Contents

- 1 Introduction to Algebraic Number Theory and Arithmetic Geometry
- 2 What is a Black Hole?
- 3 A Brief Introduction to Nonstandard Analysis
- 4 A Bit of not complex Complex Analysis
- 5 A Brief Introduction to Nonstandard Analysis
- 6 Axiomatic Number Theory and Gödel's Incompleteness Theorems
- 7 Homology: Hungry, Hungry Homology
- 8 The Discharge Method and Injective Coloring
- 9 Minimizing Convex Functions with Bregman Distance: Projection onto Simplex and Nonlinear Inverse Problems.
- 10 Minorities in Mathematics
- 11 An Introduction to Determinantal Rings
- 12 Connections between Tor and Torsion
- 13 Mathematics Education
- 14 Properties of Lelong Numbers of Currents on Complex Projective Space
- 15 Noncommutative Invariant Theory and Quantum Symmetry
- 16 Differential Forms and Integration Beyond Euclidean Space
- 17 Coxeter Groups
- 18 Low Rank Matrix Recovery via Gradient Descent
- 19 Principle Component Projection and Some Applications
- 20 F-Saturation Games
- 21 The Alexander Polynomial from the Twenties to the Teens
- 22 Liouville's Theorem and Other Fun Things
- 23 Practical Uses of Complex Analysis
- 24 "On a Certain Cooperative Hat Game"
- 25 "Lights Out: The Effect of Graph Operations on the Nullspace of the Neighborhood Matrix"
- 26 "Transcendence, Transcendence Degree, and Schanuel's Conjecture"
- 27 "A Proof from the Book: Erdõs; Proof of Bertrand's Postulate"
- 28 "Elliptic Curves, Galois Representations, and L-Functions"
- 29 "Quiver Mutations and Cluster Categories"
- 30 "The Nature of a Monster"
- 31 "Matrix Coefficients for Cuspidal Representations of "
- 32 "Geometry and Analysis on the Heisenberg Group"
- 33 "Listing the Positive Rationals"
- 34 "Morse Theory (Or: Where Multivariable Second Derivative Test Comes From)"
- 35 "A Curve of Positive Area"
- 36 "Counting Paths in a Directed Graph"
- 37 "Cut and Paste: Transversality in Dimension 3"
- 38 "Kernel-Based Approximation Method and its Applications"

##### Introduction to Algebraic Number Theory and Arithmetic Geometry

** Speaker: Caleb McWhorter (Syracuse University) **

** Friday April 20, 2018, 1:00, Carnegie 109 **

Abstract: Algebraic Number Theory (ANT) brings together the tools of Group Theory, Field Theory (especially Galois Theory), Commutative Algebra, Analysis, and Representation Theory to answer questions in Number Theory. This talk will serve to introduce the audience to the major ideas of ANT leading into the development of Arithmetic Geometry. The talk will primarily focus on the ring of integers in a number field and its properties. The talk will be concrete, using explicit examples to motivate the big theorems. Depending on interest and time, topics may include: finiteness of class ideal groups, Dirichlet's Theorem, Chebotarev Density Theorem, Quadratic Reciprocity, elliptic curves, the Class Number Formula, or introductory notions of Class Field Theory. The talk will be generally accessible to anyone who has taken (or is in) MAT 632.

##### What is a Black Hole?

** Speaker: Professor William Wylie (Syracuse University) **

** Friday April 13, 2018, 1:00, Carnegie 109 **

Abstract: Einstein's theory of general relativity makes a number of predictions about the universe that have amazingly turned out to be verified with physical experiments. With Advanced LIGO's recent detection of gravitational waves, the existence of black holes is another one of these predictions. In this talk I'll introduce the mathematics that is used to model the universe in general relativity. The goal for the talk will be to understand the mathematical definition of a black hole.

##### A Brief Introduction to Nonstandard Analysis

** Speaker: Fabian Rupp (Universität Ulm) **

** Friday March 9, 2018, 2:15, Carnegie 109 **

Abstract: The Curve Shortening Flow (CSF) is a quasilinear parabolic PDE and probably the simplest example of a geometric evolution equation. Other famous examples include the Willmore Flow, the Yamabe Flow and the Ricci Flow. We will introduce the CSF as the gradient flow for the arclength energy and examine some characteristic properties of solutions. Together with an appropriate existence and uniqueness result this allows us to completely describe the long-time behavior of the flow if the initial data is a simple closed smooth curve. If time permits, we will also look at some visualizations and see that the flow may be extended through singularities in the case of non-embedded initial data.

