Archive: Spring 2023-Spring 2024

Schedule of Talks - Spring 2024

April 22nd

Yoyo Jiang: A Geometric Introduction to Representation Theory

Abstract: What exactly does it mean for a group to represent the symmetries of an object? In this talk, I will attempt to answer this question by convincing you that the study of group actions, or representations, comes up even more naturally than the study of groups themselves. We will start by looking at Klein’s innovative solution to the problem of classifying geometric theories in the nineteenth century by looking at their symmetry groups in his 1872 Erlangen program. We will see some concrete examples in this framework and use them to motivate the study of Lie groups and their representations. We will then introduce Lie algebras as a way to turn these problems into linear algebra, and end by looking at the irreducible representations of the Lie algebra sl_2(C) to get a flavor of the general strategy for classifying the representations of arbitrary semisimple Lie algebras.

April 15th

Yash Lal: Modular Forms by Example

Abstract: In this talk, we will be exploring the bizarre and unexpected mathematics of Modular Forms, with a hands-on approach. We will first understand the basic definitions of the elements involved (modular forms, SL2(Z), fractional transformations, congruence subgroups) with lots of pictures to help us! Then, we will work through an example that shows how the coefficients of a special modular form solve the sums of four squares problem from number theory (and discuss its generalizations briefly). Finally, we will connect to the larger geometric picture that underlies the theory to conclude.

April 8th

Ben Marlin: An Introduction to the Classification of Finite Simple Groups

Abstract: With a proof spanning thousands of pages across hundreds of journals, the classification of finite simple groups is one of the most monumental achievements of 20th century mathematics. In my talk, we will explore how finite simple groups act as the building blocks for all finite groups, and discuss the general strategy of the classification of such simple groups. We will introduce Feit and Thompson’s odd order solvability theorem and show how this theorem along with the Brauer-Fowler theorem provides a possible strategy of attack on the classification theorem.

April 1st

Sean Owen: A Brief Introduction to Gödel's Incompleteness Theorems

Abstract: In 1931 Kurt Gödel proved a pair of landmark results that limited the strength of formal theories of mathematics. These "incompleteness theorems" are central to modern understandings of logic, and also to numerous misunderstandings. So today, we'll explore these theorems - how they work, how they were developed historically, and what they do (and don't) mean for mathematics as a whole. No particular mathematical background is assumed, but bring your best thinking cap and be ready to process some truly bizarre ideas.

March 25th

Dr. Rahul Dalal: What is an automorphic representation?

Abstract: Andrew Wiles' famous proof of Fermat's last theorem was crucially based on constructing a mysterious connection between two classes of mathematical objects: "modular forms" and "elliptic curves". However, if you try looking up what a modular form actually is, you'll find an elementary but very unmotivated definition involving functions on the upper-half complex plane behaving in certain ways under a strange symmetry condition. We'll discuss how to motivate this definition by looking at modular forms as a special case of something called an automorphic representation---a "localized vibrational mode" of a special high-dimensional space built out of a group of matrices and some "arithmetic" lattice inside it. We'll also discuss how these automorphic representations seem to appear in many surprising places across mathematics, including a vast generalization of Wiles' connection called the Langlands program.

March 11th

Prof. Richard Brown: A mathematical stroll through a chaotic landscape

Abstract: Hardly random and not very randomly hard, a dynamical system deemed chaotic is extremely well-ordered, yet completely unpredictable, in terms of using the system as a mathematical model to predict future behavior. In this talk, we will define the basics of chaos theory, using some elementary examples and basic analysis to see through the mess to the order. Along the way, we will learn some of the tools dynamicists use to study the nonlinear mathematical models that are completely deterministic yet useless for long term prediction.

March 4th

Kailee Lin: Functional Graph Theory

Abstract: In my talk we will see some fundamental ideas of functional graph theory. A functional graph is a directed graph where each vertex has out-degree = 1. We will explore what taking a functional approach to graph theory can reveal, and what polynomials have to do with any of this? We will use this understanding to explore problems relating to graph coloring, graceful labeling, and more! This talk will be accessible to folks of any math background.
Prerequisites: None

February 26th

Jonathan Lin: The Kissing Problem and Linear Programming bounds

Abstract: The Kissing Problem (in dimension n) asks how many blue spheres of radius one can touch a red sphere of the same size. Using what are known as "Linear Programming Bounds" we will illustrate how this problem can be elegantly resolved in dimensions 8 and 24.
Prerequisites: None

February 19th

Simone Masserini: Topological methods for solutions to semilinear equations.

