**Speaker: Sascha Troscheit, University of Oulu (Finland)**

Date: November 28, 2022

#### Time/location: 4pm @ GAB461

Title: Zoom in and enhance: Dimension theory and finer geometry of metric spaces

Abstract:

Dimension theory and fractal geometry study the local structures of geometric objects such as sets and measures. This is achieved through various 'dimensions' that encapsulate scaling behaviour. In recent years, much progress was made on the family of Assouad-type dimensions. These dimensions are linked to fine local information of geometry and have found their application in metric geometry and random geometry. I will give a tour of the last decade of research in the area and their connections to embeddability and structural theorems for random and deterministic structures.

Bio:

Sascha Troscheit is a research fellow at the University of Oulu (Finland). He completed his PhD at the University of St Andrews (Scotland) under the supervision of Kenneth J. Falconer and Mike Todd. Before coming to Oulu, he has held positions at the University of Waterloo (Canada) and the University of Vienna (Austria). In Vienna, Sascha was a Lise Meitner Fellow and is now a Marie Skłodowska-Curie Fellow funded by the Horizons Europe programme. His expertise lies in deterministic and random fractal geometry and their interactions with number theory. In his spare time, Sascha can be found sailing the waters off the British, Dutch, and German coasts.

## Speaker: Mike P. Cohen, C.H. Robinson

## Date: November 7, 2022

#### Time/location: 4pm @ GAB461

Title: Three math tracks: each alike in dignity?

Abstract: Many math Ph.D. seekers hope to find a career as either (1) a research mathematician at a doctoral granting institution; (2) a teaching professor at a liberal arts college; or (3) an industry mathematician at a Fortune 500 company. Nine years after completing my Ph.D. in math at UNT, I find myself in the weird, but perhaps not uncommon, situation of having experienced all three! I'll share some details of how I wandered through three very different environments (academic jobs at North Dakota State University and at Carleton College, and through to my current position as a data scientist at C.H. Robinson) and describe what I perceive as strengths, drawbacks, and major takeaways from these disparate career paths. The main audience for this talk is UNT math grad students, for whose benefit I will emphasize frankness, even around such shocking taboo topics as: research fatigue; two-body problems; money; what it might take to get hired (besides luck); data science bootcamps; undergraduate research mentorship; and whatever else you can think to ask about.

## This special colloquium will follow a unique format: a short talk follows by a panel discussion. The PhD graduate panelist is Ms. Jill Kaiser and the moderator is Dr. Kiko Kawamura. Cookies and coffee/tea will be served in GAB 473 prior to his talk.

## Speaker: Dominik Adolf, UNT

## Date: October 17, 2022

#### Time/location: 4pm @ GAB461

Title: Mutual Stationarity at the least singular cardinal

Abstract: Let X be any set. We say a collection of subsets of X is stationary if it contains at least one set closed under any given algebra on X. This is, perhaps, the closest we get to a universal system of measure. In this talk we will discuss what sets on the least singular cardinal can be stationary using the notion of mutual stationarity introduced by Foreman and Magidor. We will see that a rich set of sequences can consistently be mutually stationarity if we have the right set of total, non-principal 0-1 measures of a fixed uncountable additivity. This is joint work with Omer Ben-Neria.

## Speaker: Hung Tran, Texas Tech

#### Date: October 3, 2022

#### Time/location: 4pm @ GAB461

Title: Geometry and Topology: A Tale of Rigidity

Abstract: A mainstream in mathematics is to study the relation between the geometry and topology of a manifold. The geometry is about the distance, length, area, and volume and is determined by the curvature. The topology is about properties that are preserved under continuous deformations. It is generally observed that a suitable assumption on the geometry would lead to certain restrictions on the topology. In this talk, we'll discuss some recent progress in this direction, particularly from the perspective of the curvature of the second kind. There will be a resolution of a conjecture proposed by S. Nishikawa and rigidity results for Einstein metrics and gradient Ricci solitons. Much of the talk is based on joint work with X. Cao, X. M. Gursky, and E. Ribeiro Jr.

## Speaker: Harrison Gaebler, UNT

#### Date: September 19, 2022

#### Time/location: 4pm @ GAB461

**Title:** Riemann Integration and Asymptotic Structure of Banach Spaces

**Abstract:** I will introduce the topic of Riemann integration in Banach spaces in this talk and show briefly the curious property that a Banach-valued Riemann-integrable function need not be Lebesgue a.e. continuous. A Banach space X such that every X-valued Riemann-integrable function is Lebesgue a.e. continuous is said to have the Property of Lebesgue (PL). It is an open problem to characterize the PL in terms of asymptotic structure. I will detail what is currently known with respect to this problem and where the research is now headed. In addition, I will discuss at the end some related questions such as non-linear stability of the PL and whether or not a Banach space with the PL has the Radon-Nikodym property.

## Speaker: Kirill Lazebnik, UNT

#### Date: September 12, 2022

#### Time/location: 4pm @ GAB461

**Title:** A Geometric Approach to Runge's theorem

**Abstract:** We will recall Runge's theorem on approximation of analytic functions (in one complex variable) by polynomials, and remark how a lack of a global understanding of the polynomial approximant limits potential applications of this theorem in complex dynamics. We will discuss a different approach to this classical result which gives a fuller understanding of the polynomial approximants, and discuss potential applications. This talk is based on joint work with Chris Bishop.