Student Talks Session Index
Click a session title to jump to the abstracts.
Graduate Session I-A
2:15--2:51pm, BPMC 204
Speakers: Cameron Campbell, Lane D'Alessandro
Undergraduate Session II-A
2:15--3:10pm, BPMC 205
Speakers: Grant Fickes, Alexander Miller, Jacob McCann, Colin Jones, Rachel Chambers, Michelle Ly
Undergraduate Session II-B
2:15--3:10pm, BPMC 208
Speakers: Melea Roman, Garrett Bowser, Alexander Vetter, Tarang Saluja, Benjamin Warren
Undergraduate Session II-C
2:15--2:59pm, BPMC 210
Speakers: Levi C. Nicklas, Ethan Clever, Christopher Craig, Vincent Sergi
Undergraduate Session II-D
2:15--2:59pm, BPMC 211
Speakers: Bryn Woodling, Magdalena Kalinowska, Christopher Williams, Ronald Boorman Jr, Javonni Banks, Natasha Stuckey, Owen Vazquez, Chris Miller
Undergraduate Session II-E
2:15--2:59pm, BPMC 212
Speakers: Yinxi Li, Liz Dulac, Jeremy Budgeon, Yuqing Liu
Graduate Session I-A
BPMC 204
2:15, Cameron Campbell (West Chester University of Pennsylvania)
Solving the Interface Problem: An Alternating Direction Implicit Approach
View Abstract
Interface problems are a large class of problems arising in Physics, Biology, Engineering and Materials, that study the change of a physical quantity, such as heat or electrostatic potential, propagating across a material interface. Due to the irregular shape of the interface, solutions to interface problems can only be found through numerical approximation. Due to the presence of the interface, classical numerical methods fail to find accurate approximations. This case calls for development of new more accurate and efficient numerical methods. This presentation is an overview of our study of a well-tuned matched Alternative Direction Interface (ADI) method for solving the interface problems with the most general physical interface jump conditions. The ADI method for solving two-dimensional interface problems will be presented, as well as our plan to improve the methods in the aspects of efficiency, accuracy, and stability.
2:33, Lane D'Alessandro (West Chester University of Pennsylvania)
Modeling Individual Reproductive Fitness using Resource Allocation leading to a Post-reproductive Life
View Abstract
Fitness is environment-specific, and many organisms have evolved the ability to alter resource allocation based on perceived environmental cues (e.g., food/mate availability, predation risk). We are developing an optimization model that examines relative resource allocation into growth, reproduction, and defensive morphology under varying conditions. Specifically, we are investigating how reproductive investment in terms of rate and amount changes as a function of predation risk. The survival function utilizes a modified Gompertz-Makeham law for mortality. The fecundity function is the product of the reproductive schedule and output. The reproductive schedule utilizes a gamma distribution and the output is modeled exponentially. Optimizing the fitness model yields the optimal resource allocation and resulting reproductive schedule. This allows us to understand the effects of phenotypic plasticity in life-history traits on the evolution of a post-reproductive period.
Back to index
Undergraduate Session II-A
BPMC 205
2:15, Grant Fickes (Kutztown University of Pennsylvania)
Maximum proper diameter of 2-connected graphs
View Abstract
A properly colored path is a path in which no two consecutive edges have the same color. A properly connected coloring of a graph is one in which there exists a properly colored path between every pair of vertices. Given a graph \(G\) with a properly connected coloring, the proper distance between any two vertices is the length of a shortest properly colored path between them. Furthermore, the proper diameter of \(G\) is the largest proper distance between any pair of vertices in \(G\). Since the proper diameter of \(G\) is a function of its coloring, we can refer to the maximum proper diameter of \(G\), that is, the maximum value of the proper diameter across all properly connected colorings of \(G\). This talk will discuss the \(2\)-connected graphs with the largest proper diameter.
2:26, Alexander Miller (Kutztown University of Pennsylvania)
Extensions on Conway's Wizard Problem
View Abstract
Conway's Wizard Problem can be mathematically summarized in the following way. Given a sum \(s\) and a product \(p\), do there exist two \(n\)-partitions of \(s\) into distinct multisets such that both multisets have the same product \(p\)? If there are, we call \(s\) sum-admissible and \(p\) product-admissible. From this context, we define the following two functions. (1) \(f(s)=\) number of \(n\) values such that \(s\) is sum-admissible. (2) \(g(s)=\) number of \(p\) values such that \(s\) is sum-admissible; the case \(g(s)=1\) is precisely what we need to solve Conway's problem. We derive and prove the formula for \(f(s)\), and determine the value of \(s\) that gives \(g(s)=1\).
