All activities will take place on the ground floor of Skiles Classroom Building in room 005 and the adjacent atrium.
9:00-9:30 Registration & Coffee
9:30-10:30 Data Science
10:30-11:00 Tea
11:00-12:00 Knotted Surfaces
12:00-1:30 Lunch
1:30-2:30 Problem Session (Data Science)
2:45-3:45 Candice Price
3:45-4:15 Tea
4:15-5:15 Career Panel
5:30 Casual Dinner
9:00-9:30 Coffee
9:30-10:30 Data Science
10:30-11:00 Tea
11:00-12:00 Knotted Surfaces
12:00-1:30 Lunch
1:30-2:30 Problem Session (Knotted Surfaces)
2:45-3:45 Margaret Symington
3:45-4:15 Tea
4:15-4:25 Adam He
4:30-4:40 Kelly Emmrich
4:45-4:55 Emily Stamm
5:00-5:10 Rita Post
9:00-9:30 Coffee
9:30-10:30 Data Science
10:30-11:00 Tea
11:00-12:00 Knotted Surfaces
12:15-1:15 Problem Session (Data Science)
Free Afternoon
9:00-9:30 Coffee
9:30-10:30 Data Science
10:30-10:40 Group Photo
10:40-11:00 Tea
11:00-12:00 Knotted Surfaces
12:00-1:30 Lunch
1:30-2:30 Elisabetta Matsumoto
2:30-2:45 Set up posters
2:45-3:45 Problem Session (Knotted Surfaces)
3:45-4:15 Tea
3:45-5:15 Poster Session
9:00-9:30 Coffee
9:30-10:30 Graduate Panel
10:30-11:00 Tea
11:00-11:10 TBA
11:15-11:25 Shashanke Markande
11:30-11:40 Ilya Marchenko
11:45-11:55 Christopher Henson
12:00-12:10 Joshua Wang
Kelly Emmrich
Title: Sufficient Conditions for a Linear Operator on R[x] to be Monotone.
Abstract: We demonstrate that being a hyperbolicity preserver does not imply monotonicity for infinite order differential operators on R[x], thereby settling a recent conjecture in the negative. We also give sufficient conditions for such operators to be monotone.
Adam He
Title: A topological model of lasso proteins
Abstract: The presence of topologically complex structures such as knots and links in proteins has received significant attention from both theoreticians and experimentalists. In this paper, we focus on lasso proteins, a more recently discovered class of proteins that have not been very well characterized from a topological perspective. These proteins consist of a closed loop pierced by some number of chain termini. This geometric complexity is believed to influence their stability and play a role in their physiological functions. We propose a topological model of these proteins that allow us to treat lassos like other topologically complex proteins. We then show that our model allows us to distinguish between known lasso structures.
Christopher Henson
Title: Generalized Summation Methods for Divergent Series
Abstract: In recent years there have been several popularized attempts to explore assigning values to divergent series, but this is in no way a recent development, with G. H. Hardy publishing an entire book on the subject in 1949. These notes examine both with intuition and rigorous mathematics various summation methods for divergent series and briefly explore surprising practical applications.
Ilya Marchenko
Title: Pairing-Based Cryptography In Theory And In Practice
Abstract: In this talk I will give a brief introduction to pairing-based cryptography and discuss an implementation of a one-time multiplicatively homomorphic cryptosystem.
Shashank Markande
Title: A triply periodic chiral minimal surface family from space group symmetries
Abstract: Triply periodic minimal surfaces are minimal and crystallographic. We study a one parameter family of chiral triply periodic minimal surfaces, QTZ-QZD, which partition the euclidian three space into two disjoint regions, enclosing a quartz network on one side and its dual QZD network on the other. Using one of the skeletal networks any surface from the QTZ-QZD family can be numerically generated. We use symmetries and the Gauss map image of the surface to obtain a regular parametrization based on the theory of Weierstrass-Enneper representation. Most of the minimal surfaces studied so far consist of on surface symmetry elements. The QTZ-QZD family is an exception as they have three fold screw axis symmetry but lack on surface symmetry elements. The chiral pitch of these surfaces can be tuned which can be useful in developing optical meta-material devices and photonic crystals with minimal surface geometry. This is joint work with Gerd E. Schroeder-Turn, Vanessa Robins, and Elisabetta Matsumoto.
