Let $Z$ be any fixed closed connected PL 4-manifold. We give a reduction from the compressed word problem in Thompson's group $F$ to fixed-target recognition of $Z\#_2(S^2\times S^2)$. Once a finite triangulation of $Z$ has been fixed, the construction sends a straight-line program $A$ to a closed triangulated PL 4-manifold $X_{A,Z}$ such that a designated output word $\operatorname{val}(A)$ is equal to the identity in $F$ if and only if $X_{A,Z}$ is PL-homeomorphic to $Z\#_2(S^2\times S^2)$. The compressed word problem in $F$ has been shown to be PSPACE-complete, and so every such recognition problem is PSPACE-hard. In particular, fixed-target recognition of $\#_n(S^2\times S^2)$ is PSPACE-hard for every fixed $n\geq 2$.

Brain activity produces waste proteins such as amyloid-β, whose accumulation is an early hallmark of Alzheimer’s disease. Amyloid-β  dynamics are governed by cell-scale processes, namely production at neurons, diffusion and aggregation in the extracellular space, and receptor-mediated transport across the blood–brain barrier. The resulting accumulation, however, plays out at the tissue scale.

In this talk, we introduce a multi-species reaction–diffusion model in a periodically perforated domain, which is coupled to nonlinear receptor  dynamics on capillary surfaces. Since the receptors do not diffuse, they  lack spatial regularity, which makes the analysis more delicate. We  prove well-posedness with uniform L∞ bounds and, using two-scale  convergence, derive an effective macroscopic model. Its diffusion tensor  comes from a cell problem, surface processes become volume sources, and  the receptors remain a cell-scale variable. Numerical experiments show  that, at fixed porosity, the shape and curvature of cells alone can change the effective transport by tens of percent.

How large must the average degree of a graph be to force a cycle with as many chords as vertices? Chen, Erdős and Staton conjectured in 1996 that a sufficiently large constant suffices. In this talk, I will sketch a proof that sufficiently large constant minimum degree guarantees a cycle of length $\ell$ with $\Omega(\ell/(\log \ell)^C)$ chords, for an absolute constant $C>0$. This comes within a polylogarithmic factor of the conjecture and answers a related question of Dvořák, Martins, Thomassé and Trotignon for graphs of sufficiently large minimum degree.

This is joint work with Nemanja Draganić.

We propose the first return time distribution (FRTD) of a random walk as an interpretable and mathematically grounded node embedding. The FRTD assigns a probability mass function to each node, allowing us to define a distance between any pair of nodes using standard metrics for discrete distributions. We present several arguments to motivate the FRTD embedding. First, we show that FRTDs are strictly more informative than eigenvalue spectra, yet insufficient for complete graph identification, thus placing FRTD equivalence between cospectrality and isomorphism. Second, we argue that FRTD equivalence between nodes captures structural similarity. Third, we empirically demonstrate that the FRTD embedding outperforms manually designed graph metrics in network alignment tasks. Finally, we show that random networks that approximately match the FRTD of a desired target also preserve other salient features. Together these results demonstrate the FRTD as a simple and mathematically principled embedding for complex networks.

I will revisit the relation between Hilbert schemes of curves in a threefold X and spaces of stable maps to X from the perspective of wall-crossing. The expectation is that there exist proper spaces of embedded nodal curves in semistable degenerations of X that admit wall-crossings to both theories. I will sketch a construction of such spaces for a product threefold S x C, in curve classes that are non-zero on the second factor.

In 1994, Berger and Coburn showed that a Toeplitz operator on the Fock space is bounded if the heat transform of its symbol is bounded at some time 0<t<1/4, and that boundedness of the operator implies boundedness of the heat transform for 1/4<t<1. They conjectured that boundedness of the operator is equivalent to boundedness of the heat transform at the critical time t=1/4. While the conjecture holds for several classes of symbols, we show that both implications fail in general. Our counterexamples come from Baire category and closed graph arguments, which yield no explicit symbols. I will also explain a sharp positive result: trace-class membership guarantees a bounded heat transform at the critical time, whereas membership in larger Schatten classes does not.

Roughly a decade ago the Euler hydrodynamic equations were written down for integrable many-body systems. Since the number of conserved fields is extensive, the Euler equations form an infinite array of hyperbolic conservation laws. For my talk, I will focus on the Toda lattice. Since initial conditions are random, the respective Lax matrix is random. As to be explained, random matrix theory enters through determining the density of states and investigating the motion of eigenvectors.  

