Analysis Seminar
Unless specified, all seminars are
Wednesday 4-5pm at Room 250.
January 22
Guo
Chuan Thiang, Peking University
Exact fractional quantization and topological insulators
Abstract: The spectral problem for the magnetic Laplacian
on the plane with uniform magnetic field was solved almost 100
years ago, with the
lowest eigenspace being the Fock space. As this Fock space is also
realized as the kernel of an associated Dirac operator, it
possesses a certain
index-theoretic stability. I will explain the geometric-analytic
meaning of this index, and its significance in the concepts of
exact quantization and
topological insulators in modern physics. A tract formula for this
index is available, and corresponds to the famous
integer-quantized Hall
conductance discovered in experiments in the 1980s. Furthermore,
guided by the fractional quantum Hall effect, we find that Fock
space has a
hidden rational trace structure.
February
5
Danjun Zhao, Jiaxing University
Roots and logarithms of multipliers
Abstract: By deriving a differentiation formula, we show
that if a function is a non-vanishing multiplier for the
Drury-Arveson space, then its
arbitrary real power and logarithm can also serve as multipliers
under specific conditions. Moreover, by this formula the same
result holds for
spaces Hm,s of the Besov-Dirichlet type. This is a
joint work with Jingbo Xia, Congquan Yan and Jingming Zhu.
March
5
Ian Thompson, University of
Copenhagen
Residually finite-dimensional operator algebras:
universal and co-universal representations
Abstract: Beginning in the 60s, Arveson wrote
a massively influential series of papers on
subalgebras of C*-algebras. A major thrust of these
papers
came from the fact that the choice of representation
for a subalgebra plays a significant role in
deciphering its structure. Quite recently, a major
trend has been to understand which subalgebras admit
a residually finite-dimensional representation, and
to uncover their structural properties. In
opposition to the C*-algebraic setting, residually
finite-dimensional operator algebras have proven to
form a muh more flexible class of operator
algebras. Here, we will discuss structural
properties on the space of all residually
finite-dimensional representations for a fixed
operator algebra.
April
2
Yusheng Luo, Cornell University
Uniformization of gasket Julia set
Abstract: The quasiconformal uniformization
problem for fractal sets is a classical question
that has seen significant recent progress. In the
1970s,
Ahlfors provided a geometric characterization of
when a Jordan curver can be quasiconformally
uniformized to a round circle. A closely related
question--whcn a Sierpinski carpet can be
quasiconformally mapped to a round carpet--has been
extensively studied in both geometric and
dynamical setting, with key contributions from
McMullen, Bonk, and Bonk-Lyubich-Merenkov.
In contrast, the problem of determining when a
gasket can be quasiconformally mapped to a circle
packing is more subtle. In this talk, I
will discuss recent joint work with D. Ntalampekos
that provides a chracterization of when a gasket
Julia set is quasiconformally equivalent to a
circle packing. The proof builds on new results from
some joint work with Y. Zhang on renormalization
theory for circle packings.
April 9
Hanfeng Li, SUNY at Buffalo
Local entropy theory, combinatorics, and local
theory of Banach spaces
Asbtract: In 1995 Glasner and Weiss showed
that if a continuous action of a countably infinite
amenable group on a compact metrizable space X
has zero entropy, then so does the induced action on
the space of Borel probability measures on X. I will
discuss a strengthening of the
Glasner-Weiss result, in the framework of local
entropy theory, based on a new combinatorial lemma.
I will also present an application of the
combinatorial lemma to the local theory of Banach
spaces. This is joint work with Kairan Liu.
April
16
Ryo Toyota, Texas A&M University
Expanders, geometric property (T), and warped cones
Abstract: Warped cones are metric spaces
associated with dynamical systems, where their
large-scale geometric properties reflect the
dynamical
properties of the underlying actions. In this talk,
we discuss a large-scale invariant, called geometric
property (T), for warped cones and show that
if an action is ergodic, free, measure preserving
and isometric on a Riemannian manifold, then the
associated warped cone does not possess
geometric property (T). This result negatively
answers an open problem: whether the warped cone of
an ergodic action by a group with property (T)
possesses geometric property (T), and gives new
examples of (super-)expanders without geometric
property (T). This is based on a joint work with
Jintao Deng.
April
23
Xiaoqing Li, SUNY at
Buffalo
Lower bounds of the Riemann zeta function on the
line 1 and GL(3)
Abstract: In this talk, we will present a
soft method deriving effective lower bounds for the
Riemann zeta function on Re(s)=1, using the theory
of
GL(3) Eisenstein series.
Past
Analysis Seminar