Several physicists conjectured in 1985 that the height functions of 1+1 random growth models should exhibit universal fluctuations behaviors in the large time limit. Over the last 25 years several rigorous results on this universality conjecture were proved. We consider one of the most famous models, the totally asymmetric simple exclusion process (TASEP). We first discuss the known results for the TASEP on the line and then present recent results on the case when the domain is changed to a ring.
I will give essentially matching upper and lower bounds for the expected max-sliced 1-Wasserstein distance between a probability measure on a separable Hilbert space and its empirical distribution from n samples. A version of this result for Banach spaces will also be presented. From this, I will derive an upper bound for the expected max-sliced 2-Wasserstein distance between a symmetric probability measure on a Euclidean space and its symmetrized empirical distribution.
One way to understand the concentration of the norm of a random matrix X with Gaussian entries is to apply a standard concentration inequality, such as the one for Lipschitz functions of i.i.d. standard Gaussian variables, which yields subgaussian tail bounds for the norm of X. However, as was shown by Tracy and Widom in 1990s, when the entries of X are i.i.d. the norm of X exhibits much sharper concentration. The phenomenon of a function of many i.i.d. variables having strictly smaller tails than those predicted by classical concentration inequalities is sometimes referred to as «superconcentration», a term originally dubbed by Chatterjee. I will discuss novel results that can be interpreted as superconcentration inequalities for the norm of X, where X is a Gaussian random matrix with independent entries and an arbitrary variance profile. We can also view our results as a nonhomogeneous extension of Tracy-Widom-type upper tail estimates for the norm of X.
In this talk, we will discuss several results on noncommutative Hardy spaces. In his PhD thesis, Mei introduced the definition of the operator-valued Hardy space associated with a von Neumann algebra as a space of functions . The construction of these spaces relies on the noncommutative analog of the Lusin integral, which comes along with the definition of the Hardy space as the sum of a column part and a row part . As it occurs in the classical setting, numerous definitions of the Hardy space can be introduced and shown to be equivalent. For instance, a recently introduced new atomic decomposition has allowed us to obtain estimates for Calderón-Zygmund operators on . Also, we will discuss how different characterizations may yield new results regarding the boundedness of Fourier multipliers and Schur multipliers. The content of this talk is part of joint work with Éric Ricard and work in progress with Tao Mei.
Alejandro Chávez-Domínguez, University of Oklahoma
The free holomorphic operator space
Lipschitz free Banach spaces associated to metric spaces have been very prominent in Banach space theory for the past two decades. Recently there has also been interest in other free objects in the Banach space context, as evidenced by the survey by García-Sánchez, de Hevia, and Tradacete. Operator spaces, being a noncommutative version of Banach spaces, pose another natural setting in which to look for free objects.
In this talk, we show how the classical methods based on the Dixmier-Ng-Kaijser theorem can be adapted to the noncommutative world. We emphasize the free holomorphic operator space, since that is a situation where we already have better results (such as transference of operator space approximation properties), but the same ideas work in several other cases. The key is to state the Dixmier-Ng-Kaijser result in a slightly more precise form that brings convexity into play, so that we can take advantage of noncommutative notions of convexity.
This is joint work with Verónica Dimant (Universidad de San Andrés).
Valeria Fragkiadaki, Texas A&M University
Two Weight Inequalities for Dyadic Paraproduct Operators.
We will define two very important operators in dyadic harmonic analysis, namely sparse operators and dyadic paraproducts, and we will look at some relations between them. In particular, we will use a “sparse domination” argument to upper bound a composition of dyadic paraproducts by a weighted sparse operator in the two-weight setting, Lp(μ) to Lp(λ) for two Ap weights μ,λ, and get some estimate for the norm.
Paul Hagelstein, Baylor University
Current Developments in the Theory of Differentiation of Integrals
The topic of differentiation of integrals arises with the origins of calculus, yet in many respects remains mysterious with many open problems. This talk will provide a broad overview of the subject of differentiation of integrals, highlighting important milestones and emphasizing recent progress on work related to the Halo Conjecture.
