Past Talks
Spring 2024
Wednesday February 14
Aundre Wesley
Title: Remediation of Mathematics for an Algebra I Class
Abstract: The following concept of the remediation of mathematics for an Algebra I class focuses on
integrating English language arts skills into the standards outlined for Algebra I. The concept
uses handwriting as a student engagement tool through graphing parent functions and developing
the concepts of matrices. The student example notes show how fundamental components of
Algebra I can be represented. The need for remediation of all subjects is needed, especially after
the students have engaged in distance and hybrid learning for the past two years. Although tested
and taught individually, a combined approach to remediation can tackle what is an
interdependent issue as both scores of ELA and Maths are suffering.
Wednesday February 21
Zach Rail
Title: TBA
Abstract: TBA (Visual Complex Analysis)
Thursday February 29
Nicole Bonge
Title: Introduction to Psychometric Measurement Theory & Statistical Methods
Abstract: Psychometrics is a field at the intersection of psychology and mathematics, concerned with the science of assessment. This talk will give a foundational overview of psychometrics, including the two main measurement theories (classical test theory and item response theory) and an introduction to factor analysis, one of the main statistical methods used in the field, using the Modified Abbreviated Mathematics Anxiety Scale (mAMAS; Carey et al., 2017) as an example.
Wednesday April 24
Zach Rail
Title: A First Person Tour of Spacetime
Abstract: Picture yourself in a spaceship near earth. Now imagine firing up your ship's extremely powerful engines and accelerating at a tremendous rate past the earth. When you look out the window, you are shocked to see that the earth does not race away into the distance, but instead appears to flatten into a disk and then freeze in place. Despite your engine's best efforts, you appear to be no further from the earth than when you started. With the help of computer simulations, we will see that startling sight as well as other counter-intuitive effects of special relativity.
Fall 2023
Wednesday August 23
Organizational Meeting
Wednesday August 30
Eric Walker
Title: Bézout’s Theorem --- To Infinity and Beyond
Abstract: Algebraic geometry lives in the land of polynomials and their zero sets. Sometimes, you’d like to know what happens when you’ve got multiple polynomials floating around. How do they interact? In this talk, I’m going to state a theorem and then disprove it, five or six times in a row, and each time we take one more step out of Plato’s cave. Along the way, we’ll go from high school math to undergrad to grad school to tools of active research, but throughout, this talk should be beginner-friendly and accessible to a wide audience. The title is a pun in (at least) two ways.
Wednesday September 6
Will Blair
Title: An atomic representation for Hardy classes of solutions to
nonhomogeneous Cauchy-Riemann equations
Abstract: We develop a representation of the second kind for certain Hardy classes of solutions to nonhomogeneous Cauchy-Riemann equations and use it to show that boundary values in the sense of distributions of these functions can be represented as the sum of an atomic decomposition and an error term. We use the representation to show continuity of the Hilbert transform on this class of distributions and use it to show that solutions to a Schwarz-type boundary value problem can be constructed in the associated Hardy classes.
Wednesday September 13
Alex Thomas
Title: Introduction to combinatorial knot Floer homology
Abstract: This talk will have two parts. In the first part, I will describe grid homology, a self-contained combinatorial tool for studying knots and links. This description will not require much formal background, beyond comfort with finite abelian groups. Despite this, we will learn that grid homology is in fact equivalent to a deep invariant of modern knot theory: knot Floer homology. This is a very powerful invariant. For example, it provides the only general means of determining the Seifert genus of a knot.
In the second part of the talk, I will attempt to connect grid homology to knot Floer homology. In particular, we will find that the grid diagrams of grid homology are in fact Heegaard diagrams of 3 manifolds, which we shall want to realize in the language of Morse theory.