2:37, Jacob McCann (Kutztown University of Pennsylvania)
Nagata-Smirnov Metrization Theorem
View Abstract
The Nagata-Smirnov Metrization Theorem is a beautiful result of point-set topology, but remains inaccessible to many people due to unfamiliarity with point set topology. We present a brief overview of point-set topology and metric spaces, and a walk-through of a proof of the Nagata-Smirnov Metrization Theorem intended for someone with no background in topology. We will also see how the Smirnov Metrization Theorem is a corollary of the Nagata-Smirnov Metrization Theorem.
2:48, Colin Jones (Eastern University)
Vertex-Minimal Planar Graphs With Prescribed Automorphism Groups
View Abstract
In 1939, Frucht proved that for any finite group G, there exists a graph \(\Gamma\) such that the automorphism group of \(\Gamma\) is isomorphic to G. Naturally, this result gave rise to numerous extremal problems in graph theory. For instance, vertex-minimal graphs with a prescribed automorphism group are the subject of prior research by numerous authors. In this talk, I will discuss my proof of a conjecture made in 1980 by Marusic on the order of vertex minimal planar graphs with cyclic symmetry of even order. My proof completes a theorem giving the order of all vertex-minimal planar graphs with cyclic automorphism groups. I will also discuss further my proof regarding the order of vertex-minimal planar graphs with dihedral symmetry.
2:59, Rachel Chambers, Michelle Ly (Washington College)
Cobwebs, Bifurcations and Fractals
View Abstract
Actuators, sensors, and computational components may be controlled intermittently in time and, thus, inconsistently perform any given operation. The inconsistencies in control operation directly influence the time domain of the control system. It was shown that chaos can be introduced into a system by changing only the time domain. Can stochastic time domains produce this same fractal structure? We answer in the negative. Stochastic time domains appear to diminish much of the fractal nature. However, in our stochastic domains, both chaos and stability were shown. We sought to verify the relationship between bifurcation plots and chaos (as measured by the Markus-Lyapunov exponent). We verify a direct correlation between the two. Lastly, we discovered a new three-dimensional fractal with unexpected properties. Intricate and illuminating, this fractal further proves that careful choices in time domain can allow longer time steps to be taken with stability; we must choose those time steps carefully.
Back to index
Undergraduate Session II-B
BPMC 208
2:15, Melea Roman (Cedar Crest College)
Representing integers as the sum of two polygonal numbers in the ring \(Z_n\)
View Abstract
An analogue to Euler's proof of Fermat's theorem on the sums of two squares, done by Harrington, Jones and Lamarche, proved that in the ring of integers modulo n, where n is an integer greater than or equal to 2, all z in Zn can be written as the sum of two squares (4-gonal numbers), with and without allowing zero as a summand. This research extends those results to all n-gonal numbers in the ring \(Z_n\).
2:26, Garrett Bowser (Temple University)
Patterns in Collatz-Mapped Integer Trajectories
View Abstract
The Collatz conjecture is a longstanding open problem that dates back to 1937. We study the behavior of the values that arise as integers are iterated through the Collatz map. We introduce the term trajectory to describe the length of the list of integers produced by an integer's repeated iteration through the map before being disqualified as a counterexample to the conjecture. Significant subsets of the natural numbers can be disqualified as potential counterexamples to the conjecture by iterating integer forms through the Collatz map. We discuss an integer form tree that is developed using this procedure.
2:37, Alexander Vetter (Villanova University)
Reed-Müller Batch Codes
View Abstract
Batch codes, introduced by Ishai et al. encode a string \(x\in\Sigma^{k}\) into an m-tuple of strings, called buckets. In this presentation, we consider multiset batch codes wherein a set of t-users wish to access one bit of information each from the original string. We introduce a concept of optimal batch codes. Our main work provides batch properties of Reed-Müller codes. We examine locality and availability properties of first order Reed-Müller codes over any finite field. We then show that binary first order Reed-Müller codes are optimal batch codes when the number of users is 4 and generalize our study to the family of binary Reed-Müller codes which have order less than half their length.
2:48, Tarang Saluja (Swarthmore College)
Greedy Avoidance of k-term Arithmetic Progressions
View Abstract
We can construct sets of natural numbers that avoid k-term arithmetic progressions by using a "greedy algorithm." This greedy algorithm starts by placing 0 in the set. For every subsequent integer, we find that n is part of the set if and only if n does not form an arithmetic progression with k-1 terms that were already part of the set. We refer to such n as "permissible." The goal of the greedy algorithm is to populate this set as much as possible. However, another way to populate the set is to use a "lazy greedy algorithm," which adds each permissible n to the set with a probability p. This project explores how adding elements with a probability p affects the cardinality, density, and asymptotic growth rate of the set for large and small values of n.
