Virginia Tech Mathematics Colloquium – Fall 2025

One of Erdos's greatest contributions to geometry, by his own account, is his problem on distinct distances, which asks about the least number of distinct distances determined by points in the plane. In 1985 Kenneth Falconer posed an analogous question in a geometric measure theory setting. His problem, now known as the Falconer distance problem, is one of the major open problems on the interface of geometric measure theory and harmonic analysis. In this talk, I will give an introduction to the Falconer distance problem and some ideas of why such an innocent looking question has captured the attention of so many mathematicians. I will conclude by reporting on some of our very recent progress on a variant of his original problem.

CountSketch, or feature hashing, is a sparse random dimensionality reduction technique used in applications such as compressed sensing, low rank approximation, regression, second order optimization, and streaming algorithms. We study learned versions of CountSketch, where we learn both the positions and values of the hashing matrix to better adapt to the underlying data. We give theoretical and empirical results, showing improvements over classical CountSketch or earlier learned CountSketch algorithms where only the values are learned. Based on a joint work with Yi Li, Honghao Lin, Simin Liu, and David Woodruff.

Mahmudul Bari Hridoy

Christina Giannitsi

The Goldbach conjuncture for Gaussian integers

The Goldbach Conjecture is a well-known hypothesis that states that every even number bigger than 5 can be written as the sum of two primes. For the real integers, a density result for the conjecture is known. In this talk, we will introduce the setting of the Gaussian integers and present complex analogues of these density results.

Infectious Disease Dynamics in Heterogeneous Populations

Infectious diseases remain a pressing global health challenge, shaped by interactions among environmental, biological, and social factors. Demographic features such as age structure, spatial distribution, and population density further influence transmission dynamics and intervention outcomes. Mathematical models provide a powerful way to study these processes, especially when incorporating stochastic effects, heterogeneity, and real-world data. In this talk, I will highlight recent work on stochastic epidemic models that account for environmental and demographic variability, illustrating how modeling can deepen our understanding of disease dynamics and inform effective public health control.

Talia LaTona-Tequida

The Association for Women in Mathematics as a Place for Reimagining Gender Dynamics in Graduate Mathematics

This talk focuses on results from a study looking to understand the experiences of women pursuing PhDs in mathematics. I discuss the gender-inclusive practices within a graduate chapter of the Association for Women in Mathematics (AWM). Results revealed that inclusion was viewed as both a logistical necessity and a strategic move toward equity, fostering community building and identifying male allies. Additionally, tensions such as maintaining women-centered spaces and replicating gendered labor divisions also arose. AWM’s efforts reflect both an aspirational reimagining of gender dynamics and the persistent realities of societal gender-based inequities. These findings contribute to understanding how gender-inclusive social structures can both challenge and reproduce dominant gender norms.

Kirsten Morris

Decoding Challenges for Quantum LDPC Codes

Quantum Low Density Parity Check (QLDPC) codes are promising candidates for quantum error correction. To detect errors, we utilize a graph-based representation of the code and perform iterative decoding on this graph. However, decoding failure can occur not only due to the noisiness of the space, but also due to structures intrinsic to the graph itself. In this talk, we will discuss these graph structures and share results on their characterization.

Rodrigo San Jose Rubio

Evaluation codes and zeros of polynomials

Error-correcting codes have found many applications over the last few years: quantum error-correction, code-based cryptography, distributed storage, etc. In particular, evaluation codes provide a connection between the error-correction capabilities of the corresponding code and the maximum number of zeroes of polynomials. When considering polynomials of a fixed degree, this is a classical problem for both the affine and the projective spaces. For weighted projective spaces, Aubry, Castryck, Ghorpade, Lachaud, O'Sullivan, and Ram conjectured a Serre-like bound for the maximum number of zeroes. In this talk we will show these results, their connections, and discuss how the footprint bound (together with Delorme's weight reduction, and Serre's bound) can be used to prove this conjecture.