Roughly a decade ago the Euler hydrodynamic equations were written down for integrable many-body systems. Since the number of conserved fields is extensive, the Euler equations form an infinite array of hyperbolic conservation laws. For my talk, I will focus on the Toda lattice. Since initial conditions are random, the respective Lax matrix is random. As to be explained, random matrix theory enters through determining the density of states and investigating the motion of eigenvectors.  

to follow

Christian Alber will talk about 'Multiscale Spectral Generalised Finite Elements: Preconditioning and Model Reduction Beyond SPD'

Multiscale PDEs with heterogeneous, highly oscillatory coefficients can be challenging for standard numerical methods. This talk presents a framework for the design and analysis of the Multiscale Spectral Generalised Finite Element Method (MS-GFEM), a partition of unity method using optimal local approximation spaces constructed from eigenproblems. These spaces are closely related to GenEO coarse spaces in domain decomposition.  Under suitable assumptions, we establish nearly exponential local approximation rates and describe their dependence on problem parameters. We also discuss how the resulting coarse spaces can be used in two-level domain decomposition preconditioners. Applications to Helmholtz and Stokes problems illustrate the approach beyond the symmetric positive definite setting, with numerical experiments complementing the analysis.

Maria Reboredo Prado
Stratosphere-Troposphere Interactions: Understanding how the Upper Atmosphere Shapes the Weather and Climate We See.
It is well known that the stratosphere – the second layer of the atmosphere – can exert a powerful influence on weather and climate at the surface. This influence is particularly important in winter, when the strength of the stratospheric polar vortex — a belt of fast-moving winds encircling the pole — can vary dramatically. If the impacts of these variations on the lower atmosphere were better understood and more accurately represented in forecast models, they could provide early signals of winter weather patterns, improving predictions on subseasonal to seasonal timescales. Yet, the mechanisms by which changes high in the atmosphere produce a response near the surface remain poorly understood. 
In this talk, I will use an idealised atmospheric model to test how much of this downward effect can be explained by large-scale balanced dynamics. The model allows individual processes – such as atmospheric cooling, surface friction, and boundary effects – to be switched on or off, making it possible to assess their relative roles in shaping the surface response. A key improvement on previous theoretical models is that the displacement of the tropopause — the boundary between the troposphere and stratosphere – emerges naturally as part of the flow’s response to forcing. Overall, this framework provides a controlled setting for identifying the dynamical mechanisms that transmit stratospheric disturbances to the surface.

Orson Hart
Inertialess instability of viscosity-stratified Couette flow
In this talk, we discuss the stability of viscosity-stratified Couette flow. By neglecting inertia, we isolate the destabilising effect of viscosity stratification on a fluid configuration consisting of three layers of immiscible, incompressible fluid. We show how the linear stability analysis can be simplified by considering symmetric flow configurations and, by considering disturbances of arbitrary wavelength, we identify a new instability mechanism that is only induced by finite-wavelength disturbances.

Vedanta Thapar

Embedding complex networks with the random walk first return time distribution

We consider the problem of network embedding, specifically we propose the first return time distribution (FRTD) of a random walk as an interpretable and mathematically grounded node embedding. The FRTD assigns a probability mass function to each node, allowing us to define a distance between any pair of nodes using standard metrics for discrete distributions. We present several arguments to motivate the FRTD embedding. First, we show that FRTDs are strictly more informative than eigenvalue spectra, yet insufficient for complete graph identification, thus placing FRTD equivalence between cospectrality and isomorphism. Second, we argue that FRTD equivalence between nodes captures structural similarity. Third, we empirically demonstrate that the FRTD embedding outperforms manually designed graph metrics in network alignment tasks. Finally, we show that random networks that approximately match the FRTD of a desired target also preserve other salient features. Together these results demonstrate the FRTD as a simple and mathematically principled embedding for complex networks.

Chun Lam Li

Neural network augmentation of the Maxey-Riley equation for Particle-Laden Flows
Fluid-particle interactions in low-Reynolds-number confined flows govern many biological and engineering processes, including microthrombi transport in cerebral microvessels and particle separation in microfluidic devices. Fully resolved direct numerical simulations (DNS) are accurate but costly for small particle sizes and large numbers of particles, limiting the simulation and analysis of particle-laden flows. We develop a machine learning (ML) model based upon the Maxey-Riley (M-R) equation to provide fast predictions of particle motion. A key feature is that the ML model is physics-based; neural networks learn corrections to an M-R based model to improve its predictions beyond the conditions for which it was derived. We consider a single spherical particle moving between two parallel plates, with plane Poiseuille flow as the undisturbed flow. The problem is parametrized by the flow Reynolds number, particle-to-fluid density ratio, confinement ratio (particle diameter to channel height), and initial wall-normal position. The model is trained on particle velocities from DNS sampled at discrete time points, and evaluated on trajectories that were not used for training. The ML model improves prediction accuracy of particle trajectories over the uncorrected M-R baseline across the tested physical parameters, at a far lower computational cost than DNS.

work with Justin Sirignano¹, and Sarah L. Waters¹
1. Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK

The existence of minimizers in the calculus of variations is closely tied to various notions of convexity. A fundamental question in this regard is how these notions relate to each other. 

