| Date: | Monday, 6 October 2025 |
| Time: | 6:00—7:30 PM |
| Place: | Second Floor, Blocker Building |
| Organizer: | Frank Sottile, with assistance from LaKortney Hyson |
| Anders Bahrami | Validation of Material Point Method and Discontinuous Galerkin solvers for Fluid Structure Interactions | |
| Jace Castleberry Mayank Chaudhari | Stability of Discrete Point Charge Systems | |
| Cory Greger | Countable Categoricity and the Ryll-Nardzewski Theorem | |
| Joshua Im | Symmetric and sub-symmetric sums of Dirichlet L-functions | |
| Evan Kniffen | Stochastic Modeling of Bovine Respiratory Disease Using Agent-Based Modeling | |
| Jack Lao Ethan Park | Efficient Exploration of Planar Shapes Through Simple Random Walk | |
| Jennifer Mackenzie | The height function associated with a sparse collection: a Bellman function approach | |
| Ahn Viet Nguyen | R(K4-e, K8-e)=32 | |
| Aaryan Sharma | Complex Potentials Floquet Isospectral to the Discrete Laplacian | |
| Mark Shiliaev | On Unconditionality and higher-order Schreier Unconditionality |
| Anders Bahrami | Validation of Material Point Method and Discontinuous Galerkin solvers for Fluid Structure Interactions |
| The Material Point Method (MPM) is a numerical method useful for efficiently simulating large deformation solids
problems that are typically difficult for other numerical methods like Finite Element Methods (FEM). However, issues
with the original MPM formulation such as cell crossing instability and an inadequate handling of prestress often lead
to unwanted oscillations and other errors like incorrect computed wave speeds. In recent years, the Dual Domain
Material Point (DDMP) and Local Stress Difference (LSD) enhancements to MPM were introduced to mitigate both of these
issues respectively, and such enhancements have allowed the application of MPM to more difficult problems. In this poster, we present MPM simulations of an exploding steel spherical shell which make use of the aforementioned DDMP and LSD enhancements. Achieving physical fragmentation behavior for this problem has proven difficult in the past, but such simulations (when paired with a custom mesh and particle distribution) are seen to yield qualitatively similar results to an experiment. Additionally, Discontinuous Galerkin (DG) simulations are performed on the canonical flow over cylinder problem. Ultimately, these simulations are preliminary validating steps toward eventually coupling DG and MPM in fluid-structure interaction simulations (DG for the fluid, MPM for the structure). |
|
| Jace Castleberry Mayank Chaudhari |
Stability of Discrete Point Charge Systems |
| This project analyzes the behavior of critical points of discrete point charge potential systems over differing configurations. We obtain qualitative, quantitative, and numerical results describing the position and quantity of the critical points depending on the positions of the charges. This includes studying the stability of the critical points under dynamic changes to the configuration. | |
| Cory Greger | Countable Categoricity and the Ryll-Nardzewski Theorem |
| We call a set of first-order axioms countably categorical if there is only one countably infinite structure, up to isomorphism, that satisfies them. For example, the axioms describing a dense linear order without endpoints are satisfied uniquely (up to isomorphism) by the rational numbers with their usual order. We provide an exposition of the Ryll-Nardzewski theorem, which characterizes countable categoricity in three equivalent ways, connecting logical, algebraic, and topological perspectives. Along the way, we introduce several model-theoretic tools and results, including types, type spaces, Lindenbaum-Tarski algebras, and the uniqueness of prime models. | |
| Joshua Im | Symmetric and sub-symmetric sums of Dirichlet L-functions |
| We study symmetric sums of the collection L(s,\chi)_{\chi mod m} of Dirichlet L-functions induced by Dirichlet characters mod m, for Re(s)>1. We show that the coefficients of Dirichlet series obtained by any symmetric sum of Dirichlet L-functions vanish unless n\equiv 1 mod m. In addition, we introduce a method for computing the Dirichlet series coefficients in terms of the prime-power factorization of n. Furthermore, we introduce polynomials whose roots are Dirichlet L-functions for a fixed modulus m, and decompose symmetric sums over all characters into sums over subfamilies (e.g., symmetric sums over Dirichlet L-functions of a smaller modulus). | |
| Evan Kniffen | Stochastic Modeling of Bovine Respiratory Disease Using Agent-Based Modeling |
