## MATH 790-90 Minicourses in Advanced Topics

Minicourses in the Duke Mathematics department are in-depth dives into a specific advanced research topic. As these courses are very focused they only run for a few weeks of the semester. *Even though the course may not start of until the latter half of the semester it is important for students to register within the open registration period at the beginning of the semester*. Minicourses are a chance for Math faculty to showcase an interesting new development in Mathematics, share their research ideas, or delve into a peculiar subset of their interests. Minicourse topics are designed for advanced audiences, but all are welcome.

### Spring 2023 Minicourses

**MW: 1:45PM - 3:00PM**

- 790-90.01:
*01/13/2023 - 02/13/2023*Jonathan Mattingly, "Introduction to some Ergodic Theory and Stochastic Analysis" - 790-90.03:
*02/15/2023 - 03/22/2023*Giampaolo Folena, "Dynamics in Complex Landscapes" - 790-90.05:
*03/24/2023 - 04/21/2023*Jim Nolen, "PDE & Stochastic Processes"

**TuTh 10:15AM - 11:30AM**

- 790-90.02:
*01/12/2023 - 02/09/2023*Samit Dasgupta "Ribet’s method in the residually indistinguishable case" - 790-90.04:
*02/14/2023 - 03/21/2023*Sasha Kiselev "On uniqueness in equations of fluid mechanics" - 790-90.06:
*03/23/2023 - 04/20/2023*Paul Bendich "Topological Data Analysis, with Applications to Theory and Practice of Machine Learning"

**MW 10:15AM - 11:30AM**

- 790-90.07:
*01/13/2023 - 02/13/2023*Kirsten Wickelgren "A user's guide to infinity categories." - 790-90.08:
*02/15/2023 - 03/22/2023*Joe Rabinoff, Adic spaces - 790-90.09:
*03/24/2023 - 04/21/2023*Jayce Getz, Shimura varieties

#### Course Discriptions

**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 10:15-11:30

Ribet’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 pm

**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

**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 am

## Fall 2021 Minicourses

**TuTh 3:30PM - 4:45PM**

- 790-90.01: "Similarity solutions for singular dynamics in PDE's" Tom Witelski 8/30 - 9/27

**TuTh 1:45PM - 3:00PM**

- 790-90.03: "Constructible sheaves on compactified locally symmetric spaces" Les Saper 9/29 - 11/01
- 790-90.05: "Instantons and Nahm transforms" Mark Stern 11/03 - 12/06

**MWF 10:15AM - 11:05AM (or MW/MF/WF for 75 mins)**

- 790-90.04: "An introduction to contact geometry" Lenny Ng 09/28 - 10/28
- 790-90.06: "The universality phenomenon in random matrix theory (and beyond)" Nick Cook 10/31 - 12/02

## Fall 2021 Minicourses

**Tu + Th 1:45PM - 3:00PM**

- 01 08/24/2021 to 09/21/2021 Kirsten Wickelgren "L-theory, bilinear forms, and surgery"
- 03 09/23/2021 to 10/26/2021 Xiuyuan Cheng "An introduction to kernel methods in machine learning"
- 05 10/28/2021 to 11/30/2021 Tarek Elgindi "Singularity Formation in Fluids"

**M + W 10:15AM - 11:30AM**

- 02 08/23/2021 to 09/20/2021 Lenny Ng "An introduction to symplectic and contact geometry"
- 04 09/22/2021 to 10/22/2021 Jessica Fintzen "An invitation to the representation theory of reductive p-adic groups".
- 06 10/25/2021 to 11/22/2021 Dick Hain "Moduli of elliptic curves."

#### MTH 790-01: L-theory, bilinear forms, and surgery

**Instructor: Kirsten Wickelgren**

Tu + Th 1:45PM - 3:00PM

08/24/2021 to 09/21/2021*205 Physics*

L-theory is a cohomology theory formed from non-degenerate symmetric bilinear or quadratic forms modulo cobordism. So forms related by a surgery give the same element of L-theory. Examples of L-groups include the classical Witt groups of fields and rings. Poincare duality on compact oriented manifolds determines elements of L-theory of Z, and manifolds related by a geometric surgery give bilinear forms related by an algebraic surgery. The Hirzebruch signature formula can be viewed as a comparison between two orientations on the smash product of L with ordinary cohomology with Q coefficients. This minicourse will introduce L-theory and discuss these topics. It is inspired by recent work of B. Calmès, E. Dotto, Y. Harpaz, F. Hebestreit, M. Land, K. Moi, D. Nardin, T. Nikolaus and W. Steimle (https://arxiv.org/abs/2009.07223, https://arxiv.org/abs/2009.07224, https://arxiv.org/abs/2009.07225), building on work of J. Lurie.

**Topics**

- K-theory to L-theory
- Grothendieck--Witt groups and Witt groups, examples
- Witt groups and classical bilinear forms to Poincare objects (first look)
- Derived categories and stable homotopy theories
- Quadratic functors
- Poincare objects and surgery
- Geometric surgery and Poincare duality on manifolds
- L-groups, L-theory spaces and spectra
- Hirzebruch signature theorem
- Comments on recent work and future directions

Lecture notes for L1 are available here: https://services.math.duke.edu/~kgw/minicourse21-Ltheory/

#### MTH 790-02: An introduction to symplectic and contact geometry

**Instructor: Lenny Ng**

M + W 10:15AM to 11:30AM

08/23/2021 to 09/20/2021*205 Physics*

This minicourse will cover some of the basic ideas and techniques in symplectic and contact geometry. Symplectic and contact geometry is now a vast subject, with relations to dynamical systems, algebraic geometry, gauge theory, topology, and string theory, among other fields; I'll just try to hit a few highlights, with a bias toward the topological side of the subject. The minicourse should be accessible to anyone who is reasonably comfortable with smooth manifolds (along the lines of Math 620) and, not as crucially, algebraic topology (Math 611).

Rough outline of topics:

- symplectic structures; Hamiltonian vector fields; Lagrangian submanifolds
- Moser's method; Darboux's Theorem; Lagrangian neighborhood theorem
- almost complex structures; Kähler manifolds
- contact structures; Stein and Weinstein manifolds; constructing symplectic manifolds through handle attachment
- (if time permits) Arnold conjecture and Floer homology.

https://services.math.duke.edu/~ng/math790f21/

#### MTH 790-03: An introduction to kernel methods in machine learning

**Instructor: Xiuyuan Cheng**

Tu + Th 1:45PM - 3:00PM

09/23/2021 to 10/26/2021*205 Physics*

Kernel methods are fundamental tools in machine learning and high-dimensional data analysis. This course provides a mathematical introduction to the topic to help gain an understanding of the theory and applications of kernel methods, with considerations of computation and real-world applications. The mini-course consists of 9 lectures, starting from an elementary review of reproducing kernel Hilbert space and classical kernel methods in supervised and unsupervised learning. We then go to several research topics using kernels, primarily on high dimensional data: graph-based unsupervised learning (graph Laplacian), Maximum mean discrepancy and two-sample testing, predictive and generative models by neural networks. The second half of the lecture connects kernels to modern neural networks (NN), where we cover some topics in NN approximation, optimization, and applications, in all of which kernels play a role. The course goes from preliminary concepts to recent research problems in the field, featuring tentative guest lectures and in-class discussions on selected topics. The course is suitable for graduate students in applied math, statistics, computer science, and electrical engineering, as well as undergraduates with related research backgrounds. Audition welcome and registration encouraged.

#### MTH 790-04: An invitation to the representation theory of reductive p-adic groups

**Instructor: Jessica Fintzen**

M + W 10:15AM - 11:30AM

09/22/2021 to 10/22/2021*205 Physics*

This course will provide an introduction to reductive p-adic groups and their representation theory, beginning with basic definitions and culminating with recent advances on the construction of supercuspidal representations. The representation theory of p-adic groups is an active research area that has many connections to nearby areas. For example, the local Langlands correspondence connects representations of p-adic groups with Galois representations, and results in the representation theory of p-adic groups have application to the study of automorphic forms.

Possible topics include:

- brief introduction to reductive p-adic groups with examples

- smooth representations of p-adic groups

- overview of the state of the art of the construction of supercuspidal representations

- some ideas of Bruhat--Tits theory and the Moy--Prasad filtration

- depth-zero representations and epipelagic representations, if time permits

#### MTH 790-05: Singularity Formation in Fluids

**Instructor: ****Tarek Elgindi**

Tu + Th 1:45PM - 3:00PM

10/28/2021 to 11/30/2021*205 Physics*

The breakdown of solutions to the basic equations of fluid mechanics is a fundamental problem in the study of PDE. We will discuss various recent works on the breakdown problem in finite and infinite time. We will start with a survey and discussion of works on growth of solutions to the 2D Euler equation. This will take us from general arguments of Yudovich, Koch, and more recent ones found with T. Drivas to more specific techniques discovered by Denisov, Kiselev, and Sverak. We will then move on to look at various techniques to study the blow-up problem for the 3D Euler equation.

#### MTH 790-06: Moduli of elliptic curves

**Instructor: Dick Hain**

M + W 10:15AM - 11:30AM

10/25/2021 to 11/22/2021*205 Physics*

This course will be an elementary introduction to moduli spaces of elliptic curves, also known as modular curves. An elliptic curve can be defined as any of the following:

- the quotient of the complex numbers by a lattice (complex analysis)
- a flat 2-dimensional torus (differential geometry)
- a smooth cubic curve in the projective plane (algebraic geometry).

They are important in number theory, algebraic geometry, dynamical systems and mathematical physics.

Not all elliptic curves are isomorphic. The space whose points correspond to isomorphism classes of all elliptic curves is the "moduli space of elliptic curves". It is also called the "modular curve".

In this short course I will give a concrete description of the modular curve as a complex analytic orbifold (and possibly also as an algebraic stack). It is the quotient of the hyperbolic plane by SL_2(Z). This construction is elementary and is a prototype for the construction of moduli spaces in general. I will also introduce certain "generalized functions" on the modular curve, which are known as "modular forms". These are of fundamental importance in number theory and other areas, such as string theory.

This course should be useful to those studying number theory (both analytic and algebraic), as well as to those studying algebraic geometry, differential geometry and topology.

Background: students should know:

(1) basic complex analysis (at the undergraduate level)

(2) basic algebraic topology (fundamental groups, covering spaces, homology, Euler characteristic). Concurrent enrollment in MATH 611 will suffice.

The course will be based on my lecture notes: Lectures on Moduli Spaces of Elliptic Curves, in Transformation groups and moduli spaces of curves, 95–166, Adv. Lect. Math. (ALM), 16, 2011. These are available at https://arxiv.org/abs/0812.1803.

Another useful reference is Chapter VII of Serre's classic book: J. P. Serre: A course in arithmetic, Graduate Texts in Mathematics, No. 7. Springer-Verlag, 1973.