sites, and more recently Autonomous
Underwater Vehicles (AUVs) equipped with side scan sonar have been deployed
in the search for wrecks. However, the approach to determining potential wreck
site areas remains largely based on laborintensive human research into
historical archives. How can robotics based software and hardware be
advanced to enable efficient search and discovery of underwater archeological
sites? And how might robot obtained sensor information be processed to
produce 3D visualizations that effectively convey the required information to the
archeological community?
This interdisciplinary project focuses on developing novel AUV planning, control,
and visualization techniques that can be applied to a general class of
autonomous robot exploration tasks. These techniques are being applied in
actual AUV shipwreck search and mapping in coastal areas of Malta and
Sicily. We present our work on probabilistic algorithms for AUV motion planning
that maximize information gain when mapping a wreck and visualization
techniques that construct 3D models of wrecks and the ocean
environment. This talk presents the background and current state of this
ongoing joint research project between Cal Poly, Harvey Mudd College and the
University of Malta.
WIT Special Session: Topics in Homotopy Theory
(Organizers: Julie Bergner, Angelica Osorno) "Topological coHochschild homology: Tools for computations" Anna Marie Bohmann, Vanderbilt University Abstract: Hochschild homology is a classical invariant of algebras. A "topological" version,
called THH, has important connections to algebraic Ktheory, Waldhausen's Atheory, and
free loop spaces. For coalgebras, there is a dual invariant called "coHochschild homology"
and Hess and Shipley have recently defined a topological version called "coTHH," which
also has connections to Ktheory, Atheory and free loops spaces. In this talk, I'll
talk about coTHH (and THH) are defined and then discuss work with Gerhardt, Hogenhaven,
Shipley and Ziegenhagen in which we develop some computational tools for approaching coTHH. "Derived Ainfinity algebras and their homotopies" Daniela Egas Santander, Freie Universität Berlin Abstract: The notion of a derived Ainfinity algebra, introduced by Sagave, is a generalization of
the classical Ainfinity algebra, relevant to the case where one works over a commutative ring
rather than a field. Special cases of such algebras are Ainfinity algebras and twisted complexes
(also known as multicomplexes). We initiate a study of the homotopy theory of these algebras,
by introducing a hierarchy of notions of homotopy between their morphisms. In this talk I will
define these objects and describe two different interpretations of them as Ainfinity algebras
in twisted complexes and as Ainfinity algebras in split filtered cochain complexes. We use
this reinterpretation to show that this hierarchy of homotopies is an extension of the special
case of twisted complexes. This is joint work with Joana Cirici, Muriel Livernet and Sarah
Whitehouse "Galois extensions in motivic homotopy theory" Magdalena Kedziorek, EPFL Abstract: Galois extensions of ring spectra in classical homotopy theory were introduced by Rognes. In
this talk I will discuss a general formal framework to study homotopical Galois extensions and
concentrate on the applications to motivic homotopy theory. I will discuss several examples of
homotopical Galois extensions in motivic setting comparing them to the ones known from classical
homotopy theory. This is a joint project with Agn`es Beaudry, Kathryn Hess, Mona Merling and Vesna Stojanoska. "A Higher Order Chain Rule for Abelian Functor Calculus" Christina Osborne, University of Virginia Abstract: One of the most fundamental tools in calculus is the chain rule for functions.
Huang, Marcantognini, and Young developed the notion of taking higher order directional
derivatives of functions, which has a corresponding higher order iterated directional derivative
chain rule. When Johnson and McCarthy established abelian functor calculus,
they constructed the chain rule for functors which is analogous to the directional derivative
chain rule when n = 1. In joint work with Bauer, Johnson, Riehl, and Tebbe, we defined
an analogue of the iterated directional derivative and provided an inductive proof of the
analogue to the HMY chain rule. Our initial investigation of this result involved a concrete
computation of the case when n = 2, which will be presented in this talk. If time permits,
we will discuss the cartesian differential category structure, which is used for the more
general proof. "2Segal spaces and the Waldhausen construction" Angelica Osorno, Reed College Abstract: The notion of 2Segal spaces was introduced by Dyckerhoff and Kapranov as a
higher dimensional version of Rezk's Segal spaces. In this talk we will explore the motivation
for this notion, give examples, and show that it is related to a certain class of double
categories via a version of Waldhausen's construction. This is joint work with J. Bergner, V. Ozornova, M. Rovelli, and C. Scheimbauer. "Mapping Spaces for Orbispaces" Laura Scull, Fort Lewis College Abstract: Orbifolds, and more generally orbispaces, are a class of spaces which have wellbehaved
singularities. These are often modelled using topological groupoids. Using
this approach, the category of orbispaces can be described as a bicateogory of fractions
of groupoids, where a certain class of maps, the Morita equivalences, have been
inverted. Using this approach, we can define a mapping object which is another groupoid.
However, because we are dealing with a bicategory, the mapping object definition is not
completely straightforward. I will discuss the work of the WIT team (Coufal, Pronk,
Rovi, Scull, Thatcher) done at the first WIT meeting, in exploring the structure of this
mapping object. I will also present newer results, done after the WIT meeting but
following up on the foundational work done there, that allow us to give this groupoid a
topology so that it becomes another orbispace, and (with certain compactness
conditions) becomes an exponential object in the category of orbispaces. "A Homotopical Generalisation of the BestvinaBrady Construction" Elizabeth Vidaurre, University of Rochester Abstract: Using polyhedral products (X, A)
K, we recognise the BestvinaBrady
construction as the fundamental group of the fibre of (S
1
, ∗)
L →
(S
1
, ∗)
K = S
1
, where L is a flag complex and K is a one vertex complex.
We generalise their construction by studying the homotopy fibre
F of (S
1
, ∗)
L → (S
1
, ∗)
K for an arbitrary simplicial complex L and K
an (m − 1)dimensional simplex. We describe the homology of F, its
fixed points, and maximal invariant quotients for coordinate subgroups
of Z
m. This generalises the work of Leary and Saadeto˘glu who studied
the case when m = 1. "Inverting Operations in Operads" Sarah Yeakel, University of Maryland Abstract: The DwyerKan hammock localization provides a simplicially enriched model for
a homotopy category in which maps in a subcategory are inverted. In this talk, I will
define a variant of this construction which gives a localization for an operad with respect
to a submonoid of oneary operations and discuss some of its various winsome properties.
This is joint work with Maria Basterra, Irina Bobkova, Kate Ponto, and Ulrike Tillmann.
Women in Sage Math
(Organizers: Alyson Deines, Anna Haensch)
"Minimal Integral Weierstrass equations for genus 2 curves" Lubjna Beshaj, The University of Texas at Austin Abstract: We study the minimal Weierstrass equations for genus 2 curves defined over a ring
of integers OF . This is done via reduction theory and the Julia quadratic of binary sextics. We
show that when the binary sextics has extra automorphisms this is usually easier to compute.
