Talk

DAHA generally provide refined invariants of colored iterated links, which generalize the
WRT-invariants and HOMFLY-PT polynomials. In the uncolored case and for iterated knots,
they are conjectured to coincide with the stable reduced Khovanov-Rozansky polynomials (the most
powerful numerical invariants we have). The "intrinsic" DAHA conjectures are mostly verified
at the moment; these properties are generally difficult to check topologically. The DAHA super-

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.

The central theme of this talk is the question: How much of the rational homotopy type of a space can we deduce from the (co)chains on the space? More precisely, in this talk we explain what the relationship is between the singular (co)chains and various approaches to the rational homotopy theory. 

VB-groupoids can be understood as vector bundles in the category of Lie groupoids. They encompass several classical objects, such as Lie group representations, 2-vector spaces, Lie group actions on vector bundles; moreover, they provide geometric pictures for 2-term representations up to homotopy of Lie groupoids, in particular the adjoint representation. In this talk, I will attach to every VB-groupoid a cochain complex controlling its deformations and discuss some features, such as Morita invariance, as well as some examples and applications.

Let GL_n be the general linear group scheme over the integers.We show that the p-adic completion of GL_n possessescertain remarkable Frobenius lifts attached to integral symmetric matrices; these Frobenius liftscan be viewed as  arithmetic analogues, for Spec Z,  of the Levi-Civita connection attached to a metric on a manifold. We then show how these Frobeniuslifts admit algebraizations by correspondences on GL_n; the commutatorsof these correspondences can be viewed as an arithmetic analogue, for Spec Z, of Riemannian curvature. 

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

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.