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Algebra and Combinatorics Seminar: Kailash Misra, NC State, Weight multiplicities of some affine Lie algebra modules

SAS 4201

Consider the affine Lie algebra $\mathfrak{g}$ associated with the simple Lie algebra $sl(n)$ consisting of $n\times n$ trace zero matrices over the field of complex numbers. For every dominant integral weight $\lambda$ there is a unique (upto isomorphism) irreducible highest weight $\mathfrak{g}$ module $V(\lambda)$. Although there are infinitely many weights of this module, certain important…

Differential Equations and Nonlinear Analysis Seminar: Ming Chen, University of Pittsburgh, Global bifurcation for hollow vortex desingularization

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A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a perfect fluid; we can think of it as a spinning bubble of air in water. In this talk, we present a general method for desingularizing non-degenerate steady point vortex configurations into collections of steady hollow vortices. The…

Computational and Applied Mathematics Seminar: Vakhtang Putkaradze, University of Alberta, Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries

SAS 4201

Physics-Informed Neural Networks (PINNs) have received much attention recently due to their potential for high-performance computations for complex physical systems, including data-based computing, systems with unknown parameters, and others. The idea of PINNs is to approximate the equations and boundary and initial conditions through a loss function for a neural network. PINNs combine the efficiency…

Stochastics Seminar: Sayan Banerjee , UNC-Chapel Hill, Ergodicity and fluctuations of the Atlas model

SAS 4201

We investigate the long-time behavior and stationary fluctuations of an infinite-dimensional rank-based diffusion process, called the Atlas model, where particles move as independent Brownian motions, with the lowest ranked particle at any time getting a unit upward drift. The associated process of gaps between successive ranked particles possesses an uncountable collection of invariant measures. We…

Differential Equations and Nonlinear Analysis Seminar: Hakima Bessaih, FIU, Various numerical scheme for stochastic hydrodynamic models

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We will consider various models in hydrodynamic, including the 2d Navier-Stokes, Boussinesq equations, and a Brinkman-Forchheimer-Navier-Stokes equations in 3d. These models are driven by an external stochastic Brownian perturbation. We will implement space-time numerical schemes and prove their convergence. We will show some rates of convergence as well. Furthermore, we will show the difference between…

Geometry and Topology Seminar: Peter J. Olver, University of Minnesota, Structure and Generators of Differential Invariant Algebras

SAS 1216

The structure of algebras of differential invariants, particularly their generators, is based on the symbolic invariant calculus provided by the method of equivariant moving frames.  I will discuss a new computational algorithm that will, in many cases, determine whether a given set of differential invariants is generating.  As an example, we establish a new result…

Applied Math Graduate Student Seminar: John Darges, NC State, Sensitivity Analysis in Forward and Inverse Problems

SAS 4201

Global sensitivity analysis (GSA) offers a flexible framework for understanding the structural importance of uncertain parameters in mathematical models. This dissertation focuses on forward and inverse problems arising in uncertainty quantification and the computation of Sobol’ indices, measures of variance-based sensitivity. The models involved in these prob- lems are often computationally expensive to evaluate. Sensitivity…

Geometry and Topology Seminar: Adam Lowrance, Vassar College, The average value of invariants of 2-bridge knots.

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We show how to use continued fraction representations of 2-bridge knots to compute the average value of different invariants of the set of 2-bridge knots with fixed crossing number c. Examples include the Seifert genus, braid index, and the absolute value of the signature. We also mention other properties of the probability distributions of these…

Differential Equations and Nonlinear Analysis Seminar: Edouard Pauwels, Université de Toulouse, Nonsmooth differentiation of parametric fixed points

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Recent developments in the practice of numerical programming require optimization problems not only to be solved numerically, but also to be differentiated. This allows to integrate the computational operation of evaluating a solution in larger models, which are themselves trained or optimized using gradient methods. Most well known applications include bilevel optimization and implicit input-output…

Computational and Applied Mathematics – Differential Equations and Nonlinear Analysis Seminar: Alexey Miroshnikov, Discover Financial Services, Stability theory of game-theoretic group feature explanations for machine learning models.

SAS 4201

In this article, we study feature attributions of Machine Learning (ML) models originating from linear game values and coalitional values defined as operators on appropriate functional spaces. The main focus is on random games based on the conditional and marginal expectations. The first part of our work formulates a stability theory for these explanation operators…

Stochastics Seminar: Amarjit Budhiraja, UNC-Chapel Hill, Large deviations for weakly interacting diffusions and mean field stochastic control problems

SAS 4201

Consider a collection of particles whose state evolution is described through a system of interacting diffusions in which each particle is driven by an independent individual source of noise and also by a small amount of noise that is common to all particles. The interaction between the particles is due to the common noise and…