Talks will be held on Wednesdays, at 4:00pm in SCEN604. If you would like to give a talk, email *pp010* at *uark* dot *edu*

Date | Speaker | Title/Topic |

9/29/2021 | Patrick Phelps | Getting our feet wet with Navier-Stokes – An easy integral bound |

10/6/2021 | Will Blair | Atomic decomposition of holomorphic Hardy spaces on the complex unit disk |

10/13/2021 | Minh Nguyen | The inscribed rectangular problem in Jordan curve |

10/27/2021 | Patrick Phelps | A couple fluid flow solutions using the Navier-Stokes Equations |

11/3/2021 | Dr. Joshua Padgett | An exploration of approximating semigroup |

11/10/2021 | Alexander Duncan | The Mathematics of Modern Speedrunning |

TBA | Trevor Nakamura | TBA, Topic: Topology |

11/3/2021

**Speaker:** Dr. Joshua Padgett

**Title:** An exploration of approximating semigroups**Abstract:** The concept of a semigroup was originally introduced in the early 20th century in an attempt to generalize results from group theory and also to study the multiplicative properties of an algebraic ring. However, after a century or so of intense study, we now realize that these objects naturally arise in other areas of mathematics. Of particular interest is the observation that, when this purely algebraic object is endowed with additional analytic structure, semigroups can be used to describe solutions of various differential equations posed in a variety of abstract settings. Due to this fact, the task of approximating solutions to differential equations can be reformulated into the problem of approximating (certain) semigroups. The goal of this talk is to introduce some basic ideas behind how semigroups arise in the study of differential equations and then explore how one can construct appropriate approximations to these semigroups. We will motivate the idea initially by focusing on the finite-dimensional setting (i.e., the case of the matrix exponential), but then touch on how things become more complicated in the more abstract setting (which includes partial differential equations). In addition, if time permits, we will discuss how approximations for differential equations on Lie groups and for differential equations posed in high-dimensional spaces need to be more carefully treated.

11/10/2021

**Speaker:** Alexander Duncan

**Title:** The Mathematics of Modern Speedrunning

**Abstract:** As a result of constant optimization, speedruns of classic video games become increasingly more reliant on luck and potentially run-ending tricks and strategies. Therefore, world record viable speedrun routes become asymptotically more difficult to complete as speedrun categories age. In the past 5 years alone, various speedrun communities have employed a wide range of mathematical techniques in order to solve these issues and theorize new routes. In this talk, we will be analyzing the most famous applications of mathematics in Modern speedrun categories including: cover theory in the Super Mario 64 1-key TAS, statistical machine learning in The Legend of Zelda Wind Waker 100% category, and the traveling salesman problem in The Super Mario Sunshine 120 shine category.

10/27/2021

**Speaker: **Patrick Phelps

**Title:** A couple fluid flow solutions using the Navier-Stokes Equations

**Abstract:** In this talk, we will investigate solutions to some nice fluid flows. By taking advantage of simplifying factors such as axis-symmetry, planar flows, no slip boundaries, and constant pressure gradients, we can find these solutions by hand. Some examples of flows we may see are: shear forced flows between planes, Couette and Poiseuille flows through a pipe, two dimensional flows over inclined planes, and flows with (simple) interfacial instabilities. A basic understanding of differential equations should be sufficient to understand this talk!

10/13/2021

**Speaker:** Minh Nguyen

**Title:** The inscribed rectangular problem in Jordan curve

**Abstract:** Toeplitz conjecture is very easily stated: Given a Jordan curve in the plane (a non-self-intersecting curve that is continuous), there is always a square inscribed in it (4 vertices lie on the curve). The conjecture was introduced by Toeplitz in 1911, so far it is still open! In this talk, we will discuss some cases where the conjecture is verified and some of the “easier” version of the conjecture. Specifically, we will go through a rather beautiful proof proposed by Vaughan in 1981 that shows that every *continuous* Jordan curve inscribes a rectangle.

10/7/2021

**Speaker: **Will Blair

**Title:** Atomic decomposition of holomorphic Hardy spaces on the complex unit disk

**Abstract:** In this talk, I will present the results and techniques from the paper “Atomic Decompositions of Holomorphic Hardy Spaces in S^1 and Applications” by G. Hoepfner and J. Hounie that lead to a characterization of classical Hardy spaces by an atomic decomposition. Note: this will be practice for my oral exam presentation and as such will not be presented from bare fundamentals. That being said, the talk should be completely accessible to anyone who has taken the complex analysis sequence here and has some familiarity with Lebesgue spaces. Questions during the talk are highly encouraged!

9/29/2021

**Speaker: **Patrick Phelps

**Title: **Getting our feet wet with Navier-Stokes – An easy integral bound

**Abstract:** The Navier-Stokes system of partial differential equation is a 3D momentum equation which governs the movement of viscous incompressible fluids. We examine an integral bound used to in the proof of decay rates for solutions to Navier-Stokes with rough initial data. After some introduction to the system, we will dive into proving a necessary integral bound, utilizing only a basic understanding of the calculus sequence *which will be accessible to any who are interested in attending.* We will then discuss how this ties into the decay rate proof at large. This work is to be submitted for publication with Dr. Zachary Bradshaw, and will constitute a part of my dissertation work.