Rita Post
Title: Spatial trivalent graphs and their corresponding braids
Abstract: A virtual spatial trivalent graph diagram (virtual STG diagram) is a trivalent graph immersed in a plane, which contains finitely many transverse double points, each of which has information of over/under or virtual crossings. We regard virtual STG diagrams as combinatorial objects up to an equivalence relation induced by certain combinatorial moves for virtual STG diagrams. Then a virtual spatial trivalent graph is the equivalence class of a virtual STG diagram. Moreover, we say that two virtual STG diagrams are equivalent if they belong to the same equivalence class. A virtual trivalent braid is a braid similar to the notion of a classical braid, but may contain trivalent vertices and virtual crossings, in addition to classical crossings. The closure of a virtual trivalent braid with n endpoints on top and n endpoints in the bottom is a virtual STG diagram. Therefore, we can study virtual trivalent braids to gain information about virtual spatial trivalent graphs.
In this presentation we describe our method for converting any virtual STG diagram into an equivalent diagram in braid form. We also provide conditions for having two virtual trivalent braids whose closures yield equivalent virtual STG diagrams. In other words, we provide Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual trivalent braids.
Emily Stamm
Title: Lacunary Eta-quotients Modulo Powers of Primes
Abstract: An integral power series is called lacunary modulo M if almost all of its coefficients are divisible by M. Motivated by the parity problem for the partition function, p(n), Gordon and Ono studied the generating functions for t-regular partitions, and determined conditions for when these functions are lacunary modulo powers of primes. We generalize their results in a number of ways by studying infinite products called Dedekind eta-quotients and generalized Dedekind eta-quotients. We then apply our results to the generating functions for the partition functions considered by Nekrasov, Okounkov, and Han.
Joshua Wang
Title: Dessins d'Enfant: Trees and Shabat Polynomials
Abstract: In 1984, Alexander Grothendieck proposed studying the absolute Galois group of the rationals through its faithful action on certain concrete combinatorial objects known as dessins d'enfants (children's drawings). This action remains faithful when restricted to a class of particularly simple dessins d'enfants called trees. Quadratic number fields are associated to trees lying in Galois orbits of size 2. Imaginary quadratic number fields are associated to orbits consisting of a pair of trees which differ by a reflection. We will examine trees lying in orbits of size 2 whose associated quadratic number fields are real.
William Ballinger, Tynan Ochse
Title: The Prism Manifold Realization Problem
Abstract: A prism manifold is a three-manifold M whose fundamental group is a finite central extension of the dihedral group Dn. These form an infinite family P(p, q) parametrized by coprime integers p, q with p > 0. We determine, for a large range of parameters (either p > 2q or p < q) which of the prism manifolds are the result of positive integer surgery on knots in S3. We use a lattice-theoretic obstruction developed by Greene in the context of his work on the analagous realization problem for lens spaces. We find that, in this range, every prism manifold that is the result of positive integer surgery on some knot is the result of positive integer surgery on one of the Primitive/Seifert Fibered knots tabulated by Berge and Kang. This is based on joint work with Chloe Hsu, Wyatt Mackey, Yi Ni, and Faramarz Vafaee.
Erin Connelly
Title: Universality in perfect state transfer
Abstract: A continuous-time quantum walk on a graph is a matrix-valued function exp(-iAt) over the reals, where *A* is the adjacency matrix of the graph. Such a quantum walk has universal perfect state transfer if for all vertices u,v, there is a time where the (v,u) entry of the matrix exponential has unit magnitude. We prove new characterizations of graphs with universal perfect state transfer. This extends results of Cameron et al. (2014) [3]. Also, we construct non-circulant families of graphs with universal perfect state transfer. All prior known constructions were circulants. Moreover, we prove that if a circulant, whose order is prime, prime squared, or a power of two, has universal perfect state transfer then its underlying graph must be complete. This is nearly tight since there are universal perfect state transfer circulants with non-prime-power order where some edges are missing.
