We provide an iterative solution approach for the indefinite Helmholtz equation discretised using finite elements, based upon a Hermitian Skew-Hermitian Splitting (HSS) iteration applied to the shifted operator, and prove that the iteration is k- and mesh-robust when O(k) HSS iterations are performed. The HSS iterations involve solving a shifted operator that is suitable for approximation by multigrid using standard smoothers and transfer operators, and hence we can use O(N) parallel processors in a high performance computing implementation, where N is the total number of degrees of freedom. We argue that the algorithm converges in O(k) wallclock time when within the range of scalability of the multigrid. We provide numerical results verifying our proofs and demonstrating this claim, establishing a method that can make use of large scale high performance computing systems.

This talk is hosted by Rutherford Appleton Laboratory and will take place @ Harwell Campus, Didcot, OX11 0QX

The well-known Milnor-Wood inequality gives a bound on the Toledo invariant of a representation of the fundamental group of a compact surface in a non-compact Lie group of Hermitian type. While a lot is known regarding the counting of maximal Toledo components, and their role in higher Teichmueller theory, the non-maximal case remains elusive. In this talk, I will present a strategy to count the number of such non-maximal Toledo connected components. This is joint work in progress with Brian Collier and Jochen Heinloth, building on previous work with Olivier Biquard, Brian Collier and Domingo Toledo.

I will talk about the problem of recognising when a manifold admits a finite cover that fibres over the circle, with emphasis on the case of hyperbolic manifolds in odd dimensions. I will survey the state-of-art, and discuss the role that group theory plays in the problem. Finally, I will discuss a recent result that sheds light on the analogous group-theoretic problem, that is, virtual algebraic fibring of Poincaré-duality groups. The final theorem is joint with Sam Fisher and Giovanni Italiano.

I will explain how to obtain local L^\infty estimates for optimal transport problems. Considering entropic optimal transport and optimal transport with p-cost, I will show how such estimates, in combination with a geometric linearisation argument, can be used in order to obtain ε-regularity statements. This is based on recent work in collaboration with M. Goldman (École Polytechnique) and R. Gvalani (ETH Zurich).

In this talk, I will present a representation-theoretical approach to constructing a non-commutative analogue of the classical Laplace transform on Lie groups. I will begin by discussing the motivations for such a generalization, emphasizing its connections with harmonic analysis, probability theory, and the study of evolution equations on non-commutative spaces. I will also outline some of the key challenges that arise when extending the Laplace transform to the setting of Lie groups, including the non-commutativity of the group operation and the complexity of its dual space.

The main part of the talk will focus on an explicit construction of the Laplace transform in the framework of connected, simply connected nilpotent Lie groups. This construction relies on Kirillov’s orbit method, which provides a powerful bridge between the geometry of coadjoint orbits and the representation theory of nilpotent groups.

As an application, I will describe an operator-theoretic analogue of the classical Müntz–Szász theorem, establishing a density result for a family of generalized polynomials in $L^2(0,1)$ associated with the group setting. This result highlights the strength of the representation-theoretical approach and its potential for solving classical approximation problems in a non-commutative context.

to follow

Let \( G = \exp(\mathfrak{g}) \) be a connected, simply connected nilpotent Lie group with Lie algebra \( \mathfrak{g} \), and let \( H = \exp(\mathfrak{h}) \) be a closed subgroup with Lie algebra \( \mathfrak{h} \). Consider a unitary character \( \chi \) of \( H \), given by \(\chi(\exp X) = \chi_{f}(\exp X) = e^{i f(X)}, \  X \in \mathfrak{h}, \) for some \( f \in \mathfrak{g}^{\ast} \). Let \( \tau = \operatorname{Ind}_{H}^{G} \chi \) denote the monomial representation of \( G \) induced from \( \chi \).

The object of interest is the algebra \( D_{\tau}(G/H) \) of \( G \)-invariant differential operators acting on the homogeneous line bundle associated with the data \( (G, H, \chi) \). Under the assumption that \( \tau \) has finite multiplicities, it is known that \( D_{\tau}(G/H) \) is commutative.

In this talk, I will discuss the Polynomial Conjecture for the representation \( \tau \), which asserts that the algebra \( D_{\tau}(G/H) \) is isomorphic to  
\(\mathbb{C}[\Gamma_{\tau}]^{H}\),  the algebra of \( H \)-invariant polynomial functions on \( \Gamma_{\tau} \). Here, \( \Gamma_{\tau} = f + \mathfrak{h}^{\perp} \) denotes the affine subspace of \( \mathfrak{g}^{\ast} \).

I will present recent advances toward proving this conjecture, with a particular emphasis on Duflo's Polynomial Conjecture concerning the Poisson center of \( \Gamma_{\tau} \). Furthermore, I will discuss the case where \( \tau \) has discrete-type multiplicities in the exponential setting, shedding light on a counterexample to Duflo's conjecture.
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TBA

The nematic Helmholtz-Korteweg equation is a fourth-order scalar PDE modelling time-harmonic acoustic waves in nematic Korteweg fluids, such as nematic liquid crystals. Conforming discretizations typically require C1-conforming elements, for example the Argyris element, whose implementation is notoriously challenging - especially in three dimensions - and often demands a high polynomial degree. 
In this talk, we consider an alternative non-conforming C0-hybrid interior penalty method that is both stable and convergent for any polynomial degree greater than two. Classical C0-interior penalty methods employ an H1-conforming subspace and treat the non-conformity with respect to H2 with discontinuous Galerkin techniques. Building on this idea, we use hybridization techniques to improve the computational efficiency of the discretization. We provide a brief overview of the numerical analysis and show numerical examples, demonstrating the method's ability to capture anisotropic propagation of sound in two and three dimensions. 

