Fall 2021

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

9/29/2021Patrick PhelpsGetting our feet wet with Navier-Stokes – An easy integral bound       
10/6/2021Will BlairAtomic decomposition of holomorphic Hardy spaces on the complex unit disk
10/13/2021Minh NguyenThe inscribed rectangular problem in Jordan curve
10/27/2021Patrick PhelpsA couple fluid flow solutions using the Navier-Stokes Equations
11/3/2021Dr. Joshua PadgettAn exploration of approximating semigroup
11/10/2021Alexander DuncanThe Mathematics of Modern Speedrunning
TBATrevor NakamuraTBA, Topic: Topology


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.


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.


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!


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.


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!


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.


Spring 2021

If you would like to give a talk, email pp010 at uark dot edu.

Zoom link: https://uark.zoom.us/j/87060910342?pwd=VGFMU25nbDZhQ3JYTFJwajRXd01LZz09

Meeting ID: 870 6091 0342

Passcode: 0j?Uf^ZU

1/27/21Minh Nguyen2:00A discussion about some classic questions in Low dimensional topology
2/3/21Patrick Phelps4:00What function spaces go with the flow of Navier-Stokes?
2/10/21Alexander Duncan4:00Hilbert Polynomials
2/17/21Minh Nguyen4:00A brief introduction to Elliptic operators on manifolds
2/24/21Patrick Phelps4:00Data Assimilation and Determining elements for the Navier-Stokes Equations
3/10/21Thuy Scanlon2:00Monte Carlo Markov Chain Algorithms
TBAJoshua Gregory2:00TBA (Accessibility, and racial biases in math education)
TBAWill Blair4:00TBA (Generalized analytic functions)
TBAThuy Scanlon2:00Monte Carlo Markov Chain Algorithms (Continuation of previous talk)


Speaker: Minh Nguyen

Title: A discussion about some classic questions in Low dimensional topology

Abstract: In this expository talk, we shall discuss the two fundamental questions about the topology of 4-manifolds: Is there any compact topological 4-manifold that does not admit any smooth structure? And is there an instance where two compact smooth 4-manifolds are homeomorphic but not diffeomorphic? The goal of this talk is to lay down motivation and background for Seiberg-Witten invariant–a smooth 4-manifold invariant that is derived from the analysis of a system of two non-linear PDEs. This area of research is a meeting point for analysis, topology, and geometry.

Recording: https://uark.zoom.us/rec/share/F98y28UgFLIR-6XAcfyNQd4wZOhlHT-vnYZDkXzwAib8ZOtJfWqZCKfxOv_56s7Z.x5Ui5WU2j9dgv65i

Access Passcode: uY.k1^pq


Speaker: Patrick Phelps

Title: What function spaces go with the flow of Navier-Stokes?


In this talk we will explore function spaces that serve as good sources of initial data for the Navier-Stokes, we well as spaces where it can be desirable for a solution to live. Certain data spaces can guarantee well-posedness, uniqueness, and regularity, as well as permit energy methods or other techniques to analyze our solution. Spaces where short time strong existence is known include the Homogenous Sobolev Space, Lebesgue Spaces, Weighted L^p spaces, weak L^3, and Morrey Spaces. Let’s see what makes them flow!

Recording: https://uark.zoom.us/rec/share/Dr0bZW8FGUqOmtGWCrvcDmxKmtLDPrU80Vz4Px9LYEy8lxz2w4zYMGb4hBZ9rE-L._ySU9jePRIhRb4KX

Access Passcode: Rzxd3v4^


Speaker: Alexander Duncan

Title: Hilbert Polynomials

Abstract: Studying the Hilbert polynomial for a finitely generated graded module over the ring S =  k[x_0, …, x_n] is a classical problem in both commutative algebra and algebraic geometry. In this talk, we will discuss how free resolutions give rise to Hilbert polynomials and then discuss applications and techniques for calculating these polynomials including an algorithm for generating graded free resolutions of S-modules using simplicial complexes. Betti tables will also be discussed.

Recording: https://uark.zoom.us/rec/share/jFeU0D8Tqd1TRV5FjQf-K7esh48j1UxG0en3-RvOfaMEA9TEqV_E9htX2w29XwgD.uxwLhGc2VtGrbphd

Access Passcode: 7m=C2+.u


Speaker: Minh Nguyen

Title: A brief introduction to Elliptic operators on manifolds

Abstract: This is a sequel to our previous talk. The ultimate goal is to understand the applications of Seiberg-Witten equations in low dimensional topology. For us to achieve such goal, it is necessary to familiarize ourselves with the language of analysis on smooth manifolds. In this talk, we shall introduce some of the prerequisite background pertaining to the notion of covariant derivatives (connections) associated to a vector bundle, (pseudo-)differential operators, Sobolev spaces of sections, and finally (if we have time) elliptic theory of operators. The area of global analysis is vast; but we mainly provide an overview of the subject with a motivation to understand Seiberg-Witten theory.

