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Monday March 18, 2024

PDE seminar
Liouville comparison theory for breakdown of Euler-Arnold equations
    - Justin Valletta, FSU
Time: 3:05pm Room: LOV 105
Abstract/Desc: In his seminal paper of 1966, Vladmir Arnold recast Euler’s equation for the motion of an incompressible fluid as the geodesic equation of a right-invariant Riemannian metric on the group of volume-preserving diffeomorphisms. Since then, many other prominent PDEs in mathematical physics have been given an analogous geometric interpretation as geodesic equations of a right-invariant connection on a diffeomorphism group; such equations are referred to as Euler-Arnold equations. Examples include the Camassa-Holm, the Hunter-Saxton, the modified Constantin-Lax-Majda, the KdV, and the surface quasi-geostrophic equation. A certain family of PDEs, called EPDiff equations, can be realized as the geodesic equation of the right-invariant Sobolev metric of order $k$ on the diffeomorphism group of $\mathbb{R}^n$. This family of PDEs encompasses many of the aforementioned PDEs of mathematical physics. In this talk, we present a breakdown criteria for solutions to EPDiff equations, although the result is also applicable to other Euler-Arnold equations. Our approach relies on using Lagrangian coordinates to convert the EPDiff equation to an ODE on a Banach space, thereby obtaining C^1-blowup of the velocity field solution by comparison with the Liouville equation. The breakdown mechanism is that the Jacobian of the Lagrangian flow reaches zero in finite-time. This means that the flow leaves the diffeomorphism group, and hence the C^1-norm of the velocity field blows up in finite time. We demonstrate the applicability of this result by applying it to prove breakdown of solutions to various members of the EPDiff family.

PDE seminar
Liouville comparison theory for breakdown of Euler-Arnold equations
    - Justin Valletta, FSU
Time: 3:05pm Room: LOV 105
Abstract/Desc: In his seminal paper of 1966, Vladmir Arnold recast Euler’s equation for the motion of an incompressible fluid as the geodesic equation of a right-invariant Riemannian metric on the group of volume-preserving diffeomorphisms. Since then, many other prominent PDEs in mathematical physics have been given an analogous geometric interpretation as geodesic equations of a right-invariant connection on a diffeomorphism group; such equations are referred to as Euler-Arnold equations. Examples include the Camassa-Holm, the Hunter-Saxton, the modified Constantin-Lax-Majda, the KdV, and the surface quasi-geostrophic equation. A certain family of PDEs, called EPDiff equations, can be realized as the geodesic equation of the right-invariant Sobolev metric of order $k$ on the diffeomorphism group of $\mathbb{R}^n$. This family of PDEs encompasses many of the aforementioned PDEs of mathematical physics. In this talk, we present a breakdown criteria for solutions to EPDiff equations, although the result is also applicable to other Euler-Arnold equations. Our approach relies on using Lagrangian coordinates to convert the EPDiff equation to an ODE on a Banach space, thereby obtaining C^1-blowup of the velocity field solution by comparison with the Liouville equation. The breakdown mechanism is that the Jacobian of the Lagrangian flow reaches zero in finite-time. This means that the flow leaves the diffeomorphism group, and hence the C^1-norm of the velocity field blows up in finite time. We demonstrate the applicability of this result by applying it to prove breakdown of solutions to various members of the EPDiff family.

Wednesday March 20, 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
An introduction to modeling social dynamics and evolution
    - Bryce Morsky, FSU
Time: 3:05pm Room: LOV 107
Abstract/Desc: This is an introductory talk for undergrads and anyone else interested in game theory and modeling social systems. All are welcome.

Biomathematics Seminar
A territorial game between wasps
    - Townsend Porcher, FSU
Time: 3:05 Room: LOV 232
Abstract/Desc: Experimentalists have observed that male cicada killer wasps (Sphecius speciosus) will redraw the boundaries of their territories to coincide with wooden dowels randomly scattered in a field. This is an example of animals using landmarks to decide the boundaries of their territories. One hypothesis for this type of behavior is that establishing territory boundaries at landmarks provides an easy and obvious way to resolve territory disputes, rather than the potentially costly activity of fighting for territory. The legitimacy of this hypothesis partially rests on the solution to a particular game between two competing animals: accept the landmark or fight for more territory? We will build a model of this territorial game (developed by Mike Mesterton-Gibbons and Eldridge Adams) and then perform some fun computations with our model.