##### A Bit of not complex Complex Analysis

** Speaker: James Heffers (Syracuse University) **

** Friday December 1, 2017, 1:00, Carnegie 109 **

Abstract: In this talk we will explore some basics of complex analysis, and show that even the basics that we cover are powerful tools for solving other problems. To do so, we will start with what a complex number is, build up to Liouville's Theorem, and then use it to prove the fundamental theorem of algebra. While this talk will be aimed at first and second year graduate students (in an attempt to persuade the undecided masses to join the dark side, i.e., to become an analyst), it should be enjoyable for everyone (in an attempt to convert those of you who made bad life choices and did not join the dark side).

##### A Brief Introduction to Nonstandard Analysis

** Speaker: Erin Tripp (Syracuse University) **

** Friday October 26, 2017, 3:30, Carnegie 109 **

Abstract: The hyperreal numbers *R are an extension of the real numbers which include infinite and infinitesimal numbers. Both Newton and Leibniz used infinitesimals in the development of calculus, but their existence and properties were not made rigorous until the mid-20th century. Nonstandard analysis is the study of *R along with its application to real analysis via the transfer principle. In this talk we will discuss the logical basis and construction of *R and demonstrate the transfer of some familiar theorems and definitions.

##### Axiomatic Number Theory and Gödel's Incompleteness Theorems

** Speaker: Eric Ottman (Syracuse University) **

** Friday October 6, 2017, 1:00, Carnegie 109 **

Abstract: At the start of the 20th century, various mathematicians attempted to remove all paradoxes from mathematics by creating systems of "axiomatic" set theory and number theory, consisting (vaguely speaking) of a set of axioms and a strict set of rules on how these axioms may be manipulated to produce theorems. However, in 1931, Kurt Gödel proved that any sufficiently powerful such axiomatic system must be either incomplete or inconsistent - in other words, in any such system there must exist either a provable false statement or an unprovable true statement. In this talk, I will attempt to give a more precise description of what I mean by "axiomatic systems," and then to summarize the (constructive) proof of Gödel's result. It's a little more fun than it sounds and should be accessible to all graduate students.

##### Homology: Hungry, Hungry Homology

** Speaker: Caleb McWhorter (Syracuse University) **

** Friday September 13, 2017, 2:30, Carnegie 109 **

Abstract: Homology is at the heart of many areas of Mathematics and hence lies at the intersection of many fields. The talk will begin with a motivation for homology in Algebra and Topology and then discuss applications with explicit calculations. As time permits, applications to Analysis and Algebraic Number Theory will be discussed. The talk will touch on many topics covered in MAT 632, MAT 731, MAT 732, MAT 661, MAT 761, and MAT 705 so it will hopefully serve as a good 'advertisement' for these courses. However, the talk will be accessible to all graduate students regardless of background.

##### The Discharge Method and Injective Coloring

** Speaker: Jennifer Edmond (Syracuse University) **

** Friday April 28, 2017 **

Abstract: In this talk we will introduce the discharge method--a powerful technique in structural graph theory and most famously used in proving the 4-Color Theorem. The method involves assigning charge to vertices and faces and then moving that charge around to prove global results by only considering local properties. Additionally, we consider an application of the method involving the injective coloring of graphs with maximum degree 3 and girth 6.

##### Minimizing Convex Functions with Bregman Distance: Projection onto Simplex and Nonlinear Inverse Problems.

** Speaker: Yi Zhou (Syracuse University) **

** Friday March 31, 2017 **

Abstract: This talk introduces an algorithm framework based on Bregman distance for solving convex optimization problems. We start with an introduction to basic concepts of convex sets and convex functions, which then naturally leads to the definition of Bregman distance. Then we consider the problem of minimizing a convex function, and present the algorithm which iteratively approximates the original problem via Bregman distance. Finally, we demonstrate the applicability of this algorithm by solving some popular optimization problems with specific choices of Bregman distance.

##### Minorities in Mathematics

** Speaker: Caleb McWhorter (Syracuse University) **

** Friday April 17, 2017 **

Abstract: Most 'normal' people can name a theorem of Euler. Even more people can name a conjecture of Serre. [Indeed, opening a math textbook to a random page and pointing will usually get you one.] Any mathematician worth their salt will be aware of Cauchy's 'Not-So-Nice' Value Theorem and could recognize baby face Galois. But how many people have even heard of Grace Young or Grace Hooper? How many mathematicians are aware that it should rightly be called 'Noether's First Isomorphism Theorem'? Are you aware of Mary Rudin's 'Indiana Jones'-like escapades? Too often history praises the accomplishments of its white male mathematicians and forgets or outright discards the contributions of female mathematicians or mathematicians of color (even more so when one is both!) This talk will discuss some of the amazing accomplishments and stories of these forgotten mathematicians, put the importance of these recognitions into a modern context, and discuss how to re-evaluate how we think/talk about these issues in the future.

