Abstracts of Talks
One-Frequency Schrödinger operators give one of the simplest models where fast transport and localization phenomena are possible. From a dynamical perspective, they can be studied in terms of certain one-parameter families of quasi-periodic co-cycles, which are similarly distinguished as simplest classes of dynamical systems compatible with both KAM phenomena and non-uniform hyperbolicity (NUH). While much studied since the 1970's, until recently the analysis was mostly confined to ''local theories'' describing the KAM and the NUH regime in detail. In this talk we will describe some main aspects of the global theory that has been developed in the last few years.
In this talk I will give an overview of some recent results in the theory of multiple zeta values, which were first defined by Euler. After giving an informal introduction to motivic multiple zeta values, which can be viewed as a prototype for a Galois theory of certain transcendental numbers, I will then explain how they were used to prove the Deligne-Ihara conjecture, which states that the motivic fundamental group of the projective line minus 3 points spans the Tannakian category of mixed Tate motives over Z, and a conjecture on multiple zeta values due to Hoffman.
The gluing equations is a zero-dimensional system of polynomial equations which describes the complete hyperbolic structure (and its deformations) for a 3-manifold with torus boundary. The Neumann-Zagier datum consists of the solutions together with the equations themselves. Using the Neumann-Zagier datum we construct a formal power series which conjecturally (a) captures the asymptotics of the Kashaev invariant to all orders and (b) whose first order agrees with the Reidemeister-Ray-Singer torsion. We show topological invariance of the first term of our formal power series, we expand it to a series of rational function on the PSL(2,C) moduli space, and give a computer implementation of the first 3 terms confirming numerically our conjectures above--joint with T. Dimofte. In addition, we extend our results to SL(N,C) representations of 3-manifolds--joint with M. Goerner and C. Zickert.
Both the Surface Subgroup Theorem and the Ehrenpreis conjecture are proven by building immersed surfaces out of immersed pairs of pants. In the latter case there can be an imbalance of pants: there may be more pants on one side of a closed geodesic than the other. I will explain the theory, the "good pants homology" of good closed geodesics modulo boundaries of good pants, that we use to construct the correction for this imbalance.
Dehn filling is the process of attaching a solid torus to a 3-manifold M with toral boundary. There are infinitely many ways to make this attachment, which are parametrised by the essential simple closed curves s (or 'slopes') on the boundary of M. The resulting manifold is denoted M(s). When M is hyperbolic, then so too is M(s), as long as s avoids finitely many possible slopes, known as 'exceptional' slopes. This is Thurston's hyperbolic Dehn surgery theorem, and it has been highly influential. How many exceptional slopes can there be? It is a famous conjecture of Gordon that the maximal number of exceptional slopes is 10, which is realised by the exterior of the figure-eight knot. In joint work with Rob Meyerhoff, we proved this conjecture. I will give an overview of the proof, which includes some new geometric techniques combined with a rigorous computer-assisted calculation.
This talk will be a survey of my recent work with Jeremy Kahn. I will focus on the Surface Subgroup Theorem and explain its applications in low dimensional topology.
An old theorem of Fontaine-Wintenberger states that the absolute Galois groups of deeply ramified extensions of Qp are canonically isomorphic to the absolute Galois groups of fields of characteristic p. The notion of perfectoid spaces generalizes this isomorphism to a comparison of geometric objects over fields of characteristic 0 and fields of characteristic p, thereby reducing certain deep questions about mixed characteristic rings to questions in equal characteristic. We will discuss applications to the weight-monodromy conjecture, and to p-adic Hodge theory.