5/3/23
Josiah Aakre
Title: How to Not Die Or: A Josephus-Class Integer Sequence Problem
Abstract: Integer sequences show up across all different areas of Mathematics. Different Integer sequences can vary wildly in structure and so techniques used to analyze and describe sequences can be quite different from one sequence to the next. We will be exploring two specific sequences that emerge from a story problem similar to the famous(ish) Flavius Josephus Problem by looking for patterns and invariants among inputs in order to create a process and ultimately a function to describe the sequences.
4/27/23
Yifan Hu
Title: Elements of a discontinuous Galerkin method for laser wakefield acceleration modeling.
Abstract: Laser wakefield acceleration has been the pioneering idea for small-scale particle accelerators since the 1980s. In the theory and application development of this idea, numerical simulations have played a crucial role, and most well-known numerical methods have been adapted to achieve stable and accurate results. In this talk, we present the elements of a discontinuous Galerkin method to model laser wakefield acceleration by solving the relativistic Vlasov-Maxwell system. We demonstrate some key ideas to approach issues such as the high dimensionality, the timestep constraint, and the solenoidal property of the magnetic fields.
4/20/23
Mitchell Ashburn
Title: Vector-matrix algebras
Abstract: The split-octonions have a convenient representation: Zorn's vector-matrices. This allows an otherwise impractical non-associative algebra to have the feel of a typical matrix algebra. We will briefly discuss Zorn's vector-matrices before introducing a family of algebras that generalize their construction, we will call these vector-matrix algebras. We will then lay the groundwork for working with these algebras, discussing what basic properties these new non-associative algebras satisfy as well connecting them to the algebras they are built from.
4/13/23
Billy Duckworth
Title: All Topologies Come from Generalized Metrics (by Ralph Kopperman)
Abstract: Most of us started our education in topology by first studying metric spaces. We quickly learn that there are many interesting and useful topological spaces that cannot have come from a metric. Various generalizations of the concept of metrics (such as are psuedo-metrics, quasi-metrics, etc...) are used in various areas, but none of them are general enough to describe arbitrary spaces. This paper discuses a way to generalize the concept of a metric in a broad enough way to make all topological spaces "metric" spaces. Further, some basic separation properties are described in terms of additional properties of this new distance function.
4/6/23
Grad open forum: no talk given.
3/30/23
Two talks given:
Sydney Miyasaki
Title: Variations on Ramsey’s Theorem
Abstract: We present a variation on the classic Ramsey’s Theorem by adding connectivity restraints. We will explore how these additional constraints affect the corresponding Ramsey Numbers. In the case of two colors, we will completely describe what happens. We will also give a conjectured generalization for three or more colors.
Hope Pungello
Title: A Dynamical System for Gravitational Time Dilation Between Galaxy Clusters
Abstract: Einstein’s theory of relativity predicts a phenomena known as gravitational time dilation, which predicts that the stronger the gravitational field acting on some point in space, the slower time will flow at that point. For example, current estimates suggest that Earth’s crust is about 2.5 years older than its core. While a large body of work exists investigating this phenomena with a focus on objects in space within a gravitational field (such as planets, moons, and satellites), much less work has been done studying how time flows in empty — and specifically expanding — space. This work introduces a dynamical system for time dilation between two or more (theoretical) galaxy clusters, as well as a number of related questions for further investigation.
3/23/23
Chad Berner
Title: Fourier series for singular measures on the Real line
Abstract: Any square integrable function on the torus is a norm limit of its Fourier series, but what if you change the measure from Lebesgue measure to a singular measure? It turns out you will lose orthogonality of the exponentials, but by the Kaczmarz algorithm, any function that is square integrable in this new measure space has a Fourier series converging in norm. Using an operator version of the Kaczmarz algorithm along with analytic operator theory of De Branges, we discuss further results in higher dimensions as well as if and when these results extend to the real line.
3/9/23
Thomas Griffin
Title: Computation of the Future, an Introduction to Molecular Programming
Abstract: Molecular programming is an emerging interdisciplinary field that combines molecular biology, physics, chemistry, computer science, and mathematics. The goal is to apply the far-reaching ideas of computation theory to the realm of molecular reactions in order create structures, circuits, and devices for understanding and manipulating biology at the smallest scales. This talk will discuss the modeling and analysis of Tile Assembly Machines (TAMs) and Chemical Reaction Networks (CRNs) . These systems occur naturally in biological systems and can be built by design in engineered systems and will spontaneously assemble or compute. We will dive into the rules of these systems, various examples, and real world applications that already exist and possible machines of the near future.
