This Week in Mathematics


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Today:

Algebra seminar
Toric Geometry and Matroid Chern-Schwartz-MacPherson Classes
- Jeffery Liu, FSU
Time: 3:05PM Room: 232
Abstract/Desc: Toric varieties are a type of algebraic variety containing the algebraic torus as a dense open subset. There is a connection between toric varieties and rational polyhedral fans. The Chow cohomology ring of a toric variety can be deduced from the geometry of a corresponding fan. Toric vector bundles are vector bundles on a toric variety with a torus action and have Chern classes valued in its Chow ring. Matroids are a combinatorial object that encodes “dependence properties” among elements of a set; for example, linear dependence in a vector frame. Associated to each matroid is a polyhedral fan called the Bergman fan. The matroid has a characteristic class called the Chern-Schwartz-MacPherson (CSM) class which is the Chern class of a “tautological” sub-bundle associated to the matroid. This class can be used, for example, to distinguish when two matroids are non-isomorphic. We give a basic overview of the theory of toric varieties, polyhedral fans, and toric vector bundles, and use it to compute the CSM class of a matroid.

Financial Mathematics Seminar
Algorithmic Stability of Stochastic Gradient Descent with Momentum and Heavy-Tailed Noise
- Thanh Dang, FSU
Time: 3:05 Room: LOV 0231
Abstract/Desc: There have been several prior works that study algorithmic stability and generalization error of heavy-tailed SGD such as Raj, Barsbey, Gürbüzbalaban, Zhu, Şimşekli (ALT 2023) and Raj, Zhu, Gürbüzbalaban, Şimşekli (ICML 2023). In this talk, we will explain how to obtain Wasserstein-1 stability result of heavy-tailed SGD with momentum (SGDm), which implies an upper bound on the generalization error of this algorithm. Our results on algorithmic stability and generalization error cover both convex and non-convex loss functions. Then in a specific case with quadratic loss function, we are able to show a surprising fact that the stability of heavy-tailed SGD is strictly better than that of heavy-tailed SGDm, indicating that the interaction of momentum and heavy tails can be harmful for stability.

Entries for this week: 8

Monday March 31, 2025

Analysis and PDE Seminar
Self-similar instability and forced nonuniqueness
- Michele Dolce, EPFL, Switzerland
Time: 3.05pm Room: LOV 232
More Information
Abstract/Desc: Building on an approach introduced by Golovkin in the '60s, we show that nonuniqueness in some forced PDEs is a direct consequence of the existence of a self-similar linearly unstable eigenvalue: the key point is a clever choice of the forcing term removing complicated nonlinear interactions. We use this method to present a short and self-contained proof of nonuniqueness in 2D perfect fluids, first obtained in Vishik's groundbreaking result. In particular, we give a direct construction of a forced self-similar unstable vortex, where we treat perturbatively the self-similar operator in a new and more quantitative way. This is a joint work with Giulia Mescolini.

Tuesday April 01, 2025

ACM seminar
Conditional neural field latent diffusion modeling for spatiotemporal turbulence modeling
- Meet Parikh, University of Notre Dame
Time: 3:05 pm Room: 0231
Abstract/Desc: Simulating complex, unsteady turbulent flows with high fidelity is critical across science and engineering, yet traditional methods like Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) face significant computational constraints, while many deep learning surrogates struggle to capture the inherent stochasticity and complexity, particularly within irregular geometries or under varying conditions. To overcome these hurdles, we introduce the Conditional Neural Field Latent Diffusion (CoNFiLD) model, a generative framework leveraging the synergy between conditional neural fields and latent diffusion processes. This approach enables efficient, memory-conscious, and robust stochastic generation of spatiotemporal turbulent flows, flexibly adapting to diverse physical parameters and boundary conditions without retraining, facilitating applications like sparse data reconstruction and super-resolution. A particularly challenging application demanding such capabilities is the synthesis of realistic, stochastic inflow conditions for eddy-resolving simulations, where conventional recycling methods are prohibitively expensive and existing synthetic or deterministic generators often fail to replicate key turbulence structures or maintain long-term stability. The CoNFiLD framework directly addresses this need through CoNFiLD-inlet, a specialized inflow generator. Demonstrating the power of conditional generation, CoNFiLD-inlet effectively generalizes across a wide range of Reynolds numbers using a single trained model. Comprehensive validation through a-priori and a-posteriori tests within DNS and Wall-Modeled Large Eddy Simulation (WMLES) substantiates the high fidelity, robustness, and scalability of this approach. This successful application underscores CoNFiLD’s potential as a versatile, computationally efficient tool poised to advance real-time turbulence simulation and enable digital twin technologies in fluid dynamics.

