September 10, 2021: Board Game Social!

September 17, 2021: Ethan Farber

Title: Starting from square 1 (or 0?)

Abstract: Tori are everywhere. From donuts to Pac-Man’s hometown, they are abundant and natural objects. Just so, mathematicians have come to understand tori from many perspectives: as elliptic curves, as boundary components of interesting 3-manifolds, and as the only closed surfaces with vanishing f-invariant. As a result, the torus provides a dictionary between different perspectives. In this talk, I will explicate an instance of this: namely, a wonderful link between arithmetic and (Riemannian) geometry. Pictures and examples will abound!

September 24, 2021: Gage Martin

Title: The Conway knot is not slice

Abstract: In her recent Annals paper, Lisa Piccirillo showed that the Conway knot is not slice, in other words it does not bound a disk in the 4-ball. In this talk we will discuss some basics of knot theory, what made it so hard to tell if the Conway knot was slice, and give some idea of Piccirillo's strategy for her groundbreaking work.

October 1, 2021: Eric Moss (website)

Title: The Inverse Galois Problem and PGL_2(Z/nZ) (also some number theory)

Abstract: The Inverse Galois Problem asks which finite groups appear as Galois groups of Galois extensions of Q. This is an open problem, but it has been settled for many large classes of groups, including the solvable groups. One of the first classes of unsolvable groups to be proven to be a Galois group is PGL_2(Z/nZ), and the proof of this fact is fairly interesting, involving some number theory and Hilbert’s irreducibility theorem. This talk should be accessible to everyone, including those who have never seen a Galois group.

October 8, 2021: Stella Sue Gastineau

Title: The Local Langlands Correspondence

Abstract: Starting with representations of finite groups, I will present the necessary background to understanding the statement and meaning of the Local Langlands correspondence, a theorem that connects the mathematical fields of representation theory, p-adic groups, and Galois representations. From here, we will look at a first example of the LLC and how it relates to Platonic solids.

October 15, 2021: Siddharth Mahendraker

Title: An introduction to Hecke algebras, Kazhdan-Lusztig polynomials and categorification

Abstract: To any Coxeter system (W, S), one can attach a Z[v, v^-1]-algebra H(W) called the Hecke algebra. This algebra has two bases; a standard basis, coming from elements of S, and a so-called Kazhdan-Lusztig basis consisting of elements with additional symmetry. The entries in the change of basis matrix between these two bases are called Kazhdan-Lusztig polynomials, and these polynomials have remarkable connections to algebraic geometry, representation theory and combinatorics. In this talk, we explain how to compute the Kazhdan-Lusztig polynomials, and describe how many of the conjectures surrounding these polynomials were settled using a technique called categorification. No knowledge of Coxeter systems required! However, familiarity with the symmetric group on 3 letters is mandatory!

October 22, 2021: Fraser Binns

Title: What is an alternating knot?

Abstract: A long time ago Fox asked "What is an alternating knot?". Not so long ago Joshua Greene and Joshua Howie answered the question. In this talk I will discuss the question as well as the answer.

October 29, 2021: Pumpkin Carving

November 5, 2021: Zachary Gardner

Title: A p-adic Thanksgiving

Abstract: Over the past few decades, number theory has been revolutionized by a cornucopia of exotic geometric theories that can collectively be called $p$-adic geometry. Much of the story began with John Tate, who developed the notion of rigid analytic varieties. More recently, Peter Scholze and collaborators have put Roland Huber's theory of adic spaces to amazing use to construct perfectoid spaces and other such geometric objects. In this talk, we hope to give a gentle introduction to this universe of $p$-adic geometry. Prior experience with algebraic geometry is not necessary. In fact, we hope this talk is helpful for those currently learning algebraic geometry.

November 12, 2021: Qingfeng Lyu

Title: Some Non-Morse Theory

Abstract: We all know from Morse theory that a Morse function on T^2 has at least 4 critical points. But what if it’s just a smooth function, not Morse? Does there still have to be 4 critical points? - It turns out that critical points of general smooth functions on manifolds are also somehow connected to topology, much analogous to Morse theory. In this talk we’ll give a brief introduction to the so-called Lusternik-Schnirelmann category number and a sketch proof of the fundamental theorem by L-S. The content may not be very in-fashion, popular or prospective - but it’s still much fun! A little algebraic topology, if not none, is needed.

November 19, 2021: Laura Seaberg

Title: Tilings from graph-directed iterated function systems

Abstract: People have been wondering about ways to tile the plane since antiquity, inspiring years of mathematical research. Separately, the notion of an iterated function system (IFS), a collection of contractions of Euclidean space, emerged in the 80s and described the construction of many interesting sets. An IFS has a unique nonempty compact 'fixed' set which equals the union of its images under all the contractions, and this set is often fractal. Several recent papers, including one by Barnesley and Vince, take steps to synthesize these two ideas. We will define and explore a generalization of the IFS called the graph-directed iterated function system and extract some tilings therefrom. This talk will be full of pictures and examples--attached are some particularly pretty tilings!

December 3, 2021: Antony Fung

Title: Nobody can stop me from doing induction lol

Abstract: 1.5 years ago, I gave a GIST talk about the proudest moment in my entire undergrad life: Impressing my Graph Theory supervisor by solving the last part of Q13 in this worksheet using the Compactness Theorem of first-order logic.

https://www.dpmms.cam.ac.uk/study/II/Graphs/2016-2017/graph20172.pdf

But that may not be satisfying to some of you, because I didn't prove the Compactness Theorem, which relies on the Godel’s completeness theorem, which is too long to prove in a GIST talk (probably would take an entire hour). So it didn't give much insight to why things work.

Recently there're math major students asking about Graph Theory homework problem in our discord server (join it if you haven't done so). That inspires me to do this problem again without using the Compactness Theorem. According to statistics that I made up, 69% of the Graph Theory homework problems can be done with induction. So, let's use induction!

Wait...G is uncountable. I remember teachers teaching me that I can only use induction on countable things. Hmm...

HOW BOUT I DO ANYWAY?

December 10, 2021: Miguel Prado Gadoy

Title: Elliptic curves in Euclidean geometry

Abstract: I plan to prove Pascal and Poncelet theorems by using Elliptic curves. This talk will be as non-technical as possible so we can all have fun.