Analysis and Geometry Seminar

**Link to Previous Semesters**

**Spring 2021**

**
**Tuesdays, 11:00 am

On zoom (email maria.alfonseca at ndsu dot edu for the
link)

**27 April 2021**

Doğan Çömez*:* Recurrence theorems and universally good
sequences via superadditive processes

**Abstract: **Poincaré Recurrence Theorem (PRT), proved in
1893, Birkhoff Ergodic Theorem, proved in 1931, and
Wiener-Wintner Ergodic Theorem (WW-ET), proved in 1941, are
three main pillars on which modern ergodic theory stands.
Over the years all these results have been generalized to
various settings; furthermore, each paved way to new avenues
of research within ergodic theory as well as in other
fields. In this talk the focus will be on a class of
superadditive processes, whose averages are known to
converge a.e., thereby providing a generalization of
Birkhoff Ergodic Theorem. Particular attention will be on
obtaining some recurrence theorems for such processes
(connection to PRT), and study universally good sequences
generated by them (connection with WW-ET).

**Video
of the talk**

**20 April 2021**

Adam Buskirk*:* The Fourier transform of probability
measures on non-abelian groups and shuffling

**Abstract: **The process of shuffling is commonly modeled as
a random walk on the symmetric group $S_K$. In this talk, we
explain a technique introduced by Diaconis derived from
group representation theory using a version of the Fourier
transform to compute the probabilities associated with a
random walk after $n$ steps. We introduce an upper bound
lemma derived by Diaconis and Shahshahani, and use this to
obtain bounds on how rapidly the probabilities associated
with a random walk will converge to uniformity.

**Video
of the talk**

**13 April 2021**

Pratyush
Mishra*:* Groups of piecewise linear
homeomorphisms of the interval [0,1]. Talk 2.

**Abstract: **In this talk, we will continue studying groups
of PL homeomorphisms of the interval. We will mainly
focus on the study of subgroup structures of these groups.
The subgroup properties we are interested in have been well
understood for linear groups. For the PL groups of
homeomorphisms, we will see their story diverges from the
one of the linear groups in a major way, yet we will present
results which can be viewed as analogs of the results in the
case of linear groups.

**Video
of the talk **

**6 April 2021**

Pratyush
Mishra*:* Groups of piecewise linear
homeomorphisms of the interval [0,1]. Talk I: A Survey

Abstract: Group actions on manifolds have attracted
people from different fields of mathematics such as group
theory, low dimensional topology, dynamical systems, and
differential geometry. Among group action on 1-manifolds, in
this talk we will be looking at the group of piecewise linear
homeomorphisms of the closed interval I= [0,1] (denoted
PL_+(I)). We will study this infinite-dimensional group using
algebraic and dynamical methods. By the end of the first talk,
we will see a complete classification of solvable and
non-solvable subgroups of PL_+(I) as appeared in the PhD thesis
of Collin Bleak in 2005.

**Video of the talk**

** **** ** ** **

**23 March 2021**

*Jimmy
Thorne:* Finite Oscillation Stability

Abstract: This talk will explore the basics of the
Ramsey-Dvoretzky-Milman phenomenon, which is also known as
finitely oscillation stable. Intuitively this phenomenon is
the observation that uniformly continuous functions on high
dimension structure are close to constant on all but a vanishing
small measure set. This talk will start with a few basics of
uniform spaces and formally define finitely oscillation
stability. We

then move on to give examples of spaces that exhibit this
phenomenon, such as, S^\infty, S_\infty (infinite symmetric
group), and extremely amenable groups.

Video of the talk

** **

**9 March 2021**

*Lane
Morrison:* Description of Morse Homology

Abstract: Morse theory is the study of critical points of
real valued functions to understand the topology of smooth
manifolds. If V is a compact smooth manifold and f is a "nice
enough" smooth real valued function on V, then one can show V
has the homotopy type of a CW-complex where each cell
corresponds to a critical point of f. From this decomposition we
can construct the Morse homology of V and use this construction
to study the topology of V. In this talk we will define "nice
enough" and give an overview of the construction of this
homology.

**Video
of the talk**

** **** ** ** **

**2 March 2021**

Chase Reuter: **The Fifth and Eighth
Busemann-Petty Problems (Part 3)**

Video
of the talk

** ** ** **

**23 February 2021**

Chase Reuter: **The Fifth and Eighth
Busemann-Petty Problems (Part 2)**

Video
of the talk

** **

**16 February 2021**

Chase Reuter: **The Fifth and Eighth
Busemann-Petty Problems (Part 1)**

Abstract: In 1956, Busemann-Petty were studying finite
dimensional norm spaces from a geometric point of view and
sought to characterize Euclidean spaces. They posed ten
conjectures of which only one has been resolved relatively
recently, in 1999. The first talk will focus on motivating
and introducing the basic notation and machinery used to obtain
partial results in two of the remaining conjectures. Some
of the topics introduced and explored will be the Gaussian
curvature, the isotropic position, and the space of spherical
harmonics. The second and third talks will outline the solution
of the problems.

**Video
of the talk**

** **

**9 February 2021**

* *Morgan O'Brien: **A
Subsequential Individual
Ergodic Theorem on von
Neumann algebras ***
*

**Abstract: **When studying
an operator on a vector space, one
might need to know the long-term
behavior of the iterations of that
operator on a fixed element. One way
to do this is to study some type of
convergence of the sequence of
running averages that one obtains
from these iterations. However, one
might find themselves needing or
wanting to ignore a number of terms
and consider the averages of a
subsequence of the iterations - for
example, maybe an error was made
somewhere that forces some terms to
be ignored. Numerous results of this
nature are known for pointwise and
norm convergence for various types
of operators on the Lp-spaces
associated to a measure space, but
not much is known for these types of
problems in the von Neumann algebra
setting. In this talk, we will look
at the almost uniform and bilateral
almost uniform convergence of the
subsequential averages of a
Dunford-Schwartz operator acting on
noncommutative Lp-spaces associated
to a semifinite von Neumann algebra
along sequences of density 1.

**Video
of the talk**

**2
February 2021**

* *Doğan Çömez: **Quantization
for infinite affine
transformations (Part 2)**

Video of the talk

Slides

** 26 January 2021**

* *Doğan Çömez: **Quantization
for infinite affine transformations (Part
1) **

Click here for
the Abstract

**Video
of the talk**

Slides