5/2/24
Matt Burnham
Title: Extremal properties of strong tournaments
Abstract: A tournament is an orientation of the complete graph K_n. A tournament is called strong if there is a directed path from every vertex to every other vertex. In this talk, we discuss some extremal properties of strong tournaments, including the number of 3-cycles and the lexicographic ordering of out-degree sequences.
4/25/24
Billy Duckworth
Title: Topology isn't about Convergence
Abstract:'Topology ... presents the weakest structure you can establish on a set in order to have the two very important notions of convergence and continuity.' -Frederic Schuller
This sentiment has been repeated by many, but it is not quite true. This talk will show a couple of reasonable, naturally occurring examples of convergence that cannot be described topologically.
4/18/24
Kean Fallon
Title: Equiangular Tight Frames in Symplectic Vector Spaces
Abstract: Consider the problem of packing at least d lines in a d-dimensional vector space in a good way, under the condition that acute angles between lines is bad. Finding a good packing is then equivalent to minimizing a metric, called coherence, that measures the angles between lines. Taking unit-norm representatives of each line and identifying them with columns of a short, wide matrix, finding solutions to this optimization problem amounts to identifying short, wide matrices known as ETFs, or equiangular tight frames. ETFs have been studied considerably over real and complex inner product spaces and more recently in vector spaces over finite fields with analogous inner products. However, nothing is known about ETFs when working over a symplectic vector space, i.e. a vector space with a nondegenerate alternating bilinear form. In this talk, I will discuss work done with my advisor, Joey Iverson, in developing this theory in a symplectic setting, including some interesting preliminary results and plans for future research. (This talk will be a practice run of my preliminary oral exam)
4/11/24
Lillian Uhl
Title: What's a Smooth Connection?
Abstract: One of those buzzwords that mathematicians hear every now and then is "Connection", and evidently it's got something to do with manifolds and parallel transport. Dissatisfied with all definitions of a connection I'd previously seen (all using far too many indices and/or coordinates), I struck out to find an acceptable definition of a connection grounded in what differential geometry is all about: pointwise linear algebra! Join me as I explain in ⁽ʰᵒᵖᵉᶠᵘˡˡʸ⁾ simple terms using infinitesimals and projection operators!
4/4/24
Grad committee open forum: No talk given.
3/28/24
Coy Schwieder
Title: Edge-coloring the $d$-dimensional hypercube ($Q_d$) so every copy of $Q_2$ has four colors
Abstract: Given a graph $G$, a subgraph $H$ of $G$, and an integer $q$, where $1 \leq q \leq |E(H)|$, an $(H,q)$-coloring of $G$ is an edge coloring of $G$ in which every copy of $H$ in $G$ has at least $q$ colors appearing in its edges. Let $f(G,H,q)$ be the minimum number of colors needed for an $(H,q)$-coloring on $G$. Erd\"os and Gy\'arf\'as originally posed the question in 1997 for $G = K_n$ and $H = K_p$, the complete graphs on $n$ and $p$ vertices respectively, and determined a general upper bound. In 1993, Faudree, Gy\'arf\'as, Lesniak, and Schelp had already answered the case when $G = Q_d$ and $H = Q_2$, the $d$-dimensional and 2-dimensional hypercubes, and $q = 4$. In this talk, we will briefly discuss observations about the structure of $Q_d$, and how the structure gives $d \leq f(Q_d, Q_2, 4) \leq 2(d+1)$ with little effort. We will also discuss the coloring given by Faudree et. al. that proves $f(Q_d, Q_2, 4) \leq d$, which makes use of the Cartesian product of graphs, proving $f(Q_d, Q_2, 4) = d$.
3/18/24
Enrique Gomez-Leos
Title: Conflict-free hypergraph matchings and generalized Ramsey numbers
Abstract: Given positive integers n,p,q, where p≤n, 1≤q≤(p choose 2), a (p,q)-coloring of the complete graph Kn is an edge coloring of Kn in which every clique on p vertices has at least q colors appearing in its edges. Let f(n,p,q) be the minimum number of colors needed for a (p,q)-coloring on Kn. Erdős and Gyárfás originally posed the question in 1997 and determined a general upper bound. In addition to determining the linear and quadratic threshold, they also showed that 5/6(n-1) ≤ f(n,4,5) ≤ n. Recently, Mubayi and Joos introduced a new method for proving upper bounds on these generalized Ramsey numbers; by finding a “conflict-free" matching in an appropriate auxiliary hypergraph, they determined the value of f(n,4,5) to be 5/6n + o(n). In this talk, we will introduce recent improvements to f(n,5,8). Indeed, we show that f(n,5,8) ≥ 6/7(n-1) and discuss how to use the conflict-free hypergraph matching method to show that f(n,5,8) ≤ n + o(n). This is joint work with Emily Heath, Coy Schwieder, Alex Parker, and Shira Zerbib.
