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, Nicholas Wofford (firstname.lastname@example.org)
Seminars in 2018 Fall
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 dened 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-dened 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.
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, A − A must contain a subspace of codimension 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.
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 2018 Spring
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 2017 Fall
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.
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.
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.
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.
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..