2025 Undergraduate Summer Research Presentations
August 4, 2025
ALTER 108
10:00-11:30 AM
10:00 Hahn Nguyen
The COVID-19 pandemic caused unprecedented disruptions to the real estate market, with significant fluctuations in housing prices, population shift, transaction volumes, and supply-demand dynamics. This study explores the post-pandemic recovery of the real estate market in Hamilton County, Ohio, through preliminary analysis and data visualization. Using data from 2018 to 2025, we examine trends in key market indicators such as housing prices, inventory levels, mortgage rates, and unemployment. Through graphical analysis, we uncover patterns, correlations, and potential drivers of market fluctuations. While the current focus is on understanding these relationships visually, this work lays the foundation for future development of mathematical models — including deterministic and stochastic differential equations — to describe and forecast the recovery process more rigorously. The study offers early insights into the structure and volatility of the housing market, with potential implications for policymakers, investors, and urban planners interested in long-term market resilience.
10:15 Beth Howard, Reddening Sequences of Infinite Quivers
A quiver is a pair Q = (V, E) with V a set of vertices and E a multiset of arrows between two distinct vertices. Mutation is a local process of creating a new quiver from the previous. An infinite quiver Q∞ is the limit of {Qi, τi}, where Qi is an induced subquiver of Qi+1 for all i and {τi} is a sequence of embeddings. We establish a definition for mutation on an infinite quiver and demonstrate that it is well-defined. In addition, we consider reddening sequences for infinite quivers and show a constructive way to determine if an infinite quiver has such a reddening sequence.
10:30 Kadence Bouldin and Giancarlo Costanzo, The Strong Spectral Property and the
Inverse Eigenvalue Problem for Graphs
The Inverse Eigenvalue Problem on Graphs aims to establish the connection between a graph and all possible eigenvalues of symmetric matrices that are associated with the graph. For a given matrix, its spectrum is the multiset consisting of all its eigenvalues. We explore how a new tool called the Strong Spectral Property (SSP) can be utilized to obtain a deeper understanding of the relations between the spectra of matrices with a given graph and the spectra of matrices in supergraphs of the graph. Specifically, we apply the definition of the SSP to different graph families in order to see which graphs have matrices that always have the SSP, or what constraints are necessary for a graph to have matrices that have the SSP.
10:45 Renee Kovacs and Jack Palermo, Vector Space Partitions
A vector space is a set of elements that can be added together and multiplied by numbers and that satisfy certain algebraic axioms. A subspace is a non-empty subset of a vector space that is closed under vector addition and scalar multiplication. The dimension of a vector space is the number of elements that completely determine it. A finite field of order q is a finite set of numbers dependent on a prime power q for which the operations of multiplication, addition, subtraction, and division are defined, and that satisfy certain algebraic rules.
Let V (n, q) denote the vector space of dimension n over a finite field of order q and let n>= 2. A 2-spread is a collection of subspaces of V:=V (n, q), all of dimension 2, such that each nonzero vector of V appears in exactly one of the subspaces. Creating 2-spreads gets increasingly more difficult as the dimension of a vector space and q grow. We present an efficient method that we built for every q to inductively create 2-spreads of a vector space of dimension 2k+2 by using a 2-spread from a 2k-dimensional vector space.
A partial t-spread of V(n, q) is a collection of t-dimensional subspaces of V (n, q) that intersect only in the zero vector. In this talk, we will also discuss our attempts to create a partial 3-spread of size 245 of the V(8,3) vector space; this would have been larger than the known lower bound.
11:00 George Paine, Leonhard Euler's Partition Theory
A brief summary of Swiss mathematician, Leonhard Euler's papers from 1751-1768 concerning partition theory. Inspired by a question of Naudé, Euler uses clever algebraic tricks to develop mathematical machinery capable of solving these problems, as well as many others.