This Week in Mathematics


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Entries for this week: 8

Tuesday October 07, 2025

Geometry and Topology [url]
Concentration Phenomena in Frame Theory
- Ferhat Karabatman, FSU
Time: 3:05 Room: 231
More Information
Abstract/Desc: Frames are foundational in signal processing, and tight frames are particularly well-behaved, offering robustness to noise and erasures. Under the erasure model we study, epsilon-close equal-norm tight frames exhibit especially favorable behavior: the reconstruction error decreases as the number of frame vectors grows. We also establish concentration phenomena: on the space of n-tuples with prescribed norms---including the equal-norm subspace---the probability that a randomly selected frame is tight increases with n. Moreover, within the class of tight frames with a fixed frame bound, the subset of epsilon-close equal-norm tight frames concentrates as n increases. Taken together, these results show that increasing redundancy simultaneously yields robust reconstruction and prevalent structure: most frames are tight, and most tight frames are nearly equal-norm and near-optimal.

Wednesday October 08, 2025

Biomath lab meetings
Generating Functions, Rook Theory, and Problème Des Ménages (Part 2)
- Xinxuan (Jennifer) Wang, FSU
Time: 5:30 Room: LOV105
Abstract/Desc: You’re hosting a party with 5 couples attending. For the sake of helping them know each other better, you want to seat them so that men and women alternate, and nobody sits next to their spouse. This is the famous Problème Des Ménages, and it is not easy to solve at all. In this talk, I will give a straightforward solution using Rook Theory, which makes use of generating functions. I will illustrate how generating functions can reduce a complicated counting problem to simpler counting problems and power series manipulations.

Biomathematics Journal Club
Theory of Cell Fate
- Philip Asare, FSU
Time: 5:00 Room: Dirac Library

Thursday October 09, 2025

ATE / Candidacy Exam Presentation
Mathematical Model Linking Proteostasis and Metabolic Energy Balance
- Kaylie Green, Department of Mathematics, FSU
Time: 9:45am Room: LOV 204A
Abstract/Desc: This research investigates the dynamic interplay between energy metabolism and protein homeostasis (proteostasis) under conditions of energetic stress. While energy imbalance is widely regarded as the primary driver of cellular and organismal failure, this work challenges that assumption by proposing that proteostasis collapse may occur independently and even precede energy failure. Proteostasis, maintained by the proteostasis network (PN), is inherently vulnerable to instability during stress, yet its theoretical relationship with the energy network (EN) remains poorly understood. This research hypothesizes that unstable energy states can trigger proteostasis collapse, and vice versa, with cascading effects on cellular function. To test this hypothesis, we propose to develop a mathematical model linking EN and PN. We will use Principal Component Analysis (PCA) to reduce the system’s dimensionality, and separately apply the Routh-Hurwitz criteria to perform stability analysis, aiming to identify conditions under which failure in one system may propagate to the other. This model aims to define the theoretical boundaries of resilience and collapse, offering mechanistic insights into stress responses at molecular, cellular, and tissue levels.

Financial Math
Stochastic Control with Partial and Decentralized Information
- Serdar Yuksel, Queen's University
Time: 3.05 Room: 105
Abstract/Desc: Abstract : We study stochastic control with partial and decentralized information and discuss optimality, approximations, and learning theoretic results, in both discrete-time and continuous-time. We first consider partially observed stochastic control which provides a general mathematical model for many applications. The study of these has in general been established via reducing the original problem to a fully observed one with probability measure valued filter (or belief) states and an associated filtering equation forming a Markovian kernel. We establish regularity results for this kernel, involving weak continuity as well as Wasserstein regularity and contraction, and present existence results for optimal solutions for both the discounted cost and average cost criteria. Building on these, we present approximation results via either quantized measure approximations or finite memory approximations under filter stability. We present explicit conditions for controlled filter stability which are then utilized to arrive at near-optimal finite-window control policies. Finally, we establish the convergence of a reinforcement learning algorithm for control policies using these finite approximations or finite window of past observations (by viewing the quantized filter values or finite window of measurements as ‘states') and show near optimality under explicit conditions. The above will then be generalized to decentralized stochastic control involving multiple agents who have different local information. Such models are known to be difficult to study due to information structure dependent subtleties: Under a variety of information structures with an absolute continuity condition on local measurement kernels, existence (via an equivalent controlled Markovian kernel construction in a lifted space and weak continuity), near optimality of finite approximations (via weak continuity), as well as rigorous reinforcement learning results will be presented, in analogy with the noted related results on partially observable stochastic control. By establishing near optimality of discrete-time approximations of controlled diffusions with partial information or multi-agent systems with decentralized information, the above are then generalized to continuous-time systems. [Joint work with Ali D. Kara, Y. Emre Demirci, Naci Saldi, Omar Mrani-Zentar, Somnath Pradhan].

