Analysis
and Geometry Seminar
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Spring 2023
11 April 2023
Syed Husain:
Nyquist density thresholds for different Paley-Wiener spaces
Donoho and Logan used the L_1 reconstruction
method for perfect recovery of a noisy signal, where they
consider the 'window' to be a compact interval and the signals
are also bandlimited to a compact interval. In this talk, I
will discuss the general framework of the L_1 reconstruction
method. We will find concentration inequalities for a ball
window and a cube window where the signals are assumed to be
bandlimited to a cube. Consquently, we will find the Nyquist
density thresholds for the two problems and do a comparison
between the two thresholds using some approximations of Bessel
and Gamma functions. In the end, we will discuss a result for
signals that are bandlimited to a ball in dimension 2.
28 March 2023
Morgan O'Brien:
Factorization and Nikishin's Theorem (Part 2)
21 March 2023
Morgan O'Brien:
Factorization and Nikishin's Theorem
The study of L_p-spaces and the linear operators
on them is an important part of analysis. One aspect of this
is the factorization of operators between these spaces. For
example, Nikishin's Theorem states that any continuous linear
operator from an L_p-space to L_0 of a finite measure space
can be factored through a weak L_q-space; in other words, the
operator actually maps the L_p space to a weak L_q-space which
is then included in L_0 (though the measures may be
different). In this talk we will discuss this result, the
tools used to prove it, and some results that follow from it.
7 March 2023
Lane Morrison:
Handle Theory Basics (Part 2)
28 February 2023
Lane Morrison:
Handle Theory Basics
Handlebody Theory or Kirby Calculus is a
well-known tool for studying 4 manifolds. Every smooth
connected closed 4 manifold is diffeomorphic to a finite
gluing of disks. We call the disks handles and the gluing a
handlebody of the manifold. These handles can be manipulated
while still preserving the diffeomorphism type of the manifold
and give us simple objects to work with if we would like to
perform surgeries. I plan on defining what a handlebody is,
defining handle moves, looking at some examples, and then
applying these techniques.
21 February 2023
Pratyush Mishra:
Can groups be ordered like real numbers? (Part 2)
14 February 2023
Pratyush Mishra:
Can groups be ordered like real numbers?
Orderable groups (groups
admitting a translation-invariant left/right total ordering)
have attracted interests in group theory, dynamical systems, and
low-dimensional topology. While studying group actions on
manifolds, the first and the simplest case that comes to mind is
understanding groups acting on the real line. For countable
groups, left orderability is the same as admitting a faithful
action by homeomorphism on the real line.
Over the last few decades, there has been some substantial work
on this by many to understand if there are any interesting
actions of "big groups" on "small manifolds". In a series of
talks, we will try to understand this by stating some of the
related conjectures and then formulating it into the language of
left-orderable groups to better understand them. We will see a
nice proof of the fact that finite index subgroups of SL(n,Z)
for n > 2 do not have nice actions on circles and hence on
the real line (by D.W. Morris in 2008), which is a simple case
of a more general conjecture that lattices in SL(n,R) for n >
2 do not have faithful action on circles. A wide generalization
of this conjecture has been recently announced (by B. Deroin and
S. Hurtado in 2020).
7 February 2023
Azer Akhmedov: Girth
Alternative for Groups (Part 2)
31 January 2023
Azer Akhmedov: Girth
Alternative for Groups
The well-known Tits Alternative
(proved by Jacques Titis for finitely generated linear groups
over any field) is a property of a class of groups saying that
any group from this class is either virtually solvable or
contains a copy of a non-abelian free group. Tits Alternative
holds for many other classes of groups and (sometimes
interestingly) fails for some others. In my Ph.D. thesis, I
introduced the so-called Girth Alternative. The girth of a
finitely generated group is a positive integer or infinity. We
say that a class C of group satisfies Girth Alternative if any
group in C is either virtually solvable or has infinite girth.
Over the past 20 years, it has been proved or disproved (by me
and others) for various classes of groups. In one of the
recent joint works with Pratyush Mishra, we prove it for large
classes of HNN extensions and amalgamated free products.
I'll summarize the known results and list several open
questions.
24 January 2023
Seppi Dorfmeister:
Minimal Genus and Circle Sums
The minimal genus problem asks
what the minimal genus is of a connected, embedded surface S
representing the second homology class A in a 4-manifold
M.
One way to attempt to attack this problem is by constructing
submanifolds from known examples. A well-known technique
is the connected sum. Another is the circle sum, first
introduced by B. H. Li and T. J. Li.
I will describe the minimal genus problem, the two sum
techniques and try to highlight strengths and weaknesses of
each. Time permitting, I will describe how this is applied
in the case of the 4-torus.
Fall 2022
29 November 2022
Chase Reuter: Local
solutions to some uniqueness problems in convex geometry (Part
2)
15 November 2022
Chase Reuter:
Local
solutions to some uniqueness problems in convex geometry
Characterizing Euclidean spaces
was one of the goals of Busemann and Petty in the 1950's.
We will present several uniqueness problems that have been
solved locally, and survey the techniques used to obtain the
local solutions.
8
November 2022
Mariangel Alfonseca:
A
negative solution of Ulam's floating body problem (Part
2)
1
November 2022
Mariangel
Alfonseca: A negative solution of Ulam's floating body
problem
Problem 19 in the Scottish Book
was posed by Ulam and asks if a convex body of constant density
which floats in equilibrium in any orientation must be an
Euclidean ball. I will present the main ideas of the recent
counterexample by Ryabogin.