Wednesday September 20
Alexander Duncan
Title: One Does Not Simply Embed a Curve
Abstract: The most difficult and subtle setup for embedding a curve is the algebraic geometer’s setup. In this talk, we will carefully explain the subtle choices and often-forgotten conditions one is faced with when embedding a one dimensional variety into (Zariski) projective space. In particular, we will see open problems in math concerning invariants of (embeddings of) curves which will be defined and motivated with examples. An audience member can expect to walk away from this talk with the ability to understand future talks with setups needing linear series, genus, the canonical bundle, p-very ampleness, secant planes, and graded global section rings.
*** Special after GSC meeting***
At 5:00 PM on September 20 (directly after GSC), there will be a meeting in SCEN 405 to promote graduate student involvement in the Arkansas Math Discovery Day event. This is a great opportunity for you to be involved in a mathematical event for the community. I highly suggest staying after GSC and getting involved as much as you feel comfortable (there will be many different kinds of roles of varying time commitement). For first-year graduate students, this gives you another low stakes opportunity to meet other members of the department (you want to do this as often as you can). Lastly, and most importantly, there will be
pizza.
Wednesday September 27
Zach Rail
Title: Visualizing SO(3) to prove the hairy ball theorem
Abstract: In this talk we will map out SO(3), the space of all 3D rotations. Then, by relating this picture to the tangent bundle of the 2-sphere, we will develop a proof of the hairy ball theorem that seems true at a glance. This talk will assume little background knowledge and can serve as a motivated introduction to some of the most important objects in topology, including manifolds, quotient spaces, bundles, the fundamental group, and covering spaces.
Wednesday November 1
Eric Walker
Title: Cotangent complexes of derived jet and arc spaces
Abstract: Motivated by studying the birational geometry of the jet and arc schemes of a scheme X, de Fernex and Docampo (2020) described the cotangent sheaf of the jets and arcs in terms of the cotangent sheaf of X. Our work (joint with Lance Edward Miller and Roi Docampo) asks a subsequent question: how may one describe the cotangent complexes? To produce an analogous version of their theorem, one must produce a derived/animated version of the jets and arcs. We define such a construction, discuss properties, and then describe the animated version of their theorem and its consequences. (NB: This is a practice talk for a seminar talk I will give next week, so please let me know of any and all feedback you may have!)
Wednesday December 6
*** Note: two (short) talks in one meeting ***
Kailey Perry
Title: TBA
Abstract: TBA
Will Blair
Title: A New Look at the Schwarz Boundary Value Problem
Abstract: The Schwarz boundary value problem is a simplification of the famous Riemann-Hilbert problem and is a well-studied boundary value problem in complex analysis. The classical version seeks a holomorphic function on the unit disk whose real part agrees with a prescribed continuous function on the unit circle. In this talk, I will present my recent work on this problem where I generalize the solution formulas for the nonhomogeneous and higher-order Schwarz boundary value problems by lowering the required regularity of the boundary value to a distributional boundary value of a harmonic function.
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Spring 2023
Wednesday, January 18
Organizational Meeting
Friday, January 27
Shakil Rafi
Title: Modal logic and the multiverse of madness
Abstract: We want to start off the semester with something fun. While we’ve had classical logic nailed down since atleast the time of Leibniz if not the Greeks there is an alternative to our classical predicate and quantificational logic, that is, modal logic, concerned with modalities. We introduce, truth trees, multiple worlds, and relationships between worlds. We introduce the basic premise of Kripke logic and introduce a hierarchy of logics, and if time permits wax philosophical about multiple worlds.
Friday, February 10
Aiden McCue
Title: An Overview of 3-manifolds
Abstract: In this talk, we will go over brief descriptions of some of the major results that have emerged in the study of 3-manifolds in the past century. The Prime Decomposition and Torus Decomposition theorems provide us with a way to decompose any 3-manifold into pieces which are either atoroidal or Seifert fibered. We will go over each of these theorems and discuss some features of the pieces that result from the decomposition.
