The Graduate Student Seminar is put on by the Mathematics Graduate Student Association . The GSS meets approximately every other Monday.
For the Fall 2016 semester, the GSS meets at 4:00pm in Hayes-Healy 129. For the Spring 2017 semester, the GSS meets at 4:00pm in Hayes-Healy 229.
To volunteer to give a talk, or for anything else regarding the seminar, contact PJ Jedlovec.
|Monday, September 5||Jeff Diller||Anatomy of a Dynamical System|
|Monday, September 19||Adam Moreno||Isometric Actions and the Hsiang-Kleiner Theorem|
|Monday, September 26||Xiaoxiao Li||Kähler metrics with toric symmetry|
|Monday, October 3||Eric Wawerczyk||Dreams of a Number Theorist: The left side of the right-half-plane|
|Monday, October 24||PJ Jedlovec||Noncooperative Games|
|Monday, October 31||Michael Perlman||Young Diagrams and Invariant Ideals on Spaces of Matrices|
|Monday, November 7||Martin Beers||Imagination and Geometric Truth|
|Monday, November 14||Jeremy Mann||The Weyl Algebra, Quantum Mechanics, and Factorization Algebras|
|Monday, November 21||Matteo Bianchetti||Infinite Time Computations|
|Monday, November 28||Ben Lewis||Quantization Schemes on R|
|Monday, January 30||Eric Wawerczyk||A Crash Course in p-Adic Numbers and Analysis|
|Monday, February 6||Xiaoxiao Li||Riemannian convergence theory|
|Monday, February 13||Andre Jorza||Rational points on curves|
|Friday, February 17||Samantha Lee and Larry Westfall||Help with Jobs and Grants|
|Monday, March 6||David Galvin||Independent sets in regular graphs|
|Monday, March 27||Antonis Anastasopoulus||Recurrent Neural Networks for Language|
|Monday, April 10||Alex Himonas||Well-posedness of evolution equations via the unified transform method|
|Monday, April 24||Pam Urresta||An introduction to Operads via a Concrete Example|
|Monday, May 1||Job Applications Panel|
The Hsiang-Kleiner theorem is a homeomorphism classification of positively curved Riemannian manifolds whose metric admits an isometric circle action. The proof of this beautiful theorem employs many important concepts from the theory of isometric actions and Riemannian geometry. In this talk, I will provide a rundown of the material necessary to appreciate the statement and its proof with a few pictures along the way.
Euler studied the Harmonic series, the sums of reciprocals of squares, cubes, and so on. Chebyshev was able to package these together in the zeta function, defined for a real number greater than 1. Riemann had the idea of defining the function for complex numbers and found that this infinite series converges as long as the real part of the imaginary number is greater than 1. He then uses the inverse mellin transform of the Theta Function (see: modular forms) to derive the "functional equation" which gives a UNIQUE meromorphic extension to the whole complex plane outside of the point s=1 (harmonic series diverges=pole). This unique extension is called the Riemann Zeta Function. For over 150 years Number Theorists have been staring at the left side of the right half plane. We will explore this mysterious left hand side throughout the talk and its relationship to Number Theory. This will serve as our prototypical example of Zeta Functions. Finally, we will discuss generalizations of the zeta functions (Dedekind Zeta Functions, Artin Zeta Functions, Hasse-Weil Zeta Functions) and our daunting task of extending all of them to their own left sides and what mysteries lie within (The Analytic Class Number Formula, The (proven) Weil Conjectures, the Birch and Swinnerton-Dyer conjecture, The Langlands Program, and the (proven) Taniyama-Shimura Conjecture, the Grand Riemann Hypothesis)
In 1950, the late John Forbes Nash Jr. published his famous PhD thesis, titled "NonCooperative Games." The work contained in these pages was monumental for the field of game theory and eventually won him the 1994 Nobel Prize in Economics. However, few people realize just how accessible his landmark PhD thesis is, spanning just 10 pages (in modern formatting) and primarily using mathematical techniques familiar to any math graduate student. In this talk, I will go through the main results of Nash's thesis, explaining their historical and mathematical significance along the way. I will begin with the basic definitions and facts in the theory of non-cooperative, normal-form games. I will then proceed to prove Nash's main theorem, that every finite, non-cooperative, normal-form game has an equilibrium point, as well as a related theorem, that every such game has a symmetric equilibrium point. Next, I will go through some of the applications of Nash's analysis of non-cooperative games and explain their significance. Finally a few volunteers will play an extensive-form game against me, with prizes going to anyone who beats me.
The space of m x n matrices admits a natural action of the group GL_m x GL_n via row and column operations on the entries. This action extends to the coordinate ring of the space of matrices, and the invariant ideals were classified by DeConcini, Eisenbud, and Procesi in the 1980's. Their techniques were combinatorial, employing classical results on the equations for Grassmannians. In this talk, I will describe the classification theorem, with an emphasis on these methods, which fit into a broader context known as standard monomial theory.
This talk will explore – from a Thomistic perspective – the question of what it means to ascribe truth to mathematical propositions. In particular, I will focus on Aquinas' claim that the truth of geometrical judgments is grounded in the faculty of imagination. The central question motivating the talk is this: What does it mean to say that non-Euclidean geometries are true? I will look at two (mid)modern attempts to answer this question, one by an appeal to the imagination (Helmholtz) and the other by an appeal to empirical measurement (Einstein). Finally, I will explore how Aquinas himself might have answered the question, and whether he could have accepted Euclidean and non-Euclidean geometries as mathematics in the same sense.
In this talk, we will show how the Weyl Algebra encodes the data of the observables of the quantum field theory associated to the one dimensional harmonic oscillator. In order to establish this equivalence, we will articulate these objects in the language of factorization algebras. No knowledge of physics will be assumed.