3/7/24
Daniel Arreola
Title: The n=3 Case of Fermat's Last Theorem
Abstract: We will prove that the sum of two non-zero integer cubes is never a cube. The proof will involve Eisenstein integers, so we will start with an overview of this ring.
2/29/24
Alice Kessler
Title: The Basics of Infinity Category Theory: An Introduction
Abstract: Infinity categories naturally come up in homotopy theory with the "fundamental infinity groupoid" where we wanted an algebraic invariant stronger than just the fundamental groupoid. In the fundamental groupoid, we consider paths up to homotopy, which flattens higher homotopical information since two paths can be homotopic in nontrivial ways. The fundamental infinity groupoid remembers all higher-order homotopical information of the paths. In some ways, homotopy theory ends up being essentially the study of infinity groupoids. In this talk we will review some basic categorical notions, introduce the formalism needed for infinity category theory, mention the "homotopy 2-category", and draw parallels between infinity category theory and classical category theory.
2/22/24
Chad Berner
Title: Backward shift invariant subspaces of the Hardy space
Abstract: We define the Hardy space on the disk, and we discuss some of its properties. Next, we proof a representation theorem for analytic functions on the disk with non-negative real part, which gives a correspondence between finite Borel measures and bounded analytic functions. We further explore this correspondence with singular measures and the normalized Cauchy transformation, which encodes information about backward invariant subspaces of the Hardy space. Finally, we discuss a normalized Cauchy transformation in two dimensions and how this relates to another approach of backward invariant subspaces.
2/15/24
Joe Miller
Title: Constructing Non-isomorphic Real Projective Planes using Algebraically Defined Graphs
Abstract: Projective planes are often defined using three axioms which say (i) any two points lie on a unique line, (ii) any two lines intersect at a unique point, and (iii) there exists four points in which no three of them lie on the same line. This presentation will show how the standard real projective plane RP2 can be understood in this way. We will also show that "most of" RP2 can be described as a graph where adjacency between two vertices occurs if and only if a polynomial equation is satisfied. We give these graphs the insightful name of algebraically defined graphs. We ask ourselves the question of "are there any non-isomorphic algebraically defined graphs?", discuss why it's challenging to answer, and provide two nonisomorphic such graphs.
2/8/24
Sydney Miyasaki
Title: Almost Congruent Triangles
Abstract: We will present a paper of Balogh, Clemen, and Dumitrescu on almost congruent triangles. The notion of almost-congruence will be defined, and a problem about the number of such triangles stated and proved. The proof is fun and light, involving a bit of high-school geometry and a basic introduction to hypergraphs.
12/7/23
Preeti Sar
Title: Asymptotic Preserving Scheme for the Kinetic Boltzmann-BGK Equation
Abstract: The kinetic Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) collision operator describes the motion of a fluid for the simulation of gas dynamics over a wide range of Knudsen numbers with a simplified collision operator. The small scales in kinetic and hyperbolic equations lead to different asymptotic regimes which are expensive to solve numerically. Asymptotic preserving schemes are efficient in these regimes, which preserve at the discrete level, the asymptotic limit which drives the microscopic equation to its macroscopic equation. Such schemes allow the numerical method to be stable at fixed mesh parameters for any value of the Knudsen number, including in the fluid (very small Knudsen numbers), slip flow (small Knudsen numbers), transition (moderate Knudsen numbers), and free molecular flow (large Knudsen numbers) regimes. In this work, we develop an approach for solving the Boltzmann-BGK equation for achieving both arbitrary high-order and asymptotic preservation. This numerical scheme is applied to multidimensional flows and hence is an extension to previous work done in one dimension.
11/30/23
Sydney Miyasaki
Title: Computing in flag algebras
Abstract: We will give an intuitive description of flag algebras and examine how to retrieve combinatorial information from them using semidefinite programming.
11/16/23
Alice Kessler
Title: The Characteristic Function and Fusion Technique
Abstract: Transfer Systems are a combinatorial object of interest due to their applications in homotopy theory. In this talk we define transfer systems, give examples, and show some results about fusions of transfer systems and the characteristic function which we developed last summer in the eCHT REU.
11/9/23
Lillian Uhl
Title: Pointwise Differential Structure and Banach Algebraic Geometry
Abstract: In traditional algebraic geometry over commutative Noetherian rings, a key structural result in the theory of their ideals is that of primary decomposition: every ideal can be written as the intersection of finitely many primary ideals, the associated primes of which as closely related to the given ideal's radical. In sufficiently nice commutative Banach algebras, those which sit inside the algebras of continuous functions in a good way, it turns out to be the case that there is a plethora of primary ideals, to the extent that we may expect not just a (big) primary decomposition but a closed primary decomposition at that! In both of these situations, these (closed) primary ideals are intimately connected with "higher order" or "differential data" at certain points. My talk will be more of a survey and introduction to this area of investigation, with questions encouraged.