Moreover, we build a database of genus 2 curves defined over Q which contains all curves
with minimal absolute height ≤ 5 and all curves with extra automorphisms in standard form
y
2 = f(x
2
) defined over Q with height ≤ 101. "Belyi maps and effective bounds" Lily Khadjavi, Loyola Marymount University Abstract: Belyi’s theorem, mapping algebraic curves to the projective line with ramification over at
most three points, is a linchpin to deep work in algebra and number theory. These
include Grothendieck’s program to understand the structure of the absolute Galois
group and Mochizuki’s purported proof of the ABC Conjecture. (Indeed, regarding
Belyi’s theorem, Grothendieck noted, ``Never was such a profound and disconcerting
result proved in so few lines!’’) The fact that Belyi’s proof is constructive has useful
implications; we will use Sage to illustrate examples of interest. "A Census Of Quadratic PostCritically Finite Rational Functions Defined Over Q"
Michelle Manes, University of Hawaii at Manoa
Abstract: A result of Benedetto, Ingram, Jones, and Levy provides a specific
height bound on quadratic postcritically finite (PCF) rational functions defined
over Q, guaranteeing a finite number of such maps. A natural question is to find
all such rational functions. We describe an algorithm, prototyped in Sage and
implemented in both Sage and C, to search for possibly PCF maps. Using the
algorithm, we eliminate all but twelve quadratic functions, all of which are verifiably
PCF. We also give a complete description of possible rational preperiodic structures
for quadratic PCF maps defined over Q "On the Field of Definition of a Cubic Rational" Bianca Thompson, Harvey Mudd College Abstract: Using essentially only algebra, we give a proof that a cubic rational function over C with real critical points is equivalent to a real rational function. We also show that the natural generalization to Qp and number fields fails. "Constructing hyperelliptic curves of genus 3 whose Jacobians have CM" Christelle Vincent, University of Vermont Abstract: For cryptographic applications, it is convenient to be able to, given a CM field, be able
to construct an abelian variety with complex multiplication by an order in the ring of
integers of that field. It is currently wellunderstood how to do this in dimension 1, and a lot of progress has
been done in dimension 2. We discuss here the challenges of constructing an abelian threefold
with complex multiplication by the ring of integers of a sextic CM field and the work
that has been done recently in this direction, both by members of Women in Sage and Women
in Numbers projects as well as other mathematicians. "Solving the Sunit equation in Sage" Mackenzie West, Reed College Abstract: Inspired by work of Tzanakis–de Weger, Baker–W¨ustholz and Smart,
we use the LLL methods available in Sage to implement an algorithm that
returns all Sunits τ0, τ1 ∈ O×
S
such that τ0 +τ1 = 1. Portions of this code
were developed during the Women in Sage 5 workshop and at ICERM as
part of a Collaborate@ICERM project. "Arithmetic Mirror
Symmetry and Isogenies" Ursula Whitcher, Mathematical Reviews Abstract: Arithmetic mirror
symmetry is a relationship between the number of points on appropriately chosen
mirror pairs of CalabiYau varieties over finite fields. We investigate whether
arithmetic mirror relationships observed for pencils in weighted projective spaces
can be extended to mirror families obtained via the BatyrevBorisov
construction. Our results show that arithmetic mirror symmetry is controlled by
an isogeny structure. This talk describes joint work with Christopher Magyar. "Parameter space analysis for algebraic Python programs in SageMath" Yuan Zhou, University of California, Davis Abstract: A metaprogramming trick transforms algebraic programs for testing a property
for a given input parameter into programs that compute simplified semialgebraic descriptions
of the input parameters for which the property holds. Our implementation of this trick
is for Python programs (within the Pythonbased computer algebra system SageMath and using
Mathematica for semialgebraic computations). We illustrate it with an application in the
theory of integer linear optimization, the automatic discovery and proof of certain cutting
plane theorems in integer programming. Women in Government Labs
(Organizers: Cindy Phillips, Carol Woodward) "NextGeneration Adaptive Mesh Refinement" Ann Almgren, LBNL Abstract: Blockstructured adaptive mesh refinement (AMR) is a powerful tool for
improving the computational efficiency and reducing the memory footprint of
structuredgrid numerical simulations. AMR techniques have been used for over 25
years to solve increasingly complex problems. I will talk about the challenges for
designing AMR algorithms and software for solving large multiscale, multiphysics
problems on nextgeneration multicore architectures. "Optimization and Concrete Problems in Design of Complex Adaptive Systems" Natalia Alexandrov, NASA Langley Research Center Abstract: Complex adaptive systems, such as air transportation, have been managed via strict
complexity bounding, to enable control by humans. Arguably, the traditional transportation system has
reached saturation. Growing density of traffic and diversity of aircraft, including unmanned aerial
systems (UAS), require increasing reliance on hybrid humanmachine control, autonomous control, and,
potentially, a complete cleanslate redesign of the transportation system. These developments give rise
to a number of difficult unsolved problems in design, system control, and artificial intelligence. While it
is easy to develop trust in an autonomous vacuum cleaner, developing trustworthy and trusted safetycritical
systems is a much harder problem. In this talk, we discuss an approach to resolving some of
these problems via optimization. "Unique Challenges of Multiphysics High Performance Computing for DOE Labs" Anshu Dubey, Argonne National Laboratory Abstract: The Department of Energy (DOE) laboratories develop and deploy scientific software for a great deal of mission critical work. The useof such software ranges from aiding in scientific insight and discovery to development of new devices and other research prototypes. Many of these software packages model multiple physical phenomena in form of numerical components that need to interoperate with one another. These applications pose several unique challenges to their developers and users. There is often more than one kind of discretization in the same application, and several different numerical algorithms are used. The developers come from a wide range of expertise and it is critical to have some with breadth of knowledge spanning several domains. The lifecycle of the software far exceeds the lifecycles of machines or specific problems. Many aspects of the software, including discretization methods, numerical algorithms, and optimization techniques, are typically subject of ongoing research themselves. In this presentation I will discuss these and other challenges, and how they are met in the DOE laboratories with a focus on applications from high energy physics and climate modeling. "Development of novel sparse matrix algorithms and software for large scale simulations
and data analyses" Sherry Li, Lawrence Berkeley National Laboratory Abstract: Efficient solution of largescale indefinite algebraic equations often
relies on high quality preconditioners together with iterative solvers.
Because of their robustness, factorizationbased
algorithms play a significant role in developing scalable solvers.
We discuss the recent advances in highperformance sparse factorization
techniques which are used to build sparse direct solvers, domaindecomposition
type direct/iterative hybrid solvers, and approximate factorization
preconditioners. In addition to algorithmic principles, we also address
the key parallelism issues and practical aspects in order to fully utilize
the highly heterogeneous architectures of the current and future
HPC systems.
"Using Supercomputing to Solve Large Energy Grid Planning Problems" Carol Meyers, Lawrence Livermore National Laboratory Abstract: We discuss the use of supercomputing to solve energy grid planning problems,
based on work with energy stakeholders in the state of California. With the increased
introduction of renewable resources (such as wind and solar) into the electric grid, planning
models must account for increased intermittency of generation, which leads to larger and
more complex optimization problems. The underlying model in many of these instances is
a mixedinteger linear unit commitment problem, which can solve very slowly when the number
of variables and constraints are very large. In the first part of the talk we describe
our experiences in speeding execution of a commercial Windowsbased energy grid software
package via the use of improved formulations (to speed each individual instance) and supercomputing
(to enable many instances to solve at once). In the second part of the talk we describe
different parallelization strategies that we developed to solve the even larger (millions
of variables and constraints) stochastic version of the problem. "Parallel solution algorithms and modeling tools for dynamic optimization" Bethany Nicholson, Sandia National Laboratories Abstract: Dynamic optimization problems directly incorporate detailed dynamic models as constraints within an
optimization framework. Applications of dynamic optimization can lead to significant improvements in
process efficiency, reliability, safety, and profitability. A wellestablished method to solve dynamic
optimization problems is direct transcription where the differential equations are replaced with
algebraic approximations using some numerical method such as a finitedifference or RungeKutta
scheme. However, for problems with thousands of state variables and discretization points, direct
transcription may result in nonlinear optimization problems that exceed memory and speed limits of
most serial computers. In particular, when applying interior point optimization methods, the
computational bottleneck and dominant computational cost lies in solving the linear systems resulting
from the Newton steps that solve the discretized optimality conditions. To overcome these limits, we
exploit the parallelizable structure of the linear system to accelerate the overall interior point algorithm.
We investigate two algorithms which take advantage of this property, cyclic reduction and Schur
complement decomposition and study the performance of these algorithms when applied to dynamic
optimization problems. We also briefly discuss pyomo.dae, an opensource modeling framework that
enables highlevel abstract representations of dynamic optimization problems. "A Frequentist Approach to MultiSource Classification" Katherine Simonson, Sandia National Laboratories Abstract: The classification of unknown entities based on measured data is a
fundamental challenge across applications as diverse as medical diagnostics,
treaty monitoring, and electronic fraud detection. In many cases, the data
available to support classification decisions arise from multiple sources, each
with its own unique signal and noise characteristics. The method to be discussed
here, known as Probabilistic Feature Fusion (PFF), provides a means to combine
multisource classification information in a manner that is statistically
rigorous and accounts for the uncertainties associated with the constituent
sources. PFF provides final class consistency scores that are readily
interpretable within in a Frequentist framework, and allows complete
traceability back to the contributing sources. It is particularly appropriate
in applications related to high consequence decision support, where training
data may be limited, and “black box” classifiers struggle to gain trust and
cultural acceptance. The method will be
illustrated with a practical application related to the segmentation of human
skin in color imagery. "The Impact of Computer Architectures on the Design of Algebraic Multigrid Methods" Ulrike Meier Yang, Lawrence Livermore National Laboratory Abstract: Algebraic multigrid (AMG) is a popular iterative solver and preconditioner for large sparse linear
systems. When designed well, it is algorithmically scalable, enabling it to solve increasingly larger
systems efficiently. While it consists of various highly parallel building blocks, the original method also
consisted of various highly sequential components. A large amount of research has been performed
over several decades to design new components that perform well on high performance computers. As
a matter of fact, AMG has shown to scale well to more than a million processes. However, it is facing
several major challenges with future architectures: nonincreasing clock speeds are being offset with
added concurrency (more cores) and limited power resources are leading to reduced memory per core,
and highly complex heterogeneous architectures. To meet these challenges and yield fast and efficient
performance, solvers need to exhibit extreme levels of parallelism, and minimize data movement. In this talk, we will give an overview on how AMG has been impacted by the various architectures of
high performance computers to date and discuss our current efforts to continue to achieve good
performance on emerging computer architectures. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore
National Laboratory under Contract DEAC5207NA27344. LLNLABS723278.
EDGEy Mathematics: A Tribute to Dr. Sylvia Bozeman and Dr. Rhonda Hughes
(Organizers: Alejandra Avlarado, Candice Price)
"Maximum nullity, zero forcing, and power domination" Chassidy Bozeman, Iowa State University Abstract: Zero forcing on a simple graph is an iterative coloring procedure that starts by
initially coloring vertices white and blue and then repeatedly applies the following
color change rule: if any vertex colored blue has exactly one white neighbor, then
that neighbor is changed from white to blue. Any initial set of blue vertices that
can color the entire graph blue is called a zero forcing set. The zero forcing number
is the cardinality of a minimum zero forcing set. A well known result is that the
zero forcing number of a simple graph is an upper bound for the maximum nullity
of the graph (the largest possible nullity over all symmetric real matrices whose ijth
entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise).
