Research

Graduate Seminar Series
(Leader: )

Friday, November 22, 2024

Title: Quantization, Part II
Speaker: Bryce Virgin
Time: 3:00pm–4:00pm
Place: CMC 108

Abstract

Quantization is the process by which we derive our quantum prediction apparatus from the classical theories of physics that preceded them, and as well saw in last week's talk, it is an art rather than as science. This is because the most obvious constraint we could impose to determine a quantization uniquely is inconsistent with itself by the Groenewold-Van Hove Theorem. This week's talk will focus on Feynmann's Path Integral method of quantizing a classical system, the advantages of this method over the operator method, and the mathematical inconsistencies inherent in the stated approach.

Due to unforeseen circumstances, today’s talk is canceled.

Friday, November 15, 2024

Title: Probably a Good Introduction to the Probabilistic Method
Speaker: Panith Thiruvenkatasamy
Time: 3:00pm–4:00pm
Place: CMC 108

Abstract

The probabilistic method is an existence proof method that gained popularity in the mid-20th century thanks to Hungarian mathematician Paul Erd?s, who pioneered the extensive use of the probabilistic method in fields such as combinatorics and graph theory. In its essence, it is a tool that counts the total number of combinatorial objects in a space against the number of objects that don't satisfy a certain property to indicate whether an object that does satisfy the property exists in the space. Despite its simplicity, it has proven to be an essential tool in discrete mathematics with its widespread impact in combinatorics, graph theory, number theory, information theory, complexity theory, and even algebra.

I plan to give a series of talks focused on the probabilistic method that will act as a somewhat formal introduction to the subject. In the first talk, we will be introducing the first moment method and Markov's inequality. With nonzero probability, I will have time to end with a few popular applications of these concepts in different areas of mathematics, but don't set your expectations too high 😉.

Friday, November 8, 2024

Title: Quantization
Speaker: Bryce Virgin
Time: 3:00pm–4:00pm
Place: CMC 108

Abstract

Quantum mechanics is the predictive apparatus which describes the time evolution of small particles and boasts some of the smallest differences between theoretical predication and experimental outcomes in all of science. Mathematically, the subject begins with the observation that classical physics as a theory must be derivable from the more accurate quantum theory via the taking of limits over parameters which control the influence of new quantum phenomena. This talk will be concerning the art of attempting to undo this highly non-injective limiting procedure, a process known as Quantization. In particular, this talk will discuss the contradictions that can be derived from the rules of what is known as Canonical Quantization, as well as some different procedures which have been made to satisfy consistent subsets of these rules.

Friday, November 1, 2024

Title: Matrix Integrals and Enumerative Combinatorics, Part III
Speaker: Abril Arenas
Time: 2:00pm–3:00pm
Place: CMC 120

Friday, October 25, 2024

Title: Matrix Integrals and Enumerative Combinatorics, Part II
Speaker: Abril Arenas
Time: 2:00pm–3:00pm
Place: CMC 120

Friday, October 4, 2024

Title: Matrix Integrals and Enumerative Combinatorics
Speaker: Abril Arenas
Time: 2:00pm–3:00pm
Place: CMC 120

Abstract

Planar graphs are graphs that can be drawn on a plane such that their edges have no overlaps. In the early 60's Tutte wrote a "Census on Planar Maps" which seeks to enumerate planar graphs with special properties. Independently, physicists like t'Hooft recognized that certain colored graphs from QCD correspond to certain matrix integrals. From the 80's–90's special cases of these matrix integrals were computed by making "topological expansions". This expansion then provides a correspondence between matrix integrals and a generating function for planar graphs with prescribed valency.

The purpose of these talks is to explore this amusing interaction between matrix integrals and enumerative combinatorics. Since the target audience are undergraduate students, we review some fundamental facts about graphs and iterated integration before building up to matrix integrals.

Monday, September 30, 2024

Title: A Revisit of Kolmogorov's mapping theorem for Neural Network
Speaker: Alice Chen
Time: 12:30pm–1:30pm
Place: CMC 108

Abstract

The Kolmogorov-Arnold Theorem is a powerful theorem concerning the representation of arbitrary continuous functions from \(n\)-dimensional cubes to real numbers in terms of one-dimensional continuous functions. Lets revisit its limitations and dive into recent research regarding extensions and its application in the field of machine learning.

Monday, September 23, 2024

Title: Algebraic Geometry, Part 3
Speaker: Bryce Virgin
Time: 12:30pm–1:30pm
Place: CMC 108

Abstract

Classical Algebraic Geometry was the study of varieties, solutions of systems of polynomials over familiar fields like the rational, real, and complex numbers. Modern Algebraic Geometry encompasses the study of a much broader set of spaces, and in particular much of the subject concerns the investigation of schemes. In this series of talk, we conclude a discussion on the motivations of this change of perspective that began in the spring. The audience is encouraged to review the two previous videos in this series of talks, as it has been some time since the last talk, and the goal of this series is to pick up right where the last left off.

Friday, September 13, 2024

Title: Ends of Groups
Speaker: Panith Thiruvenkatasamy
Time: 3:00pm–4:00pm
Place: CMC 108

Abstract

Last time, we looked at the notion of the ends of a space and studied the properties of the corresponding end compactified space. We finished by visiting a major result derived by Freudenthal, the originator of the notion of ends. This week, we plan to look into an equivalent definition of ends for the case of geodesic metric spaces. By doing so, we aspire to associate such a space to every finitely generated group and study the ends of the space to categorize its associated group in a useful way. Time permitting, we will END by proving an important theorem in geometric group theory.

Friday, September 6, 2024

Title: Putting an end to all this s\(\dotsc\) space, Part I
Speaker: Panith Thiruvenkatasamy
Time: 3:00pm–4:00pm
Place: CMC 108

Abstract

Hans Freudenthal first introduced the notion of the ends of a space in his Ph.D. thesis in 1930. His motivation was in topological group theory. His notion of ends allowed him to associate to a given space, \(X\), a set, the set of ends of \(X\), which can be adjoined to \(X\) to compactify it. The set of ends were proven to be invariants under certain families of homeomorphisms of \(X\) after they are extended to homeomorphisms of the end compactification of \(X\). This led to results such as: a path-connected topological group has no more than 2 ends. Since then, the notion of ends of a topological space has taken on a number of definitions, some not equivalent to others. The overarching motivation is always similar: compact spaces are nicer to study since their properties can be discerned using a finite amount of local information. Therefore, we wish to generate a functor from spaces to sets such that the set associated with a space is “created” (apologies for the vagueness) using complements of compact sets and allows us to study the space as imbedded in a bigger, compactified space. In Laymen's terms, we wish to formally define all the ways to reach “infinity” in a space. In this talk, I will lightly introduce the different notions of ends and their motivations. Time permitting, we can conclude by proving some major results.