Department of

Mathematics


Seminar Calendar
for AWM Graduate Student Colloquium events the year of Tuesday, December 3, 2019.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Wednesday, January 30, 2019

4:00 pm in 245 Altgeld Hall,Wednesday, January 30, 2019

Rescheduled

Wednesday, February 6, 2019

4:00 pm in 245 Altgeld Hall,Wednesday, February 6, 2019

Systems of Calogero-Moser Type

Matej Penciak (Illinois Math)

Abstract: It is well known that many-particle systems are in general not solvable analytically. For some specific choices of interactions between particles though, a lot can be said. In this talk I aim to give an introduction to systems of Calogero-Moser type and the surprising role of algebraic geometry in their solvability. I will also give a perspective on how this subject plays a role in some hot topics in mathematics in general: Hitchin integrable systems, geometric representation theory, and the geometric Langlands philosophy.

Wednesday, February 20, 2019

4:00 pm in 343 Altgeld Hall,Wednesday, February 20, 2019

Connecting Boolean (un)satisfiability to Graph Theory

Vaibhav Karve (Illinois Math)

Abstract: Given a Boolean formula can we find consistent assignments (True or False)for variables such that the formula is satisfied? This is the Boolean Satisfiability problem, a problem of great historic value in computer science. It is the first problem that was proven to be NP-complete. In this talk, I will introduce Satisfiability and explain what the terms P, NP, NP-complete... mean. I will then demonstrate a (surprising)connection between Boolean formulas and graph theory which will help us gain a more visual understanding of when a class of formulas is satisfiable or unsatisfiable. There will be lots of small graphs in this talk.

Monday, March 11, 2019

2:00 pm in 245 Altgeld Hall,Monday, March 11, 2019

A brief survey of extremal combinatorics and some new results for (hyper)graphs

Ruth Luo (Illinois Math)

Abstract: Extremal combinatorics is a branch of discrete mathematics which studies how big or how small a structure (e.g., a graph, a set of integers, a family of sets) can be given that it satisfies some set of constraints. Extremal combinatorics has many applications in fields such as number theory, discrete geometry, and computer science. Furthermore, methods in extremal combinatorics often borrow tools from other fields such as algebra, probability theory, and analysis. In this talk, we will discuss some benchmark results in the field as well as some recent results for extremal problems in graphs and hypergraphs.

Wednesday, April 17, 2019

4:00 pm in 245 Altgeld Hall,Wednesday, April 17, 2019

From Graph Laplacian to the Stability of Coupled Oscillator Networks

Lan Wang (Illinois Math)

Abstract: There is a large amount of applied problems that can be posed as dynamical systems on a coupled oscillator network. Frequently these problems involve computing the inertia of a graph Laplacian. In this talk we will start with an overview of the properties of the Laplacian matrix and then explore how it functions in the study of the stability of fixed points of dynamical systems. Particularly, we will discuss the Kuramoto model, a classic and popular model for describing the dynamics of a large population of coupled oscillators. We will first deliberate the stability of the phase-locked solutions of Kuramoto model on single-layer networks, and then extend it to multi-layer networks by examining the Supra-Laplacian matrix.

Wednesday, September 11, 2019

4:00 pm in 245 Altgeld Hall,Wednesday, September 11, 2019

Trees and leaves where boundaries meet

Elizabeth Field   [email] (Illinois Math)

Abstract: If $H$ and $G$ are hyperbolic groups with $H\leq G$, one can ask if the inclusion map from $H$ into $G$ extends continuously to a map from the boundary of $H$ into the boundary of $G$. If such a map exists, we call this map the Cannon-Thurston map. In this talk, we will first draw inspiration from the setting originally studied by Cannon and Thurston which has a particularly nice geometry. We will then take a brief tour through the world of geometric group theory, where we will discuss the notion of a hyperbolic group and its boundary. Finally, we will return to explore various geometric, topological, and algebraic properties of the Cannon-Thurston map.

Wednesday, October 9, 2019

4:00 pm in 245 Altgeld Hall,Wednesday, October 9, 2019

An overview of Erdős-Rothschild problems and their rainbow variants

Lina Li

Abstract: In 1974, Erdős and Rothchild conjectured that the complete bipartite graph has the maximum number of two-edge-colorings without monochromatic triangles over all n-vertex graphs. Since then, this new class of colored extremal problems has been extensively studied by many researchers on various discrete structures, such as graphs, hypergraphs, boolean lattices and sets. In this talk, I would like to give an overview of some past results on this topic. The second half of this talk is to investigate the rainbow variants of the Erdős-Rothschild problem. Our first main result, confirming conjectures of Benevides, Hoppen and Sampaio, and Hoppen, Lefmann, and Odermann, completes the characterization of the extremal graphs for the edge-colorings without rainbow triangles. We also studied a similar question on sum-free sets, in which we describe the extremal configurations for the colorings of integers without rainbow sums.

Wednesday, November 6, 2019

4:00 pm in 245 Altgeld Hall,Wednesday, November 6, 2019

Framed cobordism and the $J$-homomorphism

Ningchuan Zhang

Abstract: One of the main goals in homotopy theory is to compute the stable homotopy groups of spheres. Geometrically, these groups classify cobordism classes of framed submanifolds of spheres. This is a generalization of degrees of maps between oriented manifolds. From this perspective, I will explain in this talk how to compute the first stable homotopy group of spheres. This computation leads to the $J$-homomorphism, whose image is the most well-understood part in the stable homotopy groups of spheres. I will define the $J$-homomorphism using framed cobordism. If time allows, I will also give some computational results of its image, which are related to the Bernoulli numbers.

Wednesday, December 4, 2019

4:00 pm in 245 Altgeld Hall,Wednesday, December 4, 2019

Geometry and arithmetic over finite fields

Ravi Donepudi

Abstract: We will discuss algebraic, geometric and combinatorial approaches in the study of curves over finite fields. Our goal will be to give the audience a survey of the basic tools, questions and results in this area. We will discuss topics such as: Counting points on curves, the Weil conjectures, Jacobians and the Torelli problem. We will attempt to demonstrate that there are a variety of problems in this field ranging from those requiring heavy tools from algebraic geometry to those that can be investigated by writing a few lines of computer code. No prior knowledge is assumed beyond a passing acquaintance with the field of integers modulo a prime.