Geometry and Topology Seminar [url]
Applications of nonstandard analysis in geometry
- Sam Ballas, FSU
Time: 3:05 Room: 232
More Information
Abstract/Desc: Non standard analysis is a version of analysis where the reals are replaced by the larger non-archimedian field of hyperreals, which contain infinitesimally small elements. In this talk I will introduce the hyperreals and discuss how they give a useful language for describing various degeneration phenomena in geometry.

Wednesday April 02, 2025

Biomathematics Journal Club
A Unified Mathematical Model of Thyroid Hormone Regulation and Implication for Personalized Treatment of Thyroid Disorders
- Sarah Romero, FSU
Time: 5:00 Room: Dirac Library

Thursday April 03, 2025

Algebra seminar
Toric Geometry and Matroid Chern-Schwartz-MacPherson Classes
- Jeffery Liu, FSU
Time: 3:05PM Room: 232
Abstract/Desc: Toric varieties are a type of algebraic variety containing the algebraic torus as a dense open subset. There is a connection between toric varieties and rational polyhedral fans. The Chow cohomology ring of a toric variety can be deduced from the geometry of a corresponding fan. Toric vector bundles are vector bundles on a toric variety with a torus action and have Chern classes valued in its Chow ring. Matroids are a combinatorial object that encodes “dependence properties” among elements of a set; for example, linear dependence in a vector frame. Associated to each matroid is a polyhedral fan called the Bergman fan. The matroid has a characteristic class called the Chern-Schwartz-MacPherson (CSM) class which is the Chern class of a “tautological” sub-bundle associated to the matroid. This class can be used, for example, to distinguish when two matroids are non-isomorphic. We give a basic overview of the theory of toric varieties, polyhedral fans, and toric vector bundles, and use it to compute the CSM class of a matroid.

Financial Mathematics Seminar
Algorithmic Stability of Stochastic Gradient Descent with Momentum and Heavy-Tailed Noise
- Thanh Dang, FSU
Time: 3:05 Room: LOV 0231
Abstract/Desc: There have been several prior works that study algorithmic stability and generalization error of heavy-tailed SGD such as Raj, Barsbey, Gürbüzbalaban, Zhu, Şimşekli (ALT 2023) and Raj, Zhu, Gürbüzbalaban, Şimşekli (ICML 2023). In this talk, we will explain how to obtain Wasserstein-1 stability result of heavy-tailed SGD with momentum (SGDm), which implies an upper bound on the generalization error of this algorithm. Our results on algorithmic stability and generalization error cover both convex and non-convex loss functions. Then in a specific case with quadratic loss function, we are able to show a surprising fact that the stability of heavy-tailed SGD is strictly better than that of heavy-tailed SGDm, indicating that the interaction of momentum and heavy tails can be harmful for stability.

Friday April 04, 2025

Mathematics Colloquium
Graduate Student Flash Talks
- Grad Student Council, FSU
Time: 3:05 Room: Lov 101
Abstract/Desc: Participating graduate students will give flash talks about their research in this competition.

Machine Learning and Data Science Seminar
Convolutional Neural Networks for Data Assimilation in Operational Ocean Models
- Olmo Zavala Romero , FSU
Time: 1:20 Room: Lov 106
Abstract/Desc: Deep learning models have demonstrated remarkable success in fields such as natural language processing and computer vision, powering applications like real-time translation, image classification, and anomaly detection. In ocean sciences, particularly in data assimilation (DA), recent studies have explored the use of machine learning to emulate dynamical models, accelerate assimilation steps, or enhance forecasting via hybrid surrogate models. However, many of these approaches rely on simplified ocean models of intermediate complexity. Bridging the gap to full-scale, operational ocean models remains a significant challenge. This talk introduces convolutional neural networks (CNNs) and presents recent work leveraging them to assimilate sea surface height and sea surface temperature observations into the Hybrid Coordinate Ocean Model (HYCOM) in the Gulf of Mexico. The CNNs are trained to correct model errors using a two-year, data-assimilated HYCOM run at 1/25° resolution, with assimilation performed via the Tendral Statistical Interpolation System (TSIS). We investigate the performance of CNNs through five controlled experiments designed to inform their application in systems with full primitive equations, real-world observations, and complex coastal geometries. These experiments evaluate: (1) CNN architecture and complexity, (2) the type and number of observations used as input, (3) the type and number of fields being assimilated, (4) sensitivity to the assimilation window size, and (5) the influence of coastline features on assimilation accuracy.

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This Week in Mathematics