# Caltech Math Grad Seminar

The graduate student seminar meets every other Friday at 12.00-13.00 in Linde 255.

Contact: Luis Soldevilla (lsoldevi@caltech.edu)

## Upcoming talks

January 30, 2024: Tal Hersko, The Logic of Random Graphs

In this talk I will provide a basic introduction to the theory of random graphs and its surprising connection with mathematical logic. This interesting blend of graph theory, probability and logic leads to several unexpected and beautiful results. The most classical result is Fagin's theorem about the existence of a zero-one law for first order properties in the binomial random graph G(n,p). The main purpose of the talk is to explain the meaning of Fagin's theorem and present its proof. If time permits, I will also briefly introduce some of my work in this field.

## Past talks

January 28, 2022: Jiaxin Zhang, Conformal Field Theory for SLEs

Schramm-Loewner evolution (SLE) is a family of random planar curves which are the scaling limits of a variety of two dimensional lattice models in statistical physics. Conformal Field Theory (CFT) is a conformally invariant quantum field theory which is also widely used to study the behavior of critical lattice models. There are deep connections between CFTs and SLEs. Many properties of SLE and Multiple SLEs system can be predicted and analyzed by corresponding CFTs. To reveal those connections, it is important to construct a rigorous CFT framework for SLE and Multiple SLEs System.

In this talk, I will discuss some recent progress.

December 3, 2021: Forte Shinko, Laver tables and large cardinals

The Laver table L_n is a particular self-distributive algebra on 2^n elements, with many interesting properties. Surprisingly, certain properties of Laver tables can be deduced by assuming very strong set-theoretical axioms, called large cardinals. We will investigate this interaction, along with connections to orderability of the braid group.

November 12, 2021: Giovanni Paolini, The K(π,1) conjecture

This talk will be an introduction to the K(π, 1) conjecture, a 50-year-old open problem on the topology of configuration spaces associated with Coxeter and Artin groups. If time permits, I will also give a high-level overview of combinatorial approaches to this conjecture which led to its solution in the spherical and affine cases.

October 29, 2021: Philip Easo, Supercritical percolation on finite transitive graphs

In bond percolation, we build a random subgraph of a graph G by indepedently choosing to retain each edge with probability p. We call these retained edges "open" and the rest "closed". For many examples of G, when we increase the parameter p across a narrow critical window, the subgraph of open edges undergoes a phase transition: with high probability, below the window, it contains no giant components, whereas above the window, it contains at least one giant component. Now take G to be a large, finite, connected, transitive graph with bounded vertex degrees. We prove that above the window, there is exactly one giant component with high probability. This was conjectured to hold by Benjamini in 2001 but was previously only known for large torii and expanders, using methods specific to those cases.

The work that I will describe is joint with Tom Hutchcroft.

February 19, 2021: Angus Gruen, Linking Physics and Knot Theory

The linking number is an invariant of pairs of links which can be defined in a very simple manner through via looking at individual crossings. I will discuss how we can use techniques from both classical and quantum physics to compute this and how these techniques naturally give us the ability to extend this invariant to new situations.

January 22, 2021: Alexandre Perozim de Faveri, Distribution of closed geodesics on the modular surface

This will be a survey of some classical results on prime geodesics, focusing on various ways of interpreting them from different perspectives. If time permits, I will discuss some recent results on their statistical distribution.

January 8, 2021: Tamir Hemo, Categorical Trace

The trace of a linear map can be understood and generalized in a categorical way. We will review this construction and some of its applications to geometry and representation theory.

December 4, 2020: Michael Wolman, Probabilistic Programming Semantics for Name Generation

In this talk we present a probabilistic model for name generation. Specifically, we interpret the nu-calculus, a simply-typed lambda-calculus with name generation, in the category of quasi-Borel spaces, an extension of the category of standard Borel spaces supporting both measure theory and higher-order programming. We prove that this model is fully abstract at first-order types. This is joint work with Marcin Sabok, Sam Staton and Dario Stein.