2:59, Benjamin Warren (Swarthmore College)
Modeling Anyonic Systems: Group Theoreticity in Modular Categories
View Abstract
Topological Quantum Computing harnesses swapping anyons to perform computations which are topologically protected against decoherence. We can use modular categories to model the anyon systems which we could use to make computations. In such categories, the simple objects correspond to anyon species which could be fused with each other to form superpositions. Each modular category has a representation of the braid group associated with it, which tells us how much information is preserved in the computation. We specifically studied integral metaplectic modular categories: categories with integer dimension and the same fusion rules as the quantum group category \(SO(N)_2\). We proved that such categories are group theoretical, which means that the associated braid group representation has finite image. This means that the anyon systems they correspond to can not be used to create a universal quantum computer.
Back to index
Undergraduate Session II-C
BPMC 210
2:15, Levi C. Nicklas (Shippensburg University of Pennsylvania)
Chomp: Some Winning Strategies
View Abstract
Chomp is a popular game studied in Game Theory. Chomp is a two-player impartial game which can be played with a matrix, or in our case a rectangular chocolate bar: whoever has to "eat" the top left piece of chocolate loses the game. For any \(m \times n\) chocolate bar, we call the top left position as \((1,1)\) and the bottom right position as \((m,n)\). Whoever takes the chocolate piece in the position \((1,1)\) loses. When the chocolate piece in the position \((i,j)\) is taken, every remaining piece in the position \((i',j')\) for all \(i' \geq i\) and \(j' \geq j\) is removed. In this talk, we will explore the winning strategy for the first player, invented by John Nash. We also show that the strategy can be applied and analyzed in the three-dimensional case of Chomp.
2:26, Ethan Clever (Shippensburg University of Pennsylvania)
Euclid's Windmill
View Abstract
Over 2000 years ago, the renowned Greek mathematician Euclid wrote what many consider to be the pinnacle of ancient academics: the Elements. The Elements consist of 13 books and each book provides extensive proofs of both old and new mathematical ''propositions". Book I of the Elements focuses on the most basic geometric principles but was later surrounded by controversy over a postulate known as the ''parallel postulate". Euclid's assumption of this postulate led him to prove one of the most famous geometric theorems in all of history; known as Proposition I.47, Euclid's proof of the Pythagorean theorem was the climax of Book I. Unlike other proofs of this theorem that came after him, Euclid proved it geometrically rather than algebraically. This talk will present Euclid's inspiring intuition, brilliant creativity, and imaginary ''windmill" which led him to produce one of the most ingenious proofs of his time.
2:37, Christopher Craig (Shippensburg University of Pennsylvania)
Exploring the USA MTS problem from April 1, 2018
View Abstract
The problem asked by USA Mathematical Talent Search on April 1, 2018 is as follows: Every point in the plane is colored either red, green, or blue. Prove that there exists a rectangle in the plane such that all four of its vertices are the same color. There are two separate solutions provided to answer this problem. In this talk, I suggest a modification to this problem and provide the solution.
2:48, Vincent Sergi (Ursinus College)
Ghost Series and a Motivated Proof of Bressoud's \(4k-2\) Companion to the Andrews-Gollnitz-Gordon Identities
View Abstract
We present a "motivated proof" of Bressoud's \(4k-2\) companion to the Gollnitz-Gordon-Andrews identities. Previously, a similar motivated proof of the Rogers-Ramanujan identities was given by G.E. Andrews and R.J. Baxter, and was generalized to Gordon's identities by J. Lepowsky and M. Zhu. It is expected that the steps followed in motivated proofs will give insight into representation theory of vertex operator algebras, as the steps in these proofs have interpretations in that setting. In our proof, we also introduce generating functions we call "ghost" series, originally used by S. Kanade, J. Lepowsky, M. Russell, and A. Sills in their motivated proof of Bressoud's partition identities. These ghost series are separate objects combinatorial objects and thus have their own combinatorial interpretations related to the identities studied in this work.
Back to index
Undergraduate Session II-D
BPMC 211
2:15, Bryn Woodling (Elizabethtown College)
Portfolio Theory Analysis
View Abstract
Markowitz began modern stock portfolio management in 1952 by choosing stocks to minimize the portfolio risk in terms of variance, and this has become a standard against which other theories are measured. This talk explores other possible criteria for choosing optimal stock portfolios and compares results with the traditional theory.
