Analysis Seminar

Using Bombieri-Iwaniec method, we establish an improvement on both Gauss's Circle Problem and Dirichlet's Divisor Problem. More precisely, we derive a new estimate for the first spacing problem and combine it with Huxley's results on the second spacing problem. In this talk, I will talk about several methods to deal with exponential sums on the integer ring, with focus on the Bombieri-Iwaniec method. After that, I will talk about several results in decoupling theory as well as some rough ideas in the proofs.

Quantum complexity theory is concerned with the amount of elementary quantum resources needed to build a quantum system or a quantum operation. The fundamental question in quantum complexity is to define and quantify suitable complexity measures. In this talk, combining the approach introduced by Li-Bu-Koh-Jaffe-Lloyd and well-established tools from noncommutative geometry, we propose a unified framework for resource-dependent complexity measures of general quantum channels, also known as Lipschitz complexity. This framework is suitable to study the complexity of both open and closed quantum systems. The central class of examples in this paper is the so-called Wasserstein complexity. We use geometric methods to provide upper and lower bounds on this class of complexity measures. Finally, we study the Lipschitz complexity of random quantum circuits and dynamics of open quantum systems in finite dimensional setting. In particular, we show that generically the complexity grows linearly in time before the return time. This is the same qualitative behavior conjecture by Brown and Susskind . We also provide an infinite dimensional example where linear growth does not hold.

Let G be a countable discrete group. In this talk, we are interested in the problem of completely determining G-invariant von Neumann subalgebras in L(G). If every G-invariant von Neumann subalgebra M in L(G) is of the form M=L(H) for some normal subgroup H of G, then we call this group G has the invariant von Neumann subalgebras rigidity (ISR) property. We discuss the problem of determining which group G has this ISR property and other related recent work on it. Part of this talk is based on joint work with Dr. Tattwamasi Amrutam. 

We discuss the factoriality, non-injectivity, and fullness question of q-Araki-Woods von Neumann algebras, constructed by Hiai. These algebras are non-tracial coun-terparts of q-Gaussian algebras, which combine the q-deformations of free group fac-tors due to Bozejko-Kummerer-Spiecher, and quasi-free deformations of Shlyakht-enko. We will discuss our approach using conjugate/dual variables in order to solve the mentioned problems. This is joint work with Adam Skalski and Mateusz Wasilewski.

In the 1980s, David and Journé proved the original $\mathrm{<span\; style="font-size:12px;">T</span>}\mathrm{<span\; style="font-size:12px;">1</span>}$ theorem, which gave a characterization of the ${\mathrm{<span\; style="font-size:12px;">L</span>}}^{\mathrm{<span\; style="font-size:12px;">2</span>}}$ boundedness for Calderón–Zygmund operators based on testing conditions. Soon after, using the vector-valued Calderón–Zygmund theory, Journé first established a bi-parameter $\mathrm{<span\; style="font-size:12px;">T</span>}\mathrm{<span\; style="font-size:12px;">1</span>}$ theorem. However, the $\mathrm{<span\; style="font-size:12px;">T</span>}\mathrm{<span\; style="font-size:12px;">1</span>}$ theorem to deduce compactness of multi-parameter singular integrals has been open in the last several years. The purpose of this talk is to develop a general theory of the compactness for a class of bi-parameter singular integrals.

Semigroup C*-algebras provide abundant examples in C*-algebras. For semigroups that are far from groups, we encounter completely new phenomena which are not visible in the group case. In the talk, we will introduce a special class of C*-algebras, namely semigroup C*-algebras arising from graphs of monoids. We will analyse their structral properties, including closed invariant subsets and topological freeness of the groupoids as well as simplicity, nuclearity, pure infiniteness and K-theory of the semigroup C*-algebras. Finally, we will give a classification of boundary quotients of the semigroup C*-algebras.

One of the most important questions in Banach space geometry is when does a Ba-nach space embed into another Banach space isometrically or almost isometrically.In this talk, we discuss some results regarding existence of isometric embeddings of S_m^q into S_n^p where S_n^p is the usual Schatten-p class over n × n complex matrices. We also present some results when p and q are also allowed to be less than l. We will explain several new ingredients related to perturbation theory of linear operators, namely Kato-Rellich theorem, theory of multiple operator integrals,norm-parallelism and Birkhoff-James orthogonality, followed by thorough and careful case by case analysis, which are essential to our work. This talk is based on work done in collaboration with Arup Chattopadhya, Guixiang Hong, Avijit Pal and Chandan Pradhan.

