Analysis Seminar
Unless specified, all seminars are
Wednesday 4-5pm at 250 Math Building.
August
28th
Raffael Hagger, Reading
University
Fredholm theory and localization on metric measure
spaces
Abstract
September
25th
Mariusz Tobolski, Institute of
Mathematics Polish Academy of Sciences
Towards the classification of locally trivial
noncommutative principal bundles
Abstract: A classical result from topology
states that if X is a principal G-bundle over a
paracompact space M, then there exists a map from M
to the
classifying space BG, and all isomorphic principal
G-bundles are classified by the homotopy class of
such a map. The aim of this talk is to find an
analog of this result in the realm of noncommutative
topology, where instead of topological spaces and
groups, we consider C*-algebras and
quantum groups
respectively. First, we introduce the notion of a
locally trivial noncommutative principal bundle in
the setting of compact quantum
group actions on C*-algebras. Then, for a compact
quantum group G, we define the C*-algebra of
functions on the noncommutative classifying
space C(BG) and prove that it classifies all locally
trivial noncommutative principal G-bundles.
October
2nd
Yi Wang, SUNY at
Buffalo
A
sharp inequality of Hardy-Littlewood type via
derivatives
Abstract: In this talk, I will introduce a
sharp inequality relating a parameterized set of
weighted Bergman norms and the Hardy norm on the
unit disk.
The original form of this inequality can be traced
back to 1921, when Carleman provided a complex
analytic proof of the famous isoperimetric
theorem.
In recently years, the inequality has
regained attention because of its application in
number theory. By taking a close examination of
the derivatives of
the norms with respect to the parameter, we obtain
some sufficient conditions for the inequality to
hold. This is joint work with Hui Dan and Kunyu
Guo.
October
9th Xiaocheng
Li, University of Wisconsin-Madison
An Estimate for Spherical Functions on SL(3,R)
Abstract: We prove an estimate for spherical
functions \phi_\lambda(a) on SL(3,R), establishing
uniform decay in the spectral parameter \lambda
when the group parameter a is restricted to a
compact subset of the abelian subgroup A. In the
case of SL(3,R), it improves a result by
J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan
by removing the limitation that a should remain
regular. As in their work, we estimate the
oscillatory integral that appears in the integral
formula for spherical functions by the method of
stationary phase. However, the major difference
is that we investigate the stability of the
singularities arising from the linearized phase
function by classifying their local normal forms
when the
parameters \lambda and a vary.
October
16th Quanlei
Fang, City University of New
York
Revisiting Arveson's Dirac operator of a commuting
tuple
Abstract: About twenty years ago, Arveson
introduced an abstract Dirac operator based on
Taylor spectrum and functional calculus. He showed
that
every Dirac
operator is associated with a commuting tuple. The
Dirac operator of a commuting tuple has inspired
several interesting problems in
multivariable operator theory. In this talk, we will
revisit the Dirac operator and discuss some related
problems.
October
23rd
Han Li, Wesleyan University
Masser's
conjecture on equivalence of integral quadratic
forms
Abstract: A classical problem in the theory
of quadratic forms is to decide whether two given
integral quadratic forms are equivalent. Formulated
in
terms of matrices the problem asks, for given
symmetric n-by-n integral matrices A and B, whether
there is a unimodular integral matrix X satisfying
A=X'BX, where X' is the transpose of X. For definite
forms one can construct a simple decision procedure.
Somewhat surprisingly, no such procedure
was known for indefinite
forms until the work of C. L. Siegel in the early
1970s. In the late 1990s D. W. Masser conjectured
for n at least 3, there exists
a polynomial search bound for X in terms of the
heights of A and B. In this talk we shall discuss
our recent resolution of this problem based on a
joint
work with Professor Gregory A. Margulis, and explain
how ergodic theory is used to understand integral
quadratic forms.
October
30th
Chunlan Jiang, Hebei Normal University
Similarity invariants of essentially normal
Cowen-Douglas operators and Chern polynomials
Abstract: In this talk, I will discuss our
resent work on a class of essentially normal
operators by using the geometry method from the
Cowen-Douglas
theory and a Brown-Douglas-Fillmore theorem in the
Cowen-Douglas theory. More precisely, the Chern
polynomials and the second fundamental
forms are the similarity invariants (in the sense of
Herrero) of this class of essentially normal
operators.
November
6th
Alexandru Chirvasitu, SUNY at Buffalo
Loosely
embeddable metric spaces
Abstract: Embedding finite metric spaces
isometrically into Hilbert spaces has elicited some
interest outside of pure mathematics due to
applications
to fields like
computer vision, machine learning, the structure of
networks and other such areas.
In the talk I will introduce a weaker notion of
embeddability motivated by the study of "quantum
symmetries" for metric spaces and Riemannian
manifolds. I will mention some results on the
generic behavior of "most" compact metric spaces and
list a number of open questions.
November
20th
Hui Dan, Fudan
University
The cyclicity of composite functions in the Hardy
space over the infinite polydisc
Abstract: The cyclic vector problem on the
Hardy space over the infinite polydisc is the
analytic function space version of Wintner and
Beulring's
periodic dilation completeness problem. In this
talk, I will mainly concentrate on characterizing
cyclic vectors in terms of composition operators.
Also, in order to study composition operators,
dilation theory for doubly commuting sequence of C.0
contractions is involved.
Past
Analysis Seminar