## Spring 2023

- MATH 502: ALGEBRAIC STRUCTURES II

Margaret Regan

- MATH 531: REAL ANALYSIS I

Kyle Liss

- MATH 532: REAL ANALYSIS II

Mark Haskins

- MATH 545: STOCHASTIC CALCULUS

Jonathan Mattingly

- MATH 557: INTRO TO PARTIAL DIFFERENTIAL EQNS

James Nolen

- MATH 560: ALGORITHMS

Gregory Herschlag

- MATH 577: MATHEMATICAL MODELING

Anna Nelson

- MATH 582: FINANCIAL DERIVATIVES

Xavier Mela

- MATH 585: INTRO TO ALGORITHMIC TRADING

David Ye

- MATH 602: COMMUTATIVE ALGEBRA

Kirsten Wickelgren

- MATH 612: ALGEBRAIC TOPOLOGY II

Richard Hain

- MATH 621: DIFFERENTIAL GEOMETRY

Mark Stern

- MATH 633: COMPLEX ANALYSIS

Saman Habibi Esfahani

- MATH 641: PROBABILITY

Nick Cook

- MATH 653: ELLIPTIC PARTIAL DIFFERENTIAL EQNS

Saman Habibi Esfahani

- MATH 660: NUMERICAL PARTIAL DIFFERENTIAL EQNS

Jianfeng Lu

- MATH 690-05: TOPICS IN NUMBER THEORY

Jayce Getz

- MATH 790-90: MINICOURSE IN ADVANCED TOPICS

**790-90.01**Introduction to some Ergodic Theory and Stochastic Analysis

Jonathan Mattingly*1/11-2/13/2023*; MW 1:45-3:00 pm

I will begin by giving a basic introduction to Ergodic Theory and its application to dynamics and Markov processes. I will then give some conditions which have been useful to show that the set of invariant/stationary measures is trivial. I will then introduce some basic stochastic analysis starting from basic knowledge of Stochastic differential equations. I will then use these tools to prove some interesting (to me) theorems. In particular, I will prove some ergodic theorems for S(O)DES and SPDES.

**790-90.02**Ribet’s method in the residually indistinguishable case

Samit Dasgupta*1/12-2/9/2023*; TuTh 10:15-11:30Ribet’s method is a powerful technique in number theory that allows one to construct extensions of one p-adic Galois representation by another under the assumption that an associated L-value vanishes modulo some ideal. The method has many applications, including Wiles’ proof of the Main Conjecture of Iwasawa Theory for totally real fields.

Throughout the literature, one always finds the assumption that the two representations are not congruent modulo p, i.e. that they are “residually distinguished.” The goal of my minicourse is to recall Ribet’s method and explain current joint work with Mahesh Kakde, Jesse Silliman, and Jiuya Wang, in which we show how to run Ribet’s method in the residually indistinguishable case. New topics that play a role in our study which do not traditionally appear in the study of Ribet’s method are the theory of “rational cohomology” and “matrix invariant theory.”

As an arithmethic application of our technique, if time permits we will describe how our strategy is used to complete the proof of the Brumer-Stark conjecture over Z (previously I proved the result with Kakde over Z[1/2] rather than Z, precisely because the representations involved are not distinguished at p=2). Our method also implies the Iwasawa Main Conjecture at p=2, a case that Wiles had left open, again for precisely the reason of residual indistinguishability.

As a final remark I’ll note that my collaborators and I have only handled the simplest case, when there are two representations and that are each 1-dimensional. This naturally leaves open the possibility of generalization to multiple representations of arbitrary dimension (which has important possible arithmetic applications). This is a topic that a graduate student looking for a thesis problem could explore.

**790-90.03**Dynamics in Complex Landscapes

Giampaolo Folena*2/15-3/22/2023*; MW 1:45-3:00 pmIn many research fields, from machine learning to optimization, from protein folding to statistical inference, one studies rough loss functions. A number of questions then naturally spring to mind about these "complex landscapes". How do different dynamical rules explore the landscape? Can one build some simple models to understand such complex dynamics? Is the behavior somehow universal?

