# Graduate Student Seminar

The Graduate Student Seminar is a venue that any graduate student can use to present their work, to practice for an upcoming “official” talk, or just to discuss mathematical concepts they find interesting. We also invite Faculty to introduce their research to Graduate Students.

If you are interested in giving a talk, please contact the chapter vice president, Phil Kains.

## Past Seminars

### Seminars in Spring 2021

03/25/2021
Speaker: Page Thorn
Title: The Fremlin Tensor Product on Subspaces of Continuous Functions
Abstract: The space of continuous functions on a compact Hausdorff space X, denoted C(X), is an excellent example of a Riesz space. Fremlin’s Riesz space tensor product, denoted by \overline{⨂}, enables us to combine two Riesz spaces to create a third, which is rarely accomplished by the algebraic tensor product. In this talk, we explore how various Riesz subspaces are affected in this process. In particular, we make statements of the type: “If A ⊆ C(X) and B ⊆ C(Y ) are each of type P in their respective spaces, then A \overline{⨂} B is (or is not) of type P in C(X) \overline{⨂} C(Y ).”

03/05/2021
Speaker: Derrick DeMars, Auburn University
Title: Ramsey Theory: An Introduction to Mixed Ramsey Problems
Abstract: We review basic definitions in graph theory that prepare us for concepts related to Ramsey Theory and Mixed Ramsey problems. We define the Mixed Ramsey Spectrum (from Axenovich and Choi) which is for graphs F,H, as the set of colors k, such that for some k-edge-coloring of K_n, there is neither a monochromatic copy of F nor a rainbow copy of H denoted MRS(K_n: F, H). Our result is for integers m and n, where 3 m n,

{n+2-m, . . . ,(n+( (\binom{m-1}{2})+1)-m)} ⊆ MRS(K_n: C_m, C_n). 02/26/2021
Speaker: Andrew Pham
Title: Independent Bondage Number of Planar Graphs with Minimum Degree at Least 3
Abstract: Given a simple finite graph G, a vertex subset D ⊂ V(G) is said to be a dominating set of G if every vertex v ∈ V(G) – D is adjacent to a vertex in D. The independent domination number γ^i (G) is the minimum cardinality among all independent dominating sets of G, and the independent bondage number b_i (G) is the minimum cardinality among all edge sets E of G such that γ^i (G – E) > γ^i (G). In this talk, we make use of the discharging method and provide a constant upper bound for the independent bondage number of planar graphs with minimum degree at least 3. ### Seminars in Fall 2020

10/22/2020
Speaker: Matthew Just
Title:
Fermat and the Elite Prime
Abstract: The Fermat numbers are given by Fn = 22^n + 1 for n ≥ 0. It was conjectured by Fermat that all such numbers are prime, a conjecture that was proved for all n 5. Fermat was unable to determine whether F6 was prime, and indeed it is not. An efficient test now exists to determine whether a Fermat number is prime, and it is known that Fn is composite for 6 n 32. A sequence of primes related to this test, called elite primes, are discussed in this talk.

### Seminars in Spring 2020

02/27/2020
Speaker: Qinghong Zhao
Title: Multicolor Ramsey numbers of Kipas in Gallai colorings
Abstract: A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai k-coloring is a Gallai coloring that uses k colors. Given a graph H and an integer k ≥ 1, the Gallai-Ramsey number GRk(H) is defined to be the minimum integer n such that every Gallai k-coloring of the edges of Kn contains a monochromatic copy of H. In this talk, I will present our recent results on Gallai-Ramsey numbers for some graphs with chromatic number three such as K^m for m ≥ 2, where K^m is a kipas with m+1 vertices obtained from the join of K1 and Pm, and a class of graphs with five vertices, denoted by H . We first study the general lower bound of such graphs and propose a conjecture for the exact value of GRk(K^m). Then we give a unified proof to determine the Gallai-Ramsey numbers for many graphs in H and obtain the exact value of GRk(K^4) for all k ≥ 1. Our outcomes not only indicate that the conjecture on GRk(K^m) is true for m = 4, but also imply several results on GRk(H) for some H in H which are proved individually in different papers.

02/20/2020
Speaker: Zachary Tripp
Title: Jensen Polynomials for the Riemann Xi Function
Abstract: In a recent paper by Griffin, Ono, Rolen, and Zagier, the authors discuss an equivalent formulation of the Riemann Hypothesis in terms of polynomials having real roots and give partial progress towards this conjecture. In this talk, which is intended for a general audience, we will explain basic properties of the Riemann zeta function, give the ideas behind this recent paper, and introduce the broader impact this paper has to other areas of number theory, such as partition theory. Moreover, we will discuss the extent to which the results of this paper can be made to be effective, which is based on joint work with Griffin, Ono, Rolen, Thorner, and Wagner.