Biomathematics Journals Seminar
An Indicator of Crohn's Disease Severity Based on Turing Patterns
    - Meghan Peltier, FSU
Time: 5:00 Room: Dirac library

Thursday March 21, 2024

Financial Math Seminar
Efficient Solvers for Partial Gromov--Wasserstein
    - Hengrong Du, Vanderbilt University
Time: 3:05pm Room: Lov 231
Abstract/Desc: Addressing the challenge of comparing unbalanced measures in distinct metric spaces, our work transforms the Partial Gromov-Wasserstein (PGW) problem into a more tractable variant, akin to the Gromov-Wasserstein problem. We introduce two novel solvers based on the Frank-Wolfe algorithm, offering mathematically equivalent yet computationally efficient solutions to the PGW problem. Demonstrated through shape-matching and positive-unlabeled learning applications, our solvers outperform existing methods in both computation time and effectiveness, establishing the PGW problem as a valuable metric for metric measure spaces. This is joint work with Yikun Bai, Rocio Diaz Martin, Ashkan Shahbazi, and Soheil Kolouri.

Algebra Seminar [url]
Calculus of fractions for higher categories
    - Chris Kapulkin, UWO
Time: 3:05 pm Room: LOV 107
More Information
Abstract/Desc: A central objective of (abstract) homotopy theory is to understand the localization of a category at a class of weak equivalences. While the localization is always known to exist, it is typically very difficult to compute. One case in which a workable model for the localization can be described is when the class of weak equivalences satisfies "calculus of fractions," introduced by P. Gabriel and M. Zisman in their 1967 book. I will report on joint work with D. Carranza and Z. Lindsey (arXiv:2306.02218) that generalizes calculus of fractions to higher category theory. We show that for higher categories satisfying our condition the localization can be computed via a marked version of Kan's Ex functor. These results have since been applied in several areas, including combinatorics (joint with D. Carranza and J. Kim) and string topology (A. Blumberg and M. Mandell), but we continue to look for new applications.

Friday March 22, 2024

Machine Learning and Data Science Seminar
Variograms for kriging and clustering of spatial functional data with phase variation
    - Sebastian Kurtek, Ohio State University
Time: 1:20 Room: Lov 102
Abstract/Desc: Spatial functional data arise in various applied domains including environmental sciences and medical imaging. In general, functional data often exhibit two distinct sources of variation, amplitude (y-axis) variation, and phase (x-axis) variation. Statistical analysis of functional data thus involves so-called amplitude-phase separation wherein these two sources of variation are decoupled. However, in the particular setting of spatial functional data, amplitude and/or phase are dependent in space resulting in new challenges. We describe a framework that extends amplitude-phase separation methods in functional data to the spatial setting, with a view towards performing clustering and spatial prediction. We propose a decomposition of the trace-variogram, which quantifies spatial variation, into amplitude and phase components, and quantify how spatial correlations between functional observations manifest in their respective amplitude and phase. This enables us to generate separate amplitude and phase clustering methods for spatial functional data, and develop a novel spatial functional interpolant at unobserved locations based on combining separate amplitude and phase predictions. Through simulations and real data analyses, we demonstrate advantages of our approach when compared to standard ones that ignore phase variation, through more accurate predictions and more interpretable clustering results. This is joint work with my former Ph.D. student Xiaohan Guo, and Karthik Bharath at the University of Nottingham.

Mathematics Colloquium [url]
Bayesian Data-driven Discovery of Physical Laws in a Heterogeneous Environment from Noisy Data
    - Guang Lin, Purdue University
Time: 3:05 Room: Lov 101
Abstract/Desc: The discovery of Partial Differential Equations (PDEs) is an essential task for applied science and engineering. However, data-driven discovery of PDEs is generally challenging, primarily stemming from the sensitivity of the discovered equation to noise and the complexities of model selection. In this talk, I will introduce an advanced Bayesian sparse learning algorithm for PDE discovery with variable coefficients, predominantly when the coefficients are spatially or temporally dependent. Specifically, we apply threshold Bayesian group Lasso regression with a spike-and-slab prior and leverage a Gibbs sampler for Bayesian posterior estimation of PDE coefficients. This approach not only enhances the robustness of point estimation with valid uncertainty quantification. The capability of this method is illustrated by the discovery of several classical benchmark PDEs with spatially or temporally varying coefficients from solution data obtained from the reference simulations. In the experiments, we show that the proposed approaches are more robust than the baseline methods under noisy environments and provide better model selection criteria along the regularization path.


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