Analysis   Seminar    

Unless specified, all seminars are Wednesday 4-5pm at  250 Math Building.                      

August 9th                                  Song Shao,   University of Science and Technology of China
Regionally proximal relation of order d and the maximal d-step nilfactors

By proving the minimality of face transformations acting on the diagonal points and searching the points allowed in the minimal sets, it is shown that the regionally                
                                                  proximal relation of orderd, RP[d], is an equivalence relation for minimal systems. Moreover, the lifting of RP[d] between two minimal systems is obtained, which implies
                                                  that the factor induced by RP[d] is the maximal d-step nilfactor. The above results extend the same conclusions proved by Host, Kra and Maass for minimal distal systems.

A combinatorial consequence is that if S is a dynamically syndetic subset of Z, then for each d≥ 1,
                                                                          {(n1,...,nd) ∈ Zd: n1ε1+... +ndεd∈ S, εi∈ {0,1}, 1≤ i≤ d}
                                                  is syndetic. In some sense this is the topological correspondence of the result obtained by Host and Kra for positive upper Banach density subsets using ergodic methods.

September 1st                            Organizational Meeting     
September 22nd                       Hanfeng Li,   SUNY at Buffalo
                                                Entropy for actions of sofic groups,  Part I

                                                Abstract: Classically entropy is defined for measure-preserving actions and continuous actions of countable amenable groups. The class of sofic groups includes all discrete
                                                amenable groups and residually finite groups. In 2008 Lewis Bowen defined entropy for measure-preserving actions of countable sofic groups, under the condition that the
                                                underlying space has generating partitions with finite entropy. I will give a definition of entropy for all measure-preserving actions and continuous actions of countable sofic groups,
                                                and discuss some properties of this entropy. Although the definition is in the language of dynamical systems, the proof for the well-definedness uses operator algebras in a fundamental
                                                way. This is joint work with David Kerr. 

September 29th                        Hanfeng Li,   SUNY at Buffalo
                                                Entropy for actions of sofic groups,  Part II

October 13th                           Leo Goldmakher,   University of Toronto                                                  
Sharp bounds on cubic character sums   

Abstract: A celebrated result of Halasz characterizes the multiplicative functions taking values in the complex unit disc which have non-zero mean value; recent work of Granville
                                               and Soundararajan characterizes the Dirichlet characters which have large character sums. I'll describe how one can prove a hybrid of these two theorems, and show how this leads
                                               to refinements of character sum bounds of Granville and Soundararajan. In particular, on the assumption of the Generalized Riemann Hypothesis the method yields a sharp bound on
                                               cubic character sums.
October 20th                           Byung Jay Kahng,    Canisius College
A (2n+1)-dimensional quantum group constructed from a skew-symmetric matrix

Abstract: Poisson-Lie groups are Lie groups equipped with compatible Poisson structure. They are natural candidates to perform quantization, to obtain quantum groups. In this talk,
                                                we will first discuss how some Poisson brackets arise from solutions to the classical Yang-Baxter equation (CYBE), which are often called "classical r-matrices". We will give
                                                some examples, and in particular, we will show that a certain non-linear Poisson bracket on a  (2n+1)-dimensional solvable Lie group G can be constructed from a classical r-matrix.

The Poisson bracket constructed in this way may be viewed as a cocycle perturbation of the linear Poisson bracket. From this data, we can construct a (cocycle) twisted crossed product
                                                C*-algebra that is a deformation quantization of C0(G). We will briefly indicate how to give suitable quantum group structure on the C*-algebra, which would be an example of a locally
                                                compact quantum group.

October 27th                           Peng Zhao,    Princeton University
Quantum Variance of Maass-Hecke Cusp Forms

Abstract: We discuss the quantum variance for the modular surface X. We asymptotically evaluate the quantum variance, which is introduced by Zelditch and describes the fluctuations
                                                of a quantum observable. Our approach is via Poincare series and Kuznetsov trace formula. It turns out that the quantum variance is equal to the classical variance of the geodesic flow
                                                on the unit tangent bundle of X, but twisted by the central value of the L-function associated with the Maass-Hecke form. If time permits, I will introduce some recent progress with Peter
                                                Sarnak about the quantum variance on phase space.

November 3rd                         David Larson,    Texas A&M University
Operator-valued measures, dilations, and the theory of frames

                                                Abstract:  We show that there are some natural associations between the theory of frames (including continuous frames and framings), the theory of operator-valued measures on
                                                sigma-algebras of sets, and the theory of normal linear mappings between von Neumann algebras. In this connection frame theory itself is identified with the special case in which the domain
                                                algebra for the mapping is commutative. Some of the more important results and  proofs for mappings in this case extend naturally to the case where the domain algebra is non-commutative. 
                                                This happens frequently enough, and in profound enough ways, to justify defining a noncommutative frame to be an arbitrary ultraweakly continuous linear mapping between von Neumann
                                                algebras. It has been known for a long time that a sufficient condition for a unital bounded linear map between C*-algebras to have a Hilbert space dilation to a bounded homomorphism is
                                                that the mapping is that the map be completely bounded.  Our theory shows that under suitable hypotheses even if it is not completely bounded it still has a Banach space dilation to a
                                                homomorphism, and the Banach space can be rather nice.  We view this as a generalization of the known result that arbitrary framings have Banach dilations.
November 10th                        Yonatan Gutman,   
Université Paris-Est Marne-la-Vallée

                                                Minimal Actions of Homeo(ω*) on Hyperspaces of  ω*

Let ω*=βω\ω, where βω denotes the Stone-Cech compactification of the natural numbers. This space, called the corona or the remainder of ω, has been extensively studied in  

                                                the fields of set theory and topology. Following an earlier work of Glasner and Weiss we first identify the universal minimal dynamical system of the group G=Homeo(ω*) as the sub-system

                                                of ''maximal chains'' in the hyperspace Exp(Exp(ω*)). Here Exp(Z) stands for the hyperspace comprising the closed subsets of the compact space Z, equipped with the Vietoris topology.

                                                Using the dual Ramsey theorem and a detailed combinatorial analysis of what we call stable collections of subsets of a finite set, we obtain a complete list of the minimal sub-systems of the

                                                compact dynamical system (Exp(Exp(ω*)),G). The importance of this dynamical system stems from Uspenskij's characterization of the universal ambit of G. These results apply as well to the

                                                Polish group Homeo(C), where C is the Cantor set. Joint work with Eli Glasner.

November 17th                       Piotr Nowak,    Texas A&M University
Exact groups and bounded cohomology

                                               Abstract: Exactness is a very weak counterpart of amenability for groups. It is equivalent to Yu's property A and to the existence of a topologically amenable action of the group on
                                               some compact space. Higson asked whether exactness admits a homological or cohomological characterization, similar to the ones amenable groups admit. In this talk we will give an
                                               answer to Higson's question by characterizing exact groups via vanishing of bounded cohomology (or, equivalently, of the Hochschild cohomology of the convolution algebra). This
                                               provides a vast generalization of the classical result of B.E.Johnson proved in the early 70's for amenable groups. 

December 1st                          Keiko Dow,  Canisius College
Extreme Points of Integral Families of Analytic Functions

Past Analysis Seminars