| We develop a discrete-time stochastic agent-based model (ABM) for bovine respiratory disease in which each animal follows a Markov process on the state space {S,E,I,R} (phase, location, weight). Daily infection events are generated by hazard-based Bernoulli trials: a population-level contact hazard drives SE, and a staged progression rule drives EI; both are functions of the current empirical measures (S(t), E(t), I(t), N(t)). Quarantine is a state-dependent change in the contact kernel (immediate isolation removes local transmission), and movement control alters the mixing operator by restricting spatial transitions. Infectious periods have a deterministic minimum followed by a geometric tail, producing over-dispersed generation times. Growth is a stochastic difference equation with Gompertz drift and additive Gaussian innovations; infection multiplies the drift by a penalty factor. Economic outputs are path-functionals (discounted cash flow and antimicrobial use). Uncertainty is propagated by Monte Carlo; outcome distributions (attack rate, time-to-peak, mg/PCU, NPV) are compared via quantiles rather than means. Global sensitivity uses rank-based partial correlations and variance decomposition (Sobol-style) on transmission, detection/quarantine timing, and vaccination efficacy. For calibration and interpretability, a deterministic SEIR mean-field model runs in parallel; with identical (beta, sigma, gamma) the ABM sample mean converges toward the SEIR trajectory as heterogeneity and noise vanish (law-of-large-numbers/mean-field limit), while diffusion and branching approximations explain early-phase variability and finite-population tail risk that the ODE cannot capture. |
|
| Jack Lao Ethan Park |
Efficient Exploration of Planar Shapes Through Simple Random Walk |
| The coordinate hit-and-run process is a common method to explore convex sets in R^n. We introduce a new variation, replacing the hit and run process with a simple random walk on a grid within the shape. Computer simulations of various random walks were used to investigate the mixing time of the processes in both convex and non convex shapes of defined area. We found indications that among several convex shapes in a “nice” position, the disk has the smallest mixing time. In our investigation of non-convex shapes, we explored a “bowtie” shape, which features two convex shapes connected by a thin “neck”. We compare our simulation works with some computations from the theory. We also explore other non-convex shapes and variations to the simple random walk. | |
| Jennifer Mackenzie | The height function associated with a sparse collection: a Bellman function approach |
| Sparse operators have recently emerged as a powerful method to extract sharp constants in harmonic analysis inequalities in the context of bounding singular integral operators. We investigate the level sets of height functions for sparse collections, or, in other words, weak-type (1,1) inequalities for sparse operators applied to constant functions. We utilize another notable method from dyadic harmonic analysis, which is also famous for its ability to produce sharp constants, namely the Bellman function method. In particular, we find the exact Bellman function maximizing level sets of the sparse operator associated with a binary Carleson sequence when applied to constant functions. | |
| Ahn Viet Nguyen | R(K4-e, K8-e)=32 |
| We show that R(K4-e, K8-e = 32. The method involves checking several graphs through computer programming with a theoretical framework. This method is inspired by Angeltveit and McKay's gluing algorithm, as presented in their recent work. | |
| Aaryan Sharma | Complex Potentials Floquet Isospectral to the Discrete Laplacian |
| It is known that for even values of n>2, there exist nonzero complex-valued nZ-periodic potential functions on the one-dimensional square lattice that are Floquet-isospectral to the zero potential. We conjecture that for all n>3, there exist such nonzero potential functions. We present our progress towards proving this conjecture using methods from commutative algebra and experimental data acquired via Macaulay2. | |
| Mark Shiliaev | On Unconditionality and higher-order Schreier Unconditionality |
| Let X be a complete normed vector space and (en)n=1∞ be its basis. The basis $(en)n=1∞ is called unconditional if for every vector x, the projections of x onto every subset of the basis have a smaller norm than the norm of x. For each Schreier family Sα, which is a rich collection of subsets of N, we have a weaker version of the unconditionality condition, called Sα-unconditionality. A basis is Sα-unconditional if projecting x onto sets in Sα essentially reduces the norm. Assuming the Continuum Hypothesis, we show that if a basis is Sα-unconditional for every countable ordinal α, then it is unconditional. | |
Last modified: Tue Sep 30 09:49:26 CDT 2025 by sottile