A variant of zero forcing, known as power domination (motivated by the monitoring
of the electric power grid system), uses the power color change rule that starts by
initially coloring vertices white and blue and then applies the following rules: 1) In
step 1, for any white vertex w that has a blue neighbor, change the color of w from
white to blue. 2) For the remaining steps, apply the color change rule. Any initial
set of blue vertices that can color the entire graph blue using the power color change
rule is called a power dominating set. We present results on the power domination
problem of a graph by considering the power dominating sets of minimum cardinality
and the amount of steps necessary to color the entire graph blue. "Interactions of Elastic Cilia Driven by a Geometric Switch" Amy Buchmann, Tulane University Abstract: Cilia, flexible hairlike appendages located on the surface of a cell, play an important role in many
biological processes including the transport of mucus in the lungs and the locomotion of ciliated
microswimmers. Cilia selforganize forming a metachronal wave that propels the surrounding fluid.
To study this coordinated movement, we model each cilium as an elastic, actuated body whose beat
pattern is driven by a geometric switch where the beat angle switches between two ‘traps’, driving the
motion of the power and recovery strokes. The cilia are coupled to a viscous fluid using a numerical
method based upon a centerline distribution of regularized Stokeslets. We first characterize the beat
cycle and flow produced by a single cilium and then investigate the synchronization states between
two cilia. Cilia that are initialized in phase eventually lock into antiphase motion unless a additional
velocity dependent switch is incorporated. "Regularization results for inhomogenous illposed problems in Banach space" Beth Campbell Hetrick, Gettysburg College Abstract: We prove continuous dependence on modeling for the inhomogenous illposed Cauchy problem
in Banach space. Consider the abstract Cauchy problem du(t)
dt = Au(t), u(T) = χ,
where t ≤ T, A is a denselydefined linear operator in a Banach space X, and χ ∈ X.
This final value problem is a familiar example of an inverse problem that is illposed;
that is, small differences in observed final data may lead to large differences in solutions.
For A = ∆, the Laplace operator, we have the backward heat equation that arises in
many applications. Motivated by this, we prove regularization for particular inhomogeneous
illposed problems. Written as an initial value problem, the problem is given by
du(t)
dt = Au(t) + h(t), 0 ≤ t < T, u(0) = χ, where −A generates a uniformly bounded
holomorphic semigroup {e
zARe(z) ≥ 0} and h : [0, T) → X. In the model problem, the
operator A is replaced by the operator fβ(A), β > 0, which approximates A as β goes to 0.
Here we use a logarithmic approximation introduced by Boussetila and Rebbani. Our results
extend earlier work of Karen Ames and Rhonda Hughes on the homogeneous illposed
problem. "In Pursuit of a Bayesian False Discovery Approach to Syndromic
Surveillance" Deidra Coleman, Philander Smith College Abstract: We give a procedure to detect outbreaks using epidemiological
data while controlling the Bayesian False Discovery Rate (BFDR). The procedure
entails choosing an appropriate Bayesian model that captures the spatial
dependency inherit in epidemiological data and considers all days of interest,
selecting a test statistic based on a chosen measure that provides the
magnitude of the maximum spatial cluster for each day, and identifying a cutoff
value that controls the BFDR for rejecting the collective null hypothesis of no
outbreak over a collection of days for a specified region. We use our procedure
to analyze botulismlike syndrome data collected by the North Carolina Disease Event
Tracking and Epidemiologic Collection Tool (NC DETECT). "Topological Symmetry Groups of Graphs in S
3" Emille Davie Lawrence, University of San Francisco Abstract: The study of graphs embedded in S
3 has been motivated by chemists’
need to predict molecular behavior. The symmetries of a molecule can explain
many of its chemical properties, however we draw a distinction between
rigid and flexible molecules. Flexible molecules may have symmetries that
are not merely a combination of rotations and reflections. Such symmetries
prompted the concept of the topological symmetry group of a graph embedded
in S
3
. We will discuss recent work on what groups are realizable as
the topological symmetry group for several families of graphs, including the
Petersen family and M¨obius ladders. "Regularization of nonlinear illposed problems with
applications to nonautonomous PDE’s of arbitrary even order"
Matthew Fury, Penn State Abington
Abstract: Illposed problems have a significant presence in several fields such as thermodynamics,
mathematical biology, and environmental science. The classic example is the
backward heat equation, i.e. the heat equation with a known final value. Solving this particular
problem involves the task of determining heat evolution in reverse time. A similar
problem is recovering the source of contamination within an already polluted body of water.
Because these problems are illposed with no systematic method of obtaining a solution,
several authors including Lattes and Lions, Showalter, Miller, and later Ames and Hughes
have considered approximation techniques such as the quasireversibility method.
In this talk, we show that a nonautonomous, nonlinear backward heat equation may be
regularized by replacing its model with a closelydefined wellposed model. This model follows
one first introduced by Boussetila and Rebbani and later modified by Tuan and Trong.
We first apply operator theory to gain a general result in Hilbert space and then apply our
findings to the nonlinear backward heat equation with nonconstant diffusion coefficient in
L
2
spaces. Finally, we extend these results to nonautonomous partial differential equations
of arbitrary even order. "Eigenvalue Distributions for the Hermitian TwoMatrix Model" Megan McCormick Stone, University of Arizona Abstract: The Gaussian Unitary Ensemble (GUE) is the collection of N × N Hermitian matrices
with random entries chosen from a Gaussian normal distribution. As N grows, the scaled
eigenvalues of the GUE follow a semicircle distribution. The Hermitian twomatrix model
is a generalization of the GUE. This model consists of pairs of Hermitian matrices equipped
with some probability distribution. Because the probability distribution for the twomatrix
model includes an interaction term, the techniques used to characterize the eigenvalues of
the GUE cannot be applied directly to the twomatrix model. The interaction term can be expressed, via spectral decomposition and a change of variables,
in terms of the HarishChandraItzyksonZuber (HCIZ) integral. A recent formula
due to Golden, GuayPaquet, and Novak connects the HCIZ integral to monotone Hurwitz
numbers, which count a specific class of ramified coverings of the sphere. Using the leading
order behavior of this formula, and after making assumptions about the coupling constant
used for the interaction term, the asymptotic behavior of the eigenvalues for the twomatrix
model can be characterized. In this talk, I will describe the necessary assumptions and
explain why they are reasonable assumptions to make for the twomatrix model.
"Injective choosability of subcubic planar graphs with girth 6"
Shanise Walker, Iowa State University
Abstract: An injective coloring of a graph G is an assignment of colors to the vertices of G so that any two vertices
with a common neighbor have distinct colors. A graph G is injectively kchoosable if it has an injective
coloring where the color of each vertex v of G can be chosen from any list L(v) of size k. Injective colorings
have applications in the theory of errorcorrecting codes and are closely related to other notions of colorability.
We show that a subcubic planar graph with girth at least 6 is injectively 5choosable, which improves several
known bounds on the injective chromatic number of planar graphs
"Classifying unipotent matrices in the symmetric space of SL2(Fq)" Carmen Wright, Jackson State University Abstract: Symmetric spaces for real matrix groups were orginally studided by Elie Cartan ang generalized by Berger. A generalized symmetric space is a homogeneous space Q = {gθ(g) −1 g ∈ G} where θ is an involution, an automorphism of order 2. A prposed conjecture for the symmetric space Q is that it can be decomposed into semisimple and unipotent components. We will show that this is true for SL2(Fq) and discuss the characterization of the unipotent matrices. SMPosium: A celebration of the Summer Mathematics Program for Women
(Organizers: Alissa S. Crans, Pamela A. Richardson) "Braids in Knot Theory and Contact Geometry" Diana Hubbard, University of Michigan Abstract: Braid theory is a rich area of study with many connections to different mathematical
fields. As a lowdimensional topologist, I am particularly interested in two perspectives on braids:
braids in knot theory and braids appearing in contact geometry. In this talk I will explain how
these two points of view have informed each other, and I will discuss some results that lie at their
intersection. "Models of Expander Graphs in terms of Random Maps" Angelica Gonzalez, University of Arizona Abstract: Expander graphs are graphs that are both highly connected and sparse (in terms of
the number of edges). Satisfying these two conflicting properties has proven to be useful in many
mathematical, computational, and physical contexts. For example, expander graphs are useful in the
design and analysis of communication networks, theory of error correcting codes, and convergence of
Markov chains. The quality of a graphs efficiency in this sense can be directly related to the spectrum
of its adjacency matrix. In this talk, we will explain this relationship and explore these notions for a
specific class of graphs that are directly related to oneface maps. In particular, we will see how the
genus of oneface maps plays a crucial role in this analysis. "Bounding Lengths in Hahn Fields" Karen Lange, Wellesely College Abstract: A standard power series either has finitely many terms or the terms are ordered as the
natural numbers. Thus, we can think of any standard power series as having finite length
or the length of the first infinite ordinal. Hahn fields, generalizations of standard power
series fields, consist of formal power series of arbitrary ordinal length. Given a field K and
an ordered abelian group G, let K((G)) be the Hahn field with terms of the form agt
g
, for
g ∈ G and ag ∈ K. If K is an algebraically closed field and G is a divisible ordered abelian
group, the Hahn field K((G)) is algebraically closed as well. Thus, given a polynomial
over such a Hahn field, it is natural to try to bound the lengths of its roots in terms of
the lengths of the polynomial’s coefficients. We discuss bounding results related to this
question (after briefly describing Hahn fields) and our motivation for tackling them. "Unipancyclic Matroids" Erin McNicholas, Willamette University Abstract: A uniquely pancyclic (UPC) graph is a graph on n vertices
with exactly one cycle of each size from 3 to n, The search
for such graphs was first proposed in 1973, and since that time only a handful
have been found. In this talk we broaden the search to UPC
matroids. Matroids generalize notions of dependence and independence
found in various fields including linear algebra and graph theory.
