The organizers were Kush Singhal and Leon Liu.
During the fall semester, talks were usually given in SC 232, Tuesday 11:45 am-1:00 pm.
During the spring, talks were usually given in SC 232, Tuesday 1:00 pm-2:00 pm.
Borromean Rings, Chainmaille, and Genuine Equivariant Homotopy Theory
Natalie Stewart
The topological space of chain maille weaves classifies geometric embeddings of circles into 3-space. This space of weaves, together with the evident action of the isometry group of R^3, is one of the primary morivating examples in the study of genuine equivariant homotopy theory. After developing some of the basics of this field, we prove a genuine equivariant refinement of the classical fact that the borromean rings can’t be made out of geometric circles. If enough time remains, we’ll introduce a geometricaly interesting operad, called toroidal weaves, over which the weaves form a (genuine equivariant) right-module, conditioned on a geometric conjecture. We may discuss a surprisingly simple presentation of the operad and right-module structure, conditioned on the same conjecture.
Representations of the Finite Group $SL_2(F_q)$
Wyatt Reeves
The complex representations of the finite group $SL_2(F_q)$ were first classified by Jordan and Schur in 1907. Over the next several decades, modern ideas from representation theory and algebraic geometry further elucidated the structure of these representations, culminating in a construction in a 1974 paper of Drinfeld. In this talk we will try to construct at least half of the representations of $SL_2(F_q)$. Along the way we’ll encounter important ideas and techniques such as Chevalley decomposition, local class field theory, the Bruhat decomposition, Mackey’s induction-restriction formula, Whittaker models, and the Selberg trace formula.
Can you (flat)-fold it?
Kush Singhal
Given a square paper whose side is one unit long, can you fold the paper so that, upon projecting the folded state from a certain angle, the perimeter of the projection is larger than 4? If so, how? (Hint: Yes, yes you can). The same mathematical ideas that we will use to find a solution to the above question are the same ideas used to design complex origami models! In this talk, we will discuss the mathematics of crease patterns and paper-folding.
Equipartitions of (Z/NZ)^x and Cyclotomic Polynomials
Daishi Kiyohara
In this talk we look at what happens when the set $S=\{0\le m\le n | (m,n)=1\}$ is subdivided into equal subintervals. We will prove that the parity of the size of these subsets is captured by the values of cyclotomic polynomials at roots of unity. We use this fact as a guiding principle to find nice properties about the geometry of $S$.
(Algebraic set) theory
Grant Barkley
I will do some set theory with sets of fuzzy elements. This will give short proofs of the Jordan curve theorem and the Grothendieck generic flatness lemma.
The Hegelian Taco
Taeuk Nam
We will give a brief introduction to the Hegelian dialectic and its influence on philosophical and political thought, with a focus on the central notion of sublation, or Aufhebung. Then we will discuss Lawvere’s formalization of Aufhebung using category theory.
Exceptional Isomorphisms
Sanath Devalapurkar
There are a bunch of neat coincidences in low dimensions. I’ll talk about that, and try to also say something about the historical context for some of them.
The Twistor Line
Dylan Pentland
Over C, the n’th singular cohomology group of a variety has a canonical mixed Hodge structure. If we restrict to smooth projective varieties, we get a pure Hodge structure.
It was observed by Simpson that the category of pure Hodge structures can be formulated geometrically as a category of vector bundles on a curve, and so can the association of a Hodge structure to a variety. I’ll talk about how this is done using the twistor line.
Period Three Implies Chaos
Sina Saleh
Suppose that f is a self-map on a closed real interval and that f has a point with orbit 3. Then, it turns out that f must have periodic points of every order! Moreover, there is an uncountable subset of the interval where the map behaves “chaotically”. I will talk more about what chaos means in this setting and prove the above facts i.e. the famous “period three implies chaos” theorem.
