Mean dimension is a topological invariant of dynamical systems introduced by Gromov that measures the number of parameters per iteration needed to describe a trajectory in the system. We characterize this invariant (at least for dynamical systems with the marker property, such as infinite minimal systems) using a min-max principle, where choices of both a metric on the topological space and an invariant probability measure on the system are varied.
The work I will report on is joint work with M. Tsukamoto.
As it was observed a few years ago, there exists a certain signed count of real lines on real projective hypersurfaces of degree 2n+1 and dimension n that, contrary to the honest "cardinal" count, is independent of the choice of a hypersurface, and by this reason provides, as a consequence, a strong lower bound on the honest count. Originally, in this invariant signed count the input of a line was given by its local contribution to the Euler number of an appropriate auxiliary universal vector bundle.
Repeats every week every Monday until Mon Apr 29 2019 except Mon Apr 22 2019.
4:00pm to 6:00pm
Abstract. This is a joint work with Linhui Shen.
A decorated surface is an oriented surface with punctures and a finite collection of special points on the boundary, considered modulo isotopy.
Let G be a split adjoint group. We introduce a moduli space Loc(G,S) of G-local systems on a decorated surface S, which reduces to the character variety when S has no boundary, and quantize it.
There is a general slogan according to which the limit behaviour of a one-parameter family of complex algebraic varieties when the parameter t tends to zero should be (partially) encoded in the associated t-adic analytic space in the sense of Berkovich; this slogan has given rise to deep and fascinating conjecturs by Konsevich and Soibelman, as well as positive results by various authors (Berkovich, Nicaise, Boucksom, Jonsson...).
Class field theory classifies abelian extensions of local and global fields
in terms of groups constructed from the base. We shall survey the main results of class
field theory for number fields and function fields alike. The goal of these introductory lectures
is to prepare the ground for the study of explicit class field theory in the function field case,
via Drinfeld modules.
I will talk for the first 2 or 3 times.
Repeats every week every Sunday until Sun Jun 23 2019 except Sun Apr 21 2019.
2:00pm to 4:00pm
Yun and Zhang compute the Taylor series expansion of an automorphic L-function over a function field, in terms of intersection pairings of certain algebraic cycles on the so-called moduli stack of shtukas. This generalizes the Waldspurger and Gross-Zagier formulas, which concern the first two coefficients.
The goal of the seminar is to develop the background necessary to state their formula, and then indicate the structure of the proof. If time allows, we may also discuss applications to the Birch and Swinnerton-Dyer conjecture for elliptic curves over function fields.
Repeats every week every Sunday until Sat Jun 29 2019 except Sun Apr 21 2019.
11:00am to 1:00pm
Zlil Sela and Alex Lubotzky "Model theory of groups"
In the first part of the course we will present some of the main results in the theory of free,
hyperbolic and related groups, many of which appear as lattices in rank one simple Lie groups
We will present some of the main objects that are used in studying the theory of these groups,
and at least sketch the proofs of some of the main theorems.
In the second part of the course, we will talk about the model theory of lattices in high rank simple Lie groups.
Manchester Building (Hall 2), Hebrew University Jerusalem
Abstract: If X is an object such that the notion of an automorphism of X is defined (e.g.,
an algebraic structure, a graph, a topological space, etc.), then one can define an
equivalence relation ∼ on X via x ∼ y if and only if α(x) = y for some automorphism
α of X. The equivalence classes of ∼ are called the automorphism orbits of X.
Say that X is highly symmetric if and only if all elements of X lie in the same
automorphism orbit. Finite highly symmetric objects are studied across various
mathematical disciplines, e.g. in combinatorics, graph theory and geometry. When