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Click here for information on the Fall 2023 Proof Workshop
The Columbia Undergraduate Mathematics Society has the purpose of exposing members to different topics or areas of research in mathematics that they might not otherwise encounter in class. The lectures should be accessible to all students studying mathematics or pursuing mathrelated majors. Everyone is welcome!
Date  Speaker  Title  Abstract 
September 13 
Arjun Kudinoor

Introduction to AdS/CFT: A conversation between geometry and physics 
This talk is an introduction to the AdS/CFT or gaugegravity correspondence. We will cover ideas like metric spaces, separation of scales, an introduction to quantum field theory, and a glimpse into stringy gravity. This talk is accessible to undergraduate math and physics students at all levels. It does not require prior knowledge of any of the terms mentioned above. More at https://www.arjunkudinoor.com/talks/gaugegravityqgptalk

September 20 
Andrei Okounkov

Lie theory and Langlands duality beyond Lie groups 
Lie groups play a big role in many areas of mathematics and have a beautiful structure theory with a nice classification (which involves structures like root systems). Langlands observed that a certain simplelooking involution on root system shows up in a certain very deep and mysterious way in the theory of automorphic forms and related fields. Naturally, since the completion of the classification, Lie theorists have been looking for objects that extend and generalize the classical theory. One of such potential extensions will be discussed in this lecture.

September 27 
Undergrad Talks:
Lisa Faulkner Maria Stuebner 
Sphere Packing in 8 Dimensions The quotient dimension of hyperbolic threemanifolds 
The spherepacking problem in any dimension asks how one can place nonoverlapping balls in R^d and cover the largest possible fraction of space. The aim of this talk is to give an overview of the ideas in Maryna Viazovska's solution to the sphere packing problem in 8 dimensions using modular forms. We will introduce the crucial theorem she used, then define and discuss the properties of modular forms, and finally give a sketch of her proof.
For any finitely generated group Γ, the quotient dimension of Γ refers to the minimum dimension of a complex linear group containing an infinite quotient of Γ. In his work, “How often is 84(g−1) achieved?”, Michael Larsen shows that the quotient dimension determines the asymptotics of the set of orders of finite quotient groups of Γ; furthermore, he computes the quotient dimension of various groups of hyperbolic surfaces and orbifolds. In this project, we explore further conditions for determining the quotient dimension of finitely generated groups, specifically focusing on the fundamental groups of hyperbolic threemanifolds. 
October 4 
Ioannis Karatzas

LACUNARY/HEREDITARY LIMIT THEOREMS IN PROBABILITY AND ANALYSIS

In the spirit of the celebrated Komlos theorem, we develop versions of the Weak, the Strong, and the HsuRobbinsErdos Laws of Large Numbers, which are valid along appropriate (“lacunary”) subsequences of arbitrary sequences of random variables with bounded moments; as well as along all further (“hereditary”) subsequences of said subsequences. We review also the strong connections of this subject with lacunary trigonometric series. Joint work with Walter Schachermayer, Vienna.

October 11 
Simon Brendle



October 18 



October 25 
Lucy Yang



November 1 



November 8 
Abigail Hickok



November 15 



November 22 
Academic Holiday



November 29 