The Ford-Fulkerson Algorithm, and then some
Frank Lu
The problem of maximal flow on a directed graph is a classic example of an optimization problem: if I have a directed graph with a single source and sink, and specify the capacity of each edge, how do I determine the maximal flow on a graph? This problem will turn out to be related to seemingly unrelated problems, (possibly) involving tolls, food preferences, and posets. I will start by talking about the Ford-Fulkerson algorithm, which provides an algorithm to find such a maximal flow, before moving to some diverse applications of the Ford-Fulkerson algorithm, as time permits.
A Glance at Kirby Calculus
Enrico Colon
4-manifolds are hard to visualize. Links in the 3-sphere are much easier to visualize, in fact, we can draw them (for the most part) on a 2-dimensional piece of paper. In this talk, we will learn how to get our hands dirty with 4-manifolds using the Kirby calculus, a collection of moves on framed links in $S^3$ that explicitly describe 4-manifolds and their diffeomorphisms. If time permits, we will use the Kirby calculus to detect and explicitly describe a pair of exotic (homeomorphic but not diffeomorphic) 4-manifolds.
Sylow Theorems for $\infty$-Groups
Isabel Longbottom
Taking the fundamental group gives an equivalence of categories between pointed topological spaces whose homotopy groups $\pi_n$ for $n \neq 1$ all vanish, and groups. From this perspective, group theory can be viewed as the truncation of a more general theory of pointed connected spaces. Then the natural question is: to what extent can we do group theory in this new homotopical setting (where we don’t require the higher homotopy groups to vanish)? In this talk, we will translate the Sylow theorems for finite groups to the context of finite $\infty$-groups, and use this to get a group-theoretic classification of finite nilpotent spaces, by analogy to the classification of finite nilpotent groups. We will also discuss the failure of normality for $\infty$-groups, as something of a cautionary tale.
Hat Trick
Elliot Glazer
Several players are wearing an infinite stack of black and white hats. How many can be guaranteed to successfully guess the colors of 3 of their own hats? This is, of course, independent from the standard axioms of mathematics.
4-Manifolds and Quadratic Reciprocity
Ollie Thakar
In this talk, I will discuss the amazing G-Signature Theorem of Atiyah and Singer, which computes a certain homological invariant (the g-signature) of an isometry of a manifold in terms of characteristic classes integrated over its fixed-point set. For the sake of concreteness, I will only cover the case of 4-dimensional manifolds, where the theorem simplifies to a beautiful combinatorial formula. After describing this formula and extracting simple corollaries, I will illustrate its power through a very surprising application due to Hirzebruch and Zagier to prove the quadratic reciprocity law.
Schemes as Functor of Points
Taeuk Nam
Growing up, most of us are taught algebraic geometry in the following standard way:
- The spectrum of a ring Spec(R) is the set of prime ideals of R equipped with the Zariski topology. It has a natural sheaf of rings called the structure sheaf given by localizations of R. We call these affine schemes.
- A scheme X is a locally ringed space that is locally an affine scheme. We continue to call its topology the Zariski topology and its sheaf of rings the structure sheaf O_X.
- Given a ring R and an R-module M, we can define a sheaf of modules over the structure sheaf of Spec(R) called the associated sheaf of M, given by localizations of M. A quasicoherent sheaf on a scheme is a sheaf of O_X-modules on the Zariski topology of X that is locally given by associated sheaves of modules.
We will discuss an alternative approach to algebraic geometry that does not involve locally ringed spaces. The speaker would like to express their gratitude to Elden Elmanto for rescuing them from the above indoctrination.
15, 290, and All That
Frank Lu
One fundamental line of questions in the theory of quadratic forms revolves around what integer values a quadratic form can take. In this talk, I will discuss a few related surprising results in this area, centered on the 15 Theorem. This theorem states that a positive-definite integer-matrix quadratic form represents all positive integers if and only if they represent every positive integer at most 15. I will discuss the main ideas in the proof of this theorem, as well as the ideas in the similar 290 Theorem, which applies for a slightly wider set of quadratic forms. If time permits, I will also discuss a more general statement which can be made along these lines.
