An Introduction to ART with X-ray Tomography

Ryan Hass
November 17

Abstract: X-ray tomography is used to recover a function from its line integrals and is directly related to medical imaging. An alternative to popular filtered-backprojection methods, the Algebraic Reconstruction Technique (ART) provides a robust, iterative reconstruction algorithm that is based on Kaczmarz’s method for solving linear systems. We will begin by reviewing the X-ray transform, describing a few scanning geometries used in Computed Tomography, and then have a small diversion on computing the X-ray transform of the unit square in \(\mathbf{R}^2\).

Hex

Felix Weilacher
November 10

Abstract: Hex is an abstract board game invented independently in the 1940’s by Piet Hein and John Nash. In it two players take turns placing stones in a hexagonal grid attempting to form a path between two opposite sides of a quadrilateral. The main mathematical theorem of interest about this game is that there are no ties; given a fully colored game board, exactly one player has a winning path. Much of the appeal of Hex lies not in this theorem itself but in the surprising variety of seemingly unrelated fields in which it shows up. We will prove the Hex theorem and highlight a connection to topology due to David Gale. We then discuss some variants of the game in other topological spaces and their implications.

The strange persistence of Dynkin diagrams

Vera Serganova
November 3

Abstract: I will talk about special graphs called Dynkin diagrams which appear in different areas of mathematics. We will begin with quivers and their representation theory, where Dynkin graphs correspond to finite type quivers. Then we will discuss roots systems and Coxeter groups. If time permits I will talk also about Kleinian singularities and Dynkin graphs.

Complex and p-adic uniformization

Avi Zeff
October 27

Abstract: A frequent theme in number theory is that "global" information about a number field, such as the rational numbers, can be studied in terms of its associated "local" fields, such as the p-adic numbers or the real or complex numbers. One way in which this principle manifests is by various kinds of "uniformization"; we will study some examples. An interesting feature here is that we often need to pick a distinguished "place" to use as our viewpoint, and we will try to understand how this choice affects the result.

Fermat's Last Theorem

Kenneth Ribet
October 20

Abstract: I was led to speak about Fermat's Last Theorem because a bunch of my Math 110 students asked me to explain my role in the proof of the theorem, which was completed in 1994. My talk will give an overview of the subject and its history but will contain even less hard mathematics than the previous Math Monday talks that I've given over the years. I hope that it will be fun for all of us!

Phase transitions

Vilas Winstein
October 13

Abstract: I’ll briefly discuss my experience with the Fulbright program at Budapest Semester in Mathematics/Rényi Institute and how I found my direction in research through making expository math videos. I’ll then give an introduction to phase transitions including a demo of an interactive simulation, and finish by talking about some of my current research on fluctuations in random graph models inspired by statistical mechanics

Integral formulas for the divergence operator and its generalizations

Sung-Jin Oh
October 6

Abstract: We will start with the elementary question: what does the divergence tell you about a vector field? In the first part of the talk, I will give an answer to this question by writing down formulas for "good" vector fields with a given divergence. The inspiration will come from the idea of duality in linear algebra, combined with the fundamental theorem of calculus on curves. Next, I will describe how this procedure generalizes to a large class of important under-determined differential operators, such as the linearizations of the scalar curvature in Riemannian geometry and the compatibility equations for initial data sets in general relativity.

Black hole thermodynamics

Ryan Unger
September 29

Abstract: One of the central principles of modern theoretical physics is the belief that black holes are thermodynamic objects. This principle was first formulated by Bardeen, Carter, and Hawking in 1973 as the “four laws of black hole thermodynamics,” which are precise mathematical statements within the framework of Einstein’s theory of general relativity. In this talk, I will introduce the basic concepts of general relativity, the theory of black holes, and the laws of black hole thermodynamics. I will then explain how recent advances in mathematics have led to the fall of the third “law” of black hole thermodynamics.

Irrationality and transcendence of numbers

Yunqing Tang
September 22

Abstract: In this talk, I will discuss some classical and recent results on irrationality of interesting numbers arising from number theory and geometry, including the recent joint work of Calegari, Dimitrov and myself on certain products of log values. I will also explain some famous conjectures on irrationality and transcendence of numbers coming from geometry. No prerequisite knowledge on number theory is needed for this talk.

Infinite Games

Antonio Montalbán
September 15

Abstract: Infinite two-player games have been a very useful tool to prove many results in logic and other areas. What makes them fascinating is the beautiful regularity properties one can derive from them, creating tension with unintuitive consequences of the axiom of choice. In the first half of the talk, we will describe these games and talk about their connection to the axiom of choice. In the second part of the talk, we introduce the necessary background to understand the answer-given by the author and Richard Shore-to the following question: How much determinacy of games can be proved without using uncountable objects?

From numerical semigroups to Riemann surfaces

David Eisenbud
September 8

Abstract: A numerical semigroup is a subset of the non-negative integers closed under addition. Despite the simplicity of the definition, there are many open problems both combinatorial and algebraic. I'll explain a few, and finish with an unsolved problem, posed by Hurwitz in 1892, from my current research.