##### An Introduction to Determinantal Rings

** Speaker: Laura Ballard (Syracuse University) **

** Friday March 24, 2017 **

Abstract: Determinantal rings are really nice rings that can be handy to have up your sleeve for testing conjectures. In this talk, we will define what determinantal rings are and discuss some of their nice properties. We will then use some sleight of hand to compute a system of parameters and a singular locus (these will be defined) of some determinantal rings.

##### Connections between Tor and Torsion

** Speaker: Rachel Gettinger (Syracuse University) **

** Friday February 24, 2017 **

Abstract: The absence of torsion in tensor products of modules can give useful information on the properties of the modules themselves, and thus has received much attention over the years. In this talk, we will define torsion, the Tor-functor, and explore a few of the many connections between them. We will end with a discussion of a classic open question regarding torsion of tensor products. This talk will be accessible to all graduate students.

##### Mathematics Education

** Speaker: Jordan Johnson & Sam Leitermann (Syracuse University) **

** Friday February 3, 2017 **

Abstract: Mathematical education has repeatedly found that the traditional lecture format is unsuited for communicating mathematical understandings to students. More innovative approaches to mathematics are needed to truly create mathematicians in the classroom. In this talk, Jordan Johnson and Sam Leitermann share insights they've gained from working with high school and middle school students in mathematics classrooms, including some strategies and tips to improve your mathematics instruction.

##### Properties of Lelong Numbers of Currents on Complex Projective Space

** Speaker: James Heffers (Syracuse University) **

** Friday November 11, 2016 **

Abstract: Lelong numbers can be thought of as the density of a current at a point. In this talk we will look look at the geometric properties of sets of points with "large" Lelong numbers, starting with the results of Dan Coman. We will then show that if we have at least four points with "large" Lelong numbers, then we can show that certain upper level sets will be contained in a conic with the exception of at most one point.

##### Noncommutative Invariant Theory and Quantum Symmetry

** Speaker: Joshua Stangle (Syracuse University) **

** Friday November 11, 2016 **

Abstract: Invariant theory is a branch of mathematics that extends back to before Hilbert. It is the algebraic way to think of symmetry and a great deal is known in the case of affine algebraic varieties and commutative rings. When we stop assuming our rings are commutative, much is lost. Mostly, there just are not enough group actions. In this talk we will review classical invariant theory up to the Shephard-Todd-Chevalley Theorem. We will then turn our attention to the noncommutative case. We will introduce Hopf algebras as a new source of actions and examine some attempts at generalizing the STC theorem. Examples will be easy and frequent. Lastly, we will discuss the notion of quantum symmetry and the burning question of whether or not any "genuine' Quantum Symmetries exist. This talk should be accessible to any graduate student who can define the words "group," "polynomial," and "eigenvalue."

##### Differential Forms and Integration Beyond Euclidean Space

** Speaker: Daniel Cuneo (Syracuse University) **

** Friday November 4, 2016 **

Abstract: In this talk, I will define differential forms on manifolds. I will then explore their properties and explain how they can be used to extend integration to surfaces. I will sketch of proof of Stokes' Theorem and, time permitting, define some interesting differential forms called Free Lagrangians. This talk will be accessible to all graduate students.

##### Coxeter Groups

** Speaker: Professor Mark Kleiner (Syracuse University) **

** Friday October 28, 2016 **

Abstract: This is a preview of MAT 830, “Coxeter Groups,” Topics in Modern Algebra, Spring 2017. Coxeter groups were introduced as a simultaneous generalization of several classes of groups arising in different areas of mathematics. Examples of those classes are:

1. The symmetric group of degree n.

2. The dihedral group of order 2n.

3. Finite groups generated by orthogonal reflections in the Euclidean n-space.

4. The Weyl group of a complex semi-simple Lie algebra.

5. The Weyl group of a finite graph with no loops.

We give a formal definition of a Coxeter group, show how some of the above classes fit the definition, and associate a symmetric bilinear form to each Coxeter group.

The course presents basic facts about Coxeter groups, including the classification of finite Coxeter groups, root systems, exchange condition, and Bruhat and weak orderings. The classification is based on the fact that a Coxeter group is finite if and only if its associated bilinear form is positive definite.

