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
(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[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
(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
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*
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.
December 1st
Keiko Dow, Canisius
College
Extreme Points of Integral Families of Analytic Functions
Abstract
Past Analysis Seminars