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Title: An elementary proof of Conway's theorem on Rational Tangles Speaker: Bhavneet Singh Time: 3:00pm–4:00pm Place: CMC 108

In this presentation, I will introduce the concept of knots and hence tangles and Rational tangles. We build operations on Rational tangles and associate a fraction to the Rational tangle. Conway's theorem states that two rational tangles are ambient isotopic if and only if they have the same fraction. If there's time, I'll also talk about an application of this formulation in studying recombination processes in DNA.

Title: A bit about Carlitz polynomials and the Carlitz exponential Speaker: Zach Dorson Time: 4:00pm–5:00pm Place: CMC 107

In the 1930s, American mathematician Leonard Carlitz introduced a class of polynomials over \(F_p^r [T]\), which are now sometimes called Carlitz Polynomials. We will recount some facts about these polynomials and discuss how they give rise to an analog of cyclotomic extensions of \(Q\) for the field of rational functions over a finite field. We will also mention the Carlitz exponential, a characteristic \(p\)-analog of the complex-valued exponential function, and see how it connects to Carlitz polynomials.

Title: On graphs with large girths and large chromatic numbers Speaker: Boyoon Lee Time: 3:00pm–4:00pm Place: CMC 108

A graph \(G=(V,E)\) with a large girth would locally look like a tree, which can be (properly) colored using only 2 colors. Then, one might conjecture, naively, that such a graph with a large girth would necessarily have small chromatic number. This is false, proved by Erdös using a probabilistic method.

The probabilistic method, while guaranteeing the existence of such graph with a large girth and a large chromatic number, does not give you an instruction on how to build such graph. Indeed, in general, building such a graph is quite an ordeal. However, building a graph without any triangle but with an arbitrarily big chromatic number is quite easy and doable in the given amount of time.

Title: On the tangential Fatou Theorem Speaker: Alex Stokolos, Georgia Southern University Time: 3:00pm–4:00pm Place: CMC 108

In the talk, I will address Walter Rudin's problem of extension of the Fatou theorem to tangential regions.

Title: Weyl's Law and Figuring out the Shape of a Drum Using its Sound Properties, Part II Speaker: Louis Arenas Time: 12:00pm–1:00pm Place: CMC 108

Title: Weyl's Law and Figuring out the Shape of a Drum Using its Sound Properties Speaker: Louis Arenas Time: 12:30pm–1:30pm Place: CMC 108

In 1910 Hendrik Lorentz gave a series of lectures titled "Old and New Problems in Physics". In his last lecture he mentions physicists are interested in studying the growth rate of overtones of an electromagnetic wave in a bounded domain.

In plain English, Lorentz's problem translates to the growth rate of eigenvalues of the Laplacian on a given domain. Moreover, Lorentz conjectured that the growth rate of the eigenvalues only depends on the volume of the domain and its ambient dimension.

Herman Weyl in 1912 arrived at a positive resolution of Lorentz's conjecture, with further refinements of the asymptotics in the years following Weyl's law. In this talk we begin by describing the overtones of a vibrating drumhead and compute the associated eigenvalues explicitly. Time permitting, we sketch a proof of Weyl's law.

Title: An elementary proof of Conway's theorem on Rational Tangles Speaker: Bhavneet Saini Time: 3:00pm–4:00pm Place: CMC 108

Title: Introduction to Coxeter Groups Speaker: Aniket Joshi Time: 3:00pm–4:00pm Place: CMC 108

In this talk, I will familiarise ourselves to the key concepts of Coxeter groups. Coxeter Groups, named after the renowned geometer H.S.M. Coxeter, serve as powerful tools for understanding the symmetries inherent in geometric shapes and structures. I will start with reflections, generators, and Coxeter matrices. I will move towards Coxeter diagrams and Coxeter graphs, discussing how these encode symmetry information. Next, I will highlight the applications of Coxeter Groups across various mathematical disciplines, ranging from algebra and combinatorics to theoretical physics and crystallography. Finally, I will conclude by mentioning various open problems in Coxeter Group Theory.

Title: Varieties to Schemes: Generalizing Geometric Objects Speaker: Bryce Virgin Time: 3:00pm–4:00pm Place: CMC 108

Classical Algebraic Geometry has been concerned with the solutions of systems of polynomials throughout a large part of the history of mathematics. In recent decades, a shift has occurred, where the scope of the subject of AG was enlarged to study new objects, known as schemes. This series of talks will focus on the development of the definition of an abstract scheme from the classical notion of an affine variety, as well as the reasons this leap in generality was made. Assuming that time is on our side, similar examples of generalization in the field of differential geometry will be discussed, and some of the features shared by the generalization processes will be described. A little background in point set topology and abstract algebra would benefit the audience but is certainly not necessary to attend.

Title: A Characterization of Diffeological Spaces, Part II Speaker: Bryce Virgin Time: 3:00pm–4:00pm Place: CMC 108

Title: A Characterization of Diffeological Spaces Speaker: Bryce Virgin Time: 3:00pm–4:00pm Place: CMC 108

Over the course of the past century, the subject differential topology has had its scope expanded beyond the investigation of smooth manifolds. A growing body of work in the field now treats classical ideas in the context of a variety of different generalized smooth spaces, which are categories that extend the much more familiar category of smooth manifolds. A majority of the literature in this burgeoning corner of differential topology treats the properties of particular examples of generalized smooth spaces, but efforts have been made recently to investigate common structure among, and the relationship between, these new generalized smooth spaces. In this talk, we will discuss a definition for a category whose objects can be thought of as smooth structures over topological spaces and utilize it to characterize a particular category of generalized smooth spaces known as the Diffeological Spaces.

Title: On the Proof of Gödel's Incompleteness Theorems Speaker: Claudia Salles Gallo Time: 3:00pm–4:00pm Place: CMC 108

For millennia, but especially in the last century, mathematicians have had the hope of finding an axiomatic system that formalized areas like geometry and number theory. But can we really produce one that is both consistent and complete? Can mathematics truly be reduced to a finite set of axioms and inference rules from which we can deduce all the great theorems of history and all the ones to come? In 1931, a young German mathematician from the University of Vienna publishes “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”, where he single-handedly revolutionized the field of mathematical logic by answering this very question; Kurt Gödel, among other things, proves that for an axiomatic system mirroring number theory to be consistent, it must be incomplete, meaning that it will contain true statements that we can't prove to be true. Many mathematics enthusiasts are aware of this infamous and surprising result, but due to the great technical difficulties lying within Gödel's highly-specialized field, the proof in the original paper remains to be considerably inaccessible both to the average reader, and even to the non-specialized professional mathematician. This seminar thus aims to present the general idea of the proof of Gödel's Incompleteness Theorems and its main repercussions for the field of mathematics, following the exposition of the book “Gödel's Proof” by Ernest Nagel and James Newman.