### Analysis and Geometry Seminar

#### Spring 2022

##### Seminar information

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

##### 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 performersâ€™ metacognition throughout the host course. Additionally, one outlying assessment item especially illuminates additional considerations for future application of metacognition in proof comprehension research.