Graduate Courses, Summer 2023-Spring 2024

Topics:

1. Chain complexes, resolutions (projective and injective).
2. Examples: Ext, Tor, Group cohomology, Hochschild cohomology.
3. Derived functors in abelian categories.
4. Introduction to sheaf theory in a topos.

In the “old days” one introduced ideas of triangulated derived categories. The modern thinking is to replace these by infinity categories. We won’t have time for this, but possibly, time permitting, we could introduce ideas of Simplicial Sets.

There will be homework sets assigned at each class. Assignments will not be collected and graded. Instead, at the beginning of each class, a student will present a solution to one of the homeworks.

The course is a self-contained introduction to modern probability theory. It starts from the concept of probability measures, random variables, and independence. Several important notions of convergence including almost sure convergence, convergence in probability, convergence in distribution will be introduced. Well-known limit theorems for sums of independent random variables such as the Kolmogorov's law of large numbers, Central Limit Theorem and their generalizations will be studied. Sums of independent, centered random variables form the prototype for an important class of stochastic processes known as martingales, and therefore play a major role in probability and other areas of mathematics. A part of the course will be devoted to understanding conditional probability which then forms the foundation for the theory of martingales. Brownian motion is an important example of a continuous-time martingale. If time permits, basic features of martingale theory and Brownian Motions will be briefly discussed. The course will not have any fixed text book, but some books will be cited for reference. It is important that the students take and read the class notes carefully.

In this course we introduce the nonlocal initial boundary value problem and prove convergence to a sharp fracture theory in the limit of vanishing non-locality. This is a highly nonlinear problem involving methods introduced in the past 15 years. We introduce the generalization of Sobolev space given by the Special Functions of Bounded Variation (SBV) and Special Functions of Bounded Deformation (SBD). We review their fine properties over Lebesgue measurable sets. We define and show Gamma convergence of the nonlocal energy to the classic Griffith fracture energy. Here we apply non-local methods used in the proof of the de Giorgi’s conjecture for the convergence of non-local approximations of the Mumford Shah functional developed by Gobbino. These are used in conjunction with slicing decompositions of SBV and SBD together with the integralgeometric measure of geometric measure theory. Convergence of the dynamics of intact material away from the crack is established with the aid geometric measure theory.

We conclude the course, showing how the kinetic relations of classic fracture theory, relating the speed of propagation of the crack tip singularity to the energy flowing into it are derived directly from the non-local differential equation and then passing to the local limit.

This course represents the beginning of the story. For example in the non-local differential equation the notion of regularity of solution, including the rigorous origins of crack branching, remains unanswered. These questions also lie on the frontiers of the physics describing the fracture process.

Gauge theory is often staged on 4-dimensional manifolds, both in physics and in the creation of smooth invariants for 4-manifolds. In this course, we will investigate the Seiberg-Witten equations, which are a pair of nonlinear partial differential equations on the space of connections of a line bundle and the space of sections (spinors) of a Spin^c bundle. Solutions to these equations turn out be gauge-invariant, which leads, after several steps, to the celebrated Seiberg-Witten invariants of smooth 4-manifolds—invariants that can (sometimes) detect when two homeomorphic smooth 4-manifolds are not diffeomorphic (exotic manifolds).

Along the way, we will study examples of smooth 4-manifolds and how to construct exotic 4-manifolds. This will include (fun) calculations with fundamental groups, symplectic 4-manifolds, and ``tying knots’’ into smooth 4-manifolds (Fintushel-Stern construction). For the algebra-minded student, we will show how Clifford algebras and spin representations can be and are used in gauge theory. For the analysis-minded student, we will work show how to apply Sobolev norms and Sobolev embedding theorems to understand the moduli space of solutions.

Low dimensional topologists look for new invariants by studying different gauge theories. Seiberg-Witten, in a sense, is one of the easiest gauge theories to study because it is a based upon an abelian U(1)-theory. This course will give students exposure to how such invariants are discovered and investigated. It should be interesting to students who want an overview of 4-manifold constructions and want to learn how to use advance techniques in topology, geometry, and global analysis to study smooth manifolds.