Grant Molnar
The Fox $H$-function
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In this seminar, we will explore the Fox $H$ function, a versatile tool for solving mathematical and physical problems in diverse fields. We will provide a brief historical background and outline the definition of the function. Several examples will be presented, showcasing how the $H$ function specializes to famous functions of general interest. We will also discuss various properties of the H function and illustrate how they reveal information about our specific examples. If time permits, we will highlight the broad range of applications of the $H$ function across various disciplines. This talk was inspired by "The $H$-function" by A. M. Mathai, Ram Kishore Saxena, Hans J. Haubold.
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Steve Fan
Building Finite Fields through Counting
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Finite fields, also known as Galois fields, are simply fields with finitely many elements. They are natural generalizations of the residue fields $\mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}$ and have many applications in number theory, algebraic geometry, combinatorics and cryptography. Given a prime power $q=p^n$, it is natural to ask whether there exists a finite field of order $q$. Following an elementary approach devised by Soundararajan, I will answer this very first question about finite fields by demonstrating the existence of an irreducible polynomial over $\mathbb{F}_p$ of degree $n$. As a bonus, the argument also reveals similarities in structure between $\mathbb{Z}$ and $\mathbb{F}_p[x]$. Finally, I will end my talk by giving an interesting application of finite fields to Mersenne primes.
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Richard Haburcak
Why are word problems hard?
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We won't actually be talking about word problems like you've seen them in calculus or other classes. We'll be talking about the word problem for groups, which is deciding if an element is trivial or not. We'll start with a cool example of a problem on constructing subsets of a topological space, and note that we needed to decide when two elements of a set were equal. Can we do this in general? Focusing on groups, this is the famous word problem for groups. We'll discuss more general questions, give a few reasons why you might care, and give an idea of why these problems are unsolvable.
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Matthew Ellison, Lucy Knight
Adult Coloring for Math Grads: Constructions with Ruler, Straightedge, and Origami
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Come trade your troubles for the Ancient Greeks'! Options include: construct a 5-gon with ruler and compass, construct a 17-gon with ruler and compass; construct a 7-gon with origami, and trisect an angle with origami. All materials will be provided! This GSS will help the speakers prepare for a similar activity at SK day the following weekend.
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Brian Mintz
Mathematical Art
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George Hart once said that "Math is the most abstract art," and indeed few people see the creativity and beauty in it that we do. By combining mathematical ideas with traditional media like sculpture and music, the broader public can appreciate this artistic side of math. In this talk, we'll explore the uncountable ways art has been infused with math, from using DeBrujin's theorem to count the patterns in traditional Japanese braiding, to edible polyhedra and much more!
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Ben Logsdon
Life without Limits: An Introduction to Synthetic Differential Geometry
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Synthetic Differential Geometry is a nonstandard mathematical world in which every set is implicitly a "smooth set" and every function is implicitly a smooth function. In SDG, the number line contains bona fide infinitesimals, and the derivative can be defined in terms of these infinitesimals without reference to limits. This greatly simplifies many definitions and proofs. I will not attempt to give a formal definition of SDG, but rather to give a taste of what it's like to do calculus and differential topology inside of this framework. To this end, I will define the derivative and prove some of its basic properties, define the tangent bundle on a space and prove some of its basic properties, and discuss some nonempty subset of differential forms, the Lie bracket, and integration.
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Alex Wilson
Spiders, Webs, and Representation Categories
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In my thesis, I studied representations by describing the maps between them with combinatorial diagrams. It turns out the folks down the hall studying operator algebras are interested in very similar things! In this talk, I want to give an idea of how I stumbled into this world from my work, tell a bit of the story for SL_n, and give a brief survey of what's known in the setting that most closely touches my work.
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Beth Anne Castellano
It's a hat! It's a shirt! It's an aperiodic monotile!
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In March 2023, David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss unveiled their preprint "An aperiodic monotile" on the arxiv. Within days, they gained lots of attention in the mathematical community and beyond for their exciting discovery of a tile that only admits non-periodic tilings of the plane (an "einstein tile") that has come to be known as the "hat". We'll discuss what all of this means and motivate the hype surrounding this discovery by looking at the history of aperiodic tilings. We'll also highlight important features of the two proofs given in their preprint (one of which is an entirely new approach for proving aperiodicity).
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Jinman Park
Learning Stochastic Representations of Partial Differential Equations through Deep Learning
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In recent years, there has been a growing interest in employing deep learning frameworks to deal with differential equations. This seminar aims to review the application of deep learning methods for solving partial differential equations (PDEs), with a particular focus on the stochastic approach. We will examine the derivation of neural networks and the associated loss functionals that enables obtaining of solutions. Furthermore, we will analyze the derivative-free loss method applied to elliptic PDEs and identify potential challenges that arise during experimental implementations.
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