Abstract: Introduction to topological degree theory in finite and infinite dimensional spaces; Brouwer and Leray-Shauder fixed point theorems and existence of nontrivial solutions to Partial Differential Equations.
Prerequisites: None

February 12th

Anish Chedalavada: Simplicial Homology, starting from V - E + F.

Abstract: We will discuss some of the history behind the Euler characteristic, the Betti numbers, the homology groups and ultimately the definition of simplicial homology. This story is an incredibly exciting snapshot into how mathematical intuitions come about and build on each other, and if time permits I would like to highlight how this might inform your own experiences with the field.
Prerequisites: None

February 5th

Sofia Taylor: Classification Using Support Vector Machines with Uncertainty Quantification

Abstract: Binary classification using machine learning is needed to address engineering problems such as identifying passing/failing parts based on measured features from aging hardware. In these classifications, providing the uncertainty of each prediction is essential to support engineering decision making. One popular classifier is the support vector machine (SVM). There are many variations, with the simplest being a linear division between two classes with a hyperplane. Kernel methods can be implemented for data that is not linearly separable. Adding the use of a loss function allows for control over misclassification in the training data. The many variants of SVMs and their tunable model parameters readily enable ensembling, which we use to represents model (epistemic) uncertainty. SVMs also can be used to depict intrinsic uncertainty in the data, or aleatoric uncertainty. One of the major shortcomings of SVMs is that they make predictions indiscriminately over the entire prediction space, resulting in seemingly high-confidence predictions where there is no data. In this talk, we present one way around this by combining ideas from outlier-detection methods and SVMs, with the goal being to surround each class while simultaneously separating it from the other class. A unique representation of UQ results from this “single-class” SVM strategy is presented.
Prerequisites: None

February 2nd

Prof. Su Ji Hong: Integer Decomposition Property of Lattice Polytopes

Abstract: Given a polytope (higher dimensional polygon), P, and k \geq 0, we can multiply every point of P by k to obtain another polytope kP. One of the properties P can have is the integer decomposition property: if x is a point in kP, then there exist k points in P such that they add up to x. Not all polytopes have this property. At the Graduate Research Workshop in Combinatorics, we showed that certain lattice polytopes have this property.
Prerequisites: None

January 26th

Prof. Xavier Ramos Olivé: The role of curvature in the production of sound: Music, Geometry and Beyond

Abstract: When a string vibrates, it produces different pitches depending on its length. This is how guitarists can play several notes using only one string. But what happens when we have surfaces or 3D objects that vibrate, like when playing a flute, a digeridoo, or a handpan? Beyond their size, does their shape affect the pitch? We will discuss the role that curvature plays in this question, introducing some intuition for what curvature is, and presenting some recent developments regarding how sound changes under small perturbations of the geometry of an instrument. More precisely, we will revisit classical theorems in Geometric Analysis related to the eigenvalues of the Laplacian on Riemannian manifolds, and we will introduce newer versions of these results that use integral Ricci curvature conditions. If time permits, we will also discuss new related directions of research that connect Differential Geometry, Graph Theory and Data Science.
Prerequisites: None

January 24th

Dr. Kayla Wright: Frieze Patterns

Abstract: As we are in the midst of a cold, dark winter... there is no better time to learn about frieze patterns. Frieze patterns were first defined by mathematicians to classify certain symmetries in ancient art and architure. In this talk, we will explore arrays of mathematical objects that obey these classified frieze pattern symmetries. We will start by looking at frieze patterns of integers and then show that these integers can be interpreted both geometrically and algebraically. More specifically, we will connect frieze patterns of integers to lengths of diagonals in polygons as well as generators for a combinatorially defined algebra called a cluster algebra. At the end of the talk, I will give a loose definition for these algebras and mention some important questions in cluster theory.
Prerequisites: None

January 22nd

Prof. Boya Wen: Ways to write a positive integer into sums of positive integers

Abstract: Take a positive integer, say 4; we can write it as a sum of positive integers in the following five ways: 4, 3+1, 2+2, 2+1+1, 1+1+1+1. Among these five ways, how many have only odd numbers as summands? How many of them have no repeating summands? Do the two numbers agree? Do the odd-summand ones have more summands in total, or do the distinct-summand ones? What if we replace 4, by 5, or 6, or any positive integer n? In this talk, we will explore these questions, methods to study them, and their generalizations. Time permitting, I will talk about my recent joint work with Cristina Ballantine, Hannah Burson, William Craig, and Amanda Folsom, answering a generalization of these questions.
Prerequisites: None

