Home

Welcome to the USTARS 2012 Archive.  Click USTARS 2011 if you would like to view talks from the inaugural USTARS meeting.
    


Oral Presentation Abstracts

Title: Arithmetic Progressions on Mordell Curves
Presenter: Dr. Alejandra Alvarado
Affiliation: Purdue University
Abstract: An r-term arithmetic progression (AP) is a collection of rational numbers {l1 , l2 , ..., lr } such
that there is a common difference d = li+1 −li . When we talk about an AP on the Mordell curve y 2 = x3 +k,
where k is a nonzero integer, we mean an AP in the x-coordinates. We will give a survey on what is known
about arithmetic progressions on these curves.

Presenter: Dr. Federico Ardila∗∗
Affiliation: San Francisco State University and Universidad de Los Andes, Colombia
Abstract: A cube complex X is a space built by gluing cubes together. We say that X is “CAT (0)” if
it has non-positive curvature - roughly speaking, this means that X is shaped like a saddle. CAT (0) cube
complexes play an important role in pure mathematics (group theory) and in applications (phylogenetics,
robot motion planning).
We show that, surprisingly, CAT (0) cube complexes can be described completely combinatorially. This
description gives a proof of the conjecture that any d-dimensional CAT (0) cube complex X “fits” in d-
dimensional space. It also leads to an algorithm for finding the shortest path between two points in X (and
hence to find the fastest way to move a robot from one position to another one).
The talk will describe joint work with Megan Owen and Seth Sullivant.

Presenter: Cheryl Balm
Affiliation: Michigan State University
Abstract:A cosmetic crossing in a knot diagram is a non-nugatory crossing where the overstrand and un-
derstrand can be switched without change the underlying knot type. We will demonstrate some obstructions
to the existence of cosmetic crossings in genus-one knots using Seifert matrices and basic linear algebra. As
an application, we prove the nugatory crossing conjecture (that cosmetic crossings, in fact, do not exist) for
several genus-one families of knots.

Title: Radio Numbers of Graphs of Order n and Diameter n − 2
Presenter: Katie Benson
Affiliation: University of Iowa
Abstract: A radio labeling of a simply connected graph G is a function c : V (G) → Z+ such that for every
two distinct vertices u and v of G, d(u, v) + |c(u) − c(v)| ≥ diam(G) + 1. The radio number of a graph G is
the smallest integer m for which there exists a labeling c with c(v) ≤ m for all v ∈ V (G). In this talk, we
will establish the radio numbers of graphs with n vertices and diameter n − 2. We will give an upper bound
for this radio number with a particular labeling and then outline a technique for finding a lower bound for
the radio number that matches this upper bound. This will determine the radio number of these graphs.

Presenter: Sayonita Ghosh Hajra
Affiliation: University of Georgia
Abstract: A plane field ξ on a 3 manifold M is a field of hyperplanes. A plane field ξ on M is called a
contact structure on M if there is a 1- form α (locally or globally) with ξ = Ker(α) and we have α ∧ dα is
non zero. I am going to talk about few examples of contact structures on R3 and give a natural occurrence
and application of contact structure. Also I will talk about what is a bypass triangle attachment and how it
can be used to classify overtwisted contact structure in S_2 × [0, 1].


Presenter: Dr. Edray Goins
Affiliation: Purdue University
Abstract: In 1999, it was shown by Kenter that the Euler-Mascheroni constant
can be represented as a product of an infinite-dimensional row vector, the inverse of a lower triangular
matrix, and an infinite-dimensional column vector:
Kenter’s proof uses induction, definite integrals, convergence of power series, and Abel’s Theorem. In this
paper, we recast this statement using the language of Riordan matrices. We exhibit another proof as well as a
generalization: we show that the Euler-Mascheroni constant γ and Euler’s number e can both be represented
as a product of a Riordan matrix and certain row and column vectors.
This is joint work with Asamoah Nkwanta.