Friday, February 17
Patrick Phelps
Title: An introduction to the incompressible Euler equation
Abstract: I got a new book! Bedrossian and Vicol's "The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations; An Introduction". I will give an introduction to the Euler equation, including its derivation, transport properties, vorticity formulation, and some special explicit solutions (an overview of Chapter 1). Time permitting, we will also explore symmetries and conservation laws for the system. This talk will be accessible to any student having taken PDE's, but not difficult without.
Friday, March 10
Kailey Perry
Title: Graphs of groups and Bass-Serre theory
Abstract: Van Kampen's theorem gives us a way to put together the fundamental group of a space by looking at the way subspaces are put together. We can talk about the structures of groups by thinking of a group as the fundamental group of a space, then finding subgroups and their interactions by using Van Kampen's theorem. I'll talk about these graphs of groups, and about how we can use them to think of free-by-cyclic groups and group actions on trees.
Friday, March 31
Sarah Strikwerda (NCSU ) https://sstrikwerda.wordpress.ncsu.edu
Title: Optimal Control in Fluid Flows through Deformable Porous Media
Abstract: We consider an optimal control problem subject to an elliptic-parabolic coupled system of partial differential equations that describes fluid flow through biological tissues. Our goal is to optimize the fluid pressure and solid displacement using distributed or boundary control. We first show results on the existence and uniqueness of optimal controls and then present necessary optimality conditions. The optimal controls can be approximated numerically.
Friday, April 7
Patrick Phelps
Title: Decay and approximation properties of discretely self-similar solutions to the Navier-Stokes equations
Abstract: We present recent results on spatial decay and approximation properties for the Navier-Stokes equations in 3D. We begin with decay rates for solutions and the ‘non-linear’ part of scaling invariant flows with data in the subcritical class, then show improved rates for flows with data in Hölder classes, filling in a scale of results in existing literature. Additionally, we demonstrate that solutions in the subcritical class away from the origin can be decomposed into a term with optimal decay, and a term with the expected decay which may be made small at the cost of a prefactor on the optimal term. For data in the critical Lorentz class and the supercritical Besov class, a corresponding approximation is given up to an arbitrarily small constant. In this sense, scaling invariant flows can be approximated by flows with optimal decay.
Friday, April 14
Will Blair
Title: Generalizations of the Hardy Spaces
Abstract: We consider certain classes of functions that generalize the classical Hardy spaces and present new results for these classes. The talk should be quite accessible for people that are currently in or have already taken the Complex Analysis sequence (and not too bad for those who have not).
Friday, April 28
David Crosby
Title: Bounds on Dot Product Configurations in Galois Rings
Abstract: We consider the following problems: How often can a single dot product configuration or a multiple dot product configuration occur over a Galois ring (a type of ring that encompasses both finite fields and the integers modulo a prime power)? We also find a bound on a vector matrix multiplication problem.
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Fall 2022
Wednesday, November 16 - Zach Rail
Title: Exploring 4D rotations using MagicCube4D
Abstract: MagicCube4D is a virtual 4D analogue of the Rubik’s Cube. In this talk I’ll first explain what that means and demonstrate how MagicCube4D works. In the process I’ll show how MagicCube4D gives us a tangible way of exploring the 6-dimensional space of 4D rotations. Then, I’ll discuss the rotational symmetries of the corner pieces, which (in contrast to the 3D case) form a non-abelian group of order 12. Finally, I’ll derive an invariant that shows that corner rotations must come in complimentary pairs or triplets. In particular, this means that no sequence of moves can rotate just a single corner piece without affecting any of the others.
Wednesday, November 9 - Kailey Perry
Title: Outer Space and Train Tracks
Abstract: We’ll be discussing an analogue of Teichmuller space for the group of outer automorphisms of a free group instead of the mapping class group. This outer space is a moduli space where each point is a graph that is homotopy equivalent to the $n$-rose along with the homotopy equivalence. We’ll construct the space and discuss an analogue of Thurston’s classification of surface homeomorphisms, which is then used to prove the train track theorem.
