Topics courses – syllabi
FQ26 Math 520-1, Topics in Mathematical Physics: Prof. Eric Zaslow
My goal is for us to understand the KdV integrable hierarchy from the ground up.
The KdV PDE is ut = −6uux − uxxx. Now if u(x, t = t0) is a potential for the Schro ̈dinger potential for a quantum mechanical particle, then it turms out the spectrum of the Hamil- tonian −∂2 + u remains constant if u evolves over time, t, according to KdV. In fact, there are other evolutions which are possible, an infinite number of “times,” and the evolutions commute. The system can be understood symplectically, thus endowing the whole shebang with the structure of an infinite-dimensional completely integrable system. We can discuss quantization, as well. All of this was discovered at a “hands on” level, and layers and layers of hidden structure have been unearthed over the years, including Lax pairs, inverse scattering methods, aspects of quantum groups and conformal field theory. I think that there is some- thing for any of us to learn, be we analysts, symplectic geometers, representation theorists, mathematical physicists or what-have-you. Maybe come and try to prove me wrong?
Students are expected to attend, and to give at least one talk during the quarter. I may also give quizzes, not for “gotcha” purposes but to ensure the time we are spending is well spent.
Full disclosure: I’m a novice. I will mainly be learning alongside you.
FQ26 Math 430-1, Dynamical Systems: Prof. Bryna Kra
This course is an introduction to topological dynamics and ergodic theory. We will start with problems internal to these fields, understanding when systems are conjugate or isomorphic, the mixing properties of systems, existence of invariant measures, recurrence, and covering basic convergence theorems (mean and pointwise). We will then turn to applications, including connections to combinatorics and number theory, multiple convergence and recurrence, and connections to recent advances in combinatorial and number theoretic questions.
Prerequisite: Math 410-1, 410-2 or equivalent.
FQ26 Math 517-1, Topics in Algebra: Prof. Jakub Witaszek
Title: Applications of perverse sheaves in birational geometry and commutative algebra
Description: The concept of perverse sheaves was introduced by Bernstein, Beilinson, Deligne, and Gabber in their celebrated work on the decomposition theorem. This notion has roots in the theory of intersection homology of Goresky and MacPherson and plays a fundamental role in many modern areas of mathematics, including number theory, algebra, and representation theory.
The goal of the class is to discuss the theory of perverse sheaves, with a focus on applications to birational geometry and commutative algebra. We will begin by discussing the situation in characteristic zero and its connections to Hodge theory. In the second part of the class, we will move to positive characteristic, with the goal of explaining how perverse sheaves, together with the positive-characteristic Riemann–Hilbert correspondence, have led to significant developments. Time permitting, we will briefly discuss the mixed-characteristic situation.
FQ26 Math 420-1, Partial Differential Equations: Prof. Jared Wunsch
The course is a broad intro to the field of PDE and will follow the treatment of the book by Jeff Rauch (title: Partial Differential Equations). Topics will include: analytic solutions and Cauchy-Kovalevsky, Holmgren uniqueness; Fourier transform on tempered distributions; Sobolev spaces; solutions of Schroedinger, heat, and
wave equations by Fourier methods and via fundamental solutions; solution of the Dirichlet problem by variational methods. Grades will be based on problem sets and in-class presentations.
