Singular vectors are the ones for which Dirichlet’s theorem can be infinitely improved. For example, any rational vector is singular. The sequence of approximations for any rational vector q is 'obvious'; the tail of this sequence contains only q. In dimension one, the rational numbers are the only singulars. However, in higher dimensions there are additional singular vectors. By Dani's correspondence, the singular vectors are related to divergent trajectories in Homogeneous dynamical systems. A corresponding 'obvious' divergent trajectories can also be defined.
The discovery of the Jones polynomial in the early 80's was the beginning of ``quantum topology'': the introduction of various invariants which, in one sense or another, arise from quantum mechanics and quantum field theory. There are many mathematical constructions of these invariants, but they all share the defect of being first defined in terms of a knot diagram, and only subsequently shown by calculation to be independent of the presentation. As a consequence, the geometric meaning has been somewhat opaque.
Title: Perturbations of non-normal matrices
Abstract: Eigenvalues of Hermitian matrices are stable under perturbations in the sense that the $l_p$ norm of the difference between (ordered)eigenvalues is bounded by the Schatten norm of the perturbation. A similar control does not hold for non-Normal matrices. In the talk, I will discuss
Title: "Paradoxical" sets with no well-ordering of the reals
Abstract: By a Hamel basis we mean a basis for the reals, R, construed as a vecor space over
the field of rationals. In 1905, G. Hamel constructed such a basis from a well-ordering
of R. In 1975, D. Pincus and K. Prikry asked "whether a Hamel basis exists in any
model in which R cannot be well ordered." About two years ago, we answered this positively
in a joint paper with M. Beriashvili, L. Wu, and L. Yu. In more recent joint
work, additionally with J. Brendle and F. Castiblanco we constructed a model of
The cost of a measure-preserving equivalence relation is a quantitative measure of its complexity. I will
explain what the cost is and then discuss a recent result of Tom Hutchcroft and Gabor Pete in which they construct,
for any group with property T, a free ergodic measure preserving action with cost 1.
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.
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.