Mehrdad Kalantar, University of Houston
Boundary actions of discrete groups and the structure of their C*-algebras
The representation theory of discrete groups G is closely related to the structural properties of the C*-algebras generated by G. Significant progress has been made in the latter area over the last decade, due to the striking applications of the theory of topological boundary actions, in the sense of Furstenberg, in determining the ideal and trace structure of group C*-algebras. In this talk we review these developments and present some of the main results in this context.
Denka Kutzarova, University of Illinois at Urbana-Champaign
Banach spaces with a unique subsymmetric basic sequence
Albiac, Ansorena and Wallis used Garling-type spaces to
provide the first example of a Banach space with a unique subsymmetric basis
which is not symmetric. However, that space contains a continuum of non-equivalent subsymmetric basic sequences. Altshuler constructed a Tsirelson-like space in which all symmetric basic sequences are equivalent to its symmetric basis.
We prove that the subsymmetrization of Tsirelson’s original space, T* in the notation of Figiel-Johnson, provides the first known example of a space with a unique, up to equivalence, subsymmetric basic sequence that is
additionally non-symmetric. Later, we provided more examples of such spaces. The Tirilman spaces were introduced by Casazza and Shura as variations of the spaces constructed by Tzafriri. We show that all subsymmetric basic sequences in the dual of a Tirelman space are equivalent to its canonical subsymmetic but not symmetric basis.
The talk is based on joint papers with Casazza, Dilworth, Motakis,
Sari and Stankov.
Jean-François Lafont, Ohio State University
Barycenter method in geometry & topology
The barycenter method was introduced by Besson-Courtois-Gallot in the early 1990s, and has since been used to prove a number of spectacular results in geometry and topology. In this talk, I'll give an overview of the technique, and describe some results one can prove with this method. The method relies on taking a suitable center of mass associated to certain probability measures on a space. By carefully varying the probability measures, one can define maps with desirable properties (isometries, volume control, differentiability, etc). This will be a general audience talk, emphasizing concepts and idea, and keeping prerequisites at a minimum.
Assaf Naor, Princeton University
Quantitative Wasserstein rounding.
The main focus of this talk will be to describe recent work (joint with Braverman) on the Lipschitz extension problem that obtains solutions to various natural quantitative questions by thinking about its (known) dual formulation as a question about randomly rounding an ambient metric space to its subset while preserving certain natural guarantees that are measured in terms of transportation cost. We will start by discussing the classical formulation of these old questions as well as some background and earlier results, before passing to examples of how one could reason quantitatively using the dual perspective.
Mikhail Ostrovskii, St. John's University
Quantitative results on finite determination for embeddings on locally finite metric spaces into Banach spaces
By
finite determination I mean results of the following type:
Theorem (M.O. (2012)): Let be a locally finite metric space whose finite subsets admit bilipschitz embeddings into a Banach space with uniformly bounded distortions (admit coarse embeddings into a Banach space with the same control functions for all finite subsets). Then admits a bilipschitz (coarse) embedding into .
This theorem was preceded by partial results (folklore on ultraproducts, Baudier, Lancien, M.O.)
It is natural to investigate the quantitative aspect of finite determination. In the bilipschitz case, it is the following: if distortions of embeddings into of finite subsets of do not exceed , what can be said about the optimal distortion of embeddings of into ?
The answer in speaker’s paper of 2012 was . The goal of the talk is to present new results and techniques (after a review).
Kavita Ramanan, Brown University
A probabilistic approach to the geometry of p-Schatten balls
The lp spaces are a canonical example of Banach spaces whose geometry is well understood and has been fruitfully studied using probabilistic methods. In this talk, we describe how a different set of probabilistic tools can be used to probe the geometry of their less well understood non-commutative analogs, the p-Schatten spaces of matrices, and touch upon how these results are linked with some open questions in convex geometry. Along the way we describe some new results on Haar measure on the orthogonal group (or more generally, Stiefel manifolds) that may be of independent interest. This is joint work with Grigoris Paouris.