Recording: https://uark.zoom.us/rec/share/3dTVZ8ofes00IOhJ_QybgXBPxAt1PkoHCKdWB9pXvx57oMEzyzzWAvKjxCHGSpsE.EOHM8_bSaZMLNmrf

Access Passcode: $jVze1RF

Date: 2/24/2021

Speaker: Patrick Phelps

Title: Data Assimilation and Determining elements for the Navier-Stokes Equations

Abstract:  A determining element for dissipative dynamical system is one which, if known to converge to zero for two solutions, implies they are identical. In 2-dimensions the Navier-Stokes have been shown to have a finite number of determining modes, nodes, and volume elements. Because global well-posedness is known for 2D, we may use continuous data assimilation to create approximating solutions from any type of measurement data for which an interpolation operator exists (including modes, nodes, and volume elements). This talk aims to define these determining data parameters, and give an introduction to interpolation operators through the lens of Navier-Stokes.

Date: TBA

Speaker: William Blair

Title: TBA (Generalized Analytic Functions)


Date: 3/10/2021

Speaker: Thuy Scanlon

Title: Monte Carlo Markov Chain Algorithms

Abstract: The goal of data modeling is to make predictions and inferences about parameters based on available sample data. This concept matches naturally with Bayesian statistics, which allows prior knowledge incorporated into the hierarchical models.  Much of the statistical summaries about a distribution can be estimated if one can sample from that distribution. However, the result posterior distributions of Bayesian hierarchical models are often difficult or even impossible to sample directly from. Monte Carlo Markov Chain is the most efficient and reliable sampling method in these settings.


Access Passcode: %XdW^j3Q

Fall 2020

If you would like to give a talk, email pp010 at uark dot edu.

Zoom link: https://uark.zoom.us/j/87060910342?pwd=VGFMU25nbDZhQ3JYTFJwajRXd01LZz09

Meeting ID: 870 6091 0342

Passcode: 0j?Uf^ZU

Sept 9Patrick PhelpsWeak, Strong, and Mild Solutions to the Navier-Stokes Equations
Sept 16Alexander DuncanA Summary of Cover Theory
Sept 23Shakil RafiHeegard Splittings and Basic Theorems about them
Sept 30CancelledCancelled
Oct 7CancelledCancelled
Oct 14Michael ShumateIntroduction to Morse Theory
Oct 21Patrick PhelpsSome fine properties of discretely Self-Similar Solutions (to NSE)
Oct 28Derrick WigglesworthFun with Folding in Free Groups
Nov 4William BlairHardy Spaces on R^n
Nov 11Jorge Andres Robinson ArrietaAlgebraic Topology Methods in PDE and Functional Analysis
Nov 18Meredith SargentOptimal Approximants and Orthogonal Polynomials
Nov 25Thanksgiving BreakNo talk scheduled


Speaker: Patrick Phelps

Title: Weak, Strong, and Mild Solutions to the Navier-Stokes Equations


The Navier-Stokes System of Equations is a PDE system that models viscous, incompressible fluid flow. There are two popular ways to approach the system: first, by using energy methods to force convergence of a sequence of approximate solutions (Weak), and second, to use a fixed point iteration of Duhamel’s principle and Riesz Transforms to solve a series of Heat and Laplace Equations, thereby creating a smooth solution (Mild). We introduce the mechanics of the system, some of the theory behind weak solutions, and how to prove short-time existence and regularity, using the Mild and Strong classes of solutions.


Speaker: Alexander Duncan

Title: A Summary of Cover Theory


Cover Theory is a rich, fun, and rewarding fundamental tool for studying spaces. The build up of the theory is minimal and moderately technical while the pay-off is ripe with important calculations and examples.

In this talk, we will review the fundamental group of a space and the basics of homotopy of spaces. Then, we will observe a summary of cover theory giving several examples throughout.


Speaker: Shakil Rafi

Title: Heegard Splittings and Basic Theorems about them


In this talk we get an introduction to Heegard splittings. We start with handlebodies and handle decompositions, and start with a definition of heegard splittings and classic examples of heegard splittings. We go on to give examples of heegard splittings and the fact that every closed orientable 3-manifold has heegard splittings. we then go on to describe some important structural theorems about Heegard splittings: e.g. Rademeister-Singer, Waldhausen, Bonahan-Otal, Boileau-Otal. 


Speaker: Michael Shumate

Title: Introduction to Morse Theory


Morse theory provides the foundation for many active research areas, including the various types of Floer theories. In this introductory talk, we will discuss the main ideas behind Morse theory, and point to their applications in Floer homologies.


Speaker: Patrick Phelps

Title: Some fine properties of discretely self-similar solutions


Navier Stokes has a scaling property, and in the class of scale-invariant solutions local information can be used to control global quantities. From this we prove regularity for energy solutions and solutions in Besov spaces, as well as find explicit decay rates for energy solutions.


Speaker: Derrick Wigglesworth

Title: Fun with Folding in Free Groups


We’ll discuss one or two algorithms that solve elementary decision problems in finite rank free groups.  I intend to make this talk accessible to everyone, so there will be lots of pictures and examples.  The only prerequisite is knowing what a group is…and you can probably get away without that.


