Graduate Courses, Summer 2024 – Spring 2025

1. Math 7210 (Algebra), Math 4036 (complex analysis) or equivalent are 
very helpful. You should be comfortable with meromorphic and analytic
 functions, groups theory, group actions, Galois theory, and module theory.

2. Know how to find resources and references to help you to understand the material better.

3. Know how to use computer algebra systems (such as Maple, Mathematica, MATLAB, SageMath, or Magma).

Algebraic Geometry: Notes on a Course, Michael Artin, Graduate Studies in Mathematics 222

Algebraic Geometry, Robin Hartshorne, Graduate Texts in Mathematics, Spring-Verlag, 52.

Some basic functional analysis and PDE theory (MATH 4340, or MATH 7386 or equivalent) will be reviewed. Prior exposure to numerical analysis (MATH 4065 or 4066) and Python would be useful but not mandatory.

Various Reference Texts: The Finite Element Method for Elliptic Problems, (Classics in Applied Mathematics, 2002) by Philippe G. Ciarlet;
Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edition (2007) by D. Braess;
The Mathematical Theory of Finite Element Methods, 3rd or 4th Edition (2008) by S. Brenner, R. Scott.

Additional topics will be introduced depending on the interests of the class.

More specifically: Holomorphic and meromorphic functions of one variable including Cauchy's integral formula, theory of residues, the argument principle, and Schwarz reflection principle. Multivalued functions and applications to integration. Meromorphic functions. The Fourier transform in the complex plane. Paley-Wiener theorem. Wiener-Hopf method and the Riemann-Hilbert problem. Entire functions including Jensen's formula and infinite products. Conformal mapping including the Riemann mapping theorem and the Schwarz-Christoffel integral. In the case of sufficient time, further topics include the Gamma and Zeta functions and an introduction to the theory of elliptic functions.

This is a self-contained introduction to modern probability theory. Starting from the concept of probability measures, it would introduce random variables, and independence. After a study of various modes of convergence, the Kolmogorov strong law of large numbers and results on random series will be established. Weak convergence of probability measures will be discussed in detail, which would lead to the central limit theorem and its applications. A main goal of the course is to develop the concept of conditional probability and its basic properties. Stochastic processes such as Brownian motion and martingales will be introduced, and their essential features, studied.

Luke Postle (from University of Waterloo)’s youtube recordings on Probabilistic Methods Video.

Yufei Zhao (from MIT)’s lecture notes on Probabilistic Methods PDF.

Topics: Linearity of expectations, Alteration, Second Moment Methods, Chernoff bound, Lov´asz local lemma, Correlation inequalities, Janson inequalities, Concentration of measure, Entropy method, Container method, Random graphs.

The mechanics of fracture propagation provides essential knowledge for the risk tolerant design of devices, structures, and vehicles. Ideally fracture should emerge naturally from a field theory described by an initial boundary problem. Nonlocal approaches to fracture modeling are formulated along these lines, coupling fracture caused by breaking bonds at the atomic scale with continuous and discontinuous deformation at the macroscopic scale. Both localization and emergent behavior is the hallmark of theory and simulations using nonlocal formulations. The numerical solution of the nonlocal differential equation provides for spectacular results. On the other hand the field theory while promising is in its early stages and needs to recover established results in a mathematically rigorous and systematic way.

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.

• mathematical tools: norms, projectors, Gram-Schmidt process, orthogonal matrices, spectral theorem, singular value decomposition
• error analysis: round-off errors, backward stability and conditioning
• general systems: LU decomposition, partial pivoting, Cholesky decomposition and QR decomposition
• sparse systems: the methods of Jacobi, Gauss-Seidel, steepest descent and conjugate gradient
• eigenvalue problems: power methods, Rayleigh quotient iteration and QR algorithm