Wednesday, November 2 - Qirui Peng
Title: Unified definition of Determining Wavenumber for the 3D Naiver-Stokes Equations
Abstract: We introduce a wavenumber for the solution of the 3D Navier-Stokes equation (NSE). This wavenumber is shown to be a determining wavenumber for the 3D NSE. Furthermore, we establish an upper bound for this determining wavenumber in terms of Kolmogorov’s dissipation number; the upper bound is optimal for any intermittency dimension.
Note: This will be a Zoom talk. You may join the group in SCEN 322 to watch the Zoom meeting projected, or you can access via the following link:
Join Zoom Meeting
https://uark.zoom.us/j/3916785555?pwd=QXdlQ1ptTzZ4aXNQaXNkdmpaNENHUT09
Meeting ID: 391 678 5555
Passcode: 0ff!ceHour
Dial in Passcode: 6165715785
Wednesday, October 26 - Jaredan Durbin
Title: Introduction to Special Relativity and the Mass-energy equivalence
Abstract: In this talk we will give a brief description and overview of Newtonian relativity, classical electrodynamics, and their failure. Then we move to an introduction to special relativity: including Minkowski spacetime, special Lorentz transformations, relativistic electrodynamics, and finally the mass energy equivalence.
Wednesday, October 19 - Eric Walker
Title: Video game character creation is a moduli problem
Abstract: In this talk, I want to give an introduction to the concept of a moduli problem, which in general asks us to find a space that parametrizes solutions to a given problem. I will give examples of all skill levels, including video game character creation, high school geometry, introductory topology, conic sections, algebraic geometry, and beyond. My hope is to show you that the language of moduli problems is a fun and interesting tool!
Wednesday, October 6 - Kailey Perry
Title: $\tau_I$-elasticity when $|R/I|$ is four or eight
Abstract: For a UFD $R$ with unit and with an ideal $I$, elements may have $\tau_I$-factorizations into atoms that are all congruent modulo $I$ up to a unit. These factorizations are often not unique and so one can measure the $\tau_I$-elasticity of $R$ to be the supremum of the ratios of lengths of $\tau_I$-factorizations of an element of $R$ over the elements for which such factorizations do exist. I will discuss the $\tau_I$-elasticity for quotients of order four or eight and prove that these are the smallest quotients that give certain values for the elasticity.
Friday, September 16th - Sarah McKnight
Title: Introduction to Strongly Continuous Semigroups for Linear Partial Differential Equations
Abstract: In ordinary differential equations, solutions to first order linear equations can often be expressed using an exponential function. We want to do the same thing for time-dependent partial differential equations, so we need to make sense of the "exponential" of an operator. In this talk, we will introduce the primary object for doing so, strongly continuous (s.c.) semigroups, and discuss how they apply to two different PDEs: the heat equation and the wave equation. While the two PDEs behave very differently, s.c. semigroups can address well-posedness for both of them. Time permitting, we will also discuss analyticity, a measure of "smoothness" for these semigroups, and see one reason why the behavior of the two systems is so different
Tuesday, September 6th - Lightning Talks #2
Talks from Eric Walker, Jorge Robinson Arrieta, and Kamrul Khan giving an overview of the paths they took through graduate school to work in the fields of Algebraic Geometry, Topology, and Statistics, respectively, and a quick overview of their research at UofA.
Wednesday, August 31st - Lightning Talks #1
Kailey Perry will give us a look into her work in Algebra at the University, Surya Lamichhane will talk to us about his work and research in Statistics, and Patrick Phelps will give a talk about researching in the fields of Partial Differential Equations and Harmonic Analysis. Each talk will be 15 minutes, with some time for questions in between.
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Spring 2022
5/3/2022 at 3:00 PM
Speaker: Dr. Minh Nguyen
Title: Stable (co)homotopy of a variant of Seiberg-Witten equations for multiple spinors.
