Abstract:


Title:
Effective rational approximation on spheres.

Zouhair Ouaggag

We prove an effective estimate for the counting function of Diophantine approximants on the sphere Sn. We use homogeneous dynamics on the space of orthogonal lattices, in particular effective equidistribution results and non-divergence estimates for the Siegel transform, developing on recent results of Alam-Ghosh and Kleinbock-Merrill. 



Title:
Approximation of Diagonally Invariant Measure by Tori Measures

Yuval Yifracht

We consider the family of periodic measures for the full diagonal action on the space of unimodular lattices. This family is important and natural due to its tight relation to class groups in number fields. We show that many natural families of measures on the space of lattices can be approximated using this family (in the weak sense). E.g., in our work we show that for any 0<c\leq 1, the measure cm_{X_n} can be approximated this way, where m_{X_n} denotes the Haar probability measure on X_n. Moreover, we show that non ergodic measures can be approximated. Our proof is based on the equidistribution of Hecke neighbors and on constructions of special number fields. We will discuss the results, alternative ways to attack the problem, and our method of proof.
This talk is based on a joint work with Omri Solan.


Title:
Parametric geometry of numbers and diophantine approximation

Nicolas de Saxce

The first goal of the talk will be to explain the general principles of parametric geometry of numbers, which is a way to describe combinatorially, up to a bounded error, any one-parameter diagonal orbit in a space of lattices. Then we shall discuss applications of these ideas to diophantine approximation in flag varieties, using a general version of Dani's celebrated correspondence. (Part of the talk will be based on an on-going project with Emmanuel Breuillard, Omri Solan and Nattalie Tamam.)


Title: 
An upper bound of the Hausdorff dimension of singular vectors on affine subspaces

Pengyu Yang

Abstract: We give an upper bound of the Hausdorff dimension of singular vectors on affine subspaces of the Euclidean space. This upper bound is expressed in terms of the Diophantine exponent of the parameter of the affine subspaces. The proof involves our construction of a new Margulis function on homogeneous spaces. Joint work with Shah.


Title: 
Equidistribution of primitive lattices in finite characteristic

Tal Horesh

Abstract: A primitive d-lattice in Z^m is a rank d subgroup which is maximal in the sense that it is not contained in any other rank d subgroup. In other words, primitive d-lattices correspond to d-dimensional rational subspaces in R^m. Many counting and equidistribution questions can be formulated about these objects: for example, how many primitive d-lattices are there with covolume up to X, as X tends to infinity? (When d=1, this is equivalent to asking how many primitive vectors are there with norm up to X). One can also ask about the distribution of their shapes, or of their distribution as rational points on the Grassmannian. The study of these questions dates back to Schmidt, and in the past 10 years it has received intensive treatment in dynamical methods. The goal of this talk is to introduce an analog of these results in the finite characteristic setting. This is joint work with F. Paulin.   


Title:
Equidistribution of divergent diagonal orbits in positive characteristic

Rafael Sayous

Abstract:
With $R=\mathbb{F}_q[Y]$ the polynomial ring over $\mathbb{F}_q$ and $\widehat{K}=\mathbb{F}_q((Y^{-1}))$ the local field of formal Laurent series over $\mathbb{F}_q$, we study the dynamics of the diagonal subgroup of the linear group $\operatorname{GL}_n(\widehat{K})$ on the homogeneous space of discrete $R$-lattices in ${\widehat{K}}^{\, n}$. We first give a function field version of results by Margulis and Tomanov-Weiss, characterizing the diagonal orbits that are divergent. We then investigate a function field version of a result of David and Shapira on the equidistribution of a natural family of these divergent diagonal orbits, with height given by an appropriate discriminant of the orbit. This is an ongoing joint work with N.-T.~Dang and F.~Paulin. 


Title:
Divergence of large dimensional diagonal orbits

Omri Nisan Solan

Abstract:
In a joint work with Nattalie Tammam we state that in arithmetic homogeneous spaces a divergent diagonal trajectory of the maximal possible dimension is divergent due to finite algebraic reasons.
We will discuss the geometric part of the proof (in the space of lattices).
We explain that (in some aspects) the dimension of the cusp is the Q-rank of the lattice.
We will explain how to use the fact that the dimension of an orbit is larger or equal to the dimension of the cusp to create a certain cover. We will reduce the algebraicity of the divergence to a covering question, then discuss its solution. 


Title:
On the twisted Ruelle zeta function and the Ray-Singer metric

Xenia Spilioti

Abstract:
In this talk we will present some results concerning the Fried's
conjecture, i.e., the relation of the twisted dynamical zeta function of Ruelle at zero
and spectral invariants for a hyperbolic manifold X. In particular, we consider the twisted
Ruelle zeta function twisted by an arbitrary representation of the lattice. We study then its relation to the Ray-Singer norm of the refined
analytic torsion. The refined analytic torsion  is an element of the
determinant line of the cohomology of  X with coefficients in the
flat complex vector bundle associated with the representation.


Title: Expanding cones and expanding measures on simple Lie groups

Ronggang Shi

Abstract: Let H be   a noncompact
simple Lie group and A be a diagonalizable subgroup of  H
normalizing a  unipotent subgroup U.
The expanding cone of U in A consists of elements of A which
expand all the  U-fixed nonzero vectors in non-trivial 
irreducible representations of H. If the cone is nonempty, then
 we can use it to construct  expanding measures supported on 
 AU.  Expanding measures on solvable groups
can be used to obtain rigidity and pointwise equidistribution
theorems.


Title: Random arithmetic lattices for sphere packings

Nihar Gargava

Abstract:
We consider random Euclidean lattices that have an underlying arithmetic structure coming from a number field. Such lattices have been studied to perform sphere packing, make error-correcting codes and also come to be useful in cryptography. We will describe probabilistic techniques to study random lattices like Siegel's mean value theorem, Rogers integral formula and what kind of results this can imply.
(Joint work with V. Serban, M. Viazovska, I. Viglino)