Chi Hong Chow

Mirror symmetry for flag varieties

Mirror symmetry, which originates from string theory, is a duality between Calabi-Yau manifolds. There is a version for Fano manifolds, in which their mirror spaces are Landau-Ginzburg models. In this talk, I will discuss the latter version, focusing on my work concerning a class of examples called flag varieties.

Research on undergraduate students’ engagement with mathematical proof has often focused on construction, reading comprehension, and validation of proof. While these aspects highlight important forms of proof activity, this presentation focuses on students’ evaluative reasoning in the context of mathematical proof, particularly two theoretical constructs that reflect this dimension: Logical consistency (LC), defined as an individual’s mathematical thinking characterized by the absence of logical contradictions when evaluating a mathematical statement and its accompanying argument. Proof competency (PC), defined as an individual’s evaluative reasoning skills for determining whether an argument serves as a valid (dis)prof of a mathematical statement. Whereas LC concerns the internal coherence of students’ evaluations across the three components (statement truth, argument intent, and argument validity), PC concerns the correctness of each evaluation. To investigate these constructs, my research team developed the LinC (Logical inConsistency) instrument and administered it to over 200 undergraduate students across multiple institutions (Roh & Lee, 2024). In this talk, I will present the structure of the LinC instrument and evidence that supports the construct validity of the instrument for assessing LC and PC. I will also report on the relationship between LC and LC, as well as how LC is associated with students' experiences in proof-oriented mathematics courses.

Infectious diseases cause millions of deaths globally each year. With the recent pandemic directing the focus, our understanding of infectious disease and its control increased significantly through major efforts in research, but further development of innovative efforts are needed to continue progress towards combating infectious disease spread. To this end, by developing, analyzing and simulating systems of ordinary differential equations, my work advances general epidemic models through the incorporation of mechanisms representing human behavior and its effect on single- and multi-strain disease dynamics. Specifically, I will discuss the incorporation of an endogenous behavior-transmission feedback loop, which captures how change in human behavior drives change in infection levels, which in turn drives change in human behavior. Despite its simplicity, this mechanism notably enhances fitting and forecasting capabilities and provides important insights into how adaptation of behavioral responses impacts disease prevention and control. To focus on a specific disease, I will present our deterministic, compartmental ordinary differential equation system used to model two strains of cholera with multiple transmission routes known to exhibit anti-phase cycling of the two strains. To improve understanding of mechanisms which influence the anti-phase cycling, I examined multiple iterations of this model under varying assumptions, including serotype biology, seasonality, immunity and cross-immunity, and differing assumptions on pathogen growth, and showed under what conditions a switch in dominance between the two strains is expected to occur. Revealing the underlying drivers of this cycling dynamic is of critical importance to constructing efficient disease control measures. Melding these two themes – a behavior-transmission feedback loop and a two-strain system with multiple transmission routes - has the potential to provide critical insight into optimizing effective and efficient vaccination strategies for a range of infectious diseases.

Success for students in Physics requires understanding and the application of numerous mathematical skills. Mathematical preparation (Calc I, II, III, Diff EQ, and Linear Algebra) are commonplace among required courses for physics majors. After close inspection, it is becoming clearer that the mathematics taught in Math courses, and the mathematics students are expected to use in intro- through upper-division physics courses has a number of differences that may interfere with student success. Through qualitative methods, we have unpacked distinct and potentially impactful differences in how students’ reason about integration and differentials. Most notably, is the limit vs. infinitesimal definition of integration. Interview data will demonstrate the depth and nuance of these complications, with insight into how curriculum development and instructional intervention may assist in helping students overcome these barriers.

The ring of symmetric functions is a unifying object appearing across many areas of modern mathematics. This is especially true in combinatorics, geometry, and representation theory where symmetric functions are crucial in describing the interactions between the combinatorics of partitions, the geometry of moduli spaces, and the representations of Lie groups and permutation groups. In this talk I will discuss the history of the development of symmetric function theory and its importance in modern research. Along the way I will highlight open problems in this area and discuss recent progress in generalizing the theory of symmetric functions.

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