Schedule of Talks - Fall 2023

November 27th

Yash Lal: Size and Topology

Abstract: In this talk, we will begin by motivating the definition of a topology on a set through geometric considerations. Then, we will be talking about a fundamental concept from topology: the notion of size. We will discuss it from two distinct viewpoints, that of compactness and that of filters, and combine the two to prove a hard theorem. In the end, we will touch on the more abstract implications of this theory.
Prerequisites: an informal understanding of continuity of a function and the extreme value theorem (covered in calculus 1)

November 6th

Daniel Pezzi: An introduction to the Fourier Transform and the Restriction Conjecture

Abstract: This talk will motivate the definition of the Fourier Transform and establish its connection to PDEs. From a PDE perspective we will state an equivalent form of the restriction conjecture, one of the most prominent outstanding problems in harmonic analysis.

October 23rd

Prof. Yiannis Sakellaridis: The mathematics of quantum mechanics

Abstract: The development of quantum mechanics in the early 20th century was a revolution in physics, and had a transformational impact on mathematics, as well. Even as we still grapple with the interpretations of the theory, its mathematical foundations are rigorous, and rooted in simple concepts of linear algebra. I will discuss these foundations from scratch, assuming only familiarity with vector spaces, linear operators, and eigenvalues.
Prerequisites: None

October 9th

Prof. Nathan Pennington: Why you should take Differential Equations

Abstract: A first encounter with differential equations often leaves the impression that differential equation is just integrating with extra steps. In this talk we'll show that what it means to "solve" a differential equation is more subtle (and mathematically interesting) than it initially appears. We'll also describe how these different types of "solutions" contribute to the wide applicability of differential equations.
Prerequisites: Calc 2, Some Linear Algebra, Calc 3

September 25th

Prof. Ashwin Iyengar: Elliptic Curves and Modular Forms

Abstract: If you’ve heard of Fermat’s last theorem, you might have heard that its proof involves some sophisticated geometric objects that seemingly have nothing to do with whether a^n + b^n = c^n for n > 2. The objects in question are elliptic curves and modular forms. I will attempt to explain what these are, and how despite their apparent unrelatedness (both from each other and from Fermat’s “theorem”) they are related in intricate and unexpected ways. The prerequisites for the talk will be as minimal as possible.
Prerequisites: None

September 11th

Prof. Emily Riehl: A Reintroduction To Proofs

Abstract: An introduction to proofs course aims to teach how to write proofs informally in the language of set theory and classical logic. In this talk, I'll explore the alternate possibility of learning instead to write proofs informally in the language of dependent type theory. I'll argue that the intuitions suggested by this formal system are closer to the intuitions mathematicians have about their praxis. Furthermore, dependent type theory is the formal system used by many computer proof assistants both "under the hood" to verify the correctness of proofs and in the vernacular language with which they interact with the user. Thus, there is an opportunity to practice writing proofs in this formal system by interacting with computer proof assistants such as Coq and Lean.
Prerequisites: None

Schedule of Talks - Spring 2023

March 13th

Prof. Victoria Akin: Harry Houdini’s Magic Math Trick

Abstract: What theorem is so mystifying that Harry Houdini was writing about it in his 1922 book, Paper Magic? In this talk, we’ll hint at the proof of a surprising theorem and demonstrate some magic of our own! Along the way, we’ll develop some mathematical machinery for paper folding and paper flattening. We’ll also explore some cutting, unfolding, and flattening of polyhedra. With any luck, we’ll state an open question or two.
Prerequisites: None

April 14th

Dr. Ben Dees: Buffon's Needle

Abstract:Suppose that I draw a bunch of lines parallel to each other, spaced one inch apart, and drop my standard-issue one-inch needle onto them at random. How likely is it that the needle will cross one of the lines? More generally, what if the needle is longer, shorter, or is actually a squiggly piece of uncooked pasta? There are, broadly speaking, two approaches to this. One of them involves setting up some number of integrals to figure out these probabilities. This approach is entirely valid and will work, but there’s another approach that just relies on probability theory. In this talk, we will compute no integrals, and use no calculus. Instead, we will see a marvelous display of the glorious power called “linearity of expectation,” and that’s all we’ll need.
Prerequisites: None

April 28th

Yoyo Jiang: Introduction to Monoidal Categories

Abstract: In this talk, we will introduce categories, look at some examples, and see how category theory can help us describe mathematical objects. We will attempt to reconstruct the integers with the familiar properties that we know and love using categorical language, and use that to motivate the definition of a monoidal category. We will finish by seeing some fun examples of monoidal categories. No background is required, although some exposure to linear algebra or abstract algebra will be helpful.
Prerequisites: None