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 7||Gabor Szekelyhidi
||Min-max Methods in Geometry|
|Monday, September 14||Alexander Diaz
||Peak Sets of Classical Coxeter Groups|
|Monday, September 28||Jeremy Mann
|Monday, October 12||Alan Liddell
||A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems|
|Monday, October 26||Eric Wawerczyk
||Congruences of Modular Cusp Forms|
|Monday, November 9||Gabriela Clemente
||Toric manifolds and polytopes|
|Monday, November 23||Michael Perlman
||Minimal Free Resolutions and Hilbert's Syzygy Theorem|
|Monday, December 7||Whitney Liske||Hilbert Functions of Graded Algebras|
|Monday, January 18||Eric Wawerczyk||Deformation of Sequences|
|Monday, February 8||Jeff Madsen||Parameterization and implicitization of projective plane curves|
|Monday, February 15||Xiaoxiao Li||Kahler-Einstein metrics on compact Kahler manifolds|
|Monday, February 29||Luis Saumell||D-modules, Bernstein's inequality and some of its consequences|
|Monday, March 14||Xiaoxiao Li||Kahler-Einstein metrics and stability|
|Monday, March 21||Sebastian Bozlee||A connection between recurrence sequences and affine schemes|
|Monday, April 4||Panel on Job Applications|
|Monday, April 11||Jeremy Mann||Topological Data Analysis|
|Monday, April 18||JD Quigley||Brave New Rings|
In Commutative Algebra, the Hilbert Function is an incredibly helpful tool in examining finitely graded modules in a polynomial ring. I will discuss several topics related to the Hilbert Function including bounds on how it can grow (Macaulays Theorem), and how the Hilbert Function acts under a link.
The talk begins with an introduction to a global perspective of modern number theory through the study of peculiar sequences of numbers which arise from seemingly unrelated circumstances: Fourier coefficients of holomorphic functions, counting solutions to algebraic equations over finite fields, representations of Lie Groups, and their corresponding L-functions. We will discuss the successes of this program with it's role in the proof of Fermat's Last Theorem. A useful tool for proving things about sequences of numbers is to consider them in families. We introduce Hida Families: deformation rings for p-ordinary modular forms and their generalizations.
A curve in the projective plane is the vanishing locus of a homogeneous polynomial. We refer to this polynomial as the implicit equation of the curve. Sometimes, the curve may also be described parametrically by polynomial equations. In geometric modeling, both the implicit and parametric equations are important, so it is useful to be able to convert between the two descriptions of a curve. I will explain how this conversion can be done. Along the way, we'll encounter some of the major tools used in the study of projective plane curves, such as blowing up, Bezout's theorem, linear systems, and adjoint curves.
In differential geometry, a basic problem is to find the "best" metric on a given manifold, and the most famous results are the geometrization theorems for 2 and 3-manifolds. For a compact ndimensional Kahler manifold M, one of thecandidates for the "best" metric would be the KahlerEinstein metric, i.e. a Kahler metric whose Ricci form is a constant multiple of w. Since the Ricci form represents the first Chern class c1(M), there are three different cases: c1<0, c1=0 and c1>0. In renowned works in the 1970's, the "Calabi-Yau" case (c1=0) was first solved by Yau and the negative case was settled independently by Aubin and Yau. The positive case was open for a long time and was just completely solved by Donaldson-Chen-Sun and Tian in 2013. In this talk I will focus on Yau's celebrated theorem. We will start with a very brief introduction to Kahler manifolds and then go into the Calabi's conjecture and Yau's proof. I will give the sketch of the proof and introduce the main ideas of the PDE methods that are used. Finally, time permitting, I will discuss the difficulties that arise in the positive case. Some basic knowledge of manifolds and analysis will be enough.
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!
One of the most important questions in Kahler geometry is whether a compact Kahler manifold admits any Kahler Einstein metrics. Since the Ricci form of a compact Kahler manifold represents its first Chern class, there are three different cases: c1<0, c1=0 and c1>0. It has been known since the mid-70's that c1 being zero or negative is sufficient for the existence of KE metrics. For the remaining case, it turns out that the positivity of the first Chern class is not enough and Yau conjectured that a Fano manifold admits a KE metric iff it is stable in the sense of geometric invariant theory. Tian introduced the notion of K-stability as a candidate for such a stability condition and proved that it is actually necessary. The sufficient part was proved by Chen-Donaldson-Sun and Tian in 2013. In this talk, starting with a brief introduction to finite dimensional GIT, I will explain how these ideas are fit into the story of finding extremal Kahler metrics and how they motivate the reformulation of Kstability given by Donaldson. Finally, I will discuss very briefly the ideas in Chen.
Linear homogeneous recurrence sequences are sequences (aj ) satisfying equations of the form aj = c1aj−1 + ... + ckaj−k. Common examples of such sequences are geometric sequences, which satisfy aj = raj−1, and the Fibonacci sequence, which satisfies the recurrence aj = aj−1 + aj−2. A standard result on the solutions of recurrence sequences implies that solution sets to recurrence relations admit a one-to-one correspondence to closed subschemes of the affine scheme A1 . Tantalized by this correspondence, we ask if there are similar correspondences for different recurrence problems with closed subschemes of other affine schemes. In this talk, we will answer this question in the affirmative, explore the benefits of this geometric perspective, and raise other questions.
Topological Data Analysis refers to any application of algebraic topology or abstract homotopy theory to gain insight into large or complex data sets. Topology is good for two things. First, topology has an assortment of techniques for taking a structure, and creating a "space." The "shape" of this space should somehow economically encode qualitative information about your structure. Second, topology has a toolbox (borrowed predominantly from commutative and homological algebra) for extracting computable "invariants" from these spaces. These invariants can be thought of as exotic types of clusterings, and, in many instances, they have interesting interpretations. In this talk, the speaker will attempt to give a nontechnical account of how businesses and researchers have been applying topology to data analysis. In particular, the speaker will not assume any familiarity with algebraic topology.
In first-year abstract algebra, one learns about things like commutative rings, localization, and Galois theory. I will discuss the stable homotopy theoretic analogues of these concepts: commutative ring spectra, Bousfield localization, and Galois extensions of ring spectra. I'll also mention some applications of these ideas to number theory and computing the stable homotopy groups of spheres.
MGSA - Math Department - University of Notre Dame =