##### Low Rank Matrix Recovery via Gradient Descent

** Speaker: Erin Tripp (Syracuse University) **

** Friday October 21, 2016 **

Abstract: In May 2016, a team from Cornell University and the University of Texas at Austin presented a fast algorithm for robust principal component analysis via gradient descent. I will discuss the general problem of recovering a low rank matrix from an observed matrix of corrupted data using this algorithm and its applications to image processing.

##### Principle Component Projection and Some Applications

** Speaker: Stephen Farnham (Syracuse University) **

** Friday October 7, 2016 **

Abstract: In Applied Mathematics, large data sets can be difficult to work with. Some matrices with high rank can be projected onto lower dimensional matrices without losing key information that data scientists, physicists, chemists, and engineers require for their studies. One of the operators that is particularly effective in signal noise reduction is the Principal Component Projection matrix. While extremely effective, traditional methods to compute such matrices require calculating a Singular Value Decomposition which can be time expensive as data sets increase in size. This talk will provide an introduction to Principle Component Analysis, and discuss a new method of computing the Principal Component Projection matrix, and some applications that have arisen.

##### F-Saturation Games

** Speaker: Josh Fenton (Syracuse University) **

** Friday September 23, 2016 **

Abstract: In a positional game, if both players are using optimal strategies, the game will conclude after a certain number of steps. This number will be the same every time and will only depend on the "game board." I will talk about different positional games and how the length of the game is affected by the graph being played upon.

##### The Alexander Polynomial from the Twenties to the Teens

** Speaker: Professor Douglas Anderson (Syracuse University) **

** Friday September 9, 2016 **

Abstract: This talk will give an historical survey of the Alexander polynomial from its introduction by J. W. Alexander in 1928 through its “categorification” by P. Qzvath and Z. Szebo in 2004-2014 . The talk will begin with the original definition of the Alexander polynomial given by Alexander. It will go on to describe its reformulation in the 1930s using Seifert surfaces and J. H. Conway’s refinement of it in 1969. If time permits, it will also sketch R. H. Fox’s aoproach to it via his free differential calculus. The talk will close with a brief outline of how the work of Ozvath-Szebo on Floer homology and its combinatorial relative, grid homology, recasts the Alexander polynomial as an Euler characteristic. This will be an almost entirely non-technical talk accessible to all graduate students.

##### Liouville's Theorem and Other Fun Things

** Speaker: James Heffers (Syracuse University) **

** Friday February 5, 2016 **

Abstract: Liouville's Theorem is a classical result in complex analysis which succinctly states that any bounded entire function must in fact be constant. During the talk we will investigate some basics of complex analysis that will lead up to the proof of Liouville's Theorem and then follow it up an interesting example (due to D.J. Newman) of an entire function which is bounded on every line through the origin of the complex plane, yet it is not constant. The talk is made to be accessible to all graduate students.

##### Practical Uses of Complex Analysis

** Speaker: Loredana Lanzani (Syracuse University)**

** Friday December 4, 2015 **

Abstract: The notion of conformal mapping is of fundamental importance in complex analysis. Conformal maps are used by mathematicians, physicists and engineers to change regions with complicated shapes into much simpler ones, and to do so in a way that preserves shape on a small scale (that is, when viewed up close).

This makes it possible to "transpose" a problem that was formulated for the complicated-looking region into another, related problem for the simpler region (where it can be easily solved) -- then one uses conformal mapping to ``translate'' the solution of the problem over the simpler region, back to a solution of the original problem (over the complicated region).

The beauty of conformal mapping is that its governing principle is based on a very simple idea that is easy to explain and to understand (much like the statement of Fermat's celebrated last theorem) .

In the first part of this talk I will introduce the notion of conformal mapping and will briefly go over its basic properties. I will then describe some of the many real-life applications of conformal maps, including: cartography; airplane wing design (transonic flow); art (in particular, the so-called "Droste effect" in the work of M. C. Escher).

Time permitting, I will conclude by highlighting recent work by McArthur fellow L. Mahadevan that uses the related notion of quasi-conformal mapping to link D'Arcy Thompson's iconic work *On Shape and Growth* (published in 1917) with modern morphometric analysis (a discipline in biology that studies, among other things, how living organisms evolve over time).

##### "On a Certain Cooperative Hat Game"

**Speaker: Professor Dmytro Yeroshkin (Syracuse University)**

** Friday November 13, 2015 **

Abstract: In 2010, Lionel Levine introduced a cooperative game, and posed the question of finding the optimal strategy. I will provide an overview of prior work on this

question, and describe several strategies that give the best lower bound on the probability of victory. I will also provide some discussion on finding the upper bound, and on one generalization of the problem.