Elizabeth Crow
Title: Characterizing the Gröbner Bases of Generic Ideals
Abstract: Moreno-Socías conjectured that the initial ideal of a generic ideal generated by a sequence of polynomials is weakly reverse lexicographic. Through extensive computation and observation, the reduced Gröbner bases of generic ideals can be described more explicitly, making it easier to approach the proof of Moreno Socías' conjecture in n variables.
Casandra Monroe
Title: Ordered multiplicity inverse eigenvalue problem for graphs on six vertices
Abstract: A graph $G$ consists of a vertex set $V(G)$ and edge set $E(G)$. Given $G$ with vertices $v_1,\dots, v_n$, a real symmetric matrix $M$ is in the family of matrices $\s(G)$ if: $M_{i,j}=M_{j,i}=0$ for $i\neq j$ if and only if $v_iv_j\notin E(G)$. The Inverse Eigenvalue Problem of a Graph (IEPG) asks, given a graph $G$ and a set of numbers $\textbf{L}=\{\lambda_1,\dots, \lambda_n\}$, does there exist an $M \in \s(G)$ with eigenvalues $\textbf{L}$? We can also construct a relaxation of this question by considering the eigenvalues in increasing order and preserving this order when listing the multiplicity of each value. This creates an ordered multiplicity list, and so we ask similarly if there exists a matrix $M \in \s(G)$ for a given graph G that achieves a specified multiplicity list. This poster addresses the methods used to solve the ordered multiplicity list problem for connected graphs on six vertices and the IEPG for most connected graphs on six vertices by demonstrating ways to determine if certain lists are attainable for a given graph.
Stephen Sarutto
Tite: On the classification of maps from S^3 to S^2
Abstract: The Pontryagin Construction establishes equivalence classes of manifolds modulo framed cobordism. We establish some prelimary theorems relating framed cobordism and homotopy, and use framed cobordism classes of compact 1-manifolds without boundary to classify maps from S^3 to S^2 up to homotopy.
Hunter Vallejos
Title: On Augmentations of Legendrian Knots
Abstract: We study knots, which are embeddings of the circle into R^3. We require that the knot lie tangent to a ``plane field,'' induced by the standard contact structure of R^3. Such a knot is called Legendrian. By building an algebraic structure from crossings and paths on Legendrian knots, we can extract information in hopes of building an invariant. However, this algebraic structure is usually quite complicated; a solution to this is to investigate maps from this structure to simpler ones. In this project we study maps to the integers mod n and to matrix rings.
Anna Williams
Title: An Age Structured Model of the Impact of Buffelgrass on Saguaro Cacti and their Nurse Trees
Abstract: The saguaro cactus (Carnegiea gigantea), a keystone species in the Sonoran Desert, has faced population decline in recent years. The immediate threat to the saguaro cactus is the increase in wildfires fueled by the invasive species buffelgrass (Pennisetum ciliare). The increasing rate of wildfires could result in the collapse of the Sonoran Desert ecosystem. A stage structured model is used to capture interactions between saguaro cacti, their nurse trees, and buffelgrass. In order to accurately model the impact of buffelgrass, the interactions between the saguaro growth stages, juvenile and adult, with the nurse trees, called the foothills palo verde (Parkinsonia microphylla), is studied. The later introduction of buffelgrass to the model demonstrates its influence on the natural life cycles of the saguaro and nurse tree populations. This model consists of a system of non--linear ordinary differential equations which considers commensalism between juvenile saguaros and their nurse trees and their eventual competition as juveniles mature to adulthood. The analysis of this system includes qualitative analysis of the equilibria of the system as well as a numerical analysis of the sensitivity of key parameters. In the interest of preserving the saguaro cactus population, this system will provide insight into the effectiveness of current mitigation strategies.