Dual Trimester Program: "Synergies between modern probability, geometric analysis and stochastic geometry"

10:00 - 11:00 am, Ning Zhang (Huazhong University of Science and Technology)

Title: The Aleksandrov projection theorem for convex lattice polygons

Abstract: In this talk, we will show that the examples given in Gardner-Gronchi-Zong's work are only counterexamples for the discrete analogue of the Aleksandrov projection theorem in Z^2.

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10:00 - 11:00 am, HIM lecture hall, Poppelsdorfer Allee 45,  Bram Petri (University of Sorbonne)

Title: Probabilistic methods in hyperbolic geometry

Abstract:Hyperbolic manifolds are Riemannian manifolds whose metric is complete and has constant sectional curvature equal to -1. Another way to phrase the latter property is to say that they are locally isometric to hyperbolic space (of the right dimension). The geometry and topology of such manifolds plays a role in various domains of mathematics. I will talk about how probabilistic methods can help to solve questions on the geometry and topology of hyperbolic manifolds.

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10:00 - 11:00 am, HIM lecture hall, Poppelsdorfer Allee 45, Mascha Gordina (University of Connecticut)

Title: Large deviations principle for sub-Riemannian random walks

Abstract: In this talk we consider large deviations for sub-Riemannian random walks on homogeneous Carnot groups. In addition to proving a LDP we can identify a natural rate function for such sub-Riemannian random walks, namely, as an energy functional. The diffusion processes corresponding to such random walks are hypoelliptic Brownian motions. The focus will be on a standard Brownian motion compared to the Brownian motion in the Heisenberg group.  The talk is based on the joint work with Tai Melcher and Jing Wang et al.

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10:00 - 11:00 am, HIM Lecture Hall, Poppelsdorfer Allee 45, Mira Gordin (Princeton)

Title: Vector-Valued Concentration on the Symmetric Group

Abstract:Concentration inequalities for real-valued functions are well understood in many settings and are classical probabilistic tools in theory and applications -- however, much less is known about concentration phenomena for vector-valued functions. We present a novel vector-valued concentration inequality for the uniform measure on the symmetric group. Furthermore, we discuss the implications of this result regarding the distortion of embeddings of the symmetric group into Banach spaces, a question which is of interest in metric geometry and algorithmic applications. We build on prior work of Ivanisvili, van Handel, and Volberg (2020) who proved a vector-valued inequality on the discrete hypercube, resolving a question of Enflo in the metric theory of Banach spaces. This talk is based on joint work with Ramon van Handel.

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10:00 - 11:00 am, HIM lecture hall, Poppelsdorfer Allee 45, Hermann Koenig (University of Kiel)

Title: Hyperplane sections of l_p^n for 2 < p < ∞

Abstract: tba

10:00-11:00 am, HIM lecture hall, Poppelsdorfer Allee 45, Anna Lytova (Opole University)

Title: On the  fluctuations  of the linear eigenvalue statistics  of the sample covariance matrices corresponding to data with a tensor product structure

Abstract:Consider a vector that is a tensor product of two n-dimensional copies of a  random vector Y, and the corresponding sample covariance matrix with the number of data points proportional to n. Under some additional moment conditions, we study the fluctuations of the linear eigenvalue statistics of such matrices as n goes to infinity.  In particular, we show that taking Y from the normal distribution or from the uniform distribution on the unit sphere results in different orders of fluctuations of the resolvent traces. We use this result to prove the CLT for the corresponding, properly normalized and centralized, linear eigenvalue statistics. The talk is based on the joint work with Alicja Dembczak-Kolodziejczyk.

Thursday, April 4, 10 - 11 am Imre Barany (Budapest and London)

Title: The Steinitz lemma, its matrix version, and balancing vectors

Abstract: The Steinitz lemma, a classic from 1913, states that a sequence a_1,...,a_n of (at most) unit vectors in R^d whose sum is the origin, can be rearranged so that every partial sum of the rearranged sequence has norm at most 2d. It is an important result with several applications. I plan to mention a few. I also explain its connection to vector balancing. 

In the matrix version of the Steinitz lemma A is a k by n matrix whose entries unit vectors in R^d and their sum is the origin. Oertel, Paat, Weismantel have proved recently that there is a rearrangement of row j of A (for every j) such that the sum of the entries in the first m columns of the rearranged matrix has norm at most 40d^5 (for every m). We improve this bound to 4d-2.

10:00 - 11:00 am Imre Barany (Budapest and London)

Title: The Steinitz lemma, its matrix version, and balancing vectors II

Abstract: The Steinitz lemma, a classic from 1913, states that a sequence a_1,...,a_n of (at most) unit vectors in R^d whose sum is the origin, can be rearranged so that every partial sum of the rearranged sequence has norm at most 2d. It is an important result with several applications. I plan to mention a few. I also explain its connection to vector balancing. 

In the matrix version of the Steinitz lemma A is a k by n matrix whose entries unit vectors in R^d and their sum is the origin. Oertel, Paat, Weismantel have proved recently that there is a rearrangement of row j of A (for every j) such that the sum of the entries in the first m columns of the rearranged matrix has norm at most 40d^5 (for every m). We improve this bound to 4d-2.