- 10/11/2023: Exploring Quantum Symmetries by Melody Molander
Abstract: In this talk, I will first talk about our familiar notion of symmetry. Then I will describe compact quantum spaces and groups which will then aid us in understanding what is meant by quantum symmetry. Then I will show how to better understand quantum symmetries through planar algebras. To conclude, I will relate the work of my thesis to understanding some quantum symmetries. There will be lots of pictures and lots of hand-waving. - 10/18/2023: Ultrafilters and a connection to additive combinatorics by John White
Abstract: We explore the strange new world of $\beta\bbN$, the Stone-Čech compactification of $\bbN$, and, along the way, we will prove a fundamental result of additive combinatorics. - 10/25/2023: Functional calculi of operator algebras by Quinn Kolt
Abstract: We are familiar with the matrix exponential, where we ``plug in'' a matrix into the Maclaurin series expansion of $e^x$. However, one can ``plug in'' linear operators into all sorts of functions, not just Taylor series. In this talk, I will discuss how to do this with C*- and von Neumann algebras. I will start by introducing these classes of operator algebras with examples. Then, I will prove some basic results about them using the continuous functional calculus for C*-algebras and the Borel functional calculus for von Neumann algebras. This talk will be a great blend of analysis, algebra, and topology. - 11/1/2023: How Many Scales from 12 Tones? by Alexander Sabater
Abstract: Mathematical Music Theory is the study of music through mathematical tools. In this presentation I will give a basic example of counting how many musical scales are there in 12-tone temperament. I will first discuss how best to mathematically model a scale, then use simple enumerative combinatorial techniques to list all scales. I will conclude with some remarks on further directions in the study of scales. Some basic knowledge about music is recommended. - 11/8/2023: An invitation to crystallographic groups by Choomno Moos
Abstract: History and aesthetics notwithstanding, there is much interest to be derived from crystallographic groups. For example, every compact flat manifold (constant zero sectional curvature) may be obtained as a quotient of Euclidean space by the action of a torsion-free crystallographic group. While we won't have the time to cover intricate theory, I hope to introduce the audience to the subject. - 11/15/2023: How Symmetries of the Universe Induce the Laws of Physics by Alex Bisnath
Abstract: My goal with this talk is to give the audience an introduction to the Lagrangian formalism in classical mechanics and an explanation of Noether's theorem. This theorem states that for each symmetry of the Lagrangian, there exists a corresponding conservation law. We'll begin with a brief review of basic physics so no background is needed.
Polymath Seminar
Organizers: Quinn Kolt and Choomno Moos
The Polymath Seminar is a weekly graduate student seminar which, rather than focusing on a specific subject, emphasizes sharing what we love about mathematics. Each week, a different graduate student will present any topic that interests them so long as it is math-related and accessible to other math grads.Upcoming:
- 4/3/2023: Model Theory and Ax's Theorem by Quinn Kolt
Abstract: In this talk, I will introduce some basic notions of model theory, and demonstrate the power of the field by using it to prove a theorem in algebra. Let $p_1(x_1,...,x_n), ..., p_n(x_1,...,x_n)$ be polynomials with complex coefficients, and define the function $F:\bbC^n\to\bbC^n$ by $F(x_1,...,x_n)=\begin{bmatrix}p_1(x_1,...,x_n)\\ \vdots\\ p_n(x_1,...x_n)\end{bmatrix}$. Then, Ax's theorem says that, if $F$ is injective, then it's also surjective. To prove Ax's theorem, we also prove the Lefschetz principle, which is an astounding result about the relationship between positive characteristic and zero characteristic algebraically closed fields. - 4/12/2023: Graph homomorphisms and Hedetniemi's conjecture by Choomno Moos
Abstract: We are all intimately familiar with the problem of scheduling. We deal with this each term for our courses/sections/math lab. Problems resembling scheduling are more generally called constraint satisfaction problems. As it turns out, such problems are equivalent to the problem of finding particular graph homomorphisms. After arguing for the importance of the subject, we will undergo a brief introduction to the theory of graph homomorphisms. We will conclude with a discussion of the famed conjecture of Hedetniemi, which defied graph theorists for over 50 years before a counterexample was found. - 4/19/2023: An Introduction to Interactive and Zero-Knowledge Proofs by Daniel Epelbaum
Abstract: Usually a proof of a statement provides some knowledge about why the statement is true. For example, a proof that two graphs are isomorphic might be given by simply exhibiting such an isomorphism. But what if you wanted to prove something while providing, provably, no other information about the statement? For example, can Alice prove her identity to Bob, without giving Bob enough information to impersonate Alice later? Can a government prove the results of an election without revealing information about how individuals voted? Can Russia prove to the US, or vice versa, that a particular container contains a nuclear warhead, without giving away information about its design? Can you prove that a graph is 3-colorable without giving any information about a particular coloring? In this talk, we will define the notion of a zero knowledge proof, and look at some protocols for well known problems. - 4/26/2023: Multiplex Juggling Sequences and Kostant's Partition Function by Sam Sehayek
Abstract: In 2020, Pamela Harris et al. discovered a very interesting and fun connection between mathematical juggling and vector partition functions, published in Annals of Combinatorics. In this talk, we will introduce the concepts of mathematical juggling and the underpinnings of Konstant's Partition function, and outline some of the results obtained by leveraging these ideas. To date, there isn't a known closed form for Konstant's Partition function, so if time permits, we may also discuss open problems that remain along this line of inquiry. The slides presented are borrowed with permission from Pamela Harris's keynote address at the GSCC 2023. - 5/10/2023: Factoring Quickly with Quantum Computers by Daniel Epelbaum
Abstract: This talk will serve as an introduction to quantum computing, with an eye towards one of its most talked-about applications-factoring large numbers quickly. No prior knowledge of quantum mechanics will be assumed. - 5/17/2023: Memoryless Distributions and Poisson Processes by Evan Tufte
Abstract: In this talk, I'll introduce memoryless distributions then prove the perhaps surprising fact that the only memoryless exponential distributions are the exponential distributions. After that, we'll discuss poisson processes, where they come up, and their basic properties. - 5/31/2023: Cops and Robbers on Graphs by Jeremy Khoo
Abstract: Cops and Robbers is a two-player game played on a simple connected graph, where a set of cops attempt to occupy the same vertex as a single robber. The cop number of a graph is the minimum number of cops so that there is a strategy for the cops to win in a finite number of rounds. In this talk, we will discuss some results on the cop number for some classes of graphs, including a characterization of graphs whose cop number is 1. We will introduce Meyniel's conjecture that the cop number of a connected graph with n vertices is bounded above by $O(\sqrt{n})$. We will also look at a result of Bradshaw, Hosseini and Turcotte from 2021 which exhibits a family of graphs whose cop number is $\Theta(\sqrt{n})$, suggesting that Meyniel's conjecture, if true, is the best possible bound.