Convex Projective Geometry
Charlie Reid
Convex projective geometry studies discrete subgroups of PGL(n,R) which act properly on a convex subset of RP^{n-1}. I will focus on the case when the quotient of the convex set by the subgroup is compact. Many things are known, and many things are not known about the classification of these subgroups.
The 11/8-Conjecture
Runze Yu
The 11/8-conjecture proposed by Y. Matsumoto gives an inequality that obstructs the existence of smooth structures on closed oriented spin 4-manifolds, and is closely related to the classical problem of the classification of 4-manifolds. In this talk, I will define a Pin(2)-equivariant version of the Seiberg-Witten equations and explain how M. Furuta uses it and equivariant K-theory to prove a (10/8+2)-theorem.
McKay Correspondence
Merrick Cai
There’s a very surprising and deep way to enumerate finite subgroups of SL(2,C). It turns out to be intricately linked to the geometry of the group as well.
The Beginnings of the Theta Correspondence
Daniel Hu
I will try to give a sampler of the local theta correspondence, which allows one to relate irreducible representations for certain pairs of reductive Lie groups. The story starts with the birth of modern quantum mechanics.
A Climb Up the Tower
Kush Singhal
Iwasawa theory is an area of number theory with a simple tenet: it’s easier to study the limit of a sequence of objects than it is to study the individual objects themselves. In Iwasawa theory one often sees surprising connections between complex-analytic, algebraic, and geometric objects of an arithmetic nature. This talk will be a whirlwind tour of the development of Iwasawa theory from its origins in a `failed’ attempt at Fermat’s Last Theorem, to its connections with reciprocity laws, applications to elliptic curves, and something possibly motivic.
Proving Rarity of Beautiful Things
Matt King
In analytic number theory, sieve methods give a fun, intuitive way to upper bound the number of prime numbers in a set. They do this by creating an easily manipulable upper bound for the indicator function that n is prime. Sieves have some limitations though, including the somewhat mysterious parity problem. After we prove estimates where sieve methods work quite well, we’ll see some concrete examples of the parity problem and finish by proving universal limitations of general sieves.
Borromean Rings on a Quantum Processor
Leon Liu
Topological quantum computing involves many seemingly unrelated fields, including low dim top, category theory, quantum computation, and condensed matter theory. I will explain why all those subjects are related, culminating in why people care about drawing Borromean rings on quantum processors. No physics knowledge will be required.
✂️
Rafael Saavedra
Hilbert’s Third Problem asks: given two polyhedra of equal volume, can one slice the first into finitely many pieces and then reassemble them to produce the second? I will discuss Dehn’s solution to this question and the connections between the higher-dimensional case, algebraic K-theory, and mixed Tate motives.
A Model-Independent Crash Course on ♾️-CategoryGroupoid Theory
Charles Wang
When small children (aka undergrads) learn linear algebra, they typically do everything in terms of matrices. Once one finishes with the abhorrent business of actually being able to do calculations, one learns that this is Picking a Basis and Picking a Basis is Evil and Evil is Bad. But then when these small children (aka grad students) are taught ♾️-category theory, there is always a choice of model. This is Evil. Therefore, I will do my part to improve the moral character of the country by giving a method for doing calculations in ♾️-category theory that is wholly Evil-free.
Geometric RSK, from discrete to continuous
Yuhan Jiang
The Robinson-Schensted-Knuth correspondence plays a fundamental role in the theory of Young tableaux, symmetric functions, and representation theory. By a version of Greene’s theorem, it can be defined as a piecewise linear map over the tropical semiring on matrices with non-negative integer entries. Berenstein and Kirillov extended it to the usual (+,x) algebra for real values matrices, which is called the geometric RSK. As RSK can be used to study Bernoulli walks, gRSK can be used to study Brownian motions and directed polymers. As RSK can be used to prove Cauchy-Littlewood identity for Schur functions, gRSK can be used to prove analogous identity for GL(N,R)-Whittaker functions.