Title: Toric Ideals of Hypergraphs
Presenter: Elizabeth Gross
Affiliation: University of Illinois, Chicago
Abstract: The ideal of an edge subring of a graph is an object that appears when studying statistical models
parameterized by the edges of a graph. There are many results that tell us the same beautiful story: we can
understand these ideals, if we understand the combinatorics of the underlying graph. A natural extension is
to consider the defining ideal of an edge subring of a hypergraph. In this talk we give some recent results
on the toric ideals of hypergraphs, including how to tell when the ideal is generated in a fixed degree. This
is joint work with Sonja Petrovic.

Presenter: Mela Hardin
Affiliation: San Francisco State University
Abstract: Signed graphs have a chromatic polynomial with the same enumerative and algebraic properties
as for unsigned graphs. We generalize the chromatic polynomial for signed graphs using the bivariate chro-
matic polynomial of Dohmen, Ponitz, and Tittmann and prove that specializations of this new polynomial
recover polynomials which enumerate independence and antibalance.

Title: On the adjoint representation of sln and the Fibonacci numbers
Presenter: Pamela Harris
Affiliation: University of Wisconsin, Milwaukee
Abstract: We decompose the adjoint representation of slr+1 = slr+1 (C) by a purely combinatorial approach
based on the introduction of a certain subset of the Weyl group called the Weyl alternation set associated
to a pair of dominant integral weights. The cardinality of the Weyl alternation set associated to the highest
root and zero weight of slr+1 is given by the rth Fibonacci number. We then obtain the exponents of slr+1
from this point of view.

Title: A mod four congruence in the symmetric real Schubert calculus
Presenter: Nickolas Hein
Affiliation: Texas A&M University
Abstract: Real Schubert problems satisfy #(real solutions) ≡ #(complex solutions) mod 2 since nonreal
solutions come in conjugate pairs. When the Schubert conditions exhibit certain symmetry we find a stronger
congruence modulo 4. This curious congruence occurs when complex conjugation and a natural geometric
involution act freely on the solutions. The key to understanding this phenomenon is the Hermitian Grass-
mannian, which has not been classically studied.
I will present the Hermitian Grassmannian and the real Lagrangian Grassmannian, and describe their
connection to the symmetric real Schubert calculus. I will also describe degenerate symmetric cases in which
the two involutions on the solutions do not act freely (and real solutions are only fixed mod 2).

Title: Ricci Flow on some Classes of Naturally Reductive Homogeneous Spaces
Presenter: Tanya Hepburn
Affiliation: Saint Louis University
Abstract: A homogeneous space M = G/H of Lie Groups G and H, is called naturally reductive if it admits
an ad(H)-invariant decomposition g = h + m. We will prove that the Ricci Flow on M is proportional to
the Ricci Flow on G restricted to M .

Title: Positive semidefinite zero forcing and vertex spreads
Presenter: Nicole Kingsley
Affiliation: Iowa State University
Abstract: A graph is denoted G = (V, E), where V is a nonempty set of vertices, E is a set of edges, and
each edge is two-element subset of the set of vertices V . Our research focuses on simple undirected graphs
and a type of graph parameter called a zero forcing number. A specific type of zero forcing number, positive
semidefinite zero forcing number, was introduced by Barioli, et al. in 2010. A positive semidefinite zero
forcing set for a graph G is a subset B of V , such that when B is initially colored black, all vertices of G
are colored black when the positive semidefinite color change rule is carried out to completion. We call the
minimum cardinality over all such B the positive semidefinite zero forcing number of G, and denote this
quantity Z+ (G). A positive semidefinite matrix representation of a graph G on n vertices is an n-square
positive semidefinite matrix A = [aij ], where aij is nonzero if the vertices i and j of G are adjacent and zero
otherwise. Two interesting parameters based on the family of positive semidefinite matrices representing a
graph G are positive semidefinite minimum rank and positive semidefinite maximum nullity of G, denoted
by mr+ (G) and M+ (G). It is well known that M+ (G) ≤ Z+ (G) for all graphs. We investigate the effect of
deleting one vertex of G, along with its adjacent edges, on the parameters mr+ (G), M+ (G), and Z+ (G).

