January 19th, 2024: Feedback review and games social!

January 26th, 2024: Yuzheng Yan

Title: From sum of two squares 

Abstract: The talk will start with some elementary number theory about sum of two squares. Then we will realize representation numbers of two squares as Fourier coefficients of certain modular form.  If time permits, we will try to generalize this realization via Siegel formula and beyond. 

February 2nd, 2024: Matt Zevenbergen

Title: The Dollar Auction

Abstract: The dollar auction is a model for escalatory behavior. The idea is that you have two players bidding on a dollar, but both players will have to pay out their maximum bid (and only the high bidder gets the dollar). For example, if Player 1 has bid 40 cents and Player 2 has bid 60 cents, Player 1 is incentivized to raise their bid so as to not forfeit their 40 cents. In real life trials of this game, it is common for both players to end up bidding over a dollar. We will do a game theoretic analysis of this game to see how player rationality assumptions force the course of events. This will culminate in O'Neill's Theorem.

February 9th, 2024: Siddharth Mahendraker

Title: The Cohomology of Algebraic Varieties 

Abstract: A map f : X —> Y of manifolds can be viewed as a parametrization of a family of submanifolds X_y in X, where X_y := f^{-1}(y) is the fiber of f over y in Y. A powerful feature of complex algebraic varieties, that is not shared by manifolds, is that these fibers X_y are themselves complex algebraic varieties. In the world of smooth manifolds, we know by Sard’s theorem that almost all fibers of a smooth map are smooth manifolds. A similar type of result in algebraic geometry guarantees that almost all fibers of a (sufficiently nice) algebraic map are smooth varieties. This opens up the possibility of studying a *singular* variety X_0 by embedding it within a family of varieties X_t that are smooth away from t = 0. This is desirable because many powerful results such as Poincaré duality and the Hodge and Lefschetz theorems are false for singular spaces. In this talk I’d like to give an introduction to this circle of ideas, known as the theory of perverse sheaves, with a focus on calculating the cohomology of various singular varieties and “restoring” Poincaré duality. 

February 16th, 2024: Laura Seaberg

Title: The laws of attraction, or, why we find comfort in chaos 

Abstract: It’s not often that the public finds a math concept sexy. And yet, the introduction of “chaos theory” in the 1960s has captured the imaginations of creatives and laypeople alike through manifestations such as the butterfly effect. This has always tickled and somewhat confused me—as a dynamicist, I think fractals are cool but NOT THAT DEEP. So today we unpack this phenomenon: my talk will explore examples of media which make use of ideas relating to mathematical chaos. Our aim is to understand the mathematical precursors to these depictions and analyze what they might reveal about popular perception of mathematics as well as the utility of math as a lens for philosophically making sense of the world we live in. 

February 23rd, 2024: Laura Seaberg (contd.)

March 1st, 2024: Joaquin Lema

Title: Thermodynamics and math 

Abstract: V. Arnold was a funny guy (as in "Arnold conjecture" or the A in KAM theory, among many other things). One of his many celebrated quotes is, "Any mathematician knows that it is impossible to understand any elementary course in thermodynamics," so we will take the challenge and try to learn about that in only 40 minutes! I'm sure we will do it.

This talk is motivated by another quote of his saying something like, "People don't understand contact geometry because it is a model for thermodynamics, as opposed to its symplectic sibling that is modeled in classical mechanics, and that is so so f$#*ing easy!" (may have paraphrased there). Hopefully, I'll have time to explain how Thermodynamics has fed mathematics with plenty of strange ideas to our Bourbaki-trained eyes. 

March 8th, 2024: SPRING BREAK, NO GIST

March 15th, 2024: Jordi Martinez

Title: Groups & Puzzles

Abstract: It is not surprising that some mathematicians are interested in mechanical puzzles, as they can be naturally realized as groups. In this talk, we will study some of the basic properties of two mechanical puzzles: the 15-puzzle, and the Rubik's cube. Some of the properties we will discuss are: identifying illegal moves, computing the order of the corresponding groups, and computing orders of elements (which is actually relevant to the problem of solving a Rubik's cube). If time allows, we will discuss how to solve a Rubik's cube.

Only elementary group theory will be assumed.

March 22nd, 2024: Miguel Prado

Title: 3264 bubbles

Abstract: I will go through some enumerative problems on the plane and build intersection theory tools until being able to count how many smooth conics are tangent to 5 given smooth conics. 

March 29th, 2024: EASTER BREAK, NO GIST

April 5th, 2024: Tobi Moektijono

Title: Circles, Cyclic Cohomology, and Topological Cyclic Homology.

Abstract: In this talk, I will survey some classical computations of K-groups using trace methods and also recent advances that allow computation of big perfectoid rings.

April 12th, 2024: Mira Wattal

Title: Morse-Smale functions and Morse-Smale categories: Recovering the Morse complex from a homotopy-theoretic construction

Abstract: Over the last several decades, homotopy theorists and low-dimensional topologists have worked independently and in tandem to develop constructions that better approximate geometric data. In this talk, we narrate a piece of these paralleling histories by way of the flow category C_f associated to a Morse-Smale function f : M \to R defined on a closed Riemannian manifold. It turns out that this flow category is smooth, compact, Morse-Smale, and of finite type—in a homotopy-theoretic sense, that is. It meets the sufficient amount of (pointedly named) criteria necessary to define a functor whose geometric realization realizes the associated Morse complex of f.

April 19th, 2024: Braeden Reinoso

Title: Meaning-making in math (what are we even doing?) 

Abstract: There is a tension that I think every mathematician feels at some point: math itself brings joy, but the broader systems surrounding math sometimes stymie creativity, take up too much time, or feel pointless. That tension is frustrating and discouraging, but it is also a small and important part of meaning-making. We are confronted with the reality of ourselves as more than abstract math-machines-- we are human beings working in a much larger system of production, and that system affects us as human beings. How has math influenced our perspective on the world? What values and morals are we developing through our mathematical journeys? Has being a mathematician impacted the way we interact with the people we care about? I'll draw on the thoughts of Akshay Venkatesh, C. Thi Nguyen, Neil Postman, Brian Upton, and many others, to present a perspective and a set of tools that can help us understand our own experiences of being mathematicians. Crucially, my goal is not to "reveal the answer," but to get you to consider your own continuing process of meaning-making in math. 

April 26th, 2024: Gus Schmidt

Title: Spreading Out or: How I Learned to Stop Worrying and Love Finite Fields

Abstract: There are too many numbers in infinite fields -- yuck. Luckily for us, there are many situations where we can prove (apparently difficult) theorems in characteristic 0 by reducing to (seemingly easier) problems over finite fields of positive characteristic. This may feel surprising given that there are no non-trivial field homomorphisms between these two types of fields! Nonetheless, the technique of "spreading out" often allows translation between these two very different arithmetic worlds. We will discuss some motivation, examples, and extensions of this technique.  

May 3rd, 2024: Eric Moss

Title: 

Abstract: