## 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.

### Fall 2022:

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

##### STUDENTS: It's important that you register for a course in the first two weeks of the semester, even for the later mini-courses.

**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.