Algebra seminar
Genetic algebras via Latin squares
- Louis Rubin, FSU
Time: 3:05pm Room: LOV 0231
Abstract/Desc: In this talk, we introduce the notion of a genetic algebra, which models the inheritance of traits from one generation to the next. We then consider a particular class of genetic algebras arising from a given Latin square and doubly stochastic matrix. Our main result describes the long-term behavior of a population evolving within this setting. We will outline the proof, which draws on several classical results.

Friday October 10, 2025

Data Science and Machine Learning Seminar
A Bridge Between Topological and Functional Data
- Kun Meng, FSU Statistics
Time: 1:20pm Room: Love 106
Abstract/Desc: In the 21st century, we have seen a growing availability of shape-valued and imaging data, prompting the development of new statistical methods to analyze them. Importantly, bridging the new methods and existing frameworks is advisable. In this talk, I will introduce several statistical inference methods for shapes and images based on the Euler characteristic. These methods have applications in many fields, such as geometric morphometrics and radiomics. From a statistical perspective, these methods are naturally connected to functional data analysis. From a mathematical viewpoint, they are grounded in solid foundations, bridging various branches of mathematics: algebraic and tame topology, Euler calculus, functional analysis, and probability theory. I will also briefly discuss some of my ongoing and future research directions.

Mathematics Colloquium [url]
Stochastic Kernel Topologies and Implications on Approximations, Robustness, and Learning
- Serdar Yuksel, Queen's University
Time: 3:05 Room: Lov 101
Abstract/Desc: Stochastic kernels represent stochastic processes, controlled and control-free system models, measurement channels, and control policies, and thus offer a very general mathematical framework in applied mathematics. We will first present several properties of such kernels and study several kernel topologies. These include weak* (also called Borkar) topology, Young topology, kernel mean embedding topologies, and strong convergence topologies. After a general introduction, we then study convergence, continuity, and robustness properties involving models and policies viewed as kernels, in the context of stochastic control but also related areas in applied mathematics. On models viewed as stochastic kernels; we study robustness to model perturbations, including finite approximations for discrete-time models and robustness to more general modelling errors and study the mismatch loss of optimal control policies designed for incorrect models applied to a true system, as the incorrect model approaches the true model under a variety of kernel convergence criteria: We show that the expected induced cost is robust under continuous weak convergence of transition kernels. Under stronger Wasserstein or total variation regularity, a modulus of continuity is also applicable. As applications of robustness under continuous weak convergence via data-driven model learning, (i) robustness to empirical model learning for discounted and average cost criteria is obtained with sample complexity bounds; and (ii) convergence and near optimality of a quantized Q-learning algorithm for MDPs with standard Borel spaces, which we show to be converging to an optimal solution of an approximate model, is established. In the context of continuous-time models, we obtain counterparts where we show continuity of cost in policy under Young and Borkar topologies, and robustness of optimal cost in models including discrete-time approximations for finite horizon and infinite-horizon cost criteria. Discrete-time approximations under several information structures will then be obtained via a unified approach of policy and model convergence. A concluding message is that weak kernel topologies are appropriate for policy spaces, and strong kernel topologies are suitable for studying models towards establishing very general existence, approximations, robustness and learning results. [Joint work with Ali D. Kara, Somnath Pradhan, Naci Saldi, Omar Mrani-Zentar].

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