Date: 11 February 2025
Speaker: Han Zhang
Title: Khintchine’s theorem on self-similar measures on the real line
Abstract: In 1984, Mahler proposed the following question on Diophantine approximation : How close can irrational numbers in the middle-thirds Cantor set be approximated by rational numbers? One way to reformulate Mahler’s question is to ask if Khintchine’s theorem extends to the middle-thirds Cantor set. In a joint work with Timothée Bénard and Weikun He, we prove that Khintchine’s theorem holds for any self-similar measures on the real line. In particular this applies to the Hausdorff measure on the middle-thirds Cantor set. Our result generalizes the recent breakthrough work of Khalil-Luethi in dimension one. Our proof is inspired by the work of Bénard-He regarding the semisimple random walks on homogeneous spaces.
Date: 4 March 2025
Speaker: JinCheng Wang
Title: Some geometric and dynamical properties of hyperbolic magnetic flows
Abstract: As a generalization of geodesic flows, magnetic flows trace unit-speed curves with constant geodesic curvature. We consider the magnetic flows of surfaces with negative Gaussian curvature that are nonuniformly hyperbolic. By studying its geometry on the universal covering of the surface, we show the uniqueness of the measure of maximal entropy via the Bowen-Climenhaga-Thompson machinery. This is a joint work with Boris Hasselblatt.
Date: 11 March 2025
Speaker: Michael Bersudsky
Title: Equidistribution of polynomially bounded o-minimal curves in homogenous spaces
Abstract: I will present my recent joint work with Nimish Shah and Hao Xing. We study a basic question about the limiting distribution of averages along unbounded curves in homogeneous spaces. We consider the class of curves which are definable in o-minimal structures, for example, matrix curves in SL(n,R) who’s entries are rational functions.
Our results extend Ratner’s equidistribution theorem for one-parameter unipotent flows, generalize Shah’s results for polynomial curves and generalize some of the recent results Peterzil and Starchenko. A key component in our work is the Kleinbock-Margulis $(C,\alpha)$-good property for families of functions definable in polynomially bounded o-minimal structures. The property for such functions was unknown before, and its proof relies on the tools of o-minimal structures theory.
Date: 18 March 2025
Speaker: Alon Agin
Title: The Dirichlet spectrum
Abstract: Akhunzhanov and Shatskov defined the Dirichlet spectrum, corresponding to mxn matrices and to norms on R^m and R^n. In case (m,n) = (2,1) and using the Euclidean norm on R^2, they showed that the spectrum is an interval. We generalize this result to arbitrary (m,n) with max(m,n)>1 and arbitrary norms, improving previous works from recent years. We also define some related spectra and show that they too are intervals. We also prove the existence of matrices exhibiting special properties with respect to their uniform exponent. Our argument is a modification of an argument of Khintchine from 1926.
Date: 22 April 2025
Speaker: Liyang Shao
Title: Weighted Inhomogeneous Bad is Winning and Null
Abstract: We will introduce the notion of inhomogeneous weighted badly approximable vectors. We discuss that this set can be very large (winning) in a sense and in some other sense it is very small (measure wise). In particular, we talk about such largeness and smallness via studying weighted inhomogeneous bad intersected with manifolds and support of certain measures. This is a joint work with Shreyasi Datta.
Date: 29 April 2025
Speaker: Srivatsa Srinivas
Title: Random Walks on SL_2(F_p) x SL_2(F_p)
Abstract: We will give a taste of the flavors of math that constitute the study of random walks on compact groups, followed by which we will describe the author’s work with Prof. Golsefidy in solving a question of Lindenstrauss and Varju. Namely, can the spectral gap of a random walk on a product of groups be related to those of the projections onto its factors
Date: 6 May 2025
Speaker: Emilio Corso
Title: Higher rank Furstenberg slicing
Abstract: In addition to the notoriously challenging measure rigidity conjecture for non-lacunary semigroups of toral endomorphisms, Furstenberg formulated, in the late sixties, a series of geometric rigidity conjectures aimed, as their dynamical counterparts, at capturing the heuristic principle that expansions of real numbers in multiplicatively independent integer bases are uncorrelated. Among these features the intersection conjecture, according to which the intersection of closed sets invariant under multiplicatively independent toral endomorphisms is, in dimensional terms, as small as it can be given the constraints of the system. The conjecture was settled by Shmerkin half a century after its formulation, with subsequent alternative arguments given by Meng Wu and Tim Austin. In joint work with Shmerkin, we provide a higher rank extension of the result, which handles intersections of any finite collection of invariant sets, and more generally presents uniform dimensional estimates for slices of products of such sets with arbitrary affine subspaces. As in the rank-one case, the argument hinges upon a precise understanding of Frostman exponents, via L^q spectra, for convolutions of self-similar measures.