The aim of this minicourse is to explore the interplay between dynamics and landscape geometry in mean-field (ie, simple) models of disorder. More specifically, we will apply tools from both theoretical physics and probability to these prototypical models to obtain some intuition in a subject that, despite its many applications, lacks a deep theoretical understanding. Students are expected to have basic knowledge in probability theory and

analysis.**790-90.04**On uniqueness in equations of fluid mechanics

Alexander Kiselev*2/14-3/21/2023*; TuTh 10:15-11:30 am

Uniqueness of solutions is a key property for predictive power of models. It is important to know the regularity threshold beyond which uniqueness may be lost. The course will be mainly devoted to recent construction of non-unique rough solutions to the 2D Euler equations by Vishik. Based on this construction, lack of uniqueness was proved for the Leray-Hopf weak solutions of the Navier-Stokes equation by Albritton, Brue and Colombo. In both constructions, the use of relatively rough force is necessary, but nonlinearity still plays a key role in generating non-uniqueness. The construction involves a blend of functional analysis, spectral theory and PDE estimates. There are quite a few natural open questions.

**790-90.05**PDE & Stochastic Processes

James Nolen*3/24-4/21/2023*; MW 1:45-3:00 pm

This course will develop connections between some stochastic processes and partial differential equations. Some basic material in this direction is covered in Math 545 or Math 641, but students in those courses are often interested in learning more along this direction. So, this course picks up from the end of Math 545 and Math 641. Some topics to be covered: Feynman-Kac representation for elliptic and parabolic problems, Branching processes and semilinear parabolic equations, stochastic optimal control and HJB equation, BSDE and obstacle problems, scaling limits of interacting particle systems, Models of random growth and the KPZ equation. The course is designed for students doing research in probability, PDE, stochastic dynamics.

**790-90.06**Topological Data Analysis, with Applications to Theory and Practice of Machine Learning

Paul Bendich*3/23-4/20/2023*; TuTh 10:15-11:30 am

My main goal is to make folks aware of interesting open problems and promising research directions in TDA (with a slant towards ML stuff). As a result, I'll err on the side of "broad survey with nice pictures" rather than "rigorous theorem-proof math lectures," but will certainly incorporate some of the latter when appropriate. It's totally fine if you've never coded before, and there won't be any coding assignments

Topics covered:- Persistence modules (what’s so tricky about multi-parameter persistence anyway?)
- Data science relevant examples of persistence modules
- Stability Theorems for Persistence, and cautionary tales about instabilities
- Computational issues (what is fast enough about TDA, and what is currently not fast enough?)

- TDA as front-end feature extraction method for machine-learning

- TDA as a loss function specifier within machine-learning

- TDA as a method for measuring complexity/safety of ML models

- Designing TDA-friendly model architechtures

**790-90.07**A user's guide to infinity categories

Kirsten Wickelgren*1/18-2/13/2023*; MW 10:15-11:30am

Homotopic maps are considered equal in the homotopy category.

This has many advantages. Beautiful invariants like the Euler characteristic pass through the homotopy category. On the other hand, the homotopy category loses the information of the homotopy itself, and makes notions such as the existence of (co)limits problematic. It is useful to replace the homotopy category with a homotopy *theory*. There are several ways to encode a homotopy theory. A powerful one is the notion of an infinity category. We will introduce infinity categories, and construct the homotopy theory of spaces, the stable homotopy theory of spaces, motivic versions of these constructions, and the infinity categorical versions of derived categories. The classical derived category can be recovered as the associated homotopy category.Concurrent or prior completion of a graduate course in algebra and algebraic topology is recommended.

**790-90.08**Adic spaces

Joseph Rabinoff*2/15-3/22/2023*; MW 10:15-11:30 am

Adic spaces constitute one of three main approaches to non-Archimedean geometry. The theory of adic spaces was developed in great generality by Huber, and is therefore suitable as the geometric framework underlying Schulze's theory of perfectoid spaces. The definitions rely on a careful study of the topological properties of valuative spectra. In this minicourse, we will work through the theory of valuative spectra, culminating in the definition of an affinoid adic space and its structure (pre)sheaf.

**790-90.09**Shimura varieties

Jayce Getz*3/24-4/21/2023*; MW 10:15-11:30 amThis course is an introduction to Shimura varieties. Shimura varieties are moduli spaces for certain motives generalizing the moduli space of elliptic curves.

I will define Shimura varieties, discuss their geometry over the complex numbers, and end with a discussion of canonical models for Shimura varieties over number fields. My primary reference is Milne's notes (https://www.jmilne.org/math/xnotes/svi.pdf).

Time and interest permitting I will also mention a few open problems related to the real points of Shimura varieties. The prerequisites for the course are some Lie theory and algebraic number theory.