The Whitney forms on a simplicial triangulation are piecewise affine differential forms that are dual to integration over chains.  The so-called blow-up Whitney forms are piecewise rational generalizations of the Whitney forms.  These differential forms, which are also called shadow forms, were first introduced by Brasselet, Goresky, and MacPherson in the 1990s.  The blow-up Whitney forms exhibit singular behavior on the boundary of the simplex, and they appear to be well-suited for constructing certain novel finite element spaces, like tangentially- and normally-continuous vector fields on triangulated surfaces.  This talk will discuss the blow-up Whitney forms, their properties, and their applicability to PDEs like the Bochner Laplace problem.  

Please register for the event via https://sites.google.com/view/oxwmathbio2025

Join us for an initial welcome pizza lunch to start the academic year to learn about what's happening in Mathematrix in 2025/26! Meet other students who are from underrepresented groups in mathematics and allies :) 

A Riemannian metric is said to be  Einstein if it has constant Ricci curvature. In dimensions 2 or 3, this is actually equivalent to requiring the metric to have constant sectional curvature. However,  in dimensions 4 and higher, the Einstein condition becomes significantly weaker than constant sectional curvature, and this has rather dramatic consequences. In particular, it turns out that there are  high-dimensional smooth closed manifolds that admit pairs of Einstein metrics with Ricci curvatures of opposite signs. After explaining how one constructs such examples, I will then discuss some recent results exploring the coexistence of Einstein metrics with zero and positive Ricci curvatures.

Brenier’s theorem and its Benamou-Brenier variant play a pivotal role
in optimal transport theory. In the context of martingale transport
there is a perfect analogue, termed stretched Brownian motion. We
show that under a natural irreducibility condition this leads to the
notion of Bass martingales.
For given probability measures µ and ν on Rn in convex order, the
Bass martingale is induced by a probability measure α. It is the min-
imizer of a convex functional, called the Bass functional. This implies
that α can be found via gradient descent. We compare our approach
to the martingale Sinkhorn algorithm introduced in dimension one by
Conze and Henry-Labordere.

 In the first part of the talk, after revisiting some classical models for dilute polymeric fluids, we show that thermodynamically 
consistent models for non-isothermal flows of such fluids can be derived in a very elementary manner. Our approach is based on identifying the 
energy storage mechanisms and entropy production mechanisms in the fluid of interest, which in turn leads to explicit formulae for the Cauchy 
stress tensor and for all the fluxes involved. Having identified these mechanisms, we first derive the governing system of nonlinear partial 
differential equations coupling the unsteady incompressible temperature-dependent Navier–Stokes equations with a 
temperature-dependent generalization of the classical Fokker–Planck equation and an evolution equation for the internal energy. We then 
illustrate the potential use of the thermodynamic basis on a rudimentary stability analysis—specifically, the finite-amplitude (nonlinear) 
stability of a stationary spatially homogeneous state in a thermodynamically isolated system.

In the second part of the talk, we show that sequences of smooth solutions to the initial–boundary-value problem, which satisfy the 
underlying energy/entropy estimates (and their consequences in connection with the governing system of PDEs), converge to weak 
solutions that satisfy a renormalized entropy inequality. The talk is based on joint results with Miroslav Bulíček, Mark Dostalík, Vít Průša 
and Endré Süli.

Mathematical modelling and stochastic optimization are often based on the separation of two stages: At the first stage, a model is selected out of a family of plausible models and at the second stage, a policy is chosen that optimizes an underlying objective as if the chosen model were correct. In this talk, I will introduce a new approach which, rather than completely isolating the two stages, interlinks them dynamically. I will first introduce the notion of “consequential performance” of each  model and, in turn, propose a “consequentialist criterion for model selection” based on the expected utility of consequential performances. I will apply the approach to continuous-time portfolio selection and derive a key system of coupled PDEs and solve it for representative cases. I will, also, discuss the connection of the new approach with the popular methods of robust control and of unbiased estimators.   This is joint work with M. Strub (U. of Warwick)

The question of profinite rigidity asks whether the isomorphism type of a group Γ can be recovered entirely from its finite quotients. In this talk, I will introduce the study of profinite rigidity in a different setting: the category of modules over a Noetherian domain Λ. I will explore properties of Λ-modules that can be detected in finite quotients and present two profinite rigidity theorems: one for free Λ-modules under a weak homological assumption on Λ, and another for all Λ-modules in the case when Λ is a Dedekind domain. Returning to groups, I will explain how these algebraic results yield new answers to profinite rigidity for certain classes of solvable groups. Time permitting, I will conclude with a sketch of future directions and ongoing collaborations that push these ideas further.

to follow

Join us for the inaugural session of Mathematrix book club! Have you heard that office workplaces often have the thermostat set at a temperature that is too cold for women to work comfortably? This month we will be discussing the academic articles behind concepts that often come up in conversations about gender inequality in the workplace. The goal of book club is to develop an evidence-based understanding of diversity in mathematics and academia. 

 I will present a new framework for determining effectively the spectrum and stability of traveling waves on
networks with symmetries, such as rings and lattices, by computing master stability curves (MSCs). Unlike
traditional methods, MSCs are independent of system size and can be readily used to assess wave
destabilization and multi-stability in small and large networks.

TBA 

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This talk is hosted by Rutherford Appleton Laboratory and will take place @ Harwell Campus, Didcot, OX11 0QX