# Notre Dame Math Graduate Student Seminar, 2016-2017

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.

Previous Semesters

To volunteer to give a talk, or for anything else regarding the seminar, contact PJ Jedlovec.

## Schedule Fall 2015

DateSpeakerTitle
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
Differential Cohomology
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

## Schedule Spring 2016

DateSpeakerTitle
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

## Abstracts

### September 7, 2015

Speaker
Gabor Szekelyhidi
Title
Min-max Methods in Geometry
Abstract
Min-max methods have been used recently to solve several long-standing problems in differential geometry by Fernando Coda Marques and Andre Neves.  I will explain the basic idea of the method and I will give a sketch of some of these applications.

### September 14, 2015

Speaker
Alexander Diaz
Title
Peak Sets of Classical Coxeter Groups
Abstract
Given a permutation (or more generally a signed permutation) we can "graph" it and study its "peaks."  The combinatorial study of peaks of permutations is a topic that has caught the attention of mathematicians in the past 20 years.  For example, it has been shown that the set of sums of permutations with a given peak set is a subalgebra of the group algebra.  Extending the notion of peaks to signed permutations, we can generalize some of the results for usual permutations, while some others do not admit a generalization.  In this talk I will survey some of the most relevant and beautiful results in this area, including some of my work in collaboration with Jose Pastrana and many others.

#### September 28, 2015

Speaker
Jeremy Mann
Title
Differential Cohomology
Abstract
A differential cohomology theory produces invariants of manifolds. Like a generalized cohomology theory, these invariants are in some sense “locally determined,” and give global measurements of shape. However, unlike a “regular” cohomology theory, a differential cohomology theory is not homotopy invariant. Thus, these theories can “see” more refined geometric properties of manifolds, such as the curvature of a  connection. In this talk, I will present some of the basics aspects of differential cohomology theories, their applications to physics, and their modern formulation

#### October 12, 2015

Speaker
Alan Liddell
Title
A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems
Abstract
Exceptional sets of a parameterized polynomial system are the sets in parameter space where the fiber has higher dimension than at a generic point. Such sets are arise in kinematics, for example, in designing mechanisms which move when the generic case is rigid. In 2008, Sommese and Wampler showed that one can use fiber products of bounded order to compute exceptional sets since they become irreducible components of larger systems. We propose an alternative approach using rank constraints on Macaulay matrices. This hybrid symbolic-numerical approach first symbolically constructs the appropriate matrices and then uses numerical algebraic geometry to solve the rank-constraint problem. We demonstrate the method on several examples, including exceptional RR dyads, lines on surfaces in C^3, and exceptional planar pentads.

#### October 26, 2015

Speaker
Eric Wawerczyk
Title
Congruences of Modular Cusp Forms
Abstract
Sequences of numbers {a_n} arise everywhere. We can study properties of a sequence using different functions: Arithmetic functions, Generating functions, and L-functions. If this sequence of numbers is "modular" then we can say a lot about it. Our specimen is the Ramanujan Tau  function \tau(n) which will be our prime example of a normalized eigen modular cusp form". We will discuss the duality of weight k modular forms with the weight k Hecke algebra to motivate the key functionalities of a "Hida Family" and their relationship to congruences of modular  forms.

#### November 9, 2015

Speaker
Gabriela Clemente
Title
Toric manifolds and polytopes
Abstract
We talk about the correspondence between symplectic toric manifolds and Delzant polytopes.

#### November 23, 2015

Speaker
Michael Perlman
Title
Minimal Free Resolutions and Hilbert's Syzygy Theorem
Abstract
The idea to associate a free resolution to a finitely generated module was introduced by Hilbert in the late nineteenth century. Minimal free resolutions are invariants of modules that encode algebraic and geometric information. This talk in meant to be an introduction to the theory of minimal resolutions through the computation of several examples. We will prove the Hilbert Syzygy Theorem and provide a geometric application.

#### December 7, 2015

Speaker
Whitney Liske
Title
Abstract

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.

#### January 18, 2016

Speaker
Eric Wawerczyk
Title
Deformation of Sequences
Abstract

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.

#### February 8, 2016

Speaker
Title
Parameterization and Implicitization of projective plane curves
Abstract

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.

#### February 15, 2016

Speaker
Xiaoxiao Li
Title
Kahler-Einstein metrics on compact Kahler manifolds
Abstract

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.

#### February 29, 2016

Speaker
Luis Saumell
Title
D-modules, Bernstein's inequality and some of its consequences
Abstract

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!

#### March 14, 2016

Speaker
Xiaoxiao Li
Title
Kahler-Einstein metrics and stability
Abstract

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.

#### March 21, 2016

Speaker
Sebastian Bozlee
Title
A connection between recurrence sequences and affine schemes
Abstract

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.

#### April 4, 2016

Title
Panel on Job Applications

#### April 11, 2016

Speaker
Jeremy Mann
Title
Topological Data Analysis
Abstract

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.

#### April 18, 2015

Speaker
J.D. Quigley
Title
Brave New Rings
Abstract

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.