In this talk, we study several such generalized notions of convexity for continuous integrands on the space of symmetric $3\times 3$ matrices and establish their mutual relationships.

In particular, we construct a symmetric quasiconvex integrand that is not polyconvex at symmetric matrices, and an integrand that is both symmetric rank-one convex and Hessian quasiconvex but not symmetric quasiconvex. We also establish a transfer principle that converts smooth rank-one convex functions that fail to be Hessian quasiconvex into symmetric rank-one convex integrands while preserving this failure. Applying this scheme to existing counterexamples, we obtain symmetric rank-one convex integrands that are not Hessian quasiconvex. Together with previous results, this settles all pairwise implications among the notions considered here.

Professor Rob Scheichl will talk about: 'Efficient localised model reduction for multiscale PDEs via Grassmannian Interpolation'

Multiscale Spectral Generalized Finite Element Methods (MS-GFEM) are a powerful new discretisation method for general variational problems that satisfy a Gårding-type inequality, including strongly non-Hermitian problems. The construction of optimal approximation spaces is localised and requires no a priori regularity assumptions. The global approximation error is controlled by the local errors, which are rigorously shown to decay nearly exponentially. The optimality hinges on an SVD of the local restriction operator in a suitable, coefficient-dependent inner product on an oversampled patch. Compactness of this operator in the space of a-harmonic functions guarantees spectral accuracy akin to Weyl asymptotics for the Laplacian. As such, MS-GFEM can be seen as an “hp-version” of Localized Orthogonal Decomposition (LOD). Given the coefficient function the local approximation spaces can be constructed efficiently. In this talk, I will show how we can use Grassmannian interpolation of the resulting local approximation spaces on sparse grids in high dimensions to derive localized model order reduction methods that inherit the nearly exponential spatial and parametric convergence of MS-GFEM and of sparse grids, respectively. I will show some numerical experiments confirming the theoretical results in the context of elliptic problems.

We establish non-unirational versions of Hilbert Irreducibility for all Hilbert modular surfaces which are of K3 type. As an application we prove new instances of the regular Inverse Galois Problem for the simple groups PSL2(GF(p)^2) subject to congruence conditions on p. Joint work with J. Demeio.

McKean–Vlasov stochastic differential equations (MKV SDEs) are stochastic differential equations whose coefficients depend on both the solution and its law. They arise in various applications in mathematical finance, including models of systemic risk. Their numerical approximation typically involves two discretisation parameters: the time-step size and the number of particles used to approximate the solution's law.

Multi-index Monte Carlo (MIMC) methods exploit this two-parameter structure to reduce the computational cost of Monte Carlo simulations. Haji-Ali and Tempone derived complexity bounds for MIMC approximation of MKV SDEs, assuming certain variance decay rates.

In this talk, we present a rigorous variance analysis for the MIMC method applied to MKV SDEs with additive noise. We explain the main ideas behind the analysis and show how the resulting bounds relate to the method's computational complexity. We complement our theoretical analysis with numerical experiments investigating the variance decay rates.

This talk is based on joint work with Mike Giles and Yuying Zhao.

The acquisition and maintenance of the correct cellular pattern and tissue architecture is essential for organ function in multicellular organisms. Beyond generating the required cellular diversity, developing tissues need to attain the appropriate morphology. Tissue architecture is built through symmetry breaking instabilities such as folding, branching, buckling or budding. Studying single-cell responses alone is not enough to reveal the mesoscale physical and biochemical processes that regulate tissue organization and morphology over time and scale. In our lab we investigate how the interaction between the extracellular matrix (ECM) and tissues regulates processes during development and cancer, with a focus on ECM mechanical properties. While most research in this domain has concentrated on the ECM's elasticity as a primary determinant of cell and tissue behaviour, it is important to note that the ECM possesses both viscous and elastic properties. I will present our findings demonstrating that the passive viscoelastic properties of the ECM regulate tissue architecture and patterning both during development and cancer. Specifically, we show that ECM viscoelasticity influences the spatial and temporal organization of multicellular tissues in breast and intestinal organoids. Overall, our work highlights the critical role of viscoelasticity in driving morphological symmetry breaking instabilities, a fundamental process in morphogenesis and oncogenesis, and suggests ways of controlling tissue through ECM mechanics.

Exotic phenomena in 4-manifold topology include exotic 4-manifolds, diffeomorphisms,  and surfaces.  A famous result of Wall implies that homeomorphic simply-connected, smooth 4-manifolds are stably diffeomorphic, meaning that they become diffeomorphic after connected sum with enough copies of S^2 x S^2.  In other words, there are no stably exotic simply-connected 4-manfolds. I will discuss the extent to which stable exotica are known to exist in other contexts, with a focus on the case of surfaces. 
 

