The isoperimetric inequality has a long history in mathematics, going back to the legend of Queen Dido. It is closely related to other important inequalities in analysis, such as the Sobolev inequality and the Brunn-Minkowski inequality. Many different proofs of the isoperimetric inequality have been found. These proofs employ a variety of techniques, including symmetrization, optimal mass transport, the ABP technique, and ideas from the calculus of variations. In this lecture, I will discuss some of these techniques, and explore connections to minimal surface theory.

Given four general lines in three dimensional space, how many lines intersect all of them? We will explore the field of algebraic geometry, using this question to introduce the theory of Schubert calculus. Along the way we will glimpse key concepts such as moduli spaces, Chow rings, and intersection theory.

Spaces are complicated objects and so mathematicians started using algebraic gadgets to study them. As time went on, we actually realized that we can use spaces instead to do algebra. This eventually led to the stable category, where instead of doing algebra over the integers, we do algebra over the sphere.

This week, we will be having a game night! Available games include Uno, Exploding Kittens, multidimensional tic-tac-toe, and a standard card deck for your games of choice.

This week will just be a social dinner; we hope this gives you all a chance to decompress from midterm season!

I'll talk about some weird connections between physics, math, and puzzles. This will be interactive!

I will describe some of the basics of symplectic topology, and try to explain some kinds of problems that people work on today.

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Grothendieck introduced K-theory to give meaning to formally subtract one object from another. For example, while you can take the (direct) sum of two vector spaces, there is no natural way to subtract one vector space from another. However, it turns out that simply allowing such formal operations is extremely useful. This notion (along with its “higher” analogues) has been generalized to various contexts; in particular, algebraic K-theory studies the K-theory of commutative rings. Surprisingly, despite its topological origins, algebraic K-theory has deep and unexpected connections--some conjectural, some proven--with number theory. I will explain what (algebraic) K-theory is and describe some of these connections.

Continuous and smooth topology are the same in small dimensions, and the difference between them in high dimensions is governed by algebraic topology. I'll discuss the funny "middle dimension" 4, why it's so much less pleasant than its bigger or smaller siblings, and what this has to do with theoretical physics.

A dynamical system is a mathematical framework for studying “chaos”. This talk is a gentle introduction to the mathematical area of “dynamics” and how it interacts with geometry through explicit examples and constructions. We will find and certify chaotic behavior by just looking at matrices!

In this talk, we’ll explore how to represent mathematical knots - loops in 3-dimensional space that can twist and tangle but never break - in a simple, combinatorial format called a grid diagram. We’ll then build an invariant called grid homology, which captures deep geometric and topological information about knots, but can be read directly from our diagram. Along the way, we’ll briefly touch on the broader theory of Heegaard Floer homology and see how grid homology offers an accessible gateway into this powerful area of modern mathematics. We’ll also discuss how the grid approach is computationally advantageous without sacrificing any of the power of the invariant. This talk is aimed at a general undergraduate audience, with no prior experience in topology, algebra, or knot theory needed.

In recent years, certain ideas from mathematical logic, namely o-minimality, have found their way into parts of number theory in unexpected ways. In this talk, we will give a brief overview of o-minimality and explain how it can be used to get at cool number-theoretic problems.

Motivated by the classical Brachistochrone problem, we will explore some basic techniques of the Calculus of Variations and their evolution to deal with more complex minimization questions. We will touch upon free boundary problems which beautifully describe natural phenomena such as the formation of snowflakes or the melting of the ice.