Abstract: We consider a variant of the Seiberg-Witten equations for multiple spinors defined on a closed smooth 4-manifold. Multiple spinors can be thought of as $E-$valued spinors, where $E$ is some SU(n)-bundle over the base manifold ($n>1$). A generalization of the classical Seiberg-Witten equations has been considered by Taubes, Walpuski, Doan, and Zhang, etc., in dimension 3 and dimension 4. Our variant of the equations differs from theirs in the way we define our moment map $\mu$ to always be proper. With the properness of the moment map, the technique of Bauer-Furuta in associating a functional of a system of non-linear elliptic PDEs to a certain stable (co)homotopy class of spheres carries over naturally. Such a (co)homotopy class is an invariant of the underlying manifold. This is a work in progress.
4/27/2022
Speaker: Shakil Rafi
Title: Why Knot Theorists are Unsure about Tying Shoelaces, Part 2: Return of the Shoelace
Abstract: We follow-up on the talk given earlier this semester on the problem of additivity of crossing numbers. Whereas Daio introduced an invariant and proved additivity of crossing number for torus knots, this time we explore a different approach followed Tetsuya Ito. This paper (arXived in late 2020) explores braid-foliation technology pioneered by Birman and Menasco to give us an asymptotic bound on the crossing number of connect sums.
4/20/2022
Speaker: Jorge Robinson
Title: The mapping class group of connect sums of S^2 x S^1
Abstract: We review the paper: "The mapping class group of connect sums of S^2 x S^1" where it is studied a short exact sequence involving the mapping class group of S^2 x S^1, the group of outer automorphisms out(F_n), and the group K generated by some sphere twists.
4/13/2022
Speaker: Ryan Holley
Title: Robust Simulations of Turbulent Mixing due to Richtmyer-Meshkov Instability of an Air/SF$_6$ interface using Front Tracking
Abstract: Turbulent mixing due to hydrodynamic instabilities occurs in a broad spectrum of engineering, astrophysical and geophysical applications. Theory, experiment, and numerical simulation help us to understand the dynamics of hydro-dynamically unstable interfaces between fluids. In this talk, an increasingly accurate and robust front tracking method for the numerical simulations of Richtmyer-Meshkov Instability (RMI) is used to simulate the growth rate is presented. Front tracking is an adaptive computational method where the front (interface) between fluids is explicitly followed. Front tracking represents interfaces as lower dimensional meshes moving through a rectangular grid. All the states (pressure, density, and velocity) on the center of each grid cell are updated using the classical fifth order weighted essentially non-oscillatory (WENO) scheme of Jiang and Shu along with Yang's artificial compression. The strength of this method is shown through simulation of the single mode Mach 1.11 and Mach 1.2 shock tube experiments of an air/SF_6 interface by Collins and Jacobs (2002). We observe a very good agreement of early time amplitude and displacement of the Mach 1.11 experiment and 2% discrepancy compared to the Mach 1.2 experiment when 512 grid points per initial perturbation wave length is used.
4/6/2022
Speaker: James Burton
Title: A Random Choice Method of the Glimm's Scheme
Abstract: Numerical methods for the solution of hyperbolic partial differential equations concerns shock formation
and propagation. In order to solve the Euler equations of compressible fluid dynamics, stable, accurate and robust algorithms for shock computations are needed. In our numerical simulations of compressible multiphase flows in 1D, we use Glimm's scheme because of its good algorithmic properties, especially near discontinuities. However, this scheme is difficult to extend to multidimensional hyperbolic problems. Glimm's scheme, using the random choice method (RCM), is revisited to investigate convergence properties using low-discrepancy sampling methods. A set of van der Corput sampling sequences are examined to determine the optimal choice in sampling sequence. Numerical solutions on the various meshes using different sequences are performed to determine the optimal sampling choice with a good convergence rate.
