A deep result of Furstenberg from 1967 states that if Γ is a lattice in a semisimple Lie group G, then there exists a measure on Γ with finite first moment such that the corresponding harmonic measure on the Furstenberg boundary of G is absolutely continuous. I will discuss some of the history of this result and some recent generalizations.
I'll give an overview of recent progress in the structure and classification of simple amenable C*-algebras, making parallels to the Connes-Haagerup classification of amenable von Neumann algebras and drawing examples from group actions.
Motivated by a discovery by Radchenko and Viazovska and by a work by Ramos and Sousa, we find conditions sufficient for a pair of discrete subsets of the real axis to be a uniqueness or a non-uniqueness pair for the Fourier transform. These conditions are not too far from each other. The uniqueness theorem can be upgraded to the frame bound and an interpolation formula, which in turn produce an abundance of Poisson-like formulas (a.k.a. 'crystalline measures').
Many notions in classical Fourier analysis have beautiful extensions to discrete quantum theory. We give an overview of this rapidly developing area.
We introduce conformal Bach flow and establish its well-posedness on closed manifolds. We also obtain its backward uniqueness. To give an attempt to study the long-time behaviour of conformal Bach flow, assuming that the curvature and the pressure function are bounded, global and local Shi's type L2-estimate of derivatives of curvatures are derived. To make the talk more accessible, we will spend some time to survey on high-order parabolic curvature flow.
In this talk I will introduce the notion of self-similarity for compact quantum groups. I will start by looking at the quantum automorphism group of an infinite homogeneous rooted tree. Self-similar quantum groups are then certain quantum subgroups of these quantum automorphisms. I will then look at a class of examples called finitely-constrained self-similar quantum groups, and I will describe a subclass as quantum wreath products by subgroups of the quantum permutation group.
Many combinatorial objects (or similar) give rise to a C-algebra with a distinguished C-subalgebra. Examples of such which have been studied a lot in the past decade are étale groupoids which give rise to a so-called Cartan pair, consisting of the groupoid C-algebra and a commutative C-subalgebra. Other examples include discrete groups acting on compact Hausdorff spaces, for which the reduced crossed product C-algebra contains the reduced group C-algebra as a C- subalgebra. I will talk about invariants and properties for such inclusions of C- algebras. This talk will be somewhat non-standard (for me, at least) and will only contain few (if even any) theorems, but should more be thought of as a conversation starter/brainstorm for what invariants one can associate to certain mathematical objects.
The category of Hilbert spaces and bounded linear functions forms the mathematical basis for quantum theory. But why? What physical principles enforce this mathematical structure? This category is also where C*-algebra theory lives, as it's the universal C*-category. But why? What properties does it have that accommodate this mathematical structure? As a first answer, this talk provides axioms that guarantee a category is equivalent to that of Hilbert spaces and bounded linear functions. The axioms are purely categorical and do not presuppose any analytical structure such as complex numbers, continuity, dimension, convexity, probabilities, etc. We will also discuss variations, such as linear contractions, finite-dimensional Hilbert spaces, and Hilbert C*-modules.
Let SO3(ℝ) be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure μ. Confirming a conjecture by Breuillard and Green, we show that if A is an open subset of SO3(ℝ) with sufficiently small measure, then μ(A2) > 3.99 μ(A).
Let L be a lattice in ℝn and let K be a convex body. The covering volume of L with respect to K is the minimal volume of a dilate rK, such that L+rK = ℝn, normalized by the covolume of L. Pairs (L,K) with small covering volume correspond to efficient coverings of space by translates of K, where the translates lie in a lattice. Finding upper bounds on the covering volume as the dimension n grows is a well studied problem in the so-called 'Geometry of Numbers', with connections to practical questions arising in computer science and electrical engineering. In a recent paper with Or Ordentlich (EE, Hebrew University) and Oded Regev (CS, NYU) we obtain substantial improvements to bounds of Rogers from the 1950s. In another recent paper, we obtain bounds on the minimal volume of nearly uniform covers (to be defined in the talk). The key to these results are recent breakthroughs by Dvir and others regarding the discrete Kakeya problem. I will give an overview of the questions and results.
The Zimmer programme asks how lattices in higher-rank semisimple Lie groups may act smoothly on compact manifolds. Below a certain critical dimension, the recent proof of the Zimmer conjecture by Brown-Fisher-Hurtado asserts that, for SLn(ℝ) with n ≥ 3 or other higher rank ℝ-split semisimple Lie groups, the action is trivial up to a finite group action. In this talk, we will explain what happens in the critical dimension for higher rank ℝ-split semisimple Lie groups. For example, non-trivial actions by lattices in SLn(ℝ), n ≥ 3, on (n-1)-dimensional manifolds are isomorphic to the standard action on ℝPn-1 up to a finite quotient group and a finite covering.