November 20, 2020: Frid Fu, Kazhdan-Lusztig polynomials and the BGG category

Kazhdan-Lusztig polynomials are of fundamental importance in representation theory. In this talk, we explain two beautiful formulae that relates the Kazhdan-Lusztig polynomials to extension groups in the BGG category. One of the formulae is an inequality and is still yet to be fully understood. If time permits, I will also comment on the combinatorial aspects of the Kazhdan-Lusztig polynomials.

November 6, 2020: Forte Shinko, A dichotomy for Polish modules

Given two vector spaces, it's not hard to check whether one embeds into the other, since it suffices to compare their dimensions. However, such an embedding may not always be "definable"; for instance, it may require the axiom of choice. In the spirit of descriptive set theory, it is thus natural to consider Borel embeddings of Polish vector spaces, and more generally, Polish modules. We show under the Borel embedding preorder that there is a minimum uncountable-dimensional Polish vector space, and that there is a countable set of uncountable Polish abelian groups such that every uncountable Polish abelian group contains one of these. This is joint work with Joshua Frisch.

October 23, 2020: Nathaniel Sagman, The Plateau problem and symmetries of Riemann surfaces

In its most basic form, the Plateau problem asks: given a Jordan curve Gamma in R^3, is there a surface of minimal area among all surfaces with boundary Gamma. I will discuss the problem, its solution, and some of the mathematics that grew out of it. If time allows I will explain a more recent and semi-related result.

March 4, 2020: Adam Artymowicz, The Toda Lattice

The Toda lattice is a chain of particles which interact through a certain nonlinear potential. It is a so-called integrable system, which means that its time evolution can be explicitly solved. I will talk about some aspects of the solution and its surprising connections on one hand with the theory of algebraic curves, and on the other hand with the theory of semisimple Lie algebras.

February 19, 2020: Joshua Lieber, A comparison between model categories for concurrency

This talk will largely be an introduction to the theory of model categories as a setting for homotopy theory. Towards the end, a discussion of two topological models of concurrent computing will be given, and it will be shown that they do not yield equivalent homotopy theories by standard means.

February 5, 2020: Yuhui Jin, Central limit theorem on conjugacy class measure of the symmetric group

In the 1990s, Kerov first studied a central limit theorem on the character ratio of the Plancherel measure of the symmetric group. Character ratio is an important quantity to understand random walks on the symmetric group generated by transpositions. Here, we study a similar measure called conjugacy class measure on the symmetric group and prove a CLT on the character ratio, using the method of moments.

January 22, 2020: Sunghyuk Park, 3-manifolds and q-series

In 2016, Gukov, Putrov and Vafa conjectured the existence of invariants of 3-manifolds which are q-series with integer coefficients. They are expected to have a categorification in a sense of Khovanov homology. More recently, in 2019, Gukov and Manolescu studied the analogue of GPV invariants for knot complements and their behavior under Dehn surgery. In this talk, I will review what is known about these conjectural invariants, recalling the results by Gukov-Pei-Putrov-Vafa and Gukov-Manolescu, and then I will discuss a generalization of these results to arbitrary gauge group. If time allows, I will also talk about a HOMFLY-PT analogue of F_K, which is a work in progress with Gruen, Gukov, Kucharski and Sulkowski.

December 4, 2019: Bowen Yang, Quantum spin systems

Quantum spin systems are theoretical models commonly seen in mathematical physics. They allow us to study the physical world rigorously with tools such as operator algebras, complex analysis, K theory and so on. This talk aims to introduce the mathematical formulation of quantum spin models. We will emphasize the important role correlation functions play in the study.

November 20, 2019: Andrei Shubin, Lattice points on the sphere

The integers which are sums of two or three squares (n = x^2 + y^2 or n = x^2 + y^2 + z^2) can be considered from a geometric point of view as the points (x, y) or (x, y, z) lying on a circle or the surface of a 2-sphere of radius √n, respectively. We will talk about the distribution of such points and how these questions relate to L-functions.