3/30/2022
Speaker: Minh Nguyen
Title: Finite dimensional approximation and Pin(2)-Equivariant property for Rarita-Schwinger-Seiberg-Witten equations
Abstract: Rarita-Schwinger operator Q was initially proposed in the 1941 paper by Rarita and Schwinger to study wave functions of particles of spin 3/2, and there is a vast amount of physics literature on its properties. Roughly speaking, 3/2−spinors are spinor-valued 1-forms that also happen to be in the kernel of the Clifford multiplication. Let X be a Riemannian spin 4−manifold. Associated to a fixed spin structure on X, we define a Seiberg-Witten-like system of non-linear PDEs using Q and the Hodge-Dirac operator d ∗ + d + after suitable gauge-fixing. The moduli space of solutions M contains (3/2-spinors, purely imaginary 1-forms). Unlike in the case of Seiberg-Witten equations, solutions are hard to find or construct. However, by adapting the finite dimensional technique of Furuta, we provide a topological condition of X to ensure that M is non-compact; and thus, contains infinitely many elements.
3/2/2022
Speaker: Patrick Phelps
Title: Space asymptotics for self-similar Navier-Stokes flows
Abstract: The Navier-Stokes equations are a system of partial differential equations governing viscous, incompressible fluid motion. Despite having widespread applications and being extensively studied, solutions evolving from a finite energy state are not known to be unique from a rigorous perspective. Indeed, non-uniqueness has been proven in certain classes, and non-uniqueness for the steady state Navier-Stokes equations can be seen physically in Taylor-Couette flows. Scaling invariant solutions are compelling candidates for non-uniqueness because they have a stationary quality, similar to Taylor-Couette flow. In this talk, we discuss results in a submitted paper, wherein we establish spatial decay rates for scaling invariant solutions under very general conditions. This work improves results in the existing literature in several directions in that we (1) loosen the assumptions on the data required to obtain the optimal decay rate, (2) establish new decay rates for Holder continuous data, and (3) provide decay rates for solutions with rough data. As an application, we obtain a new upper bound on how rapidly non-unique scaling invariant solutions can separate away from a singularity.
2/23/2022 CANCELLED DUE TO INCLEMENT WEATHER
2/16/2022
Speaker: Sam Whitmire
Title: An Aperiodic Set of 11 Wang Tiles
Abstract: We provide an exposition on the above 2021 paper by Emmanuel Jeandel and Michael Rao in this talk. This paper incorporates results from a wide array of mathematical subfields (including discrete geometry, theoretical computer science, computational mathematics, analysis, and combinatorics) to demonstrate the nonexistence of an aperiodic set of ten or fewer Wang tiles as well as the existence of three sets of eleven Wang tiles that are aperiodic. In our presentation, we begin with a basic overview of the tiling problem and then translate these ideas into the language of transducers. After a brief discussion on the results for sets of ten or fewer, we then prove that one of the sets of eleven Wang tiles is aperiodic.
2/07/2022
Speaker: Shakil Rafi
Title: Torus knots and their additivity
Abstract: We will do an expository talk from Daio (2004) where he defines the deficiency of a knot and uses it to prove among other things, that crossing number is additive for torus knots.
2/02/2022 - CANCELLED DUE TO SNOW
1/26/2022
Speaker: Eric Walker
Title: UARK GSC What is...? #1: Tropical Geometry
Abstract: Ever since you were in the womb, you've known that addition and multiplication form primary building blocks for so much of mathematics: we can discuss arithmetic, polynomials, curves, vector spaces, rings, etc. To deviate from this must seem like pure blasphemy. But in tropical geometry, we do exactly this, and remarkably enough, don't throw the baby out with the bathwater. In this talk, we'll see a beginner-friendly exposition into what happens when you replace addition x+y with min{x,y} and multiplication x*y with x+y (and also maybe why the heck you'd wanna do such a thing). Based off of notes by Speyer and Strumfels.
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