This talk will be about computability in an infinitary setting. The structure N = (N, + , - , 0, 1) is computable in the classic sense, while the structure R = (R, + , - , <, 0, 1) is not. However, becomes computable if we define computability in terms of infinite time Turing machines (ITM). I will define what ITM are and I will discuss their legitimacy as a means to defining computability. I will argue that they have unpleasant features. I will argue that there are weaker variants of them, which I call ITM-, which are enough to make computable and with less unpleasant features. Then I will present some results about ITM- and I will compare them with ITM. I will not presuppose familiarity with computability theory.
The axioms of quantum mechanics stipulate that there is a method to turn quantities from classical mechanics (e.g. energy, position, momentum) into operators on a quantum Hilbert space, giving us a subtle connection between classical mechanics and quantum mechanics. This method is called quantization. We will derive the position and momentum operators as well as the Hamiltonian operator. We will then set our sights on methods of quantizing more complicated functions on the phase space, concluding with Groenewold's “No Go” theorem and its implications for quantum mechanics. No experience with quantum mechanics will be assumed. Both mathematicians and physicists are encouraged to attend!
In 1972, Dedekind and Cantor independently constructed the Real numbers we all know and love. In 1897, Hensel constructed the more mysterious metrics on the Rational numbers whose completions we call the p-adic numbers. In 1916, Ostrowski proved to us that these were the only ways of defining a metric on the rationals (up to homeomorphism). This talk serves as a crash course in p-adic numbers and the awesome powers of p-adic analysis. The goal, if time permits, is to discuss two of the works of John Tate. After our crash course, we will briefly state the main results of Tate's 1950 cult classic Ph.D thesis: "Fourier Analysis in Number Fields and Hecke's Zeta Functions" and the Langlands Program. Afterwards, our real goal: we will discuss Tate's 1961 p-adic Uniformization of elliptic curves (with bad reduction) which is the origin of Rigid Analytic Geometry, an ever advancing industry of math which is responsible for some of the largest strides in Number Theory over the last 50 years.
Our notion of D - Module will be that of a module over the nth Weyl Algebra. We will look at filtrations and use this to define a notion of dimension and multiplicity for D - Modules. Soon after this, Bernstein's inequality makes its grand entrance. Before going into the proof, we will see some consequences among which we discuss the category of Holonomic D - Modules (it will turn out to be Abelian and Artinian) and the existence of the Bernstein-Sato polynomial. After convincing ourselves (hopefully) that Bernstein's inequality is useful, we delve into its proof! Here there will be two roads to take: a purely algebraic one or a more "geometric" one using Characteristic Varieties (here Algebraic Geometry and (some) Symplectic Geometry make their clear appearance). Depending on time I will choose one or another... To conclude, I want to mention the connection to Perverse Sheaves via the Riemann-Hilbert Correspondence and finally say something about some questions I am thinking about!
Any curve of genus at least 2, defined by equations with rational coefficients, will have finitely many points with rational coordinates. The proof of this result, known as Mordell's conjecture, earned Faltings a Fields medal. Faltings' proof is in no way effective, theoretically or computationally. For certain curves an effective result was obtained in 2015 based on an intuitive approach of Chabauty from the 1940s (completely different from the technical one of Faltings). The idea is to realize the rational points on the curve as zeros of suitably chosen analytic functions and then to count zeros of analytic functions. I'll present Chabauty's method (as reinterpreted by Coleman in the 1980s). I'll explain or at least present as a black box everything I'll use.
Come hear about the many different resources that Notre Dame makes available to graduate students to help them plan their career, begin their professional lives, and apply for jobs, grants, fellowships, etc.
A sum-free set in a group is a subset A of elements with the sum of any two elements of A falling outside A. In 1988 Andrew Granville asked "at most how many sum-free sets can a group of order n have?'' One approach is to transform the question into one about independent sets (sets of mutually non-adjacent vertices) in a graph, and ask "at most how many independent sets can a regular graph with n vertices have?'' This question is related not just to combinatorial group theory, but also coding theory, statistical physics, and even Dedekind's question on the number of monotone Boolean functions. It took twenty years for the independent set question to be fully answered (by Yufei Zhao, at the time an undergraduate), but in the interim there were some lovely partial results, using techniques from probability, combinatorics, and information theory (entropy). The question is still an active one, with two new proofs of the main result having been found recently, and with lots of associated open questions. In this talk I'll review the history, sketch some of the proof methods, and mention some of my favourite open problems.
The field of Natural Language Processing (NLP) and Machine Learning in general has seen a resurrection in the use of Neural Networks. Deep Learning is now achieving state-of-the-art results in numerous tasks, such as Machine Translation, Language Modelling, Speech Recognition, etc. I'll do a review of sequence-to-sequence Recurrent Neural Network models, on how a network is constructed, what are attention mechanisms, and how the networks are trained, with a focus on natural language applications.
The unified transform method was introduced in late nineties as the analogue of the inverse scattering transform machinery for integrable nonlinear equations on the half-line. It was later understood that it also has significant implications for linear initial-boundary value problems. In this talk, this method is used for showing well-posedness of nonlinear dispersive equations on the half-line with data in appropriate spaces. The nonlinear Schr\"odinger (NLS) equation will serve as the basic model for demonstrating this method.
An operad is an algebraic structure consisting of a sequence of sets satisfying certain composition properties and an action by the symmetric group. We will delve into the formal definition by exploring the endomorphism operad, a prototypical example.
Come hear all about the job application process after mathematics graduate school and what you can do to make the process a smooth and successful one! A panel of current graduate students (and one post-doc) will speak to us about their experiences with the job application process and answer any questions you may have.
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