3/1/23
Laura Gamboa Guzman
Title: Machine-Assisted Proofs and Proof Assistants: A quick overview
Abstract: Human and computer interaction is something we take for granted nowadays when it comes to making certain tasks easier, like doing our taxes, writing formal documents, or solving a specific math/logic problem by encoding it in our favorite programming languages, but when it comes to developing mathematical theories, most people don't think of much other than a few algorithms to solve some numerical problems or maybe a search for some integer sequences. However, since a mathematical proof is nothing but a sequence of facts where each one is a hypothesis or follows from some of the previous ones, verifying the correctness of a proof is a process that a machine can carry out. This process is known as Proof verification, which is not a way to produce machine-assisted proofs, but it inspired the creation of tools capable of finding proofs automatically or interactively. In this talk, I will briefly overview what we understand about machine-assisted proofs and how to work with an interactive theorem prover. We will see examples of what computers can do by themselves with today's technology and examples of digital libraries that contain formalizations of many relevant mathematical theories. Additionally, we will see the main challenges one typically encounters when formalizing complex mathematical theories. I hope this talk will convince you to use a proof assistant to formalize some of your favorite theorems or look at what these tools are capable of.
2/23/23
Lillian Uhl
Title: Convenient Categories for Functional Analysis: Cast Out From Paradise Into Abstract Nonsense
Abstract: Mainstream analysis tends be resistant to highly abstract means of thinking, at least in the common conception. Nonetheless, analysis, especially functional analysis throughout 20th century mathematics, has been the principle inspiration behind some of the most abstract areas in mathematics which exists today; one should not forget that Cantor was investigating the pointwise convergence of Fourier series, nor that Grothendieck was a functional analyst before anything else! Though it may seem unfashionable to take a highly abstract approach to analysis, the power of such is undeniable: several fundamental theorems underpinning modern analysis are the direct fruits of these labours. The goal, as one must always keep in mind when working at such a level of abstraction, is to strike a good balance between generality and specificity: our desire as mathematicians, first and foremost, is a convenient setting within which we may investigate our own questions. Thus, join me on an expedition through some modern abstract nonsense (born of functional analysis and returning home once more), to find a convenient framework within which we may reside. Along the way, we shall encounter categories with various amounts of structure, consider what it means to have a "topological structure" & tour through a small collection of such structures, and conclude with finding a remarkably convenient category within which to perform analysis.
2/15/23
Joshua Rice
Title:
Abstract: A standard graded C-algebra R is said to be Koszul if the minimal graded free R-resolution F is linear; by linear, we mean in every differential matrix of F, every entry is a linear form. The definition of a Koszul algebra seems restrictive, but as it turns out, Koszul algebras possess remarkable numerical and homological properties and are ubiquitous in commutative algebra. Thus, it is an interesting question to determine families of rings that admit the Koszul property. In this talk, we introduce some geometrical concepts in projective space, free resolutions, summarize some classical examples of Koszul algebras, discuss certain numerical invariants of the coordinate ring R of a generic set of m lines in complex projective n-space, as well as when these rings admit the Koszul property.
2/9/23
Enrique Gomez-Leos
Title: Perturbed random 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. Recall that the Erdos-Renyi random graph $G_{n,p}$ consists of the vertex set $[n]$ where each edge is present, independently, with probability $p$. In this regime, a key question is to establish the threshold for which $G_{n,p}$ contains a spanning subgraph. Bohman, Frieze, and Martin introduced the perturbed random graph model which connects the two questions together. In this talk, we will make the necessary preparations for introducing this graph model and present recently developed theory.
2/2/23
Zac Brennan
Title: Wasan - Traditional Japanese Mathematics
Abstract: During the Edo period (1603-1868), governmental isolationist policy (sakoku) fostered the independent development of mathematics in Japan, called wasan, or "traditional calculations." This time saw mathematics in Japan, independent of and parallel to Europe, grow from a basic understanding of arithmetic to achievements such as calculating pi to dozens of digits of precision, an imagining of calculus and its methods to solve real-world problems, and the development of the determinant (nearly ten years before Leibniz) to solve arbitrary systems of equations. This talk attempts to give a brief survey of the history of wasan and some of its more remarkable aspects while providing societal context for these developments and the mathematicians behind them.