November 6, 2019: Sunghyuk Park, F_K

F_K is a two-variable series for knot complements introduced recently by Gukov and Manolescu. It is the knot complement analog of \hat{Z} and is believed to be categorifiable. In this talk, I will review some of those recent developments and explain how to generalize F_K for arbitrary gauge groups.

October 23, 2019: Forte Shinko, How large is the outer automorphism group?

For a countable group Gamma, what are the possible sizes of Out(Gamma)? From the point of view of cardinality, the answer is not so interesting: if it is uncountable, then it has size continuum. There is a more suitable notion of Borel cardinality, coming from the theory of countable Borel equivalence relations in descriptive set theory. We will see that although there are many Borel cardinalities of varying complexity, Out(Gamma) is always among the lower end of them; namely, Out(Gamma) is always hyperfinite. This is joint work with Joshua Frisch.

May 30, 2019: Todd Norton, Spinors and Dirac operators

After recalling what the Dirac operator on R^n is, we will discuss how and when it's possible to extend that notion to a Riemannian manifold. To do so we'll need Clifford algebras and spinors, so we'll say what those are. We'll then square the Dirac operator and compare to the Laplace operator on spinors. This talk won't assume much background beyond knowing what a metric on a manifold is. (This is relevant background for the Index Theorem, noncommutative geometry, quantum field theory, Seiberg-Witten theory, and much more. If time permits, we'll mention some of these applications.)

May 19, 2019: Jim Tao, A twisted local index formula for curved noncommutative two tori

We consider the Dirac operator of a general metric in the canonical conformal class on the noncommutative two torus, twisted by an idempotent (representing the K-theory class of a general noncommutative vector bundle), and derive a local formula for the Fredholm index of the twisted Dirac operator. Our approach is based on the McKean-Singer index formula, and explicit heat expansion calculations by making use of Connes' pseudodifferential calculus. As a technical tool, a new rearrangement lemma is proved to handle challenges posed by the noncommutativity of the algebra and the presence of an idempotent in the calculations in addition to a conformal factor.

May 5, 2019: Joshua Frisch, Random walks, harmonic functions, 0-1 laws, and the Poisson boundary of groups

Random walks are probabilistic processes where you walk in a random direction in each time step. I'll discuss random walks on d-dimensional spaces and explain why every event which is asymptotic either always or never occurs. Surprisingly, based on a connection pointed out by Furstenberg, this turns out to be intimately related to discrete harmonic functions. Groups and convex analysis also play some intriguing roles.

April 19, 2019: Sunghyuk Park, Introduction to homological blocks

$\hat Z_b$, known as the homological block or WRT block, is an invariant of 3-manifolds first introduced in the works of Gukov, Pei, Putrov and Vafa based on physics. While a full-fledged definition of $\hat Z_b$ is yet to be discovered, physics predicts that these homological blocks provide a way to categorify WRT invariants, thereby generalizing Khovanov homology. In this talk I will first review how to compute \hat Z_b for some classes of 3-manifolds, and then move on to discuss some recent progress revealing interesting properties of $\hat Z_b$.

April 5, 2019: Praveen Venkataramana, Path counting and rank gaps in differential posets

We study the gaps $\Delta p_n$ between consecutive rank sizes in r-differential posets by introducing a projection operator whose matrix entries can be expressed in terms of the number of certain paths in the Hasse diagram. We strengthen Miller's result that $\Delta p_n\ge 1$, which resolved a longstanding conjecture of Stanley, by showing that $\Delta p_n\ge 2r$. We also obtain stronger bounds in the case that the poset has many substructures called threads. (Joint work with Christian Gaetz)

March 14, 2019: Angus Gruen, Computing modular data of pointed fusion categories

A well known invariant of a modular category is the modular data (S,T) which give a projective unitary representation of the modular group. Knowing this invariant vastly aids the understanding of the corresponding category but in general only a small number of these have been explicitly computed. Specializing to the family of modular categories which arise as Drinfel'd centers of pointed fusion categories, I'll discuss how this invariant can be efficiently computed using Galois symmetries and central extensions.

