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

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.

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

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.

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

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.

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

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.

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

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.