The natural parallels between the four big geometric disciplines in mathematics,

- algebraic geometry,

- complex geometry,

- differential geometry,

- topology,

can be explained by the fact that these disciplines appear as a visual image when studying the very same reality with the help of different observation tools.

A construction that formalizes this idea is called an envelope. This is a special kind of functor in category theory, generated by a class of morphisms chosen as the observation tools. In the case of the mentioned geometric disciplines, the common reality they reflect is the theory of topological (more precisely, stereotype) algebras, and the functors that transform this reality into the last three disciplines are respectively

- the holomorphic envelope, where the observation tools are the homomorphisms into Banach algebras,

- the smooth envelope, where the observation tools are the so called differential homomorphisms into C*-algebras with the joined self-adjoint nilpotent elements, and 

- the continuous envelope, where the observation tools are the homomorphisms into C*-algebras. Each of these functors generates, apart from the corresponding geometric discipline itself, a special kind of duality in it, which is called stereotype duality, and which generalizes the famous Pontryagin duality for locally compact Abelian groups (to some class of not necessarily commutative groups).

This leads to an intriguing picture, where it becomes possible to compare these geometries as disciplines, to find common features, differences, generalizations, new examples, and so on. In my talk I'm going to give accurate definitions and discuss some details of this picture.

The Novikov conjecture is a prominant problem in differential topology. The operator K-theoretic approach provided by noncommutative geometry has yielded some of the best results that verify this conjecture for vast classes of groups. A natural class of groups for which the conjecture remains largely mysterious is that of countable groups of diffeomorphisms on smooth manifolds. In an upcoming joint paper with Sherry Gong, Zhizhang Xie, and Guoliang Yu, we prove that the (rational strong) Novikov conjecture holds for geometrically discrete countable subgroups of the group of diffeomorphisms of any closed smooth manifold. This removes the volume-preserving condition in a previous joint paper with S. Gong and G. Yu.           

In this talk, I will introduce the notion of principal symbol. Firstly, in the archaic context of pseudo-differential operators, then in the general C*-algebraic context. Examples to be provided are (a) rather old ones, such as Euclidean spaces and tori (and their non-commutative analogues) and (b) very new ones, such as stratified groups (in particular, Heisenberg groups). I will also demonstrate how does the principal symbol mapping behave under the natural action of diffeomorphisms (in the Euclidean case) and Heisenberg diffeomorphisms (in the Heisenberg group case). This is needed to extend the notion of the principal symbol to the manifolds (from Euclidean case) and to the contact manifolds (from the Heisenberg case). Corresponding globalization theorems will be provided. As an application, I will present the Connes Trace Theorem (including, in particular, the spectrally correct sub-Riemannian volume).

Topological full groups, as an algebraic invariant, were introduced to study continuous orbit equivalence relations by Giordano, Putnam, and Skau. Then, these groups have been found applications to geometric group theory by providing interesting examples with certain properties such as simplicity, soficity, amenability, and LEF-ness. In this talk, we will show new methods of establishing the soficity and LEF-ness for topological full groups. Moreover, we will explain how can one obtain amenability from the sofic approximations when the acting group is amenable and the action is free and distal.

L_p-Fourier multipliers form a central tool in harmonic analysis on Euclidean spaces as well as the torus. In this talk we are concerned with natural generalisations of such multipliers on non-abelian groups and focus on the special case of SL(n,R). We show how to construct such L_p-Fourier multipliers by requiring sufficient smoothness of the symbols in the spirit of Hormander-Mikhlin and Calderon-Torchinsky. Our methods in particular give new results if p approximates 2. If time permits I will also report on versions of the De Leeuw restriction theorem for  SL(n,R) and  SL(n,Z). The latter is joint work with Bas Janssens, Amudhan Krishnaswamy-Usha and Lukas Miaskiwskyi.

In first part of the talk we discuss basic examples and properties of orthogonally additive operators in vector and Banach lattices emphasizing on connections with other areas of the modern mathematics. In the second part we present some open problems concerning order and topological properties of orthogonally additive operators in Banach and vector lattices.  

Let f be an unknown function on the n-dimensional discrete hypercube. How many values of f do we need in order to approximately reconstruct the function? In this talk, we shall discuss the random query model for this fundamental problem from computational learning theory. We will explain a newly discovered connection with a family of polynomial inequalities going back to Littlewood (1930) which will in turn allow us to derive sharper estimates for the query complexity of this model, exponentially improving those which follow from the classical Low-Degree Algorithm of Linial, Mansour and Nisan (1989), while maintaining a running time of the same order. Time permitting, we will also show a matching information-theoretic lower bound. Based on joint works with Paata Ivanisvili (UC Irvine) and Lauritz Streck (Cambridge).