Talk

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their
structural properties. The course is aimed to be accessible by graduate students and young researchers.

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their
structural properties. The course is aimed to be accessible by graduate students and young researchers.

Time: Tuesdays, 4.30 - 6 pm
Place: MPIM Lecture Hall, Vivatsgasse 7
First lecture: on April 2, 2019, end on July 2

The theory of arithmetic jet spaces has been developed by A. Buium using rings with a delta structure for a fixed prime p. In this arithmetic setting, the delta structure on a ring plays the analogous role of a derivative operator in geometry. One naturally associates a lift of Frobenius morphism to such a delta structure. As an example, the delta structure on the ring of integers is the Fermat quotient operator and the identity map is the associated lift of Frobenius mod p.

In this talk we describe a new proof of a classical result and a number of new results that can be proved along similar lines. At the heart of this approach are the verification of the assumptions of F"urstenberg's theorem about products of random matrices and a way to parlay the output of this theorem via large deviation estimates into statements commonly referred to as ``Anderson localization.'' The various settings we consider include quantum evolution in random environments and orthogonal polynomials on the unit circle with random recursion parameters.

About two decades ago, Ivan Dimitrov and I defined  ind-varieties of generalized flags. These varieties are nothing but
G/P for G=GL(infty) and a parabolic subgroup P of G. In this talk I would like to explain that the ind varieties of
generalized flags can be alternatively defined in a purely algebraic-geometric fashion as direct limits of linear
embeddings of usual flag varieties. The fact that the so-defined ind-varieties are homogeneous ind-spaces for
GL(infty) is then a consequence of the comparison theorem.

Serre-Tate proved in the 60's that an ordinary abelian variety E over a finite field k can be canonically lifted to an ordinary abelian variety over the Witt vectors of k. Expressed in terms of the moduli stack M_ord of ordinary abelian schemes, this is precisely a universal lifting property with respect to the Witt vectors (of finite fields).