### Analysis and Geometry Seminar

#### Spring 2023

##### Seminar information

**Location:**South Engineering 120**Time:**Tuesdays, 11:00-11:50 am**Organizer:**Maria Alfonseca-Cubero

##### 4 April 2023

**Syed Husain: **Nyquist density thresholds for different Paley-Wiener spaces.

**Abstract: **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

**Abstract:** 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

**Abstract: **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?

**Abstract:** 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

**Abstract: **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

**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. 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

##### Seminar information

**Location:**Morrill 109**Time:**Tuesdays, 11:00-11:50 am**Organizer:**Maria Alfonseca-Cubero

##### 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

**Abstract: **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 to Ulam's floating body problem (Part 2)

##### 1 November 2022

**Mariangel Alfonseca:** A negative solution to Ulam's floating body problem

**Abstract:** 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

**Abstract:** 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.