This Week in Mathematics


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Today:

Biomath Seminar
Dynamics Shape the Perception of Network Structure / Modeling Disease Spread with the Fisher–KPP Equation
- Callie Reid / Varkshini Maheswaran, FSU
Time: 3:05pm Room: Love 232
Abstract/Desc: Reid: Dynamical processes can fundamentally reshape how network structure is perceived, with functional organization emerging even when topological modularity is weak or absent. We investigate this phenomenon in nonlinear random walks—mean-field models of diffusion with volume exclusion that are nonlinear on heterogeneous networks and linear otherwise. For connected undirected graphs, we analyze the rate of convergence to equilibrium for n interacting walkers and introduce a localization-based reduction exploiting the spontaneous concentration of Laplacian eigenvectors on specific nodes in large random graphs. We show analytically, and validate numerically, that in scale-free networks the Jacobian spectrum of the nonlinear random walk undergoes a characteristic spectral deformation: eigenvalues associated with peripheral, low-degree nodes are compressed toward the origin, while those linked to hubs are systematically displaced outward. This nonlinear spectral reorganization reveals how heterogeneity and crowding jointly redefine the network's effective dynamical structure, offering a principled mechanism for dynamic modularity in complex systems. Building on this perspective, we introduce weighted-node paths in which nodes are weighted according to their crowding capacity, allowing dynamical proximity to be quantified through path-based measures that naturally extend classical closeness and betweenness centrality to account for congestion effects induced by the dynamics. Maheswaran: The spread of infectious diseases can be modeled using reaction–diffusion equations that capture both local growth and spatial movement. In this study, we investigate disease propagation using the general Fisher–KPP equation, a nonlinear partial differential equation that describes the interaction between diffusion and logistic growth. Using the finite element method, we compute numerical solutions to examine how spatial heterogeneity and boundary conditions influence the dynamics of infection spread. In particular, we explore how variations in growth parameters and mixed boundary conditions affect persistence, extinction, and wave propagation behavior. Our simulations reveal threshold-like dynamics, where the balance between growth and diffusion determines whether the infection spreads or dies out. We also observe that regions with higher local growth can dominate the overall dynamics, even in the presence of boundary-induced suppression. These results highlight how mathematical modeling can provide insight into structured disease dynamics and demonstrate the role of spatial effects in shaping the evolution of infectious diseases.

Biomathematics Journal Club
Power in Networks: The Medici
- Dana Hughes, FSU
Time: 5:00 Room: Dirac Library

Entries for this week: 9

Tuesday April 07, 2026

Topology seminar
Algebraic vector bundles of rank 2 over smooth affine fourfolds
- Morgan Opie, Northwestern University
Time: 3:05PM Room: Zoom
Abstract/Desc: To what extent do Chow-valued Chern classes determine the isomorphism class of an algebraic vector bundle? In this talk, I'll discuss some progress on this question for algebraic vector bundles of rank 2 over smooth affine fourfolds. These results imply some concrete cohomological classification results (e.g., over the complex numbers, there are exactly 9 isomorphism classes of rank 2 vector bundles over the complement of a smooth degree 3 hypersurface in P^4). I'll also highlight some possible computations that, if completed, would shed further light on this problem. This is joint work with Thomas Brazelton and Tariq Syed.

Wednesday April 08, 2026

Biomath Seminar
Dynamics Shape the Perception of Network Structure / Modeling Disease Spread with the Fisher–KPP Equation
- Callie Reid / Varkshini Maheswaran, FSU
Time: 3:05pm Room: Love 232
Abstract/Desc: Reid: Dynamical processes can fundamentally reshape how network structure is perceived, with functional organization emerging even when topological modularity is weak or absent. We investigate this phenomenon in nonlinear random walks—mean-field models of diffusion with volume exclusion that are nonlinear on heterogeneous networks and linear otherwise. For connected undirected graphs, we analyze the rate of convergence to equilibrium for n interacting walkers and introduce a localization-based reduction exploiting the spontaneous concentration of Laplacian eigenvectors on specific nodes in large random graphs. We show analytically, and validate numerically, that in scale-free networks the Jacobian spectrum of the nonlinear random walk undergoes a characteristic spectral deformation: eigenvalues associated with peripheral, low-degree nodes are compressed toward the origin, while those linked to hubs are systematically displaced outward. This nonlinear spectral reorganization reveals how heterogeneity and crowding jointly redefine the network's effective dynamical structure, offering a principled mechanism for dynamic modularity in complex systems. Building on this perspective, we introduce weighted-node paths in which nodes are weighted according to their crowding capacity, allowing dynamical proximity to be quantified through path-based measures that naturally extend classical closeness and betweenness centrality to account for congestion effects induced by the dynamics. Maheswaran: The spread of infectious diseases can be modeled using reaction–diffusion equations that capture both local growth and spatial movement. In this study, we investigate disease propagation using the general Fisher–KPP equation, a nonlinear partial differential equation that describes the interaction between diffusion and logistic growth. Using the finite element method, we compute numerical solutions to examine how spatial heterogeneity and boundary conditions influence the dynamics of infection spread. In particular, we explore how variations in growth parameters and mixed boundary conditions affect persistence, extinction, and wave propagation behavior. Our simulations reveal threshold-like dynamics, where the balance between growth and diffusion determines whether the infection spreads or dies out. We also observe that regions with higher local growth can dominate the overall dynamics, even in the presence of boundary-induced suppression. These results highlight how mathematical modeling can provide insight into structured disease dynamics and demonstrate the role of spatial effects in shaping the evolution of infectious diseases.

