Professor Kaibo Hu will be talking about: 'Structure-Preserving Finite Element Methods for the Einstein Equations'

Some of the most successful vector-valued finite elements in computational electromagnetics and fluid mechanics, such as the Nédélec and Raviart-Thomas elements, are recognized as special cases of Whitney’s discrete differential forms. Recent efforts aim to go beyond differential forms and establish canonical discretizations for more general tensors. An important class is that of form-valued forms, or double forms, which includes the metric tensor (symmetric (1,1)-forms) and the curvature tensor (symmetric (2,2)-forms). Like the differential structure of forms is encoded in the de Rham complex, that of double forms is encoded in the Bernstein–Gelfand–Gelfand (BGG) sequences and their cohomologies. Important examples include the Calabi complex in geometry and the Kröner complex in continuum mechanics.
These constructions aim to address the problem of discretizing tensor fields with general symmetries on a triangulation, with a particular focus on establishing discrete differential-geometric structures and compatible tensor decompositions in 2D, 3D, and higher dimensions.
 

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. 
 

In this talk, we are interested in neural network approximations for Hamilton–Jacobi–Bellman equations.These are non linear PDEs for which the solution should be considered in the viscosity sense. The solutions also corresponds to value functions of deterministic or stochastic optimal control problems. For these equations, it is well known that solving the PDE almost everywhere may lead to wrong solutions. 

We present a new method for approximating these PDEs using neural networks. We will closely follow a previous work by C. Esteve-Yagüe, R. Tsai and A. Massucco (2025), while extending the versatility of the approach. 

We will first show the existence and unicity of a general monotone abstract scheme (that can be chosen in a consistent way to the PDE), and that includes implicit schemes. Then, rather than directly approximating the PDE -- as is done in methods such as PINNs (Physics-Informed Neural Networks) or DGM (Deep Galerkin Method) -- we incorporate the monotone numerical scheme into the definition of the loss function. 

Finally, we can show that the critical point of the loss function is unique and corresponds to solving the desired scheme. When coupled with neural networks, this strategy allows for a (more) rigorous convergence analysis and accommodates a broad class of schemes. Preliminary numerical results are presented to support our theoretical findings.

This is joint work with C. Esteve-Yagüe and R. Tsai.

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