This Week in Mathematics


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Entries for this week: 5

Wednesday November 12, 2025

Biomathematics Journal Club
Theoretical Reconstruction of Field Potentials and Dendrodendritic Synaptic Interactions in Olfactory Bulb
- David Wharton, FSU
Time: 5:00 Room: Dirac Library

Thursday November 13, 2025

Algebra seminar
Tropical Abel--Jacobi theory
- Dan Corey, Embry-Riddle Aeronautical University
Time: 3:05pm Room: LOV 0231
Abstract/Desc: The intermediate Jacobians and Abel–Jacobi map are higher-dimensional generalizations of the classical Jacobian and Abel–Jacobi map for curves. They are powerful tools for studying higher-dimensional algebraic varieties. For example, they may be used find obstructions to rational and algebraic equivalence of cycles. In this talk, I will explain how to extend these constructions to tropical varieties of arbitrary dimension. As in the classical setting, the tropical versions yield obstructions to rational and algebraic equivalence. We will apply this framework to the Ceresa cycle—a canonical nullhomologous cycle associated to an algebraic curve that has been the subject of significant recent interest. I will present a combinatorial formula for the Abel–Jacobi image of the tropical Ceresa cycle and discuss its relationship with the complex and l-adic Ceresa classes. This is joint work with Omid Amini and Leonid Monin.

Financial Math
Branched Signature Model
- Munawar Ali, Florida State University
Time: 3.05 Room: 105
Abstract/Desc: In this paper, we introduce the branched signature model, motivated by the branched rough path framework of [Gubinelli, Journal of Differential Equations, 248(4), 2010], which generalizes the classical geometric rough path. We establish a universal approximation theorem for the branched signature model and demonstrate that iterative compositions of lower-level signature maps can approximate higher-level signatures. Furthermore, building on the existence of the extension map proposed in [Hairer-Kelly. Annales de l'Institue Henri Poincar\'e, Probabilit\'es et Statistiques 51, no. 1 (2015)], we show how to explicitly construct the extension of the original paths into higher-dimensional spaces via a map $\Psi$, so that the branched signature can be realized as the classical geometric signature of the extended path. This framework not only provides an efficient computational scheme for branched signatures but also opens new avenues for data-driven modeling and applications.

Friday November 14, 2025

Data Science and Machine Learning Seminar
The world of world models
- Mao Nishino, FSU Mathematics
Time: 1:20pm Room: Love 106
Abstract/Desc: This review surveys recent developments in world models. In the broader pursuit of artificial general intelligence (AGI), a frequently cited missing piece between current large language models (LLMs) and AGI is a grounded understanding of the external world—a “world model.” Following the survey by Ding et al. and related work, I organize the literature into two strands: (1) representations of the world and (2) prediction of the future. For (1), I discuss model-based reinforcement learning techniques, such as Dreamer, alongside evidence for emergent world models in LLMs. For (2), I cover video-generation approaches (e.g., Sora and Genie) as well as research on embodied AI.

Mathematics Colloquium [url]
The Molecular Dynamics Underlying Intracellular Phase Separation
- Kelsey Gasior, University of Notre Dame
Time: 3:05 Room: Lov 101
Abstract/Desc: An emerging mechanism for intracellular organization is liquid-liquid phase separation (LLPS). Found in both the nucleus and the cytoplasm, liquidlike droplets condense to create compartments that are thought to localize factors, such as RNAs and proteins, and promote biochemical interactions. Many RNA-binding proteins interact with different RNA species to create droplets necessary for cellular functions, such as polarity and nuclear division. Additionally, the proteins that promote phase separation are frequently coupled to multiple RNA binding domains and several RNAs can interact with a single protein, leading to a large number of potential multivalent interactions. This work focuses on a multiphase, Cahn-Hilliard diffuse interface model to examining the RNA-protein interactions driving LLPS. Using a ‘start simple, build up’ approach to model construction, these models explore how the molecular interactions underlying protein-RNA dynamics and RNA species competition control observable, droplet-scale phenomena. Numerical simulations reveal that RNA competition for free protein molecules contributes to intra-droplet patterning and the emergence of a heterogeneous droplet field. More in-depth analysis using combined sensitivity analysis techniques, such as Morris Method screening and Sobol’ method, highlights the complicated relationships underlying protein-RNA interactions and the results we can measure. Finally, droplet-level patterns are complicated when the initial conditions are considered. Under in vitro conditions, this model shows how experimental set up and initial conditions can produce complex droplet Turing patterns at phase separation. In-depth analysis using numerical simulations, shape analysis, and sensitivity analyses show that these systems are also susceptible to changes in the protein-RNA binding dynamics. The addition of a second RNA species competing for free protein introduces an element of asymmetry to the system and alters the emerging patterns at the onset of phase separation. But both systems show that certain initial conditions can produce sustained multi-droplet patterns as t goes to infinity. Ultimately, this targeted approach to intracellular LLPS begins to peel back the layers of complex molecular dynamics controlling observable LLPS phenomena that contribute to droplet regulation and, ultimately, cellular function.

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This Week in Mathematics