### Seminars in Fall 2019

12/03/2019
Speaker: Ayla Gafni
Title: Job Talk Demonstration
Abstract: The “job talk” is one of the most important aspects of your academic job search, but it’s also one of the most daunting to prepare.  It is your chance to introduce yourself and your research to the entire department.  A good job talk presents a clear and cohesive research program of past results and future projects, in a way that is understandable to all mathematicians.  In this session, Dr. Gafni will demonstrate the job talk she used in interviews last year.  After the talk, there will be an informal discussion about her experience on the academic job market.

11/08/2019
Speaker: Joe Lopez
Title: An introduction to Knot Theory
Abstract: When looking at DNA within a cell, how do we study the structure? One answer is to unknot the DNA. In order for cellular replication to occur, quickly knotted DNA must unknot itself. We can study and answer this question in Knot theory. A Knot is a smooth embedding  f : S1 R3. Knot theory is the study of loops in 3-dimensional space. Knot theory is a modern branch of mathematics and a subbranch of Topology. The largest objective of Knot Theory is classifying all Knots. In this talk, we will go over the following: What is a Knot? Classical Knot Invariants. Computing Knot Invariants. Conjectures in Knot Theory.

09/19/2019
Speaker: Carla Cotwright-Williams, Department of Defense, Women in Mathematics Society
Title: Secret Lives of Mathematicians
Abstract: Mathematics can be more than just a subject in school; it can be a career. The government is the number one single employer of mathematicians in the country. Many of those mathematicians end up at the National Security Agency, where they find careers in research, information assurance, and cryptanalysis. This talk will be an introduction to the roles of mathematicians at NSA, as well as basics of cryptography.

### Seminars in Fall 2018

11/13/2018
Speaker: Meghann Moriah Gibson
Title: Algebraic Properties and Geometric Applications of Fibonacci Numbers
Abstract: The Fibonacci number sequence is famous for its connection to the Golden Ratio and its appearance within natural objects. There also exist Fibonacci-like sequences that share the same recursive definition without possessing its same connection to the Golden Ratio. We will explain a method for constructing such Fibonacci-like sequences, and subsequently give an example of the relationship between the Golden Ratio and the Fibonacci sequence using a sunflower.

09/23/2018
Speaker: Christopher Seaton, Rhodes College, Department of Mathematics and Computer Science
Title: Distinguishing between symplectic quotients by SU2 using invariant theory
Abstract: Let K be (closed) subgroup of the group of n×n unitary matrices, considered as a group of linear transformations from Cn to itself. Forgetting the complex structure on Cn, the underlying real vector space isomorphic to R2n is an example of a symplectic manifold. The action of K admits a moment map, which in this case is a collection of quadratic polynomials, and the symplectic quotient is denoted to be X = Z/K where Z is the set of points on which the moment map vanishes. While X is usually singular, it has several structures, including well-defined notions of what it means for a function X→R to be smooth or polynomial. The algebra R[X] of polynomial functions on X can be described using the real polynomial invariants associated to the group K. We will describe methods from invariant theory that have been used to study the properties of a symplectic quotient X via R[X]. We will in particular focus on the case K = SU2, the 2×2 unitary matrices with determinant 1, and discuss recent results regarding distinguishing between symplectic quotients.

09/20/2018
Speaker: Zhenchao Ge
Title: Subspaces in difference sets in vector spaces
Abstract: A common theme in additive combinatorics states that if A is a subset of positive density of a vector space Fpn, then the difference set A A must contain a large subspace. A result of Sanders says that when p=2 and A has density at least 1/2 − c/√n, AA must contain a subspace of co-dimension 1. We generalize this result to all p while giving a simpler and elementary proof. Our proof is based on an argument of Wirsing. This is joint work with Thai Hoang Le.

09/14/2018
Speaker: Brad Cole, Senior Mathematician for BEN Markets
Title: A Q&A with Brad Cole about his role at BEN Markets
Abstract: In 2011, local resident Jason Finch founded Binary Event Network, a start up, in New York City to combine public opinion, news and data to produce live odds on global event outcomes. Former graduate students Brad Cole and Adam Gray accepted positions with BEN Markets, and Mr. Cole is now Senior Mathematician on staff. Mr. Cole will discuss the day to day life of a sports quant, how he learned of the opportunity, and how recent changes to sports gaming laws may change the industry.