I will discuss probabilistic and analytic aspects of two-dimensional Coulomb gases confined to Jordan domains, curves, and arcs. In particular, I will describe the asymptotic behavior of particle distributions as the number of particles tends to infinity. I will also explain how the Loewner energy and related quantities arise in the asymptotic expansion of the free energy. A key analytical tool throughout is the Grunsky operator. 

The talk is based on joint works with Kurt Johansson.

Stars are commonly modelled as a spherically symmetric gas under the influence of either Newtonian or relativistic gravitational forces, giving rise to either the Euler-Poisson or Einstein-Euler equations. In certain regimes, it is known that smooth initial data may give rise to blow-up solutions, corresponding to the collapse of a star under its own gravity. In the relativistic setting, in certain regimes, the blow-up creates a so-called naked singularity. In this talk, I will present the nonlinear stability of a particular family of these blow-up profiles, the (relativistic) Larson-Penston solution. This is based on joint works with Yan Guo, Mahir Hadzic, and Juhi Jang.

We introduce a systematic framework for constructing emergent local commuting Hamiltonians in (d+1) dimensions from symmetries defined on a d-dimensional boundary. Starting from a generic d-dimensional symmetry, we gauge it by introducing layers of gauge fields carrying dual symmetries, which can in turn be gauged. The fixed points of this iterative gauging procedure yield (d+1)-dimensional stabilizer Hamiltonians built from local Gauss-law constraints. These Hamiltonians encode the emergent bulk order while realizing the original symmetry on the boundary.

I will show how, starting in one dimension, this construction reproduces familiar two-dimensional topological codes, such as the toric code, and naturally realizes all gapped boundary conditions as different one-dimensional symmetric phases at the boundary. The framework extends to higher-form and subsystem symmetries in arbitrary dimensions, yielding, for example, three-dimensional fracton models.

If time permits, I will also discuss how these symmetries can be classified within the framework of tensor networks.

Based on arXiv: 2403.07575, 2410.09044, and 2203.12563.
 

Given a p-adic group G, one would like to find a systematic way of constructing all its irreducible representations. As a first approximation, one can parabolically induce irreducible representations of Levi subgroups of G, however, the resulting representations are often reducible. A more refined approach is to consider (analytic) intertwining operators between such induced representations, whose images turn out to be irreducible in many cases. These intertwining operators have proven to be highly useful in a variety of problems, yet their general behavior remains far from fully understood. In the case of general linear groups, conjectures relate their behavior to the geometry of the cotangent bundle of the space of Langlands parameters.

This talk offers a gentle introduction to these topics and reports on recent progress towards these conjectures.

TBA

We motivate and define the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Frączyk and Sam Mellick, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group (for higher rank semisimple Lie groups, the minimum number of generators in a lattice is sublinear in the covolume). In this talk we give some ideas of the proof. Time permitting, we will discuss related percolation results, joint with Matteo D'Achille, Jan Grebik, Ali Khezeli, and Konstantin Recke.

“Sparsification” is the act of reducing a network to a subset of its edges while approximately preserving its properties: either to reduce the computational cost of solving problems about it, or to identify which edges are the most important in some sense. Computer scientists have developed beautiful techniques for sparsifying a graph using physics-related ideas like the effective resistance. However, while these methods preserve the spectral properties of the Laplacian, it is not obvious to what extent they preserve the behavior of nonlinear dynamical systems. Using a mobility network from the United States as an example, I’ll show that they do very well for the SIR epidemic model, including the probability each node becomes infected and its distribution of arrival times, even when the sparse network includes less than 10% of the original edges. Choosing edges using purely topological methods, or by thresholding edge weights, does not perform nearly as well. I will end by discussing the possibility of using sparsification to “denoise” networks from bioinformatics, and present some preliminary results on the Kuramoto model of coupled oscillators.

 

This is joint work with Alexander Mercier (Harvard School of Public Health), Emmie Fitz-Gibbons (Brown), and Sam Scarpino (Northeastern).

to follow

TBA

We extend the construction of BPHZ renormalised models in regularity structures via the spectral gap inequality to the fractional heat operator. Its Green’s function fails to be smooth across the entire time-zero hyperplane, violating the classical assumption that kernels are smooth away from the origin. The proof requires a delicate balance between the gain in regularity provided by the spectral gap approach and the limited spatial decay of the kernel. 

Joint work with L. Broux and R. Steele.

to follow

Your chance to ask Mathematrix DPhil students about the process of applying to PhD programs, including written stages and interviews!