This Week in Mathematics


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Entries for this week: 5

Tuesday January 21, 2025

ACM seminar [url]
Pure-quartic solitons with PT-symmetric nonlinearity
- Savvas Sardelis, FSU
Time: 3:05 pm Room: 0231
Abstract/Desc: The idea of having solitary waves in Kerr nonlinear media arising in the presence of only quartic dispersion was briefly theoretically considered in the early 90's and then almost forgotten until its experimental discovery in 2016. These so-called pure-quartic solitons (PQS) were observed in a silicon photonic crystal waveguide where quartic dispersion was the dominant dispersion effect and all the other dispersion orders were negligible. In this talk, we present a new class of soliton based on the interaction of parity-time (PT) symmetric nonlinearity and quartic dispersion or diffraction. This novel kind of soliton is related to the recently discovered PQS, that arises from the balance of Kerr nonlinearity and quartic dispersion, through a complex coordinate shift. We find that the PT-symmetric PQS are linearly stable and present important differences with respect to its Hermitian (Kerr) counterpart, including a nontrivial phase structure, a skewed spectral intensity, and a higher power for the same propagation constant.

Wednesday January 22, 2025

Biomathematics Journal Seminar
Lotka-Volterra Predator-Prey Model With Periodically Varying Carrying Capacity
- Bhargav Karamched, FSU
Time: 5:00 Room: Dirac Library

Mathematics Colloquium [url]
An Adversarial Deep Learning approach using Natural Gradients for solving Partial Differential Equations
- Shu Liu, UCLA
Time: 3:05 Room: 101
Abstract/Desc: We propose a scalable, preconditioned primal-dual algorithm for solving partial differential equations (PDEs). By multiplying the equation with a test function, we reformulate it as an inf-sup problem, resulting in a loss function involving lower-order differential operators. To address this saddle point problem, we employ the Primal-Dual Hybrid Gradient (PDHG) algorithm. By introducing suitable preconditioning operators to the metric terms in PDHG proximal steps, we obtain an alternative natural gradient ascent-descent optimization scheme for updating primal and adversarial neural network parameters. These natural gradients are efficiently computed using the Krylov subspace iteration. An a posteriori convergence analysis is established for the time-continuous version of the proposed method. The algorithm is tested on various types of linear and nonlinear PDEs, scaling seamlessly to 50 dimensions. Numerical experiments highlight the method's improved accuracy, efficiency, and stability in convergence when compared to conventional deep-PDE solvers.

Thursday January 23, 2025

Financial Mathematics Seminar
Kakutani’s walk on spheres and efficient methods for solving Laplace and Poisson equation in domains with complex geometry
- Arash Fahim ,
Time: 3:05 Room: LOV 0231
Abstract/Desc: Kakutani’s walk on spheres (WoS) has proven to improve the efficiency of Feynman-Kac formula in solving linear elliptic equations with constant coefficients in bounded domains where the geometry is challenging. Such problems have applications in rendering in computer graphics, simulations in nuclear reactors, and evaluation of capacitance in semi-conductors. Motivated by the recent advancement in solving partial differential equations via deep learning, we introduce a new method, WoS-NN, by integrating WoS into a deep learning framework. Our study shows accurate field estimations, reducing 76.32% errors while using only 8% of path samples compared to the conventional WoS method via Feynman-Kac formula, which saves abundant computational time and resource consumption.

Algebra seminar
Solving linear recurrence relations in terms of solutions of lower order recurrences
- Mark van Hoeik, FSU
Time: 3:05PM Room: 232
Abstract/Desc: We use difference Galois theory to classify recurrence relations of orders 3 and 4 that are solvable in terms of solutions of recurrence relations of order 2. We also develop algorithms that can decide such solvability, and if so, express the solutions in terms of solutions of one or more recurrences of order 2. This is joint work with Heba Bou Kaedbey and Man Cheung Tsui.

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Department of Mathematics

This Week in Mathematics