Christian Rosendal, University of Maryland
Coordinate systems in Banach spaces and lattices
Using methods of descriptive set theory, in particular, the determinacy of
infinite games of perfect information, we answer several questions from the
literature regarding different notions of bases in Banach spaces and lattices.
For the case of Banach lattices, our results follow from a general theorem
stating that (under the assumption of analytic determinacy), every
σ-order basis (en) for a Banach lattice X=[en] is a uniform basis,
and every uniform basis is Schauder. Moreover, the notions of order and
σ-order bases coincide when X=[en]. Regarding Banach spaces, we
address two problems concerning filter Schauder bases for Banach spaces, i.e.,
in which the norm convergence of partial sums is replaced by norm convergence
along some appropriate filter on ℕ. We first provide an example of a
Banach space admitting such a filter Schauder basis, but no ordinary Schauder
basis. Secondly, we show that every filter Schauder basis with respect to an
analytic filter is also a filter Schauder basis with respect to a Borel filter.
This is joint work with A. Aviles, M. Taylor and P. Tradacete.
Gideon Schechtman, Weizmann Institute of Science
Approximate identities in Ideals of the algebras of bounded operators on Lebesgue spaces.
A net {Tα} in a Banach algebra is called a left approximate identity if Tα S tends to S for all S in the algebra. Right approximate identity is defined in a similar way.
I’ll present a recent work with Bill Johnson addressing the question of which closed ideals of the algebras of bounded operators on Lebesgue spaces have left or right approximate identity.
Nikhil Srivastava, University of California, Berkeley
Many Nodal Domains in Random Regular Graphs
A nodal domain of a Laplacian eigenvector of a graph is a maximal connected component where it does not change sign. Sparse random regular graphs have been proposed as discrete toy models of "quantum chaos", and it has accordingly been conjectured by Y. Elon and experimentally observed by Dekel, Lee, and Linial that the high energy eigenvectors of such graphs have many nodal domains.
We prove superconstant (in fact, nearly linear in the number of vertices) lower bounds on the number of nodal domains of sparse random regular graphs, for sufficiently large Laplacian eigenvalues. The proof combines two different notions of eigenvector delocalization in random matrix theory as well as tools from graph limits and combinatorics. This is in contrast to what is known for dense Erdos-Renyi graphs, which have been shown to have only two nodal domains with high probability.
Joint work with Shirshendu Ganguly, Theo McKenzie, and Sidhanth Mohanty.
Joseph Slote, California Institute of Technology
Dimension-free Remez Inequalities and Norm Designs
What global properties of a function can we infer from local information? Remez-type inequalities offer one answer: they show the supremum norm of a polynomial can be controlled by its (absolute) supremum over a small subset of the domain. While Remez-type inequalities enjoy widespread use in analysis and approximation theory, their multivariate versions are often limited by a strong dependence on dimension.
In this talk we show that for many domains and test sets a dimension-free Remez inequality is in fact available. The main idea is a probabilistic technique for tensorizing one-dimensional inequalities without paying a dimension-dependent price. Applications to Bohnenblust–Hille inequalities and learning theory will also be discussed.
Based on the joint work arXiv:2310.07926 with Becker, Klein, Volberg, and Zhang.
Jinmin Wang, Texas A&M University
Stoker problem and index theory on manifolds with polyhedral boundary
The Stoker problem states that the dihedral angles of a convex Euclidean polyhedron determine the angles of each face. In this talk, I will introduce an index theory of Dirac-type operators on polyhedral manifolds, which leads to an affirmative answer to the Stoker problem. Our approach is based on the analysis of conical singularities and conical operators. The talk is based on joint works with Zhizhang Xie and Guoliang Yu.