The last decade has witnessed a revolution in the circle of problems concerned with proving sharp moment inequalities for exponential sums on tori. This has in turn led to a better understanding of pointwise estimates, but this topic remains extremely challenging. One way to bridge the gap between global and pointwise behaviour is to study the restriction of exponential sums to submanifolds of the torus.
In my talk I will explore the behavior of lower dimensional Weyl sums and toral Laplace eigenfunctions restricted to hypersurfaces. The proofs will involve decoupling, a Fourier analytic tool that has recently found a broad range of applications in many areas of mathematics.
The classical Trudinger-Moser inequality is a borderline case of Sobolev inequalities and plays an important role in geometric analysis and PDEs in general. Aubin in 1979 showed that the best constant in the Trudinger-Moser inequality can be improved by reducing to one half if the functions are restricted to the complement of a three dimensional subspace of the Sobolev space H1, while Onofri in 1982 discovered an elegant optimal form of Trudinger-Moser inequality on sphere. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without mass center constraints.
One such inequality, for example, incorporates the mass center deviation (from the origin) into the optimal inequality of Aubin on the sphere, which is for functions with mass centered at the origin. The main ingredient leading to the above inequalities is a novel geometric inequality: Sphere Covering Inequality.
Efforts have also been made to show similar inequalities in higher dimensions. Among the preliminary results, we have improved Beckner's inequality for axially symmetric functions when the dimension n = 4, 6, 8. Many questions remain open.
The celebrated product theorem says if A is a generating subset of a finite simple group of Lie type G, then |AAA| ≫ min ( |A|1+c, |G| ). In this talk, I will show that a similar phenomenon appears in the continuous setting: If A is a subset of a compact simple Lie group G, then μ(AAA) > min ( (3+c)μ(A), 1 ), where μ is the normalized Haar measure on G. I will also talk about how to use this result to solve the Kemperman Inverse Problem, and discuss what will happen when G has high dimension or when G is non-compact.
The restriction conjecture, one of the most central problems in harmonic analysis, studies the Fourier transform of functions defined on curved surfaces; specifically, it claims that the level sets of such Fourier transforms are relatively small. The Mizohata-Takeuchi conjecture further studies the shape of these level sets, and in particular the extent to which they can avoid clustering on lines. In this talk we will present a small improvement on the Mizohata-Takeuchi conjecture. This is joint work with Anthony Carbery and Hong Wang.
The sup norm problem concerns the size of L2-normalized eigenfunctions of manifolds. In many situations, one expects to be able to improve upon the general bound following from local considerations. The pioneering result in that direction is due to Iwaniec and Sarnak, who in 1995 established an improvement upon the local bound for Hecke-Maass forms of large eigenvalue on the modular surface. Their method has since been extended and applied by many authors, notably to the 'level aspect' variant of the problem, where one varies the underlying manifold rather than the eigenvalue. Recently, Raphael Steiner introduced a new method for attacking the sup norm problem. I will describe joint work with Raphael Steiner and Ilya Khayutin in which we apply that method to improve upon the best known bounds in the level aspect.
It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=ZM of a finite cyclic group. Coven and Meyerowitz (1998) proposed a characterization of all finite tiles in terms of the cyclotomic divisors of associated mask polynomials, and proved it when the tiling period M has at most two distinct prime factors. In joint work with Itay Londner, we extended it to the case when M=(pqr)2, where p,q,r are distinct primes. The methods we developed can be applied to other questions that hinge on cyclotomic divisibility, ranging from number theory to harmonic analysis and geometric measure theory. In particular, Caleb Marshall and I were able to use cyclotomic divisibility methods to prove new Favard length estimates for product Cantor sets. The talk will provide an introduction to this group of problems.
In many recent works, analysis and number theory go beyond working side by side and team up in an interconnected back and forth interplay to become a powerful force. Here I describe two distinct meetings of the pair, which result in sharp counts for equilateral triangles in Euclidean space and statistics for how often a random polynomial has Galois group not isomorphic to the full symmetric group.
Morrey’s conjecture arose from a rather innocent-looking question in 1952: is there a local condition characterizing 'ellipticity' in the calculus of variations? Morrey was not able to answer the question, and indeed, it took 40 years until first progress was made with V. Sverak’s ingenious counterexample. Nevertheless, the case pertaining to planar maps remains open despite much progress, and has fascinated many through its interesting connections to complex analysis, geometric function theory, harmonic analysis, probability and martingales, differential inclusions and the geometry of matrix space. In the talk, I will give an overview of some of these connections and some of the recent progress.
In the absence of measures fully invariant with respect to a group action, this role can be to a certain extent played by the measures "invariant on average", with respect to a certain fixed distribution on the group. These measures are called stationary, and they naturally arise as harmonic measures of random walks. I will provide several partial answers to the general question about the dependence of harmonic measures on the underlying step distributions on the group and discuss counterexamples related to the Minkowski and Denjoy measure classes on the boundary of the classical modular group.