Results presented in this talk include work done over three summers as part of
the Willamette Math Consortium REU. "Genus one knots and their derivatives" Carolyn Otto, University of WisconsinEau Claire Abstract: In this talk we will discuss the relationship of genus one knots and their derivatives.
Specifically, we will prove that if a knot is algebraically slice and genus one, we will
always be able to find a derivative of the knot with an arf invariant of zero. Using this
result, we will be able to show that there are families of knots, that are created by an
infection operator, that admit a derivative with vanishing arf invariant. We will end by
showing how this result generalizes to higher genus knots. "Doppelgangers: Bijections of Plane Partitions" Rebecca Patrias, Universite du Quebec a Montreal Abstract: We introduce the Kjeu de taquin algorithm of Thomas and Yong and use it to give a bijection between certain plane partitions, thereby solving a 30yearold combinatorial mystery. This is joint work with Zachary Hamaker, Oliver Pechenik, and Nathan Williams. "Quantum ZeroKnowledge Protocols" Mariel Supina, University of California, Berkeley Abstract: Zeroknowledge protocols (ZKPs)
are interactive protocols in which a prover convinces a verifier of the truth
of a statement without revealing any information to the verifier other than the
fact that the statement is true. First developed by Goldwasser, Micali, and
Rackoff in 1985, such protocols have many practical uses. ZKPs are particularly
applicable to cryptography (for example, when one party wishes to prove that
they possess a private key without conveying any information about that key). Proofs
that a given protocol is zeroknowledge often fail if we allow the verifier to
be a quantum computer, since many such proofs involve “rewinding” (reverting
the verifier to a previous internal state), which is impossible on a quantum
computer. Watrous (2009) and Unruh (2015) have developed techniques to
circumvent this issue. In this talk, I will discuss such techniques and explore
some open problems related to quantum zeroknowledge protocols. "Using Data from
Longitudinal Observations to Portray How Motor Development Impacts Infants’
Sleep" Calandra Tate Moore, US Government Abstract: Sleep functions to consolidate newly learned information and
skills into memory. Researchers of motor
development aim to understand developmental changes that contribute to adaptive
control of motor actions such as sitting, crawling, and walking. Infant sleep patterns vary greatly for many
reasons but seldom do we contemplate the influence of sleep on development. Prior work indicates that for some motor skills,
sleep measures such as the number of nighttime wakings, may indicate changing
motor ability.
This talk describes a microgenetic approach to studying the relationship between motor development and sleep using data from a larger longitudinal study to document infants’ sleep experiences and motor abilities on a daily basis. Details of this longitudinal study will be discussed, as well as the role of statistical methods in datamining, relationship discovery, and timeseries analysis to appropriately classify patterns that characterize the complex relationship between sleep and motor development. The many facets of statistics  applied, pure and BIG
(Organizers: Monica Jackson, Jo Hardin) "Measuring Teacher Effectiveness Using ValueAdded Models" Anna Bargagliotti, Loyola Marymount University Abstract: Measuring quality of teaching is a difficult task. Education policy has pushed to find direct measures of teacher effectiveness. Two such measures include teacher observations and looking at outcomes of a teachers’ students. The latter methods looks to model the value that a teacher has added over the course of a year. This talk will discuss the nuts and bolts of valueadded models and, with a toy data set, show how valueadded measures for a set of teachers can be estimated."Race and causal
inference in health disparities research: are we just going around in circles?" Emma Benn, Icahn School of Medicine at Mount Sinai Abstract: While researchers examining racial/ethnic disparities in
health are often wellintentioned, the underlying conceptual framework used to
conduct their studies do not always get us closer to finding a mechanistic mode
of intervention. This stems from the fact that the common approach to these
types of studies may lack some core statistical principles of causal inference,
thus resulting in circular conclusions that merely reinforce the notion that
racial/ethnic differences in health outcomes exist. In this talk, I intend to
discuss some important statistical considerations for health disparities research
that might move us away from describing racial/ethnic differences and closer to
eradicating them. "Base
Calling, Binning, SNP Calling on Metagenomic Sequencing Data" Xinping Cui, University of California, Riverside Abstract: Recently,
the emerging new field of metagenomics facilitated by the advent of
nextgeneration sequencing (NGS) technology enables genome sequencing of
unculturable and often unknown microbes in natural environments, offering
researchers an unprecedented opportunity to delineate biodiversity of any
microbial organism. While the sequencing technologies are evolving at
unprecedented speed, researchers engaged in this enterprise
are facing major computational, algorithmic and statistical challenges in the
analysis of the massive metagenomic data. In this talk, I will introduce a new
integrated statistical and computational pipeline empowered by high performance
computing that consists of (1) basecalling; (2) binning; and (3) SNP detection
on NGS sequencing data. "Modelfree Knockoffs for Highdimensional Controlled Variable Selection" Yingying Fan, University of Southern California Abstract: Many contemporary largescale applications involve building interpretable models linking a large set of
potential covariates to a response in a nonlinear fashion, such as when the response is binary. Although
this modeling problem has been extensively studied, it remains unclear how to effectively control the
fraction of false discoveries even in highdimensional logistic regression, not to mention general highdimensional
nonlinear models. To address such a practical problem, we propose a new framework of
modelfree knockoffs, which reads from a different perspective the knockoff procedure (Barber and
Candes, 2015) originally designed for controlling the false discovery rate in linear models. The key
innovation of our method is to construct knockoff variables probabilistically instead of geometrically.
This enables modelfree knockoffs to deal with arbitrary (and unknown) conditional models and any
dimensions, including when the dimensionality p exceeds the sample size n, while the original knockoffs
procedure is constrained to homoscedastic linear models with n ≥ p. Our approach requires the design
matrix be random (independent and identically distributed rows) with a covariate distribution that is
known, although we show our procedure to be robust to unknown/estimated distributions. To our
knowledge, no other procedure solves the controlled variable selection problem in such generality, but
in the restricted settings where competitors exist, we demonstrate the superior power of knockoffs
through simulations. Finally, we apply our procedure to data from a casecontrol study of Crohn's
disease in the United Kingdom, making twice as many discoveries as the original analysis of the same
data. "Correlation induced by missing spatial covariates" Monica Jackson, American University Abstract: Residual spatial correlation in linear models of
environmental data is often attributed to spatial patterns in related
covariates omitted from the fitted model. We connect the nonunique
decomposition of error in geostatistical models into trend and covariance
components to the similarly nonunique decomposition of mixed models into fixed
and random effects. We specify spatial correlation induced by missing spatial
covariates as a function of the strength of association and (spatial)
covariation of the missing covariates. "Integrating Mathematics and Statistics into the Data Science Curriculum" Stacey Hancock, Montana State University Abstract: With the rise of “big data,” the past few years have seen the rapid growth of
undergraduate, graduate, and professional programs in data science. Indeed, there is a need for
such programs. The widely quoted McKinsey Global Institute Study on Big Data in 2011 reports
that “the United States alone faces a shortage of 140,000 to 190,000 people with deep analytical
skills as well as 1.5 million managers and analysts to analyze big data and make decisions based
on their findings. The shortage of talent is just beginning.” Since the definition of “data science”
is still evolving, the core courses in data science curricula range from business to mathematics
and statistics to foundational computer science. This talk will explore how mathematics and
statistics play a role in data science curricula and how to leverage the core analytical skills honed
in mathematics and statistics to educate the next generation of data scientists. "Statistical Approaches in Personalized Medicine using Nonparametric
Parameter Estimation" Alona Kryshchenko, California State University Channel Islands Abstract: Modeling drug behavior is a very complicated task since every person responds to a
drug in his or her own unique way. Pharmacokinetics is the study of drug behavior,
from the moment that it is administered up to the point at which it is completely
eliminated from the body. Pharmacokinetic population models are very complex
and high dimensional. Most methods that are developed in this area use parametric
approaches to estimate distributions of population mixture models. These methods
limit the search for estimates only under assumptions of specific types of
distributions. The nonparametric methods do not make any underling assumptions
on distributions and allow users to estimate multimodal and longtailed
distributions, which commonly occur in populations with different genotypes. In
this talk, I will describe nonparametric methods for estimating distributions of
parameters of various population models and their applications. "Don’t Count on
Poisson: Introducing a flexible
alternative distribution to model count data" Kimberly Sellers, Georgetown University Abstract: The Poisson
distribution is a popular model for count data. Its constraining
equidispersion assumption (where the variance and mean equal), however, limits
its usefulness. The ConwayMaxwellPoisson (CMP) distribution, instead, is a
flexible alternative count distribution that accommodates data over or
underdispersion (where the variance is larger or smaller than the mean), capturing
three classical distributions as special cases. This talk will introduce the
statistical properties of this distribution, and survey the diverse methods
work that has been developed with this model as motivation. "TimeDynamic Profiling with Application to Hospital Readmission Among Patients on
Dialysis" Damla Senturk, University of California, Los Angeles Abstract:Standard profiling analysis aims to evaluate medical providers, such as hospitals,
nursing homes or dialysis facilities, with respect to a patient outcome. The outcome, for
instance, may be mortality, medical complications or 30day (unplanned) hospital readmission.
