Current Graduate Courses

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.

In 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.

• 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

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.

This 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.