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The seminars of the Columbia Undergraduate Mathematics Society are currently held virtually via Zoom on Wednesdays at 7:30 PM EDT unless otherwise noted. The talks have 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 
January 20 
Simon Brendle

Minimal surfaces and the isoperimetric inequality 
The isoperimetric inequality has a long history, going back to the legend of Queen Dido. In this lecture, I will discuss the isoperimetric inequality and how it relates to the calculus of variations, the notion of mean curvature, and minimal surface theory.

January 27 
Francesco Lin

You cannot hear the shape of a drum 
The goal of spectral geometry is to understand the relation between geometric shapes and the frequencies at which they can vibrate. After providing a general introduction to the subject, I will discuss concrete examples of planar domains that are isospectral, or, more informally, sound the same.

February 3 
Michael Miller

ell^p is not ell^q 
It is an important fact that all finitedimensional normed spaces are equivalent  meaning that you can bound each norm x_1 and x_2 in terms of the other, c x_1 ≤ x_2 ≤ Cx_1 for some 0 < c < C < infty. This is an important and useful fact in analysis. However, not all normed spaces are isometric, so that there is no invertible matrix A so that Ax_2 = x_1 for all vectors x. We will show that the finitedimensional normed spaces ell^p_n (that is, R^n equipped with the l^p norm) are not isometric for any p =/= q unless {p, q} = {1, infty}. Even better, we will discuss and calculate a quantitative measure of how far apart these spaces are. On the way, we will pass briefly through probability and convex geometry, and these ideas give rise to nonobvious results in the theory of infinitedimensional normed spaces.

February 10 
Evan Warner

The padic zeta function 
I'll sketch the construction of an object that deserves to be called a "padic zeta function" and admire it a bit, starting with what "padic" means.

February 17 8:30pm 

Undergraduate town hall on diversity, equality and inclusion in mathematics 
On Wednesday February 17 at 8:30pm, you are invited to participate in a zoom town hall meeting for Barnard/Columbia undergraduates who would like to discuss diversity, equality and inclusion (DEI) in mathematics. This is a chance for students to voice their perspectives, experiences and ideas. This event is cosponsored by the math department DEI committee and the Undergraduate Math Society.

February 24 

No meeting  Midterm/final exams 

March 3 

No meeting  Spring break 

March 10 
Sam Mundy

The infinitude of primes 
Circa 300 BCE, Euclid proved for the first time that there are infinitely many prime numbers. Euler reproved this fact in 1737 using a completely different argument, and in 1955, Furstenberg gave a variant on Euclid's proof using elementary pointset topology. In my talk, I will start by reviewing these proofs, and then I will recast Furstenberg's proof in a way which, as far as I'm aware, is new, and which will shed some light on Euler's proof as well.

March 17 
Nick Salter

Life After Galois 
The famous AbelRuffini theorem asserts that there is no formula for expressing the roots of a general fifthdegree polynomial using only radicals. Rather than being the definitive end to a story, it turns out that there are many further aspects of rootfinding left for more modern (and indeed future) mathematicians to discover. In this talk, I will discuss some more modern perspectives on rootfinding that showcase the roles played by topology and by dynamics.

March 24 
Alan Zhao

Modular Forms and RiemannRoch 
It is wellknown that the set of modular forms is ring isomorphic to the set of complex polynomials in two variables. I will first go over the traditional method by which this is obtained. I then take a more geometrical approach to the result, viewing the objects involved in the traditional method as Riemann surfaces.

March 31 
Henri Roesch

The Geometry of Light 
In Einstein's theory of general relativity, light rays weave together into a peculiar geometry measuring zero length. In this talk we will explore a class of these geometries and, in the presence of a Black Hole, use them to prove the famous positive energy theorem.

April 7 
Raymond Cheng

Seeing Sylow's Theorems 
The three theorems of Sylow presented in an introductory group theory course have always been a mystery to me. That is until recently, when I learned of a fundamentally geometric interpretation of these theorems. I shall explain this, and also use this to draw some analogies and unify some seemingly disparate concepts.