### Seminars in Spring 2018

01/26/2018
Speaker: Irina Ilioaea, Georgia State University
Title: Finding Infeasible Cores of a Set of Polynomials using the Gröbner Basis Algorithm
Abstract: This talk investigates an algorithmic approach to identify a small unsatisfiable core of an ideal I in K[x1, . . . , xn], where K is a field and the ideal I is found to have an empty variety. The main aim of the talk will be to introduce the fundamental notions and to illustrate the concepts we use by examples. We identify certain conditions that are helpful in deciding whether or not a polynomial from the given generating set is a part of the unsat core. Our algorithm cannot guarantee a minimal unsat core; hence the talk discusses opportunities for refinement of the identified core.

### Seminars in Fall 2017

11/03/2017
Speaker: Sumeet Kulkarni, Department of Physics
Title: Use of Randomised Techniques in Gravitational Wave Searches
Abstract: The recent joint discovery by LIGO-Virgo of Gravitational waves (GW) emitted from a pair of coalescing Neutron Stars, and the consequent observations of the system across the Electromagnetic spectrum by numerous observatories has kicked off the era of multi-messenger astronomy. To enable more such observations in the future, it is important to detect incoming gravitational wave signals as soon as possible to generate alerts for astronomers worldwide to follow-up. Gravitational wave detection involves cross-correlating the detector data with millions of modeled templates, each sampled at millions of templates. Given the magnitude of the databases, data reduction techniques provide extremely useful solutions to speeding up GW searches. In the past, singular value decomposition (SVD) has been applied to eliminate degeneracies that exist within a template bank. Weaddress some of its limitations by employing the technique of Random Projections, inspired by the Johnson-Lindenstrauss lemma (1984). Popular in other avenues of data science for dimensionality reduction, this method proves to be surprisingly helpful for reducing GW templates. I’ll present an overview of GW data analysis through this perspective and discuss the use of Randomised techniques in order to reduce the complexity of the search operations.

10/27/2017
Speaker: Zhenchao Ge
Title: Primes in Arithmetic Progression
Abstract: Euler proved the existence of infinitely many primes by showing that the sum of reciprocals of primes is divergent. By the same idea, Dirichlet proved that for every coprime a and q, the arithmetic progression a, a + q, a + 2q, … contains the infinitely many primes. In this talk, we sketch Euler and Dirichlet’s proof. Also, we introduce Dirichlet characters and character orthogonality.

10/13/2017
Speaker: Sashwat Tanay, Department of Physics
Title: Bivariate Power Series via a Perturbative Approach
Abstract: Power series are one of the most important inventions in mathematics and have been employed to solve a myriad of problems in the physical science. There are multiple ways to obtain a power series (Taylor’s formula, perturbative approach etc.) but the uniqueness theorem guarantees that all such methods yield the same answer. In this talk, the perturbative approach to obtain the power series (in two variables) is discussed. The perturbative approach is one where we first take into account only the leading order input information to obtain the leading order solution and gradually process the higher order terms of the input information to churn out the higher order terms of the solution as a power series. This happens order by order. This method is first demonstrated for the case of a much simpler univariate sine series because it captures the essence of the problem, which is then employed to the more complicated bivariate case. Finally, the physical context of this mathematical procedure is discussed briefly. The method finds application in simulating gravitational waves, which is crucial to their detection. The 2017 Nobel prize in physics was awarded to three key members of the LIGO collaboration (Profs. Weiss, Thorne and Barish) for their contributions to gravitational wave detections.

10/06/2017
Speaker: Khazhakanush Navoyan
Title: Connected Spaces
Abstract: Let X be a topological space. A separation of X is a pair of U, V of disjoint nonempty open subsets of X whose union is X. The space X is said to be connected if there does not exist a separation of X. Connectedness is obviously a topological property, since it is formulated entirely in terms of the collection of open sets of X. Said differently, if X is connected, so is any space homeomorphic to X. The definition of connectedness for a topological space is a quite natural one. One says that a space can be “separated” if it can be broken up into two “globs”- disjoint open sets. Otherwise, one says that it is connected.

09/08/2017
Speaker: Khazhakanush Navoyan
Title: On Spectrum and Resolvent Set, Part 1
Abstract: In this talk we deﬁne resolvent set and spectrum of a linear norm bounded operator T in a complex Banach space V. In particular, we deal with the case that V is a Banach lattice and the operator T is positive. We show that the spectrum is a closed and bounded set in the complex plane C and discuss the properties of spectral radius, which is the radius of the smallest circle having as its centre the origin and containing the spectrum..