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Professor Ana Djurdjevac will talk about; 'The Dean–Kawasaki Equation: Theory, Numerics, and Applications'

The Dean–Kawasaki equation provides a stochastic partial differential equation description of interacting particle systems at the level of empirical densities and has attracted considerable interest in statistical physics, stochastic analysis, and applied modeling. In this work, we study analytical and numerical aspects of the Dean–Kawasaki equation, with a particular focus on well-posedness, structure preservation, and possible discretization strategies. In addition, we extend the framework to the Dean–Kawasaki equation posed on smooth hypersurfaces. We discuss applications of the Dean–Kawasaki framework to particle-based models arising in biological systems and modeling social dynamics.

Speaker Estefania Loayza Romero will talk about:  A Riemannian Approach for PDE-Constrained Shape Optimization Using Outer Metrics

In PDE-constrained shape optimisation, shapes are traditionally viewed as elements of a Riemannian manifold, specifically as embeddings of the unit circle into the plane, modulo reparameterizations. The standard approach employs the Steklov-Poincaré metric to compute gradients for Riemannian optimisation methods. A significant limitation of current methods is the absence of explicit expressions for the geodesic equations associated with this metric. Consequently, algorithms have relied on retractions (often equivalent to the perturbation of identity method in shape optimisation) rather than true geodesic paths. Previous research suggests that incorporating geodesic equations, or better approximations thereof, can substantially enhance algorithmic performance. This talk presents numerical evidence demonstrating that using outer metrics, defined on the space of diffeomorphisms with known geodesic expressions, improves Riemannian gradient-based optimisation by significantly reducing the number of required iterations and preserving mesh quality throughout the optimisation process.

This talk is hosted at RAL. 

TBA

Professor Nick Trefethen will speak about: 'Quadrature = rational approximation'

Whenever you see a string of quadrature nodes, you can consider it as a branch cut defined by the poles of a rational approximation to the Cauchy transform of a weight function.  The aim of this talk is to explain this strange statement and show how it opens the way to calculation of targeted quadrature formulas for all kinds of applications.  Gauss quadrature is an example, but it is just the starting point, and many more examples will be shown.  I hope this talk will change your understanding of quadrature formulas. 

This is joint work with Andrew Horning. 
 

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I will discuss the paper "Construction of Gross-Neveu model using Polchinski flow equation" by Pawel Duch (https://arxiv.org/abs/2403.18562).