Title: On the quasidiagonality of group C_∗ -algebras
Presenter: Jose Luis Lugo
Affiliation: Purdue University
Abstract: The notion of amenability for groups was introduced by Jon von Neumann in his study of the
Banach-Tarski paradox. To each group G, we can associate to it its reduced group C ∗ -algebra Cr (G), which
is a certain completion of the group ring C[G]. In 1987 Jonathan Rosenberg proved that if G is at most
countable and Cr (G) is quasidiagonal, then the group G is necessarily amenable. Whether the converse is
true is still an open question. We discuss this and a special class of groups for which the converse holds.

Title: Two families united as one: A Symmetric Functions Tale
Presenter: Dr. Aba Mbirika
Affiliation: Bowdoin College
Abstract: A fundamental tool used to study various algebraic and topological objects is symmetric func-
tions. They arise in a variety of areas from algebraic, combinatorial, and geometric perspectives. Symmetric
functions are polynomials in the ring Z [x1 , x2 , . . . , xn ] that are fixed by a natural action of the symmetric
group Sn on the variables {x1 , x2 , . . . , xn }. Truncated symmetric functions, on the other hand, are polyno-
mials symmetric in a subset of these variables. The main objects of our interest are truncated elementary
symmetric functions (TESF) and truncated complete symmetric functions (TCSF). In this talk, we give a
number of identities involving these two different families of functions, culminating in a remarkable identity
relating TESF to TCSF. From the family of TESF, we construct ideals that generalize the Tanisaki ideal
which arises in Springer theory. From the family of TCSF, we build Gr¨bner bases for this family of gener-
o
alized Tanisaki ideals. The corresponding polynomial quotient rings easily yield the Betti numbers for the
cohomology rings of an important generalization of the Springer variety, called regular nilpotent Hessenberg
varieties.

Title: Spectral functions in the presence of background potentials
Presenter: Pedro Morales
Affiliation: Baylor University
Abstract: In this talk, we present how the presence of background potentials and the geometry of a man-
ifold can affect the behavior of spectral functions, as well as their relation with topological invariants such
as the heat kernel coefficients.

Title: Noether’s Problem for Alt(n)
Presenter: Kevin Mugo
Affiliation: Purdue University
Abstract: In the 1930’s, while studying the inverse Galois problem, Emmy Noether proposed the following
question: Let a finite group, G, act faithfully by permutations on a a finite set of indeterminates x1 , ....., xn .
Is the fixed field F (x1 , ..., xn )G a purely transcendental extension of F ? We discuss a strategy for solving
Noether’s problem when G = Alt(n), the Alternating group on n letters, for certain n, by considering an
embedding of F (x1 , ..., xn ) into F (E[N ]), where E[N ] are the N -division points of some elliptic curve, E.

Title: The Homology of Filtered Algebras and A-infinity Structures
Presenter: Cris Negron
Affiliation: University of Washington
Abstract: Koszul algebras are a special class of N-graded algebras, over a field k, which are generated in
degree one and have relations generated in degree 2. Koszul duality is a relationship between a Koszul algebra
and its “Koszul dual” algebra ExtA (k, k). This relationship exists as a certain equivalence of categories, but
has other more tangible interpretations as well.
It has recently become apparent that Koszul duality can be extended beyond the class of Koszul algebras
by equipping Ext algebras with a higher structure called an A∞ structure. This new Ext A∞ algebra becomes
a kind of “A∞ Koszul dual”. A type of Koszul duality was also proposed by Positselskii for filtered algebras
U with Koszul associated graded algebras. I will talk about how Positselskii’s filtered Koszul duality and this
new A∞ Koszul duality can be used together to form a theory of Koszul duality for a large class of filtered
algebras. I will also talk about how this broader duality can be harnessed to gain a new understanding of
the homological algebra of filtered algebras.

