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This Week in Mathematics


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Today:
Mathematics Colloquium [url]
Loss of Synchrony to Silencing in Networks of Excitable Cells: Impact of Cell and Coupling Heterogeneity in Small Network Examples
    - Brad Peercy, University of Maryland
Time: 3:05 Room: Lov 101
Abstract/Desc: Experiments on pancreatic islets have raised a question about the potential unitary impact of certain cells in islet synchrony. Previous modeling to corroborate these findings under the suggested conditions proved unfruitful, but wide parameter searches did identify cases where silencing or ablating individual beta cells could completely or nearly completely silence islet behavior. This includes three cell beta cell networks and a small 2D grid network of simpler two-variable excitable cells. We find examples of "switch" behavior in each case and look to generalize.

Entries for this week: 10
Monday March 25, 2024

PDE seminar
Stability of Shear Flows & Sum-of-Squares Polynomial Optimization
    - Elizabeth Carlton, Caltech
Time: 3:05pm Room: LOV 105
Abstract/Desc: The stability of steady states for complex dynamical systems are notoriously difficult problems, even for the seemingly simplest cases. For example, it is expected that the standard steady state shear profile for 2D Couette flow is globally stable for all Reynolds numbers, however the state-of-the-art analysis using standard energy method approaches is decades old and only proves the stability for relatively low Reynolds numbers or small initial data. In recent years, a promising computational approach uses polynomial sum-of-squares optimization to find Lyapunov functions based on low-mode projections onto an orthogonal basis. The choice of this basis can technically be arbitrary, but the natural basis comes from solving a numerical eigenvalue problem uncovered by solving a calculus of variations problem. One can solve this numerical eigenvalue problem (and similar eigenvalue problems) exploiting the inherent geometry. The solutions to these eigenvalue problems moreover yield heuristic insight into the difficulty of this problem and the sensitivity of this approach for determining global stability for Reynolds numbers greater than the current state-of-the-art critical value given by analysis.

Tuesday March 26, 2024

Topology and Geometry Seminar [url]
Conjugating Representations in PGL(k, C) into PGL(k, R)
    - Jared Miller, FSU
Time: 3:05 Room: LOV 107
Abstract/Desc: Properties of the space of representations of a surface group into a given simple Lie group is a very active area of research and is particularly relevant to higher Teichmüller theory. In this talk we study representations of finitely generated groups into PGL(k, C) and determine necessary and sufficient conditions for such a representation to be conjugate into PGL(k, R). In this way, we identify representations in the larger representation variety which are conjugate in PGL(k, C) to a representation in hom(pi_1 (S), PGL(k, R))/PGL(k, R).

Financial Math Seminar
Lower estimates for SDEs driven by fractional Brownian motion
    - Xi Geng, University of Melbourne
Time: 4:05pm Room: Lov 102
Abstract/Desc: Stochastic differential equations (SDEs) driven by fractional Brownian motion arise as natural non-Markovian models in financial mathematics. While upper estimates for the distribution of solution have been extensively studied in the literature, very little is known about lower estimates. In this talk, we discuss two types of sharp lower estimates (local and global) including a surprising lack-of-Gaussian-tail phenomenon which does not occur in the diffusion case. This talk is based on two earlier joint works (one with C. Ouyang and S. Tindel 2022 and the other with H. Boedihardjo 2023).

Wednesday March 27, 2024

Applied and Computational Math Seminar -- Stochastic Computing and Optimization
Stochastic Computing and Optimization
    - ACM/Fin Math students,
Time: 3:05PM Room: LOV 0231
Abstract/Desc: Students from ACM and Financial Math will present their research progress. Some invited speakers may also present their research.

Undergraduate Math Major Seminar
What is Biomathematics?
    - Richard Bertram, FSU
Time: 3:05 Room: LOV 107
Abstract/Desc: Everyone is welcome to attend this talk for undergraduates.

