### Analysis and Geometry Seminar

#### Spring 2022

##### Seminar information

**Location:**Minard 302**Time:**Tuesdays, 11:00-11:50 am**Organizer:**Maria Alfonseca-Cubero

##### 3 May 2022

**Michael Roysdon (Tel Aviv University): **Measure theoretic inequalities and projection bodies

**Abstract:** 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 (Part 2)

**Abstract:** : 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

**Abstract:** 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

**Abstract:** 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

**Abstract: **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 (copy 1)

**Azer Akhmedov: **The geometry of Banach spaces (Part 2)

##### 15 February 2022

**Azer Akhmedov: **The geometry of Banach spaces

**Abstract****:** 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

##### Seminar information

**Location:**Ladd 114**Time:**Tuesdays, 11:00-11:50 am**Organizer:**Maria Alfonseca-Cubero

##### 16 November 2021

**Josef Dorfmeister: **Minimal Genus in Rational Manifolds (Part 2)

##### 9 November 2021

**Josef Dorfmeister: **Minimal Genus in Rational Manifolds

**Abstract:** 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. There are very few manifolds for which this problem has a complete solution, and only a few more for which estimates of any kind are known.

I will describe the problem in detail, describe techniques that have been used to study this problem and then describe recent results for rational manifolds. On such manifolds, there exists a well-understood group action whose fundamental domain is partially known (Tits cone of a Kac-Moody algebra); I will describe the structure of the full fundamental domain. This involves reducing the problem to solving a Diophantine set of equations. Finally, studying the minimal genus question for this fundamental domain leads to new results.

##### 2 November 2021

**Pratyush Mishra:** Growth of groups (Part 2)

##### 26 October 2021

**Pratyush Mishra:** Growth of groups

**Abstract:** The study of growth rate dates back to one of the famous theorems of Milnor & Schwarz in Riemannian geometry, which states that "The volume growth of the universal cover of a complete Riemannian Manifold M is equivalent to the growth rate of the fundamental group of M". Growth rates are of great interest in geometry, group theory and dynamical systems. In these talks, we shall study how the growth rate is slow for some class of groups while its fast for another class. We shall see some concrete examples of groups arising from geometry and compute their growth rate.

##### 19 October 2021

**Frankie Chan: **Profinite Rigidity and Fuchsian Groups (Part 2)

##### 12 October 2021

**Frankie Chan: **Profinite Rigidity and Fuchsian Groups

**Abstract: **Inspired by a result from Bridson--Conder--Reid, my work produces an effective construction for distinguishing the collection of finite quotients of a triangle group with that of a non-isomorphic Fuchsian lattice. With an aim of balancing motivation and technical details, I will begin with some background and exposition on profinite groups and surface theory. This is joint work with Ryan Spitler (Rice University).** **

##### 5 October 2021

**Doğan Çömez: **Ergodic Theorems in Fully Symmetric Banach Spaces (Part 2)

##### 28 September 2021

**Doğan Çömez: **Ergodic Theorems in Fully Symmetric Banach Spaces

**Abstract: **Fully symmetric Banach spaces are large function spaces that include classical Banach spaces. Some well-known examples of spaces of this kind are Orlicz Spaces, Lorentz Spaces and Marcinkiewicz Spaces. Although these spaces are studied extensively in the functional analysis literature, investigation of convergence of ergodic averages in such spaces is fairly recent. In this talk, first we will provide a brief review of symmetric Banach spaces, then we will give necessary and sufficient conditions for almost uniform convergence of ergodic averages. These, in turn, will be utilized in extending several ergodic theorems to the setting of fully symmetric Banach spaces.

##### 21 September 2021

**Morgan O'Brien: **A Non-Commutative Return Times Theorem (Part 3)

##### 14 September 2021

**Morgan O'Brien: **A Non-Commutative Return Times Theorem (Part 2)

##### 7 September 2021

**Morgan O'Brien: **A Non-Commutative Return Times Theorem

**Abstract**: Weighted and subsequential ergodic theorems are frequent subjects of study in ergodic theory. An important version of these types of results are the Wiener-Wintner type ergodic theorems, which allow for averages of operators weighted by a collection of sequences to be considered simultaneously. In classical ergodic theory, these types of results are known for numerous classes of weights, while the noncommutative setting thus far has required all sequences to be bounded. In these talks, we will discuss a version of the Banach principle that has been specialized for proving such theorems for various types of weights in the von Neumann algebra setting. Using this, we will prove some Wiener-Wintner type ergodic theorems for some broad classes of Hartman sequences.