Presenter: Long Nguyen
Affiliation: Brigham Young University
Abstract: A weak Cayley table isomorphism is a bijective map f : G → G satisfying two conditions: 1)If
g and h are conjugate then f (g) and f (h) are conjugate. 2)For all g, h in G, f (gh) is conjugate to f (g)f (h).
A weak Cayley table isomorphism is a generalization of an automorphism. Let W (G) denote the group of
all weak Cayley table isomorphisms. Any automorphism is an element of W (G). The inverse map given by
I(x) = x−1 is also an element of W (G). These are called trivial weak Cayley table isomorphisms. Given a
group G, in general, there are a lot of nontrivial weak Cayley table isomorphisms. The research question
that I worked on for my dissertation is “For what groups G does W (G) consist of only trivial weak Cayley
table isomorphisms?”

Presenter: Olivier Olela Otafudu
Affiliation: University of Cape Town
Abstract: In this talk, we discuss the concept of hyperconvexity that is appropriate to the category of
T0 -quasi-metric spaces (called di-spaces in the following) and nonexpansive maps. Moreover, we provide an
explicit construction of the corresponding hull (called Isbell-convex hull or, more briefly, Isbell-hull) of a
di-space.

Presenter: Jolie Roat
Affiliation: Iowa State University
Abstract: The maximum positive semidefinite nullity of a multigraph G is the largest possible nullity over
all real positive semidefinite matrices whose (i, j)th edge (for i = j) is zero if i and j are not adjacent in G,
is nonzero if {i, j} is a single edge, and is any real number if {i, j} is a multiple edge. The definition of the
positive semidefinite zero forcing number for simple graphs is extended to multigraphs; as for simple graphs,
this parameter bounds the maximum postive semidefinite nullity from above. The tree cover number T(G)
is the minimum number of vertex disjoint induced simple trees that cover all the vertices of G. The result
that M+ (G) = T(G) for an outerplanar multigraph G is extended to show that Z+ (G) = M+ (G) = T(G) for
a multigraph G of tree-width at most 2.

Presenter: Gautam Sisodia
Affiliation: University of Washington
Abstract: Let A be a free algebra on n generators modulo a single quadratic relation of maximal rank. Zhang
shows that these are the connected-graded Gorenstein algebras of global dimension two. A noncommutative
version of Beilinson’s Theorem establishes a derived equivalence between the category of representations of
the n-Kronecker quiver Q and the quotient category qgrA of the finitely-presented graded A-modules by the
finite-dimensional ones. Noncommutative algebraic geometry then suggests studying the moduli space of Q
as the noncommutative space with homogeneous coordinate ring A. We show that the Grothendieck group
of qgrA is isomorphic as an ordered abelian group with order unit to the subgroup of the reals generated by
the minimal positive pole of the Hilbert series of A.

Presenter: Anastasiia Tsvietkova ∗
Affiliation: University of Tennessee
Abstract: Thurston demonstrated that every link in S 3 is a torus link, a satellite link or a hyperbolic link
and these three categories are mutually exclusive. It also follows from work of Menasco that an alternating
link represented by a prime diagram is either hyperbolic or a (2, n)–torus link.
A new method for computing the hyperbolic structure of the complement of a hyperbolic link, based on
ideal polygons bounding the regions of a diagram of the link rather than decomposition of the complement
into ideal tetrahedra, was suggested by M. Thistlethwaite. The method is applicable to all diagrams of
hyperbolic links under a few mild restrictions. The talk will introduce the basics of the method. Some
applications will be discussed, including a surprising rigidity property of certain tangles, a new numerical
invariant for tangles, and formulas that allow one to calculate the exact hyperbolic volume, as well as com-
plex volume, of 2–bridged links directly from the diagram.

Title: 2-Selmer Groups of Elliptic Curves Associated with Cunningham Numbers
Presenter: Jaime Weigandt
Affiliation: Purdue University
Abstract: The factorization properties of integers of the form bn ± 1 are the subject of a great deal
theoretical and computational effort. This work is motivated by its relevance to ancient problems concerning
perfect numbers and the construction of regular polygons as well as its use as a testing ground for modern
factorization algorithms and primality tests.
In this talk, I’ll illustrate a connection between some of these questions and the study of rational points on
elliptic curves, another subject of immense theoretical and computational efforts with equally ancient origins.

