Mon, 19 Jan 2026

16:30 - 17:30
L4

Towards Computational Topological (Magneto)Hydrodynamics: long term computation of fluids and plasma

Kaibo Hu
((Mathematical Institute University of Oxford))
Abstract
From Kelvin and Helmholtz to Arnold, Khesin, and Moffatt, topology has drawn increased attention in fluid dynamics. Quantities such as helicity and enstrophy encode knotting, topological constraints, and fine structures such as turbulence energy cascades in both fluid and MHD systems. Several open scientific questions, such as corona heating, the generation of magnetic fields in astrophysical objects, and the Parker hypothesis, call for topology-preserving computation. 
 
In this talk, we investigate the role of topology (knots and cohomology) in computational fluid dynamics by two examples: relaxation and dynamo. We investigate the question of “why structure-preservation” in this context and discuss some recent results on topology-preserving numerical analysis and computation. Finite Element Exterior Calculus sheds light on tackling some long-standing challenges and establishing a computational approach for topological (magneto)hydrodynamics.

 
Mon, 02 Feb 2026
14:15
L4

TBA

Spandan Ghosh
((Mathematical Institute University of Oxford))
Mon, 09 Feb 2026
14:15
L4

Biharmonic maps on conformally compact manifolds

Marco Usula
((Mathematical Institute University of Oxford))
Abstract

In this talk, I will present a result proved in my recent paper arXiv:2502.13580. I will discuss biharmonic maps between (and submanifolds of) conformally compact manifolds, a large class of complete manifolds generalizing hyperbolic space. After an introduction to conformally compact geometry, I will discuss one of the main results of the paper: if S is a properly embedded sub-manifold of a conformally compact manifold (N,h), and moreover S is transverse to the boundary and (N,h) has non-positive curvature, then S must be minimal. This result confirms a conjecture known as the Generalized Chen’s Conjecture, in the conformally compact context.

Mon, 19 Jan 2026
14:15
L4

TBA

Johanna Bimmerman
((Mathematical Institute University of Oxford))
Mon, 02 Mar 2026
14:15
L4

Metric wall-crossing

Ruadhai Dervan
(University of Warwick)
Abstract
Moduli spaces in algebraic geometry parametrise stable objects (bundles, varieties,...), and hence depend on a choice of stability condition. As one varies the stability condition, the moduli spaces vary in a well-behaved manner, through what is known as wall-crossing. As a general principle, moduli spaces admit natural Weil-Petersson metrics; I will state conjectures around the metric behaviour of moduli spaces as one varies the stability condition.
 
I will then prove analogues of these results in the model setting of symplectic quotients of complex manifolds, or equivalently geometric invariant theory. As one varies the input that determines a quotient, I will state results which explain the metric geometry of the resulting quotients (more precisely: Gromov-Hausdorff convergence towards walls, and metric flips across walls). As a byproduct of the approach, I will extend variation of geometric invariant theory to the setting of non-projective complex manifolds.
Mon, 23 Feb 2026
14:15
L4

A toric case of the Thomas-Yau conjecture

Jacopo Stoppa
(SISSA)
Abstract

We consider a class of Lagrangian sections L contained in certain Calabi-Yau Lagrangian fibrations (mirrors of toric weak Fano manifolds). We prove that a form of the Thomas-Yau conjecture holds in this case: L is isomorphic to a special Lagrangian section in this class if and only if a stability condition holds, in the sense of a slope inequality on objects in a set of exact triangles in the Fukaya-Seidel category. This agrees with general proposals by Li. On
surfaces and threefolds, under more restrictive assumptions, this result can be used to show a precise relation with Bridgeland stability, as predicted by Joyce. Based on arXiv:2505.07228 and arXiv:2508.17709.

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