Speaker: William Blair

Title: Hardy spaces on R^n

Abstract: We will define the Hardy spaces on R^n and present some characterizing results.  All material is based on the treatment given in Harmonic Analysis : Real-Variable Methods, Orthogonality, and Oscillatory Integrals by Elias M. Stein.


Speaker: Jorge Andres Robinson Arrieta

Title: Algebraic Topology Methods in PDE and Functional Analysis

Abstract: In this talk we are going to talk about how generalized cohomology theories such as homology and K-theory can help us to prove that a general map between Banach spaces or Banach algebras has a zero. This is a common situation in PDE, where you translate the fact that a PDE having a solution with certain properties is the same as having a zero of a map between spaces. The central idea of the talk is to make the participant notice that homotopy theory could be a natural setting for studying general PDE’s.  Unfortunately, there are more questions than answers, so the talk would be rather an invitation to dig deeper in these topics. 


Speaker: Dr. Meredith Sargent

Title: Optimal Approximants and Orthogonal Polynomials

Abstract: We will begin with an introduction to some of the ideas in functional analysis, including defining some of the relevant function spaces used later in the talk and discussing some of the motivations. Then I will discuss my recent work (with Alan Sola) on multivariable optimal approximants: polynomials that best approximate a function that is NOT in the function space of interest. I will finish by mentioning some open problems.

Spring 2020

Wednesdays at 4:10pm in SCEN 604

If you would like to give a talk, email jskeyton at uark dot edu.

Jan 29Shakil RafiTopology and Computation
Feb 5Patrick PhelpsExistence for the Perturbed Stokes System
Feb 12Minh NguyenYet two more proofs of The Fundamental Theorem of Algebra using differential topology
Feb 19Caleb ParksA Generalized Voltera-Type Operator Between Hardy Spaces and Multiplication Operators Between Analytic Tent Spaces
Feb 26Alexander DuncanAlgebraic Groups
Mar 11Eric WalkerWhat’s on the mind of air traffic controllers and algebraic geometers?
Cancelled for remainder of semester due to COVID-19

Fall 2019

Location: SCEN 406

Time: 4:10pm

Sep 4Rachel LehmanAn Introduction to Dehn Surgery and its
Importance to Hyperbolic 3-Manifolds
Sep 11Jesse KeytonUnimodality of Pure O-sequences
Sep 18Jean Pierre M.Insolvability of the Quintic
Sep 25Caleb ParksAn Introduction to Bergman Spaces:
The Bergman Kernel, I
Oct 2Caleb ParksAn Introduction to Bergman Spaces:
The Bergman Kernel, II
Oct 9Sam CowgillSolving the Functional Equation of e^x
Oct 16Minh NguyenU(n) Seiberg-Witten Equations
Oct 23Emily FossThe Ohsawa-Takegoshi Extension Theorem
Oct 30Caleb ParksInterpolation in analytic tent spaces
Nov 6Trevor NakamuraClassification of Group Extensions with Abelian Kernel by Cohomology Classes
Nov 13Rachel LehmanObtaining a Good 3-Orbifolds From a Bad 3-Orbifold
Nov 20Sam CowgillExploring Tetration in the Complex Plane
Dec 4Alexander DuncanProperties of Cohen Macaulay Schemes
Dec 11Yoav Rieck

Spring 2019

Jan 30Minh NguyenSpin geometry and Dirac
eigenvalues (I)
Feb 6Minh NguyenSpin geometry and Dirac
eigenvalues (II)
Feb 13Rachel LehmanOrbifolds and the 3-orbifold
Feb 20Patrick PhelpsA discussion about matroids
Feb 27Minh NguyenSpin geometry and Dirac
eigenvalues (III)
March 6Michael ShumateKnot Floer Homology
March 13Caleb ParksInterpolations in tent spaces
March 27Jean Pierre MutanguhaWhat is… residually finite
April 10Jesse KeytonConditional Independence Models
and Primary Decomposition
April 17Daniel Levine (PSU)Moduli of Sheaves
April 24Mike Miller (UCLA)
James Cook (LU)
Triangulations and Homology cobordism (Mike)
From Algebra to Calculus and Back again (James)

Fall 2018

Aug 29Minh Nguyen Introduction to Clifford Algebra: Why it
matters in Analysis and Differential topology
Sep 12Michael ShumateIntroduction to Knots, Why They Matter,
and Some Knot Invariants
Sep 19Caleb ParksTopological vector spaces and Distribution theory
Sep 26Jean Pierre Mutanguha A Hyperbolic Space Odyssey
Oct 3Jesse KeytonAlgebraic Linkage and Hilbert functions
Oct 10Rachel LehmanIntroduction to Thurston’s Geometrization theorem
Oct 17Michael DuffyGradient Estimates for Elliptic PDE’s
Oct 24Dr. Lance MillerQ&A Session
Oct 31Jean Pierre Mutanguha \\
Jesse Keyton
Hyperbolic Immersion of Free Groups \\
Non-sequentially bounded Licci Ideals
Nov 7Michael ShumateKnots and not-Knots
Nov 14Muhenned AbdulsahibHartogg Domain and Diederich-Fornaess Index