Poster Presentation Abstracts

Title: Conic Sections Containing Rational Distance Sets of Three Points
Presenter: Jonathan Blair
Affiliation: Purdue University
Abstract: Euler showed that there are infinitely many rational points on the unit circle having pairwise
rational distances. Campbell showed that the parabola and Goins and Mugo showed that the hyperbola
each contain a rational distance set of at least four points. In this project we seek to generalize these results
to an arbitrary conic section.
In this talk, I’ll illustrate a connection between some of these questions and the study of rational points on
elliptic curves, another subject of immense theoretical and computational efforts with equally ancient origins.

Title: Distance to uncontrollability with Hermitian matrices
Presenter: Eugene Cody
Affiliation: University of Kansas
Abstract: Controllability is a concept that plays a fundamental role in systems and control. If a system,
(A, b), where A is a square matrix and b is a column vector, is controllable, how large a perturbation is
necessary so that the resulting system is uncontrollable? This can algebraically be expressed by the distance
to uncontrollability, which is a minimization problem over the complex field.
We consider the distance problem with a special case when the matrix A is Hermitian. In this case,
the system (A, b) is reduced the pair (Λ, z), where Λ is a real diagonal matrix. By using the real diagonal
matrix structure we prove that when A is Hermitian, the search field for the minimization problem of the
distance to uncontrollability is reduced to the real field. We observe the behavior of the secular equation
and study the relationship between Λ and z to determine the minimizer using a combination of two methods.

Title: Propagation Time for Zero Forcing of a Graph
Presenter: My Huynh
Affiliation: Arizona State University
Abstract: Zero forcing (also called graph infection) on a simple, undirected graph G is based on the color
change rule: if each vertex of G is colored either white or black and vertex v is a black vertex with only one
white neighbor u then v forces u to become black. A minimum zero forcing set if a set of black vertices of
minimum cardinality that can color the entire graph black using the color change rule. The propagation time
of a graph G is the minimum amount of time that it takes to force all the vertices of G black using a minimum
zero forcing set and performing independent forces simultaneously. The study of propagation times of graphs
is related to the study of control quantum systems. Examples that demonstrate various features of the propa-
gation time of a graph are introduced and results on graphs having extreme propagation times are presented.

Title: 2-Selmer Groups of Elliptic Curves Associated with Cunningham Numbers
Presenter: Blanche Ngo Mahop
Affiliation: Howard University
Abstract: The factorization properties of integers of the form bn ± 1 are the subject of a great deal
theoretical and computational effort. This work is motivated by its relevance to ancient problems concerning
perfect numbers and the construction of regular polygons as well as its use as a testing ground for modern
factorization algorithms and primality tests.
In this talk, I’ll illustrate a connection between some of these questions and the study of rational points on
elliptic curves, another subject of immense theoretical and computational efforts with equally ancient origins.

Title: The Effect of Graph Operations on the Positive Semidefinite Zero Forcing Number
Presenter: Arianne Ross
Affiliation: Iowa State University
Abstract: The positive semidefinite zero forcing number Z + (G) of a graph G was introduced by Barioli
et. al..The effect of various graph operations on positive semidefinite zero forcing number and connections
with other graph parameters are studied.

Title: Exploring Dessins d’Enfants
Presenter: Anika Rounds
Affiliation: Purdue University
Abstract: Suppose there are three cottages, and each needs to be connected to the gas, water, and electric
companies. Using a third dimension or sending any of the connections through another company or cottage
are disallowed. Is there a way to make all nine connections without any of the lines crossing each other?
To answer such a question, we explore the properties of planar graphs. It is natural to generalize to graphs
which can be embedded into Riemann surfaces, such as the sphere and the torus. In this talk, we discuss
how to draw such graphs using Grothendieck’s concept of a Dessin d’Enfant. This is based on joint research
with Edray Goins through the Zoltners Summer Undergraduate Research Fellowship.

Title: 2-Isomorphism Theorem for Hypergraphs
Presenter: Eric Taylor
Affiliation: California State University, San Bernardino
Abstract: Whitneys 2-isomomorphism theorem characterizes when two graphs have isomorphic cycle ma-
troids. This paper will generalize Whitneys theorem to hypergraphs.