11/2/23
Kimberly Hadaway
Title: Directional Derivative of Kemeny's Constant
Abstract: In a connected graph G, Kemeny's constant gives the expected time for a random walk from an arbitrary vertex x to reach a randomly-chosen vertex y. Kemeny's constant is a measure of how well a graph is connected, and it is unknown how the addition or removal of edges will affect Kemeny's constant. Inspired by using the directional derivative of the normalized Laplacian as a centrality measure, we derive the directional derivative of Kemeny's constant for several graph families in this presentation. In addition, we find sharp bounds for the directional derivative of an eigenvalue of the normalized Laplacian and for the directional derivative of Kemeny's constant. We end with a discussion of some fun examples and pose some problems for future exploration.This is joint work with C. Albright, A. Holcombe Pomerance, J. Jeffries, K. Lorenzen, and A. Nix.
10/26/23
Grad open forum: no talk given
10/19/23
Mitchell Ashburn
Title: An equivalence of anticommutative formed algebras
Abstract: Beginning with a formed algebra (A,\beta), we can construct an algebra Z(A,\beta). Determining whether two formed algebras become isomorphic under this operation can be reduced to a question of the formed algebras themselves. This gives us an equivalence relation on the class of anticommutative formed algebras with non-degenerate invariant symmetric bilinear forms. Using this as motivation, we will explore what equivalence classes look like under this equivalence relation, focusing mostly on examples concerning Lie algebras. We will give a couple of concrete examples, specifically concerning the Lie algebras sl(2) and so(3) over the reals.
10/12/23
Enrique Gomez-Leos
Title: Tiling perturbed multipartite graphs
Abstract: Embedding problems form a central part of both extremal and random graph theory. Many results in extremal graph theory concern minimum degree conditions that guarantee the existence of some spanning substructure. In the random setting, a key question is to establish the threshold for which $G_{n,p}$ contains a spanning subgraph. The perturbed random graph model combines these problems. In this talk, we’ll discuss recent joint work with Ryan Martin.
10/5/23
Chad Berner
Title: Fourier Dextroduals
Abstract: Given a finite Borel measure on the torus, can every element of its L^2 space be expressed as a Fourier series? We will discuss examples and properties of measures on the torus that admit Fourier frame-like expansions for their corresponding L^2 spaces.
9/28/23
Two talks given:
Dauda Gambo
Title: Role of extended memory effect on the flow of magnetized fractional Maxwell fluid
Abstract: Mass transport phenomenon of an electrically conducting viscoelastic fluid in an annular domain has been examined. The generalized memory effect on the hydrodynamic behavior of the fluid is described by the time-fractional derivative based on the Prabhakar model. The fractionalized momentum equation governing the flow is resolved semi-analytically by a two-step process. The use of the Prabhakar operators shows the possibility of a convenient choice of fractional parameters. Consequently, limiting the Prabhakar derivative renders the fractionalized model in the Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) sense respectively. Impact of the pertinent parameters on the flow formation has also been considered.
Lillian Uhl
Title: Parallel Mathematics
Abstract: At the very heart of mathematics, there's a duality which very few seem to exploit or even recognize. In fact, the only subject of study where I've ever seen this emphasized was in electrical engineering: given two passive two-terminal electrical circuits, we may either combine them in series with one another or in parallel with one another. In the mathematical representations of the circuits, these two methods of combination correspond to addition for series combination and to the reciprocal of the sum of the reciprocals for parallel combinations.My talk with be a more exploratory, less formal talk, where I introduce this operator and talk about some of the mathematical aspects of it. Time permitting, we would touch on such diverse topics as convex analysis analysis, dimensional analysis, and projective geometry.
9/21/23
Daniel Arreola
Title: Basis-free Dimension
Abstract: Is it possible to do linear algebra without ever talking about vectors? Perhaps not entirely, but a surprising number of linear algebraic notions can be described (and hence studied) categorically, i.e., without reference to vectors. In this talk I will describe the Frobenius-Perron dimension, which extends the notion of vector space dimension to "generalized vector spaces" (objects in tensor categories). We will see that there exists a vector space-like object whose dimension is the golden ratio. Though the application is category theoretic, anyone familiar with basic linear algebra and ring theory should be able to follow along.
9/14/23
Agil Muradov
Title: No title given
Abstract: I will present an alternative way of constructing the set of real numbers, different from the Dedekind cuts and from the Eudoxus reals. Effectively I will be constructing the set of real numbers from just two integer numbers {0,1} but skipping explicitly constructing N, Z and Q.