Profiling analysis involves regression modeling of a patient outcome, adjusting for patient
health status at baseline, and comparing each provider's outcome rate (e.g., 30day
readmission rate) to a normative standard (e.g., national ``average''). To date, profiling
methods exist only for non timevarying patient outcomes. However, for patients on dialysis, a
unique population which requires continuous medical care, methodologies to monitor patient
outcomes continuously over time are particularly relevant. Thus, we introduce a novel timedynamic
profiling (TDP) approach to assess the timevarying 30day readmission rate. TDP is
used to estimate, for the first time, the riskstandardized timedynamic 30day hospital
readmission rate, throughout the time period that patients are on dialysis. We develop the
framework for TDP by introducing the standardized dynamic readmission ratio as a function of
time and a multilevel varying coefficient model with facilityspecific timevarying effects. We
propose estimation and inference procedures tailored to the problem of TDP and to overcome
the challenge of highdimensional parameters when examining thousands of dialysis facilities. "Linear mixed effect models and genegene interaction: Something
old, something new ..." Janet Sinsheimer, University of California, Los Angeles Abstract: Linear mixed effect
models (LMMs) have a long history in genetics, going back at least as far as
when R. A . Fisher proposed the polygenic model. However, quite recently
LMMs surged in popularity for omic studies and in particular for genome wide
association studies. In my talk, I will review what makes these models so
popular now in genomics, discuss my groups’ recent work with LMMs to detect
maternal gene by offspring gene interactions, and then touch on some open
questions. "Intuition
to Modern Statistical / Machine Learning: An Illustration in a BIG Problem" Zhaoxia Yu, University of California, Irvine Abstract: We humans use intuition in our daily life. For example, INTUITIVELY,
students with similar performance spend similar effort in learning. This
SimilarXSimilarY (SXSY) intuition, when blended with rigorous statistical
modeling, mathematical derivations, and computational algorithms, can become a
powerful modern learning tool to understand the interplay between multiple sets
of massive, complexly structured, and highdimensional data. As an
illustration, I will present how the SXSY intuition can be applied to investigate
the connection between Brain, Imaging, and Genetics (BIG). Our preliminary
results suggest that a person’s neuroimaging profile, like his or her human
genome, is a signature and is associated with the person’s genetic profile.
History of Mathematics
(Organizer: Janet Beery) "G.H. Hardy and the Reform of Mathematics Education at Cambridge circa 1910" Brenda Davison, Simon Fraser University Abstract: Mathematics training at Cambridge prior to 1907 was centered on preparing students to sit
examinations called the mathematical Tripos. Highly competitive, students were ranked by their
performance and their future career prospects depended on their results. Dissatisfaction with this
system – particularly with the order of merit – led to a sweeping reform in 1910, a reform in which
Hardy played a major role. The Tripos system, Hardy claimed, when at its zenith in terms of notoriety, difficulty and
complexity, was the very time when English mathematics was at its lowest ebb. I will discuss the
historical context of this change, and the role of G.H. Hardy, in abolishing a system that put the
mathematical training to equip a student to become a research mathematician completely secondary
to examination preparation. "It’s All for the Best”: Optimization, Theology, Calculus, and Science" Judith Grabiner, Pitzer College Abstract: Many problems, from optics
to economics, are solved mathematically by finding the highest, the quickest,
the shortest – the best of something.
This has been true from antiquity to the present. We’ll look at why scientists started looking
for such explanations, examples of how the approach progressed from optics,
mechanics, economics, and theology, and the roles played by Heron of
Alexandria, Fermat, Leibniz, Maclaurin, and Adam Smith. "Incorporating Contributions of Women and Minorities in Classrooms:
David Blackwell, Evelyn Boyd Granville and Mary Gray" Sarah Greenwald, Appalachian State University Abstract: Stories of mathematicians and statisticians and their contributions can
help students connect to mathematics and inspire them. We’ll discuss
how to incorporate these into a variety of classes including linear
algebra, senior capstone, and general education courses. We’ll
examine the benefits and challenges in addition to student responses
as we look at examples related to David Blackwell, Evelyn Boyd
Granville and Mary Gray. Interviews abound in the existing literature,
and I’ve also personally communicated with each of these individuals
(David Blackwell is deceased but I communicated with him in the early
2000s). For more information, see
http://cs.appstate.edu/~sjg/history/wmm.html "Learning and Teaching
Mathematics in World War II Poland: Experiences of Three Daring Women" Emelie Kenney, Siena College Abstract: Poland is known as having had the largest underground during
World War II, with this underground involving vibrant, determined women and
men. In the area of mathematics, we find clandestine teaching at all levels of
education in addition to a secret focus on earning degrees at the gymnasium,
undergraduate, and graduate levels. In this talk, I would like to present the lives
and accomplishments of three specific women: Zofia Krygowska, who worked as a
teacher, student, and organizer during the war, as well as a founder of
didactics afterwards; Zofia Szmydt, a student and later a successful
differential equations specialist; and Irena Golab, a teacher of mathematics,
who ran clandestine classes in her home for younger students. Everything that
such women did during World War II put their lives at risk at the hands of the
Nazis, but no such terrible risks prevented them from their quests to educate
themselves and others. "Thou Shalt not envy: A Sperner’s Lemma Guide to fair division" Deborah Kent, Drake University Abstract: In 1928, Emmanuel Sperner proved an elegant, graphtheoretic result: Every properly colored
simplicial subdivision contains a cell whose vertices have all different colors. Sperner arrived at
this surprisingly useful result while studying dimensionality of Euclidean spaces. In this talk,
Sperner’s Lemma will provide a neat solution to Gametheoretic questions of equitable and
envyfree division. This lemma is also central to a proof of the Nash Equilibrium Theorem and
the result that the game Hex will never end in a tie. "Certain Modern Ideas and Methods: Charlotte Angas Scott's Philosophy of Mathematics" Jemma Lorenat, Pitzer College Abstract: While mostly known for her role in breaking gender barriers at Cambridge, educating
doctoral students at Bryn Mawr, and systematizing analytic geometry, Charlotte Angas
Scott intersected with many of the most prominent figures in the philosophy of mathematics
at the turn of the twentieth century. She vetted Bertrand Russell for his 1896
lecture series at Bryn Mawr. She reported on both David Hilbert and Henri Poincar´e’s
methodological addresses at the International Congress of Mathematicians in Paris for the
American Mathematical Society in 1900. And at her endowed chair celebration in 1922,
Alfred North Whitehead delivered the address. Far from a mere bystander to these events,
Scott’s outspoken and multifaceted writing delves into philosophical questions. This talk
will consider three different manifestations of Scott’s philosophy: standards for women’s
mathematics education, the relationship between algebra and geometry, and the ontology
of imaginary points. This talk focuses on one individual, but it also raises the broader
question of who counts as a philosopher in the history of mathematics. "The Enduring Legacy of Mario Pieri (18601913)" Elena Marchisotto, California State University Northridge Abstract: Mario Pieri
(18601913) has been called “a true bridge” between the two most prestigious
Italian schools of mathematics which flourished at the University of Turin at
the turn to the twentieth century  the
research groups of Corrado Segre and Giuseppe Peano. Pieri left a legacy of results in algebraic
and differential geometry, vector analysis, foundations of mathematics
(elementary, projective and inversive geometries, as well as arithmetic), logic
and the philosophy of science.
In my talk, I plan to present a brief synopsis of the panorama of Pieri’s work, providing a picture of the context in which it was produced. In focusing on some of his more noteworthy results, I hope to convey the depth and breadth of Pieri’s mathematics as well as the challenges I have encountered in researching his work, for more than two decades, and attempting to give it its rightful place in the history of mathematics. "The KriegerNelson Prize Lectureship"Laura Turner, Monmouth University Abstract: The KriegerNelson Prize Lectureship honours outstanding research by women
members of the Canadian mathematical community. First awarded in 1995, it is named after
Cecilia Krieger (1894–1974), the first woman to earn a Ph.D. in mathematics from a Canadian
university, and Evelyn Nelson (1943–1987), a prolific researcher in universal algebra. In this talk,
we explore the origins and early history of this prize, from the contributions of its namesakes to
the motivations behind the prize itself.
