Session Index
Click a session title to jump to the abstracts.
Faculty Session 1
1:10pm--2:10pm,
Speakers: Chris Catone, Ken Monks, Alexander Diaz-Lopez
Faculty Session 2
1:10pm--2:10pm,
Speakers: Melissa M Fuentes, Michael Carrion, Zaphenath Joseph
Faculty Session 3
1:10pm--2:10pm,
Speakers: Benjamin Pentecost, Rommel G. Regis
Faculty Session 4
1:10pm--2:10pm,
Speakers: Susanna Molitoris-Miller and Brian Kronenthal, Samantha Pezzimenti, Wing Hong Tony Wong
Faculty Session 1
1:10pm, Chris Catone (Albright College)
Multiplicative Functions: A play in two acts.
View Abstract
A multiplicative function is a function on the natural numbers such that $f(mn)=f(m)f(n)$ whenever $\mathrm{gcd}(m,n)=1.$ We investigate the group of these functions, and discuss the pedagogical benefits of adding this infinite group to your students' repertoire in the Abstract Algebra curriculum.
Close Abstract1:30pm, Ken Monks (University of Scranton)
Proof Verification with Lurch
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Would your students benefit from an easy-to-use, open-source, web-based word processor that could check their assigned mathematical proofs? In this talk we introduce Lurch, our software project designed specifically for this purpose. In particular, we will discuss how Lurch was integrated into our undergraduate introduction to mathematical proof bridge course during the Spring 2024 semester. Additionally, we will explain how you can use this software and accompanying course materials, and customize it for your own purposes.
Close Abstract1:50pm, Alexander Diaz-Lopez (Villanova University)
Using AI models in a proof-based course
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In this interactive talk, we will showcase different uses of generative AI models in upper level proof-based courses. We will do a live demonstration of the use of AI models, so we encourage the audience to come to the talk with questions and things you want us to try.
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Faculty Session 2
1:30pm, Melissa M Fuentes (Villanova University)
Graph-theoretic Extensions of the Erdős-Ko-Rado (EKR) Theorem
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The Erdős-Ko-Rado (EKR) Theorem is a fundamental result in extremal finite set theory that states that any family of $k$-subsets of a finite set containing at least $2k$ elements which is intersecting—meaning that no two subsets in the family are disjoint—cannot exceed the size of the family of all $k$-subsets that include a fixed element $x$. This latter family, known as a star centered at $x$, is trivially intersecting. In this discussion, we explore recent advancements in a graph-theoretic generalization of the EKR theorem, focusing on intersecting families of subgraphs within the perfect matching graph on $2n$ vertices, the graph consisting of $n$ disjoint edges.
Close Abstract1:50pm, Michael Carrion, Zaphenath Joseph (Villanova University)
Well-covered Erdos--Ko--Rado graphs
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The Erdos–Ko–Rado (EKR) theorem is a foundational result in extremal finite set theory that states any family of r-subsets of an n-element set (with $r\leq n/2$) can be no bigger than the family of all r-subsets containing a fixed element x, which is trivially intersecting and called a star centered at x. We consider recent work on a graph-theoretic generalization of the theorem, partly inspired by a conjecture of Holroyd and Talbot (2005), that any graph G will have the EKR property for $1 \leq r \leq \mu(G)/2$, where $\mu(G)$ is the size of the smallest maximally independent set in G. If a counterexample to this conjecture exists, it is suspected to be well-covered. We discuss new results proving that a certain class of well-covered graphs have the EKR property, specifically graphs constructed by adding a clique pendant to every vertex in a complete graph.
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Faculty Session 3
1:10pm, Benjamin Pentecost (West Chester University of Pennsylvania)
An Enhanced Augmented Matched Interface and Boundary (AMIB) method for solving elliptic and parabolic problems on irregular 2D domains
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A new method called Augmented Matched Interface and Boundary (AMIB) has been developed to solve partial differential equation models, such as the heat equation, over irregular two-dimensional domains. The original AMIB method features unique numerical treatments to solve problems with various boundary conditions and shapes, resulting in highly accurate and efficient numerical solutions. However, recent numerical experiments have revealed that the original AMIB method can fail when dealing with sharply curved boundaries. To address this issue, new numerical techniques have been introduced in our latest work to enhance the robustness of the AMIB method. These techniques have been numerically verified to improve the accuracy and efficiency of the AMIB method when solving various problems with sharply curved boundaries. Due to its success, we plan to use this method to simulate Magnetic Fluid Hyperthermia (MFH) cancer treatment by solving the Pennes Bioheat Equation on a domain that consists of a tumor and the surrounding healthy tissue.
Close Abstract1:30pm, Rommel G. Regis (Saint Joseph's University)
Cosine Measures and Uniform Angle Subspaces
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A set of vectors S positively spans an n-dimensional real vector space if every vector in that space is a nonnegative linear combination of elements of S. The cosine measure of S is the cosine of the largest possible angle between a nonzero vector and S. The notion of cosine measure is used in derivative-free optimization, and it provides a way of quantifying the positive spanning property of a set of vectors. Next, given a finite set S containing at least two nonzero vectors, the uniform angle subspace of S is the set of nonzero vectors that make uniform angles with every vector in S. This talk will discuss some of the properties of the cosine measure, uniform angle subspaces, and their connections. It will also explore possible connections between spherical Voronoi diagrams and the solution to the cosine measure problem.
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Faculty Session 4
1:10pm, Susanna Molitoris-Miller and Brian Kronenthal (Kutztown University)
CATANbinatorics and the Probabilty of Constructing a Legal Board
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The popular board game Catan requires players to construct a new board, within certain parameters, every time they play. In this talk we use combinatorial techniques to count how many boards could be constructed with and without observing the restriction that no two red numbers (6 or 8) are placed on adjacent tiles. We then use these results to determine the probability that a player who ignores this often overlooked rule will actually construct a board which adheres to it.
Close Abstract1:30pm, Samantha Pezzimenti (Penn State Brandywine)
The Fish, the Crab, and the Kraken: Knot Mosaics
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Knot projections can be constructed by arranging a set of mosaic tiles into a square grid. These knot mosaics, which were first introduced by Lomonaco and Kauffman in 2008, are useful in tabulating knots and defining invariants, such as the mosaic number. In this talk, we will discuss a modified set of tiles that can be used to represent front projections of Legendrian knots, which are knots that have some additional geometric structure affecting their shape. We define variations of the mosaic number in this context and explore the implications. This work was joint with Tony Wong and our REU students.
Close Abstract1:50pm, Wing Hong Tony Wong (Kutztown University of Pennsylvania)
Nonisomorphic affine planes over $\mathbb{R}$ arising from algebraically defined graphs
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Given a function $f:\mathbb{R}^2\to\mathbb{R}$, a two‐dimensional algebraically defined graph $\Gamma_{\mathbb{R}}(f)$ is a bipartite graph where each partite set is a copy of $\mathbb{R}^2$, and two vertices $(a,a_2)$ and $[x,x_2]$ are adjacent if and only if $a_2+x_2=f(a,x)$. It is known that the point-line incidence graph of the classical real affine plane is isomorphic to $\Gamma_{\mathbb{R}}(XY)$, but it was unknown whether there exists a polynomial $f\in\mathbb{R}[X,Y]$ such that $\Gamma_{\mathbb{R}}(f)$ gives rise to a nonisomorphic affine plane over $\mathbb{R}$. We answer this question affirmatively in this talk.
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