# Research

## Graduate Seminar Series (Leader: Nathan Hayford <nhayford (at) usf.edu>document.write('<a href="mai' + 'lto:' + 'nhayford' + '&#64;' + 'usf.edu' + '">Nathan Hayford</a>');)

### Friday, April 15, 2022

Title: Differentiation in real topological vector spaces
Speaker: Bryce Virgin
Time: 2:00pm–3:00pm
Place: Zoom Meeting

#### Abstract

Differentiation is a fundamental operation in the subject of analysis, which systematizes the process of approximating sufficiently regular functions and shapes with linear substitutes near points of interest. Many of the classical results of analysis deal with the properties of differentiation for maps between finite dimensional, Euclidean vector spaces, where the theory of differentiation is well-behaved. There is little difficulty in extending the concept of differentiation to quite general topological vector spaces, but in this case, derivatives will no longer act as expected. The talk will explore the generalization of the standard derivative to general real topological vector spaces, which important theorems and properties it retains from the Euclidean case, and which are lost in the generalization.

### Friday, April 1, 2022

Title: Extremal Graph Theory and the Problem of Zarankiewicz, Part II
Speaker: Eion Mulrenin
Time: 12:30pm–1:30pm
Place: Zoom Meeting

### Friday, March 25, 2022

Title: Extremal Graph Theory and the Problem of Zarankiewicz
Speaker: Eion Mulrenin
Time: 12:30pm–1:30pm
Place: Zoom Meeting

#### Abstract

Extremal Combinatorics is an area of mathematics which seeks to maximize properties of a structure given certain restrictions placed upon it. One goal of Extremal Graph Theory, then, as one might expect, is to find the maximum number of edges in a graph which does not contain a given subgraph. In this talk, we will introduce the subject by examining its origins in several classic theorems and by exploring several old and interesting open problems in the field, primarily the problem of Zarankiewicz. No background in graph theory or combinatorics will be assumed.

### Friday, March 18, 2022

No seminar this week — Spring Break.

### Friday, March 4, 2022

Title: Introductory Lectures on Quantum Field Theory, Part IV
Speaker: Razvan Teodorescu
Time: 2:00pm–3:00pm
Place: Zoom Meeting

### Friday, February 25, 2022

Title: Introductory Lectures on Quantum Field Theory, Part III
Speaker: Razvan Teodorescu
Time: 2:00pm–3:00pm
Place: Zoom Meeting

### Wednesday, February 16, 2022

Title: Weighted Composition Operator Over the Mittag-Leffler Spaces of Entire Functions
Speaker: Himanshu Singh
Time: 12:00pm–1:00pm
Place: Zoom Meeting

#### Abstract

We provide the operator theoretic characterizations for the Weighted Composition Operator over the Mittag-Leffler RKHS. The important reference during the study is Weighted Composition Operators on the Mittag-Leffler Spaces of Entire Functions by Hai and Rosenfeld.

### Friday, February 11, 2022

Title: Introductory Lectures on Quantum Field Theory, Part II
Speaker: Razvan Teodorescu
Time: 2:00pm–3:00pm
Place: Zoom Meeting

### Friday, February 4, 2022

Title: Introductory Lectures on Quantum Field Theory
Speaker: Razvan Teodorescu
Time: 2:00pm–3:00pm
Place: Zoom Meeting

#### Abstract

Quantum field theory (QFT) is the mathematical formulation of our current theoretical description of nature's fundamental laws. It relates deeply to functional analysis, operator theory, Lie groups representation theory, and PDE, and is known for open problems such as the 4D Yang–Mills Millenium problem.

The presentation will introduce the main elements of QFT by reviewing Einstein's postulates of special relativity, irreducible representations of the Poincare group, the Pauli–Lubanski theorem and classification of free fields, Emmy Noether's theorem, internal symmetry groups, minimal connections and gauge theory, leading to the derivation of the action functionals for electro-weak and QCD theories.

The continuation would require a detour through Gaussian random field theory and the Feynman–Kac theorem, before returning via Wick calculus to QFT, 't Hooft–Veltman dimensional regularization, the Migdal–Kadanoffrenormalization (semi)group and Feynman diagrammatic expansions for standard applications such as the Lamb precession, CPT theorem, asymptotic freedom and the Goldstone–Higgs theorem.

### Friday, January 28, 2022

Title: Statistical Mechanics and the Tutte Polynomial
Speaker: Nathan Hayford
Time: 2:00pm–3:00pm
Place: Zoom Meeting

#### Abstract

The Tutte polynomial is a well-known graph invariant, from which many important graph-theoretic properties can be read off. We discuss the equivalence of this polynomial to an important model in statistical physics: the $$q$$-Potts model, which is a generalization of the simplistic model of a magnet (the Ising model). We use this equivalence to develop the well-known high-temperature expansion of the Potts mode. Furthermore, we discuss what we can learn about graph theory from statistical mechanics, and vice versa.