Commutative Algebra
(Organizers: Emily Witt, Alexandra Seceleanu) "Free complexes on smooth toric varieties" Christine Berkesch Zamaere, University of Minnesota Abstract: Given a module M over the Cox ring of a smooth toric variety, one can consider free complexes
that are acyclic for M modulo irrelevant homology. These complexes have many
advantages over minimal free resolutions over smooth toric varieties other than projective
spaces. We develop this in detail for products of projective spaces. This is joint work with
Daniel Erman and Gregory G. Smith. "Integrable Derivations of Some Hypersurfaces in Characteristic p>0" Eleanore Faber, University of Michigan Abstract: Let k be a commutative ring and A a commutative kalgebra. A klinear derivation
δ of A is called nintegrable, where n is a positive integer or n = ∞, if it extends up
to a Hasse–Schmidt derivation of A over k of length n. In this talk let k be a field of characteristic p > 0. While over a field of characteristic
0 any derivation is integrable, this question is much more delicate over positive
characteristic fields. We study IDerk(A; n), the module of nintegrable derivations
along X = Spec(A), for some classes of quasihomogeneous hypersurfaces X. We can
describe when elements of IDerk(A; n) are no longer n+1integrable and show that for
our classes of singularities the chain of inclusions IDerk(A; n) ⊇ IDerk(A; n+1) always
becomes stationary, that is, all nintegrable derivations are ∞integrable for n . 0.
In particular, we can explicitly determine the integers n, for which socalled jumps
appear, that is, IDerk(A; n) ) IDerk(A; n + 1). These jumps seem to be interesting
new invariants of A. This is joint work with Ang´elica Benito. "Asymptotic Behavior of Certain Koszul Cohomology Modules" Patricia Klein, University of Michigan Abstract: Let (R, m) be a local ring, M a finitely generated module over R, and
f1, . . . , fd a system of parameters on M. Lech’s limit formula states that as mini ti → ∞
`(M/(f
t1
1
, . . . , ftd
d
)M)
t1 · · ·td
−→ e(f1, . . . , fd  M),
the multiplicity of (f1, . . . , fd) on M. One may ask whether powers of a fixed sequence
of parameters may be replaced in this formula by any sequence of parameter ideals In
such that In ⊆ mn
. Recalling that the multiplicity may be realized as the alternating
sum of the lengths of Koszul cohomology modules and that Hn
(f
t1
1
, . . . , ftd
d
 M) ∼=
M/(f
t1
1
, . . . , ftd
d
)M, we may rewrite Lech’s limit formula as follows
Pn
j=0(−1)n−j
`(Hi
(f
t1
1
, . . . , ftd
d
; M))
`(Hn(f
t1
1
, . . . , ftd
d
))
−→ 1.
From this point of view, it is also natural to ask in the case when dim M = dim R = d
for which i < d we have `(Hi
(In; M))/`(R/InR) → 0. In this talk, we will consider
the latter question. The main result is that when M is faithful, the M satisfying the
condition that `(Hi
(In; M))/`(R/InR) → 0 for all i < d are exactly those M that are
CohenMacaulay on the punctured spectrum. "Trace ideals of modules and algebras over
commutative rings" Haydee Lindo, Williams College Abstract: I will present some new results regarding trace ideals of modules and algebras
over commutative rings. This continues the project begun in arXiv:1603.08576 relating
the center of the endomorphism ring of a module M, over a commutative noetherian ring,
to the endomorphism ring of the trace ideal of M. "Adams operations for matrix factorizations and a conjecture of Dao and Kurano" Claudia Miller, Syracuse University Abstract: Using an idea of Atiyah from 1966, we develop Adams operations on the Grothendieck groups of perfect complexes with support and of matrix factorizations using cyclic group actions on tensors powers. In the former setting, Gillet and Soule’ developed these using the DoldKan correspondence and used them to solve Serre's Vanishing Conjecture in mixed characteristic (also proved independently by P. Roberts using localized Chern characters). Their approach cannot be used in the setting of matrix factorizations, so we use Atiyah's approach, avoiding simplicial theory altogether. As an application, we prove a conjecture of Dao and Kurano on the vanishing of Hochster's theta pairing for pairs of modules over an isolated hypersurface singularity in the remaining open case of mixed characteristic. Our proof is analogous to that of Gillet and Soule’ for the vanishing of Serre's intersection multiplicity. This is joint work with Michael Brown, Peder Thompson, and Mark Walker."The Frobenius Complexity of Hibi Rings" Janet Page, University of Illinois at Chicago Abstract: Cartier algebras and their duals, rings of Frobenius operators, have come up in the study of Frobenius
splittings, which have been useful in many topics ranging from singularity theory in algebraic geometry to
representation theory. When R is a local ring of characteristic p > 0, the Cartier algebra C(R), which is the
ring of all potential Frobenius splittings of R, is dual to the ring of Frobenius operators (p
e
linear maps) on
the injective hull of the residue field. This ring of Frobenius operators need not be finitely generated over R,
which led Enescu and Yao to define Frobenius complexity as a measure of its nonfinite generation. In their
examples Frobenius complexity is not always even rational, but its limit as p → ∞ is an integer. Few other
examples have been computed. In this talk, I will discuss a method to compute limit Frobenius complexity
for Hibi rings, which are a class of toric rings defined from finite posets. I will show that this computation
can be read directly from the defining poset in nice cases. "Intersection Algebras of Noetherian Rings and Their Properties" Sandra Spiroff, University of Mississippi Abstract: The intersection algebra of a commutative Noetherian ring R with
respect to two ideals I, J is BR(I, J) = L
r,s∈N
I
r∩J
s
. It was defined by J. B. Fields
in 2002, who showed that it is finitely generated when I and J are monomial ideals
in a polynomial ring in finitely many variables over a field. In the case that I and
J are principal monomial ideals, we obtain explicit formulæ for certain invariants
of BR(I, J), dependent upon families of parameters. Specifically, we will discuss
the HilbertSamuel and HilbertKunz multplicities, the divisor class group, and the
Fsignature. This is joint work with F. Enescu at Georgia State. "Generalized mixed
multiplicities" Yu Xie, Pennsylvania State, Altoona Abstract: Mixed multiplicities of 0dimensional ideals can be traced back to Teissier’s Carg`ese paper in
1973, where he and Risler used this notion to interpret the Milnor numbers which were used to control
the Whitney equisingularity of families of complex analytic hypersurfaces having only isolated
singularities. Kirby and Rees as well as Kleiman and Thorup generalized mixed multiplicities to
0dimensional modules, namely mixed BuchsbaumRim multiplicities, to study families of complex
analytic complete intersections having only isolated singularities. In 2001, Trung extended
the mixed multiplicities to the case where one ideal is 0dimensional and the other one is arbitrary,
and therefore he generalized the classical Milnor numbers to complex analytic hypersurfaces having
nonisolated singularities. In this talk, I will show how we extend the concept of mixed multiplicities
to arbitrary ideals and modules, how we computes these numbers, their properties and applications.
Biological Oscillations Across Time Scales
(Organizers: Stephanie Taylor, Tanya Leise)
"Piecewise smooth maps for the circadian modulation of sleepwake dynamics" Victoria Booth, University of Michigan Abstract: The timing of human sleep is strongly modulated by the 24 h circadian rhythm, and desynchronization of
sleepwake cycles from the circadian rhythm can negatively impact health. We have developed a
physiologicallybased mathematical model for the neurotransmittermediated interactions of sleeppromoting,
wakepromoting and circadian rhythmgenerating neuronal populations that govern sleepwake
behavior in humans. To investigate the dynamics of circadian modulation of sleep patterns and of
entrainment of the sleepwake cycle with the circadian rhythm, we have reduced the dynamics of the
sleepwake regulatory network model to a onedimensional map. The map dictates the phase of the
circadian cycle at which sleep onset occurs on day n + 1 as a function of the circadian phase of sleep
onset on day n. The map is piecewise continuous with discontinuities caused by circadian modulation of
the duration of sleep and wake episodes and the occurrence of rapid eye movement (REM) sleep
episodes. Analysis of map structure reveals changes in sleep patterning, including REM sleep behavior, as sleep
occurs over different circadian phases. In this way, the map provides a portrait of the circadian
modulation of sleepwake behavior and its effects on REM sleep patterning. Using the map, we can
analyze bifurcations of the sleepwake regulatory network model to understand how variations in REM
sleep propensity and the homeostatic sleep drive affect human sleep patterning. Tanya Leise, Amherst College
Abstract: Circadian clocks track
internal time in most organisms on earth and are generated by feedback loops of
clock gene expression. We’ll take a look at analysis of circadian oscillations in
behavioral and molecular records of mice, fruit flies, and brown bears,
employing a variety of methods ranging from autocorrelation to wavelet
transforms. In mice and flies, we can track expression of a key clock gene,
while in brown bears we have records of activity and body temperature rhythms.
Our data for these noisy biological oscillators often include relatively few
cycles, so that reliable estimation of period can be quite challenging. The
phase relationships between different rhythms in the same organism, e.g.,
between temperature and activity or between intracellular calcium levels and
clock gene expression, are also of interest, as well as transient changes in
relative phase following a disruption, potentially yielding insight into how
such rhythms might be coupled. "Hippocampal sleep rhythms and memory
reactivation: a computational study" Paola Malerba, University of California San Diego Abstract: During slowwave sleep, memories are consolidated
in a dialogue between cortex and hippocampus. Although the specific mechanisms
of sleepdependent consolidation are not known, the reactivation of specific
neural activity patterns – replay – during slow wave sleep has been observed in
both structures and is believed to represent a neuronal substrate of
consolidation. In the hippocampus, replay happens during sharp wave –
ripples, short bouts of excitatory activity in area CA3 which induce high
frequency oscillations in area CA1. Despite the importance of replay within the
broader phenomenon of sleepmediated memory consolidation, the neural mechanisms
underlying hippocampal sequence replay are still unknown.
