5 – Lorenzo Sadun
Title: Random Graphs and Statistical Mechanics
Abstract: Suppose we have a random collection of water molecules whose total energy and volume are fixed. What else can we say about the water molecules? Would it matter if we changed the energy and volume a little? Questions like these are answered via statistical mechanics. The energy and volume (or the dual parameters temperature and pressure) control everything about the system, and at certain values of the parameters there are sharp phase transitions, where the nature of the material suddenly changes — ice, water, and steam are all different.
Now instead of water molecules, suppose we have a random graph with 1000 vertices, 300,000 edges, and 35,900,000 triangles. What else can we say about the graph? Would it matter if we changed the number of triangles to 36,100,000? It turns out that much of the formalism of statistical mechanics applies to this problem, and our usual cast of characters, including entropy, phases, and phase transitions, play familiar roles.
In this talk I’ll explain how statistical mechanics and random graph theory fit together, go over the key ideas developed elsewhere (“graphons” and “large deviations”), and sketch the current research being done here at UT by Charles Radin, Kui Ren and myself.
19 – Richard Hughes
Title: Geometric and physical aspects of symmetry
Abstract: In this talk we will explore the idea that symmetry is a fundamentally geometrical concept. In particular we will consider the ways in which symmetry is exploited in physics, and how geometry helps us organize and understand physical symmetries. We will conclude with an informal discussion of some symmetries which do not have obvious geometrical interpretations and the ways in which mathematicians and physicists approach them.
26 – Michael Hott
Title: An Application of GGT to Algebra
Abstract: I present a proof of the Nielsen–Schreier Theorem, which states that every subgroup of a free group is free. Though the result is purely algebraic in its statement, the proof makes use of topology. Knowledge of a group is required, and knowledge of free groups and graphs is useful. Those of you who haven’t worked with group theory very much might want to sit this one out.
10 – Altha Rodin
Title: Linear Feedback Shift Registers
Abstract: Pseudorandom numbers have many applications and a linear feedback shift register (LFSR) provides one well-understood way to generate them. We will look at some of the properties of LFSR’s and consider their strengths and weaknesses when it comes to cryptographic applications.
24 – Leon Liu
Title: Algebraic Geometry and Bézout’s Theorem
Abstract: This talk will be an introduction to modern Algebraic Geometry from the perspective of Scheme theory. Our focus will be developing a famous classical result known as Bézout’s Theorem. Algebraic Geometry, in its barest form, is the study of zeros of polynomials over a field. In classical Algebraic Geometry, the objects of study are called varieties, which are points in n-dimensional vector spaces over R or C. We will go over the limitations of classical Algebraic Geometry, how a scheme generalizes the notion of a variety, and how it surpasses the limitations. Then we will focus on Bézout’s Theorem, using tools such as Cech Cohomology, Hilbert Polynomials, and Weil Divisors to prove Bézout’s Theorem.
31 – Michelle Chu
Title: Triangle Groups
Abstract: Triangle groups are realized geometrically by sequences of reflections along the sides of a triangle. These groups are classified into 3 types: Euclidean, spherical, and hyperbolic. This talk will be a gentle introduction to triangle groups and will include some discussion on 2-dimensional geometry. If time permits, we will discuss a particular hyperbolic triangle group, the (2,3,7) triangle group, and its connection to the Hurwitz surfaces.
14 – Sebastian Munoz
Title: The Many Approaches to Optimal Transport
Abstract: The main idea of this talk is to set up the basic optimal transport problem, in increasing levels of generality. First we will talk about the original Monge problem and the motivation for optimal transport, then set up the more general Monge-Kantovorich problem. From this we will formulate questions which illuminate the different perspectives on optimal transport. Specifically, we will see the sorts of questions we can ask about the relationsip between optimal transport and the Monge-Àmpere Equation, the optimal transport map and curvature on a Riemannian Manifold, and finally state an open problem about the Levy-Gromov Inequality.
28 – Kartik Chitturi
Title: Classifying Endomorphism Algebras of Elliptic Curves
Abstract: We will talk about endomorphism algebras of elliptic curves. For elliptic curves defined over finite fields we will examine the differences between ordinary and supersingular curves. For elliptic curves over the complex field, we will talk about the relation to lattices and the ideal class group.
Title: Disk packings of the upper half plane & invariant measures
Abstract: The study of disk packings is an important part of the mathematics of aperiodic order, an active area of mathematics with close ties to physics and other areas of mathematics. In particular, the classic problem of densest disk packings of euclidean space has been solved in several dimensions; most recently in dimensions 8 and 24 by Maryna Viazovska, using methods from Fourier analysis.
For this talk, we will discuss the disk packing problem in hyperbolic space from a measure-theoretic viewpoint to understand how this setup differs from the same problem in euclidean space. By examining the Borozcky packing, we see why the notion of density of disk packings was long believed to be ill-defined for the hyperbolic case. With examples from the 1-D euclidean case, we build intuition on how to construct invariant measures on spaces on packings and how the density of a packing and the density of a measure is defined. Armed with these definitions, we consider results of Lewis Bowen et. al. that demonstrate some of the known properties of disk packings of the hyperbolic plane.
5 – Ewin Tang
Title: Totally Non-Negative Matrices (A Romp Through Algebra, Topology, and Combinatorics)
Abstract: A totally nonnegative (positive) matrix is a matrix whose minors are all nonnegative (positive). Matrices of these forms have arisen in areas from combinatorics to dynamics and probability, and have developed a deep and interesting theory surrounding them. We will explore the question of how to parametrize totally nonnegative matrices, and discover their interesting connections to topology and Coxeter groups. If time permits, we will describe a cute combinatorial interpretation of total positivity, and briefly discuss my work on a generalization of total nonnegativity, which was done at the University of Minnesota Twin Cities REU.