Math 191 Spring 2019

Section 1 - Topics in perturbative quantum field theory and the p-adic holographic correspondence (Parikh)

This course will discuss aspects of partition functions, path integrals and renormalization group flow in statistical mechanics, quantum mechanics and scalar quantum field theories. Topics expected to be covered include: functional integrals, Feynman diagrams, Feynman-Kac formula, correlation functions, lattice systems, renormalization group flow, p-adic (conformal) field theories, Bruhat-Tits tree as discrete anti-de Sitter space, toy model of p-adic holographic correspondence.

This class is appropriate for undergraduates (and graduate students) with prior knowledge of calculus and classical mechanics. Familiarity with statistical and quantum physics, as well as abstract algebra and p-adic numbers will be useful, but not strictly imposed.

Section 2 - Topics in Number Theory (Ramakrishnan)

 This course will introduce some basic conjectures on rational points on varieties over number fields, and then test them on certain simple modular examples. Different perspectives, algebraic, analytic and geometric, will be explained, and the deep connections to L-functions. Depending on the time everything takes, there many also be a discussion of integral points.

The prerequisites are some basic knowledge of Algebra, Analysis and Topology; it will suffice to have had Ma 120, 110 and 151, the basic first year graduate courses. Some background in Number theory will also be needed.

Section 3 - Intersection Theory on Shimura Varieties (Amir-Khosravi)

The course will be about the geometry of special cycles on Shimura varieties, and the modularity properties of their generating series. We'll focus on Arakelov cycles on PEL Shimura varieties, their moduli descriptions, integral models and compactifications. We'll discuss aspects of the recent proof of modularity by Bruinier-Howard-Kudla-Rapoport-Yang, as well as the conjectural relations between arithmetic intersection numbers and Fourier coefficients of derivatives of incoherent Eisenstein series. Rather than give a broad survey of results, we will focus on developing the tools needed to carry out research in the area, mostly on the algebraic geometry side.

Expected Background:
Some familiarity with PEL Shimura varieties, basic algebraic geometry, and classical theory of modular forms.

Section 4 - Noncommutative Algebraic Geometry and Special Functions (Rains)

Since an algebraic scheme can be recovered from the category of quasicoherent sheaves on that scheme, it is natural to ask whether there are other families of categories that behave similarly. For instance, the category of quasicoherent sheaves on an affine scheme is just the category of modules over a commutative ring, and dropping the word "commutative" gives a category one may view as the category of sheaves on a noncommutative affine scheme.  The main topic for the course is noncommutative projective geometry, and in particular noncommutative projective surfaces, their construction, and their similarities (and differences) to the commutative case.  I will also discuss a motivating application, to wit that many classification problems involving differential equations or discrete analogues have a natural translation into analogous questions about sheaves on noncommutative projective surfaces.

 Background:  A solid understanding of commutative algebraic geometry (at the level of Hartshorne or Vakil, in particular cohomology of sheaves) will be essential.  Familiarity with curves and surfaces would be helpful, as would familiarity with derived categories.

Section 5 - Topics in Harmonic Analysis (Krause)

Syllabus TBD

Background - Measure theory is expected.

Section 6 - Derived Algebraic Geometry (Campbell)

This course will cover recent developments in derived algebraic geometry, with an eye toward applications in geometric representation theory. We'll follow A Study in Derived Algebraic Geometry by Gaitsgory and Rozenblyum, emphasizing concepts and examples. The topics covered will include: basics of higher category theory and higher algebra, derived schemes and Artin stacks, ind-coherent sheaves, deformation theory, inf-schemes, crystals, derived Lie theory, and infinitesimal differential geometry.

 Although the course will develop the theory from first principles as much as possible, some familiarity with the corresponding "classical" topics will be very helpful. These include algebraic geometry at the level of Hartshorne's book, as well as category theory and homological algebra as treated in e.g. the book by Gelfand and Manin. Overall a high level of mathematical maturity will be assumed.