##### "Lights Out: The Effect of Graph Operations on the Nullspace of the Neighborhood Matrix"

**Speaker: Laura Ballard (Syracuse University)**

** Friday October 16, 2015 **

Abstract: Based on a puzzle by Tiger Electronics, “Lights Out” can be formulated as a problem in graph theory and linear algebra. The objective of Lights Out is to turn off all of the lights (vertices), in which case a graph has been “won.” In our research, we worked with generalized Lights Out puzzles in which each light is in one of several states, one of which is designated as “off.” We not only studied winnable states of graphs, but also investigated the null space of the neighborhood matrices of these graphs, and how the null space changes when a graph is altered. In my talk, I will give an overview of how the Lights Out Problem is built up, and will talk about some of our results. Our goal is that these results lead to a more complete understanding of how the null space (and hence, the number of winnable states) changes as graphs are built up from paths and cycles, or as subgraphs are removed.

You may also find the presentation for the talk at this link.

##### "Transcendence, Transcendence Degree, and Schanuel's Conjecture"

**Speaker: Joshua Stangle (Syracuse University)**

** October 10, 2015 **

Abstract: Most mathematician's know that there are only countably many algebraic numbers. Thus, there must be many transcendental numbers. How many can you name? This talk will give some insight into how one might look for transcendental numbers or prove that a number is transcendental. We will also discuss Schanuel's conjecture, which would solve many questions in number theory, including the relationship between and .

##### "A Proof from the Book: Erdõs; Proof of Bertrand's Postulate"

**Speaker: Dr. Jeffrey Meyer (Syracuse University)**

** April 10, 2015 **

##### "Elliptic Curves, Galois Representations, and L-Functions"

**Speaker: Caleb McWhorter (Syracuse University)**

** October 24, 2014 **

Abstract: The theory of elliptic curves is perhaps the most eclectic field of mathematics, bring together number theory, complex analysis, topology, algebraic geometry, and representation theory. These theories and their related functions and groups have answered some of the oldest problems in mathematics including Fermat's Last Theorem and the Congruent Number Problem. However, there are many open questions such as the famous question of Birch and Swinnerton-Dyer.

The purpose of the talk is twofold. First, introduce the theory of elliptic curves by introducing some of the fundamental theorems governing their structure, especially their torsion points: the theorems of Mordell-Weil, Nagell-Lutz, Serre, and Siegel. Finally, the talk will discuss a bit about the connection between elliptic curves, L-functions, and Galois representations with a focus on Wiles' proof of Taniyama-Shimura and the conjecture of Birch and Swinnerton-Dyer.

##### "Quiver Mutations and Cluster Categories"

**Speaker: Stephen Hermes (Wellesley College)**

** October 17, 2014 **

##### "The Nature of a Monster"

**Speaker: Claudio DiMarco (Syracuse University)**

** October 10, 2014 **

##### "Matrix Coefficients for Cuspidal Representations of "

**Speaker: Carl Ragsdale (Syracuse University)**

** April 4, 2014 **

Abstract: If is a group, a complex representation of is a pair , where is a complex vector space and is a group homomorphism. This talk will introduce complex representations and some of the basic results used to study them. These techniques will then be used to classify the irreducible representations of the group , where is an arbitrary finite field. Many (but not all) irreducible representations can be constructed by considering representations of the upper-triangular subgroup of . Irreducible representations that cannot be constructed in this way are called cuspidal representations. The last portion of the talk will be devoted to describing an explicit model for these cuspidal representations, which can be used to explicitly compute the matrix coefficients of any cuspidal representation of .

##### "Geometry and Analysis on the Heisenberg Group"

**Speaker: Alex Austin (University of Illinois at Chicago)**

** February 28, 2014 **

##### "Listing the Positive Rationals"

**Speaker: Jack Graver (Syracuse University)**

** February 21, 2014 **

##### "Morse Theory (Or: Where Multivariable Second Derivative Test Comes From)"

**Speaker: Margaret Doig (Syracuse University)**

** November 15, 2013 **

##### "A Curve of Positive Area"

**Speaker: Claudio DiMarco (Syracuse University)**

** November 8, 2013 **

##### "Counting Paths in a Directed Graph"

**Speaker: Thomas Howard (UC Santa Barbara)**

** October 11, 2013 **

##### "Cut and Paste: Transversality in Dimension 3"

**Speaker: Peter Horn (Syracuse University)**

** October 4, 2013 **

##### "Kernel-Based Approximation Method and its Applications"

**Speaker: Qi Ye (Syracuse University)**

** September 20, 2013 **