Thu, 19 Jun 2025

12:00 - 12:30
L4

TBA

Astrid Herremans
(KU Leuven)
Abstract

TBA

Thu, 12 Jun 2025

12:00 - 12:30
L4

TBA

Boris Andrews
(Mathematical Institute (University of Oxford))
Abstract

TBA

Thu, 05 Jun 2025

12:00 - 12:30
L4

TBA

Kate Zhu
(Mathematical Institute (University of Oxford))
Abstract

TBA

Thu, 08 May 2025
16:00
Lecture Room 4, Mathematical Institute

Uniform Equidistribution of Quadratic Polynomials via Averages of $\mathrm{SL}_2(\mathbb{R})$ Automorphic Kernels

Lasse Grimmelt
(University of Oxford)
Abstract

In recent joint work with J. Merikoski, we developed a new way to employ $\mathrm{SL}_2(\mathbb{R})$  spectral methods to number-theoretical counting problems, entirely avoiding Kloosterman sums and the Kuznetsov formula. The main result is an asymptotic formula for an automorphic kernel, with error terms controlled by two new kernels. This framework proves particularly effective when averaging over the level and leads to improvements in equidistribution results involving quadratic polynomials. In particular, we show that the largest prime divisor of $n^2 + h$ is infinitely often larger than $n^{1.312}$, recovering earlier results that had relied on the Selberg eigenvalue conjecture. Furthermore, we obtain, for the first time in this setting, strong uniformity in the parameter $h$.
 

Thu, 29 May 2025

12:00 - 12:30
L4

TBA

Taejun Park
(Mathematical Institute (University of Oxford))
Abstract

TBA

Thu, 22 May 2025

12:00 - 12:30
L4

TBA

Arselane Hadj Slimane
(ENS Paris-Saclay)
Abstract

TBA

Thu, 15 May 2025

12:00 - 12:30
L4

Fast solvers for high-order finite element discretizations of the de Rham complex

Charlie Parker
(Mathematical Institute (University of Oxford))
Abstract

Many applications in electromagnetism, magnetohydrodynamics, and pour media flow are well-posed in spaces from the 3D de Rham complex involving $H^1$, $H(curl)$, $H(div)$, and $L^2$. Discretizing these spaces with the usual conforming finite element spaces typically leads to discrete problems that are both structure-preserving and uniformly stable with respect to the mesh size and polynomial degree. Robust preconditioners/solvers usually require the inversion of subproblems or auxiliary problems on vertex, edge, or face patches of elements. For high-order discretizations, the cost of inverting these patch problems scales like $\mathcal{O}(p^9)$ and is thus prohibitively expensive. We propose a new set of basis functions for each of the spaces in the discrete de Rham complex that reduce the cost of the patch problems to $\mathcal{O}(p^6)$ complexity. By taking advantage of additional properties of the new basis, we propose further computationally cheaper variants of existing preconditioners. Various numerical examples demonstrate the performance of the solvers.

Thu, 08 May 2025

12:00 - 12:30
L4

Computing complex resonances with AAA

Nick Trefethen
(Harvard University)
Abstract

A beautiful example of a nonlinear eigenvalue problem is the determination of complex eigenvalues for wave scattering. This talk will show how nicely this can be done by applying AAA rational approximation to a scalarized resolvent sampled at a few real frequencies.  Even for a domain as elementary as a circle with a gap in it, such computations do not seem to have been done before. This is joint work with Oscar Bruno and Manuel Santana at Caltech.

Thu, 01 May 2025

12:00 - 12:30
L4

High-order finite element methods for multicomponent convection-diffusion

Aaron Baier-Reinio
(Mathematical Institute (University of Oxford))
Abstract

Multicomponent fluids are mixtures of distinct chemical species (i.e. components) that interact through complex physical processes such as cross-diffusion and chemical reactions. Additional physical phenomena often must be accounted for when modelling these fluids; examples include momentum transport, thermality and (for charged species) electrical effects. Despite the ubiquity of chemical mixtures in nature and engineering, multicomponent fluids have received almost no attention from the finite element community, with many important applications remaining out of reach from numerical methods currently available in the literature. This is in spite of the fact that, in engineering applications, these fluids often reside in complicated spatial regions -- a situation where finite elements are extremely useful! In this talk, we present a novel class of high-order finite element methods for simulating cross-diffusion and momentum transport (i.e. convection) in multicomponent fluids. Our model can also incorporate local electroneutrality when the species carry electrical charge, making the numerical methods particularly desirable for simulating liquid electrolytes in electrochemical applications. We discuss challenges that arise when discretising the partial differential equations of multicomponent flow, as well as some salient theoretical properties of our numerical schemes. Finally, we present numerical simulations involving (i) the microfluidic non-ideal mixing of hydrocarbons and (ii) the transient evolution of a lithium-ion battery electrolyte in a Hull cell electrode.

Understanding and misunderstanding cell counts of the human brain: the crux of biological variation
Goriely, A Brain (15 Apr 2025)
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