18 October 2022
Michael Preheim: Student Confidence and Certainty in
Comprehension of a Proof by Induction
Researchers typically utilize response correctness to
interpret student proficiency in proof comprehension. However,
student metacognition offers important information about
performance behavior but has not been simultaneously analyzed
alongside correctness to determine student competency in proof
comprehension. The primary objective of this study is to
investigate the accuracy of student confidence and certainty
levels at local and holistic aspects of proof comprehension
regarding a proof by induction. Students were given a
three-factor proof comprehension assessment at the beginning and
end of an undergraduate transition-to-proof course that
collected student confidence, correctness, and certainty at each
tier of an established proof comprehension framework. Results of
this study highlight a critical distinction between high and low
performers metacognition throughout the host course.
Additionally, one outlying assessment item especially
illuminates additional considerations for future application of
metacognition in proof comprehension research.
Spring
2022
3
May 2022
Michael Roysdon (Tel
Aviv University): Measure theoretic
inequalities and projection bodies
This talk will detail two recent papers
concerning the
Rogers-Shephard
difference
body
inequality and
Zhang's
inequality for
various
classes of
measures. The
covariogram of
of a convex
body w.r.t. a
measure plays
an essential
role in the
proofs of each
of these
inequalities.
In particular,
we will
discuss a
variational
formula
concerning the
covariogram
resulting in a
measure
theoretic
version of the
projection
body operator.
If time
permits, we
will discuss
how these
results imply
some reverse
isoperimetric
inequalities.
Joint work
with: 1) D.
Alonso-G, M.
H. Cifre, J.
Yepes-N, and
A. Zvavitch
and 2) D.
Langharst and
A. Zvavitch.
26
April 2022
Adam Erickson: A
Rokhlin lemma for noninvertible totally-ordered
measure-preserving dynamical systems
Suppose (X,ℱ(<),μ,T) is a non-invertible
MPDS with σ-algebra ℱ generated by the intervals
in the total order <. In this second of two
talks, we describe a paper by the speaker,
recently submitted to Real Analysis Exchange,
proving that the Rokhlin lemma applies to such a
system if a strengthened form of aperiodicity is
assumed, utilizing an adaptation of the technique
introduced by Heinemann and Schmitt to prove
Rohklin's lemma for non-invertible
measure-preserving systems on separable nonatomic
aperiodic spaces.
19
April 2022
Adam Erickson: A
survey of the history of the Rokhlin lemma and
its variants
The Rokhlin lemma tells us that many
measure-preserving
dynamical
systems
(X,ℱ,μ,T) can
"almost" be
viewed as a
disjoint union
of preimages
of some E ∈ ℱ.
Beyond its
original use
in the theory
of generators,
this idea and
its variants
turns out to
have
tremendous
utility
throughout
ergodic
theory. In
this first
talk of two,
we will
introduce the
many victories
of the Rokhlin
lemma in
constructing
examples and
proving a wide
variety of
ergodic
theorems.
12
April 2022
Pratyush Mishra: Girth
alternative for HNN extension (Part 2)
5
April 2022
Pratyush Mishra: Girth
alternative for HNN extension
The notion of a girth was first introduced by S.
Schleimer in
2003. Later, a
substantial
amount of work
on the girth
of finitely
generated
groups was
done by A.
Akhmedov,
where he
introduced the
so-called
Girth
Alternative
and proved it
for certain
classes of
groups, e.g.
hyperbolic,
linear,
one-relator,
PL_+(I) etc.
Girth
Alternative is
like the
well-known
Tits
Alternative in
spirit;
therefore, it
is natural to
study it for
classes of
groups for
which Tits
Alternative
has been
investigated.
In this talk,
we will
explore the
girth of HNN
extensions of
finitely
generated
groups in its
broadest sense
by considering
cases where
the underlying
subgroups are
either full or
proper
subgroups. We
will present a
sub-class for
which Girth
Alternative
holds. We will
also produce
counterexamples
to show that
beyond our
class, the
alternative
fails in
general. The
talk will be
based on joint
work with Azer
Akhmedov.
29
March 2022
Morgan O'Brien: Dilations
of contractions on Hilbert and L_p-spaces (Part
2)
22 March 2022
Morgan O'Brien: Dilations
of contractions on Hilbert and L_p-spaces
Contractions are a very common and one of the
most basic
types of
bounded
operators on a
Banach space,
since they are
exactly those
operators in
the unit ball
of all bounded
operators on
the space.
Though
sometimes this
may be enough
for some
purposes,
unsurprisingly
one frequently
imposes more
conditions on
the operators
they use to do
things. For
example,
unitary
operators on a
Hilbert space
are also
contractions,
but they are
also
invertible
operators
whose adjoint
is their
inverse, and
so the methods
of spectral
theory may be
applied to
them. Although
unitary
operators seem
much more
restrictive to
work with, the
two types of
operators are
surprisingly a
lot more
related than
one would
think.
In the first
part of this
talk, we will
discuss a
dilation
theorem for
arbitrary
contractions
on a Hilbert
space that
allows one to
study a
contraction by
extending it
to a unitary
operator on a
larger Hilbert
space. This
will then be
used to prove
some mean
weighted
ergodic
theorems for
such
operators. In
the second
part, we will
a similar type
of result that
extends
positive
contractions
on L_p-spaces
to positive
isometries on
larger
L_p-spaces,
and discuss
how this can
be used to
prove some
pointwise
ergodic
theorems.
8 March 2022
Azer Akhmedov: The
geometry of Banach spaces (Part 3)
1 March 2022
Azer Akhmedov: The
geometry of Banach spaces (Part 2)
15 February 2022
Azer Akhmedov: The
geometry of Banach spaces
This will be a series of 2 (or 3) talks
aimed at a general audience. In the first talk,
I'll review some classical results from theory of
Banach spaces. We will take a (perhaps) somewhat
non-traditional view, concentrating on the study
of topology of Banach spaces. In the second talk,
I'll discuss recent results on the geometry of
Banach spaces.
Fall
2021