In this work, we develop a model of hippocampal spike sequence replay during sleep. We represent CA3 and CA1 activity with a simplified network model of synaptically coupled pyramidal and basket cells. Noiseinduced activation of CA3 pyramidal cells triggered an excitatory cascade, with size controlled by the spread of recurrence in the network. Sharp waves in CA3 resulted in strong excitatory input to area CA1, inducing local ripples. Sharpwave ripples occur stochastically in the model, and their location and size depend on the convergence between pyramidal cells connections within CA3. The projections from CA3 to CA1 – Schaffer Collateral – induce coordination between spiking regions in CA1 and CA3, so that localized sharpwave CA3 events produce consistently localized CA1 ripples. In our model, we study the spontaneous reactivation of CA1 and CA3 pyramidal cells. "Datadriven models of human brain dynamics"Sarah Muldoon, University at Buffalo, SUNY Abstract: Understanding the brain as a complex network of interacting components allows for useful insights into brain function, and computational modeling provides a controlled environment to test theoretical predictions of brain network structure. In this talk, I'll describe work using datadriven computational modeling of brain dynamics to examine individual differences in brain activation and task performance. The computational model is built on structural brain networks derived from diffusion spectrum imaging data, and regional brain dynamics are modeled using biologically motivated nonlinear WilsonCowan oscillators. We find that, based on the global versus local spread of activation throughout the brain and/or taskspecific subnetworks, we are able to predict individual performance across three different language tasks. Thus, by emphasizing differences in the underlying structural connectivity, our model serves as a powerful tool to examine structurefunction relationships in dynamic brain networks. "A Model for Menstrual Cycle Follicle Waves with Applications"Nicole Panza, Francis Marion University Abstract: Ovarian follicle waves have been reported in women by Baerwald et al. (2003).
Typically two or three waves occur per cycle. Two nonlinear differential equation
models which represent the hormonal regulation of the menstrual cycle for two
and three follicle waves per cycle are presented. The model exhibits waves of
antral follicles during a woman’s cycle using a Follicle Stimulating Hormone
threshold function. The model is used to explore phenomenon such as
superfecundation. "The flexible coordination of hippocampal neurons in rhythms" Lara Rangel, University of California, San Diego Abstract: During successful computation, brain regions must have an efficient method for filtering information from multiple sources and coordinating communication with other regions. A great circuit for examining this is the hippocampus, a brain structure critical for learning and memory that must integrate and associate information arriving from multiple sources. Research conducted by Dr. Lara Maria Rangel suggests that the successful processing of information from multiple afferents in the hippocampus is dependent on coordinated oscillatory activity, and more specifically the engagement of hippocampal cells in their surrounding rhythmic circuits. Dr. Rangel is a systems neuroscientist, whose work characterizes the temporal dynamics of crossregional oscillatory interactions and the flexible participation of neurons in local rhythmic networks during behavior. "Analysis and Models of Spontaneous Activity in the Lateral Line of Zebrafish" Nessy Tania, Smith College Abstract: Temporal patterns of spontaneous activity may vary between sensory systems such
as the auditory, vestibular, and lateral line systems due to differences in physiology
at the level of hair cells. In the absence of stimuli, hair cells display spontaneous
synaptic vesicle fusion and neurotransmitter release, which lead to action potential
(spike) generation in innervating afferent neurons. We will discuss properties of the
distribution of interspikeintervals (ISI) from spontaneous spiking data recorded
from the lateral line of zebrafish (collected by the lab of Josef Trapani, Biology,
Amherst College). Additionally, successive ISI's in the lateral line recordings tended
to have positive serial correlation, i.e., successive ISI pairs were either short/short
or long/long. This pattern contrasts previous findings from the auditory system
where ISI's tended to have negative serial correlation presumably due to the effects
of synaptic depletion. We have built a computational model of spike generation that included the calciumdependency
of neurotransmitter release at the ribbon synapse of hair cells. The
model can generate ISI distributions consistent with experimental data. Numerical
simulations suggest that fluctuations in total calcium channel activity, including
both the number and cooperativity of channels in the population, are a primary
contributor to serial correlations in haircell evoked spike trains. Given the
difference in innervation pattern between auditory and vestibular/lateral line hair
cells, we further modeled the effects of single versus multiple synapses on the
temporal patterns of spontaneous spike trains. Altogether, our findings provide
evidence for how physiological similarities and differences between the auditory,
vestibular, and lateral line systems can account for differences in spontaneous
activity. "Using Augmented PhaseAmplitude Oscillators to Infer Directed Connections between Regions of the Mouse Circadian Clock" Stephanie Taylor, Colby College Abstract: The master clock that controls the daily rhythms in mouse behavior is a multioscillator composed of thousands of individual oscillators that synchronize via intercellular communication. The oscillators can be separated into two regions  the shell and core. The network which links oscillators within the regions and between the regions is the subject of ongoing study. Data are too sparse to infer connections between individual oscillators, but, with the aid of mathematical modeling, are not too sparse to infer the direction of connections between the two regions. We augmented a traditional phaseamplitude oscillator model to simulate experiments in which the communication network of a mouse circadian clock was destroyed and restored. Using experimental evidence and simulations of the dynamics upon restoration, we infer that the core entrains the shell. In this talk, we describe the model, why the augmentation was necessary, and our results. Geometric Group Theory
(Organizers: Pallavi Dani, Tullia Dymarz, Talia Fernos) "Understanding some reducible outer automorphisms of the free group" Radhika Gupta, University of Utah Abstract: In analogy to the action of the mapping class group on the curve complex, the group of
outer automorphisms of the free group acts on the free factor complex. A fully irreducible outer
automorphism acts with positive translation length on the free factor complex but a reducible
element acts elliptically. I will discuss some spaces on which the action of reducible elements is
more interesting. "Splittings of mapping tori of linearly growing
automorphisms of free groups" Natasa Macura, Trinity University Abstract: Guirardel and Levitt Guirardel defined a tree of cylinders
Tc for a tree T with an action of a finitely generated group G. This tree
only depends on the deformation space of T, and is invariant under the
automorphisms of G if T is a JSJ splitting. We discuss cyclic splittings of
mapping tori of linearly growing automorphisms of free groups and describe
trees of cylinders for these splittings. This is joint work with C. Cashen. "Quasiisometric Boundary Swapping" Molly Moran, Colorado College Abstract: Bestvina formalized the concept of a group boundary by introducing
the notion of a Zstructure on a group. In his initial paper,
Bestvina proved a boundary swapping theorem that can be applied
to a group G with a finite K(G, 1). He also suggested that a generalized
version of boundary swapping should hold for two groups that
are quasiisometric. We will present a generalization of this result
and discuss some of the implications. This is joint work with Craig
Guilbault. "Normal Forms for DiestelLeader Groups" Anisah Nu'Man, Ursinus College Abstract: DiestelLeader graphs where intially introduced in 2001 by Diestel and
Leader as a potential answer to the following question posed by Woess: “Is every
connected, locally finite, vertex transitive graph quasiisometric to some Cayley
graph? Let Γd(q) denote the group whose Cayley graph, with respect to a certain
finite generating set Sd,q, is the DiestelLeader graph as constructed by Bartholdi,
Neuhauser and Woess. In the case when d = 2 these groups are the well known
lamplighter groups Lq = Zq o Z whose Cayley graph is the horocyclic product of
two trees of valence q + 1. Metric properties of DiestelLeader groups have been
studied by Stein and Taback, in which they provide a method for computing word
length in Γd(q) with respect to the generating Sd,q. In this discussion, we will build
upon Stein and Taback’s use of word length to construct a set of normal forms for
the DeisetelLeader group Γd(q) with respect to the generating set Sd,q. "Algebraic and topological properties of big mapping class groups" Priyam Patel, University of California, Santa Barbara Abstract: The mapping class group of a surface is the group of homeomorphisms of the surface
up to isotopy (a natural equivalence). Mapping class groups of finite type surfaces have
been extensively studied and are, for the most part, wellunderstood. There has been a
recent surge in studying surfaces of infinite type and in this talk, we shift our focus
to their mapping class groups, often called big mapping class groups. In contrast to the
finite type case, there are many open questions regarding the basic algebraic and topological
properties of big mapping class groups. Until now, for instance, it was unknown whether
or not these groups are residually finite. We will discuss the answer to this and several
other open questions after providing the necessary background on surfaces of infinite
type. This work is joint with Nicholas G. Vlamis. "Geodesics in Outer Space" Catherine Pfaff, University of California, Santa Barbara Abstract: Outer automorphisms of free groups are studied via their action on CullerVogtmann
Outer Space. I will introduce Outer Space and how geodesics in Outer Space
resemble and differ from geodesics in hyperbolic spaces and Teichm¨uller space. "Obstructions to Riemannian smoothings of a locally CAT(0) manifold" Bakul Sathaye, The Ohio State University Abstract: In this talk I will discuss obstructions to Riemannian smoothings of a locally CAT(0) manifold.