Biomathematics Journal Club
Power in Networks: The Medici
- Dana Hughes, FSU
Time: 5:00 Room: Dirac Library

Thursday April 09, 2026

Algebra seminar
Tropical correspondence theorems for plane curve counts over arbitrary fields
- Sabrina Pauli, TU Darmstadt
Time: 3:05pm Room: Zoom
Abstract/Desc: We study the problem of counting rational curves of fixed degree on a toric del Pezzo surface subject to point conditions. Over algebraically closed fields, this count is invariant under the choice of point conditions. Over non-algebraically closed fields, however, the invariance fails. For real numbers, Welschinger's groundbreaking work introduced a signed count of real curves that restores invariance. Building on this, Levine and Kass-Levine-Solomon-Wickelgren have developed curve counts over arbitrary fields that not only generalize Welschinger's signed counts and classical counts over algebraically closed fields, but also encode much richer arithmetic information. In this talk I will survey these different approaches to counting rational curves with point conditions and discuss a recent joint result with A. Jaramillo Puentes, H. Markwig, and F. Röhrle. We establish a tropical correspondence theorem for curve counts over arbitrary fields, identifying the count of algebraic curves with point conditions with a weighted count of their tropical counterparts with point conditions. The latter are combinatorial objects and there are several purely combinatorial methods to find all tropical curves with point conditions.

Financial Math
Optimization of win martingales.
- Xin Zhang, New York University
Time: 3.05 Room: LOV 231
Abstract/Desc: Prediction market is a market where people can trade based on outcomes of future events. It is widely used in sports games, elections, and pricing of digital options. In math finance, prediction markets can be modeled by the so-called win martingales, which are continuous time martingales that end up with Bernoulli distributions. In this talk, choosing different divergences as objective functionals, we will solve a class of optimal win martingales. In some cases, we will get explicit formulas of optimizers, and make connections to Schrödinger, filtering problems, Wright-Fisher diffusion, and the problem of identifying most exciting games.

Friday April 10, 2026

Ph.D. Defense
Model-Data Integration in Complex Dynamical Systems: Applications in Materials Synthesis
- Ruth Lopez Fajardo, Florida State University
Time: 1:15PM Room: LOV107
Abstract/Desc: Mathematical models governed by differential equations are widely used to describe the evolution of complex physical systems. In many scientific and engineering applications, however, these models contain uncertain parameters, incomplete knowledge of governing mechanisms, and rely on measurements that are indirect, partial, and contaminated by noise. These challenges are particularly evident in emerging frameworks for autonomous materials synthesis, where real time experimental diagnostics must be integrated with physics based models to guide and control growth settings. This thesis investigates sequential Bayesian data assimilation methods for integrating dynamical models with experimental observations in settings where measurements are collected over time and model data integration must be performed online. This problem is commonly referred to as filtering in the data assimilation literature. The work focuses on ensemble based filtering approaches for state estimation and joint state parameter estimation in dynamical systems. First, we investigate filtering methods for systems governed by partial differential equations (PDEs), where spatial discretization leads to high dimensional state spaces and significant computational challenges. In particular, the recently proposed Ensemble Score Filter (EnSF) is studied and evaluated on representative PDE benchmark problems, including phase field models of pattern formation. Through identical twin experiments under partial and noisy observations, we analyze the stability, accuracy, and computational scalability of the method and compare its performance with state-of-the-art filtering algorithm across different uncertainty regimes. Building on these methodological investigations, the thesis considers a physics based kinetic model describing nucleation and growth processes during two dimensional thin film deposition with Pulsed Laser Deposition (PLD) technique. The model is combined with sequential data assimilation to estimate latent surface coverage states and unknown kinetic parameters from in situ optical reflectivity measurements. This approach enables the online reconstruction of film growth dynamics and supports the development of data driven monitoring and closed loop control strategies for thin film synthesis. Overall, this work investigates the integration of differential equation models with streaming data using sequential Bayesian data assimilation methods, with particular emphasis on data driven monitoring of thin film synthesis processes.

Mathematics Colloquium [url]
Math Honors Day
- Monica Hurdal, FSU
Time: 3:05 Room: 101
Abstract/Desc: Faculty, staff, and students are invited to join us as we celebrate and award our graduate and undergraduate students for their achievements and excellence in academics, teaching, and service.

Data Science and Machine Learning Seminar [url]
Multiparameter persistence landscapes
- Pan Fang, FSU
Time: 1:20 Room: Lov 106
Abstract/Desc: Persistent homology studies the evolution of homological features across a filtration. In the one-parameter setting, barcodes provide a simple and powerful summary. Multiparameter persistence is more complicated, and unlike in the one-parameter setting, there is no direct barcode analogue in general. Multiparameter persistence landscapes offer one approach to understanding this richer setting. In this talk, we introduce the directional persistence landscape and integrated persistence landscape. We study metric properties among them and present stability and instability results, including results related to the erosion distance and the interleaving distance. We also show that the interior of the landscape interval is determined by finitely many points, and describe the general form of the integrated landscape curves.

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Department of Mathematics

This Week in Mathematics