2:26, Magdalena Kalinowska, Christopher Williams (University of the Sciences in Philadelphia)
Password Security
View Abstract
Modern technology comes with advanced settings, improved privacy, and more ways to connect across the world. As social media and the internet becomes a more integral part of our lives, techniques to hack into people's accounts have become more advanced. This calls for stronger password security. This presentation will be discussing the mathematical concepts behind passwords and password security such as combinations, permutations, probability, and password entropy.
2:37, Ronald Boorman Jr, Javonni Banks, Natasha Stuckey (University of the Sciences in Philadelphia)
Secrets of Origami
View Abstract
Origami has many properties that can be applied to the real world through the use of mathematics and opens the door to many new discoveries using simple methods. The types of folding that can be achieved through origami are more complex then we give them credit for, spanning the 3rd and 4th dimensions. A small review of the history of origami and some examples of its applications will be discussed in this presentation.
2:48, Owen Vazquez, Chris Miller (University of the Sciences in Philadelphia)
Keeler's Theorem
View Abstract
Our presentation will focus on Keeler's Theorem, a theorem based on Group Theory that describes the solution to a unique problem portrayed in the episode "The Prisoner of Benda" of the show Futurama. The problem is that a machine exists that allows two individuals to switch consciousness, however the machine cannot exchange the consciousness of the same two individuals more than once. The theorem describes the solution to this problem with a specific regard to the number of individuals found within the group.
Back to index
Undergraduate Session II-E
BPMC 212
2:15, Yinxi Li (Franklin and Marshall College)
Motif Detection and Music Visualization
View Abstract
This talk describes a project that uses numerical methods to identify and visualize similarities of thematic sections within a music score. This project uses modular arithmetic and Matlab codes. Aiming for precision, we will start by identifying short recurring motives (a few-measures-long) as small units and later group them and classify them into thematic sections of resemblance. Furthermore, we use the data to visualize what is going on in the music by illustrating how the appearances and overlaps of the motives fit into and interact with the overall shape of the piece. We specifically focus on contrapuntal pieces such as Baroque Fugues.
2:26, Liz Dulac (Millersville University of Pennsylvania)
Modelling a Human Skeleton using Physical Constraints
View Abstract
A combination of joints and bones can accurately model a human skeleton. The joints must store information about their orientation and their relationship between linked joints. By leveraging a hierarchical structure, knowing the position of one joint and the joint's relationship with linked joints is enough to calculate the position of the entire skeleton. Bones can store the length between joints and the weight of the bone to aid in mimicking realistic physical properties such as torque, elasticity, and static friction for collision detection. By setting up physical constraints, this skeleton will have a fairly realistic environment with which to interact. Much of this research will be focused on the design of the skeletal system so as to be able to freely add physical constraints. This project would be best understood visually, so there is need for a graphics component in addition to a physics component.
2:37, Jeremy Budgeon (West Chester University of Pennsylvania)
Predation Risk's Effect on Snail Survivability and Fecundity
View Abstract
In this research experiment, snails were exposed to different levels of predation, and data was collected on the reproductive output of the snails. Snails detect chemical cues from their predators and alter their behavior based on these cues. Thus, levels of predation were simulated by varying the amounts of predator chemical that was put into the snails' tanks. For this experiment, there were three different levels of predation: no predation, 50% exposure, and 100% exposure. For each level of predation, data was collected on the frequency of snail reproduction, clutch size, and the reproductive lifespan of the snails. Using this data, a statistical analysis comparing different reproductive variables was conducted. In the analysis, the reproductive intervals of the snails were compared with the ages, predation levels, and clutch sizes of the snails. Based on these findings, we were able to predict and simulate the final reproductive interval of the snails. The final reproductive interval gets interrupted by a period of no reproduction before the snails die. Thus, we were interested in trying to predict the post reproductive lifespan based on the data collected on reproductive intervals and the other factors that contributed to the reproductive intervals.
2:48, Yuqing Liu (Ursinus College)
Equivalence of discrete Morse functions using persistent homology
View Abstract
The past 15 years have witnessed an explosion of methods and applications in computational topology, including image processing, data analysis, and sensor networks. One method known as persistent homology relies on simple linear algebra techniques and takes as input a data set and outputs a bar code's visual representation of how the topology of the data changes over time in an attempt to determine what features are important in the data and what features are simply "noise." Closely related to persistence is discrete Morse theory, which has been used as a tool to improve certain homology computations. Several connections between persistence and discrete Morse theory have been made. In this talk, we will continue making connections between discrete Morse theory and persistent homology by introducing a new notion of equivalence of discrete Morse functions based on the persistence diagram induced by the sublevel complexes from a discrete Morse functions. We share several properties of this equivalence and compute some examples.
Back to index