Biomathematics Journals Seminar
Molecular Switch Architecture Determines Response Properties of Signaling Pathways
    - Christopher Ryzowicz, FSU
Time: 5:00 Room: Dirac library

Biomathematics Seminar
Structure and dynamical behavior of non-normal networks
    - Haoyang Qian, FSU
Time: 3:05 Room: LOV 232
Abstract/Desc: In the last two decades, network science has become known as a critical model for understanding complex systems across a wide range of fields. At its core, network science considers a system as a collection of nodes that can be connected directly by an edge or indirectly by a series of edges, generating channels of interaction. To bridge the gap between network structure and dynamics, a nonlinear dynamical model on the nodes is typically defined, with the process controlled by a matrix corresponding to the underlying network's adjacency matrix. Furthermore, essential elements of the system, such as its stability and distinctive time scales, are typically characterized by its spectrum. The spectrum is the classic way to characterize a linearized system, however it is unreliable when the linear operator is not normal. Our key finding is based on the fact that a huge number of empirical networks in a variety of disciplines are strongly non-normal, demonstrating that strong non-normality is common in network science. Furthermore, we examine their global structure and demonstrate that asymmetry is insufficient, and that specific network designs must be chosen to establish a strong non-normality. Finally, we construct algorithms to explain their growth.

Thursday March 28, 2024

Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis
    - Giulia Livieri, London School of Economics
Time: 3:05pm Room: Zoom
Abstract/Desc: Causal operators (CO), such as various solution operators to stochastic differential equations, play a central role in contemporary stochastic analysis; however, there is still no canonical framework for designing Deep Learning (DL) models capable of approximating COs. This paper proposes a "geometry-aware" solution to this open problem by introducing a DL model-design framework that takes suitable infinite-dimensional linear metric spaces as inputs and returns a universal sequential DL model adapted to these linear geometries. We call these models Causal Neural Operators (CNOs). Our main result states that the models produced by our framework can uniformly approximate on compact sets and across arbitrarily finite-time horizons Hölder or smooth trace class operators, which causally map sequences between given linear metric spaces. Our analysis uncovers new quantitative relationships on the latent state-space dimension of CNOs which even have new implications for (classical) finite-dimensional Recurrent Neural Networks (RNNs). We find that a linear increase of the CNO's (or RNN's) latent parameter space's dimension and of its width, and a logarithmic increase of its depth imply an exponential increase in the number of time steps for which its approximation remains valid. A direct consequence of our analysis shows that RNNs can approximate causal functions using exponentially fewer parameters than ReLU networks.

Algebra Seminar [url]
An explicit generating function for the Betti numbers of the moduli space of stable, n-pointed rational curves
    - Paolo Aluffi, FSU
Time: 3:05 pm Room: LOV 107
More Information
Abstract/Desc: The variety $\overline\mathcal M_{0,n}$ parametrizes stable rational curves with n marked points. This is a central object in algebraic geometry, as the most studied and best understood moduli space of curves. Explicit constructions of this variety have been known for several decades, and recursion formulas for its betti numbers were obtained more than 30 years ago, but (to our knowledge) a more explicit expression for the betti numbers was not available. We obtain just such an expression, in the form of an explicit generating function for the class of $\overline\mathcal M_{0,n}$ in the Grothendieck group of varieties. As an application, we prove an asymptotic form of log concavity for the Poincaré polynomial of $\overline\mathcal M_{0,n}$.

Friday March 29, 2024

Mathematics Colloquium [url]
Loss of Synchrony to Silencing in Networks of Excitable Cells: Impact of Cell and Coupling Heterogeneity in Small Network Examples
    - Brad Peercy, University of Maryland
Time: 3:05 Room: Lov 101
Abstract/Desc: Experiments on pancreatic islets have raised a question about the potential unitary impact of certain cells in islet synchrony. Previous modeling to corroborate these findings under the suggested conditions proved unfruitful, but wide parameter searches did identify cases where silencing or ablating individual beta cells could completely or nearly completely silence islet behavior. This includes three cell beta cell networks and a small 2D grid network of simpler two-variable excitable cells. We find examples of "switch" behavior in each case and look to generalize.


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