I will focus on obstructions in dimension = 4 given by DavisJanuszkiewiczLafont and show how
their methods can be extended to construct more examples of locally CAT(0) 4manifolds that do not
support Riemannian metric with nonpositive sectional curvature. Further, the universal cover of such
a manifold satisfies the isolated flats condition and contains a collection of 2dimensional flats with
the property that their boundaries at infinity form a nontrivial link in the boundary of the universal
cover. "Coarse and fine geometry of the Thurston metric" Jing Tao, University of Oklahoma Abstract: I will present results of a recent collaboration with Anna Lenzhen, David Dumas, and Kasra
Rafi in which we study the geometry of Thurston's metric on Teichmuller space. This is an
asymmetric metric based on the Lipschitz constants of maps between hyperbolic surfaces.
We study the coarse properties of Thurston metric geodesics in general, and some finer
properties (local isometric rigidity, quantitative nonuniqueness of geodesics) in the case of
the punctured torus.
Recent progress in Several Complex Variables
(Organizers: Purvi Gupta, Loredana Lanzani) "On the dimension of the Bergman space for some unbounded, pseudoconvex
domains" AnneKatrin Gallagher, Oklahoma State University Abstract: A sufficient condition for the infinite dimensionality of the Bergman space of a
pseudoconvex domain is given. This condition holds on any pseudoconvex domain that
has at least one smooth boundary point of finite type in the sense of D’Angelo. This is
joint work with T. Harz and G. Herbort. "Studying Hyperbolicity of Complex Domains" Samangi Munasinghe, Western Kentucky University Abstract: We are studying some aspects of hyperbolicity related to certain complex
domains. Our goal is to find some useful information about the
domains. "On the instability of an identity involving the Menger curvature" Malabika Pramanik, University of British Columbia Abstract: The Menger curvature is a geometric quantity that has proved
to be surprisingly useful in analytical problems. For instance, a symmetrization
identity involving the Menger curvature has led to new connections with
the Cauchy transform (a widely studied operator in real and complex analysis)
and analytic capacity (a concept in geometric measure theory). On
the other hand, it is believed that such identites are rare, and extremely
sensitive to changes in the underlying symmetrized kernel. In this talk, I
will report on ongoing work with Loredana Lanzani, where we provide a
quantification of this instability. "ChoquetMongeAmp`ere Classes" Sibel Sahin, Ozyegin University Abstract: In this talk we will consider a special class of quasiplurisubharmonic functions,
namely ChoquetMongeAmp`ere classes on compact K¨ahler manifolds.
These classes become a useful intermediate tool in the analysis of Complex
MongeAmp`ere operator with their small enough asymptotic capacity. We will
first characterize these classes through Choquet energy and then compare them
with the finite energy classes. We will see that over different singularity types
the comparison between ChoquetMongeAmp`ere classes and the finite energy
classes yields totally different characteristics. "The CauchyRiemann Equations in Complex Manifolds" MeiChi Shaw, University of Notre Dame Abstract: The purpose of this talk is to discuss the recent progress on the CauchyRiemann
equations in complex manifolds. We will examine the strong Oka’s Lemma and its role in
existence and regularity for ∂¯. Recent results on the L
2
closed range property for ∂¯ on an
annulus between two pseudoconvex domains will be reported. In particular, we show the
Hausdorff property of the L
2 Dolbeault cohomology group on a domain between a ball and
a bidisc, the socalled Chinese Coin problem. Characterization of Lipschitz domains with
holes through their Dolbeault cohomology groups will also be discussed. Thus one can hear
pseudoconvexity for domains with holes using L
2 Dolbeault cohomology. (joint work with
Debraj Chakrabarti, Siqi Fu, and Christine LaurentThi´ebaut). "On Torsion and Cotorsion of Differentials on Certain Complete Intersection Rings" Sophia Vassiliadou, Georgetown University Abstract: I will discuss some results, old and new, on the vanishing/nonvanishing of torsion and cotorsion
of K¨ahler differentials on certain complete intersection rings. Some geometric consequences of these
results will also be discussed. This is joint work with Claudia Miller.
"Parabolic skewproducts and parametrization" Liz Vivas, Ohio State University Abstract: It is a classical result that parametrizaion of unstable manifolds
on hyperbolic holomorphic maps can be obtained by a limit of iterates of
the map composed with an appropriate inverse action. In this talk I will
generalize this result for skewproduct invariant holomorphic maps that are
parabolic. I will first give an overview of the results known in one and several
complex dimensions.
Research in Collegiate Mathematics Education
(Organizers: Shandy Hauk, Paosheng Hsu) Specifically
designed for people who have advanced degrees in the mathematical sciences,
session activities will touch on what research suggests about thinking and
learning across the college curriculum from college algebra to calculus,
combinatorics, proof, and more. Speakers will communicate the landscape of
current research in undergraduate mathematics education as well as offer useful
information for present and future faculty members. The goal is to generate lively conversations about
the foundations and implications of collegiate mathematics education research. "Unearthing students’ problematics through proof scripts" Stacy Brown, California State Polytechnic University, Pomona Abstract: In this talk I will share findings from a study that explored students’ reasoning about the “within
argument contradictions” that arise from logically degenerate cases by analyzing the
problematics noticed in students’ proof scripts. Drawing on the exploratory findings, I will
report on a framework for students’ noticed proof problematics and explores the viability of the
proof script methodology as a mechanism for identifying difficulties experienced by students but
unseen by experts. In the case of logically degenerate cases, findings indicate students held
conceptions of proofs by cases that inhibited students’ reasoning about the encountered
contradictions, as well as students’ difficulties correctly reasoning with logical conjunctions. "Leveraging Our Bodies When We Learn" Nicole Infante, West Virginia University Abstract: Every
concept in mathematics can be represented multiple ways. A function is a graph,
a formula, a set of points. A critical aspect of learning and understanding
mathematical concepts is the ability to use and move between different
representations of a common idea. Connections between varied representations
and concepts form the foundations of advanced mathematics. Helping students
make these connections is a key component of our profession as teachers. Here,
we explore how we can assist students in making these connections and deepening
their understanding. In particular, we take an embodied cognition approach: our
understanding of concepts is shaped through our bodily experiences such as gesture.
We examine how instructor gesture can aid student learning and will showcase at
least two student centered activities. "How and when do high school math teachers have the opportunity to learn to mathematics
that benefits their teaching?" Yvonne Lai, University of NebraskaLincoln Abstract: "The more mathematics a teacher knows, the better"  this truism has dictated teaching
licensure exams and mathematics requirements for centuries. However, it was not until
the past decade and a half that we have seen studies that show that there is mathematical
knowledge that is specifically involved in teaching, and that this knowledge may not be
found in typical mathematics major coursework such as abstract algebra or real analysis.
I will begin with brief survey of the research that leads to this conclusion, for both
elementary and high school teaching. I will then discuss some recent results on how policy
tends to be more consistent with these findings at the elementary level than at the high
school level, and why this may be. Finally, the talk will conclude with some future directions
and open questions about the mathematical preparation of high school teachers, and the
potential role of mathematicians in contributing to addressing these questions. "Examining Students’ Combinatorial Reasoning: The Case of the
Multiplication Principle" Elise Lockwood, Oregon State University Abstract: Combinatorics is a rich and accessible topic, but counting problems are
difficult for students to learn and for teachers to teach. In this talk, I
present some of my research interests, focusing on one particular area of
study: undergraduate students’ reasoning about the multiplication principle. This principle is fundamental to combinatorics,
underpinning many standard formulas and counting strategies. I will
present a categorization of statement types found in a textbook analysis, and I
will incorporate excerpts from a reinvention study that sheds light on student
reasoning. Findings from both studies
reveal surprisingly subtle aspects of the multiplication principle. I
conclude with a number of potential mathematical and pedagogical implications
of the research, as well as some future research directions. "Findings from a National
Study of Calculus Programs" Chris Rasmussen, San Diego State University Abstract: In this talk I
present findings from a national study of Calculus programs, which included
both a national survey and case studies of institutions identified as having a relatively
successful calculus program. Based on survey results I first present
characteristics of STEM intending students who begin their post secondary
studies with Calculus and either persist or switch out of the calculus
sequence, and hence either remain or leave the STEM pipeline. I then present
case study findings from five doctoral degreegranting institutions, including
technical universities and medium to large public institutions. Understanding
the features that characterize exemplary calculus programs at doctoral degree
granting institutions is particularly important because the vast majority of
STEM graduates come from such institutions. Analysis of over 95 hours of
interviews with faculty, administrators and students reveals seven different
programmatic and structural features that are common across the five
institutions, including substantive graduate teaching assistant training,
coordination across sections, and the use of active learning. A community of
practice and a socialacademic integrations perspective are used to illuminate
why and how these seven features contribute to successful calculus programs. "An Example of Inquiry in Linear Algebra: The Roles of Symbolizing and Brokering" Michelle Zandieh, Arizona State University Abstract: In this presentation we address practical questions such as: How do symbols
appear and evolve in an inquiryoriented classroom? How can an instructor connect
students with traditional notation and vocabulary without undermining their sense of
ownership of the material? We tender an example from linear algebra that highlights the
roles of the instructor as a broker, and the ways in which students participate in the
practice of symbolizing as they reinvent the diagonalization equation A = PDP−1.

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