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

August 9th Song Shao, University of Science and Technology of China

(Monday) Regionally proximal relation of order d and the maximal d-step nilfactors

Abstract: 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

that the factor induced by RP

A combinatorial consequence is that if S is a dynamically syndetic subset of Z, then for each d≥ 1,

{(n

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

(4:30pm)

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 C

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^{*}

Abstract: 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.

Extreme Points of Integral Families of Analytic Functions

Abstract

Past Analysis Seminars