Analysis and Geometry Seminar

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:
T
his 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)