February 14, 2019: Alexandre de Faveri, Small gaps between primes

One of the most famous open problems in number theory is the twin prime conjecture, which says that there are infinitely many prime numbers at distance two. I will introduce some of the tools used to deal with this problem, such as the Selberg sieve and the Bombieri-Vinogradov theorem, and outline the new ideas that led to the breakthroughs of Zhang and Maynard in 2013, who independently proved that the gap between primes is bounded infinitely often. If time permits, I will touch on the known obstructions towards twin primes, such as the parity problem in sieve theory.

January 31, 2019: Bowen Yang, Linear reductive algebraic groups

If you don't have a solid education in linear algebraic groups, you might nevertheless encounter the term "reductive groups" now and then. People keep telling you to think about GL_n as an algebraic group, and that's a good first approximation. If you want to go a step further, some confusion can happen. I want to give some very short explanations on how to think about reductive groups and semisimple algebraic groups.

January 25, 2019: Nathaniel Sagman, Harmonic maps and Teichmüller theory

I will introduce the subject of harmonic maps between Riemannian manifolds and discuss some applications to rigidity problems in geometry. I will then outline some basic aspects of Teichmüller theory and its interactions with the study of harmonic maps between surfaces. Time permitting, I will discuss some recent developments: universal Teichmüller space and the Schoen conjecture.

December 7, 2018: Angad Singh, Asset pricing in the cross-section of stocks

Did you know that from 1926-1979 the stocks of small companies earned on average about 3 times more per year than the stocks of large companies? More specifically, small company stock prices grew 15% per year on average, whereas large company stock prices grew only 5% per year on average. What a huge difference!

How on earth is this possible? Surely over the course of 50 years people would have noticed this pattern. Surely they would have all run to buy small company stocks and run to sell big company stocks. Surely this would have caused market prices to adjust and made the pattern disappear. Evidently this is not what happened.

It turns out that there are tons of patterns like these in the stock market. Asset pricing theory is the area of finance that tries to understand why these patterns exist. I’ll present an introduction to the field, focusing on two models: the Capital Asset Pricing Model (CAPM) and the Fama-French 3 Factor Model (FF3). I’ll dabble in both the theory (math) and the empirical facts (statistics and data).

No background is needed to understand this talk.

November 20, 2018: Angus Gruen, String diagrams, a graphical calculus for monoidal categories

Given a monoidal category, we can represent morphisms in the category using beads on strings and further structure on the monoidal category (e.g. braided, rigid, ribbon, ...) can be encoded as geometrical properties on these strings. Given a ribbon monoidal category, its string diagrams naturally give rise to a family of link invariants indexed by objects in the category. I will give a construction of the Temperley-Lieb category whose string diagram calculus produces the Jones polynomial.

November 9, 2018: Victor Zhang, Affine Grassmannian with examples

We introduce the affine Grassmannian and its Schubert cells. We give examples and relate them to classical objects in algebraic geometry.

November 2, 2018: Zachary Chase, Triangles in Cayley graphs

https://arxiv.org/abs/1809.03729

October 26, 2018: Forte Shinko, Choice puzzles

There are some well-known puzzles whose solution requires some form of the Axiom of Choice. We'll discuss these along with some variants not requiring choice, and maybe sneak in some descriptive set theory as well.

October 19, 2018: Tamir Hemo, An introduction to perfectoid spaces

Perfectoid spaces, introduced by Scholze, transformed the way we think about p-adic geometry. In this talk, after a review of the basic ideas, we will try to explain what perfectoid spaces are and what kind of results they have been used to prove.

October 12, 2018: Sunghyuk Park, What is ... Chern-Simons theory?

This is an informal talk on some aspects of Chern-Simons theory. The emphasis is on general overview and intuition, not on any rigorous proof.

July 7, 2018: Nathaniel Sagman