Past events

This presentation focuses on the study of the regulartiy of random wavelet series. We first study their belonging to certain functional spaces and we compare these results with long-established results related to random Fourier series. Next, we show how the study of random wavelet series leads to precise pointwise regularity properties of processes like fractional Brownian motion. Additionally, we explore how these series helps create Gaussian processes  with random Hölder exponents.

On homogeneous spaces, solutions to the prescribed Ricci curvature equation coincide with the critical points of the scalar curvature functional subject to a constraint. We provide a complete description of Palais--Smale sequences for this functional. As an application, we obtain new existence results for the prescribed Ricci curvature equation, which enables us to observe previously unseen phenomena. Joint work with Wolfgang Ziller (University of Pennsylvania).

The tremendous success of the Machine Learning paradigm heavily relies on the development of powerful optimization methods, and the canonical algorithm for training learning models is SGD (Stochastic Gradient Descent). Nevertheless, the latter is quite different from Gradient Descent (GD) which is its noiseless counterpart. Concretely, SGD requires a careful choice of the learning rate, which relies on the properties of the noise as well as the quality of initialization.

 It further requires the use of a test set to estimate the generalization error throughout its run. In this talk, we will present a new SGD variant that obtains the same optimal rates as SGD, while using noiseless machinery as in GD. Concretely, it enables to use the same fixed learning rate as GD and does not require to employ a test/validation set. Curiously, our results rely on a novel gradient estimate that combines two recent mechanisms which are related to the notion of momentum.

Finally, as much as time permits, I will discuss several applications where our method can be extended.

Oxford University Innovation, the University’s commercialisation team, will explain the support they can give to Maths researchers who want to generate commercial impact from their work and expertise. In addition to an overview of consulting, this talk will explain how mathematical techniques and software can be protected and commercialised.

In this possibly speculative talk I will try to outline a way to define analytic D-modules, using (higher) category theory and the ``six operations" on quasicoherent sheaves as the main tools. The aim is to follow the successful approach of Andy Jiang in the algebraic setting, who obtained such a theory without using stacks or formal schemes (as in Gaitsgory-Rozenblyum's approach). By using local cohomology, Jiang was able to avoid enlarging the category of algebras beyond the usual ones. We believe that an analytic variant of local cohomology can be used to recover the Ardakov-Wadsley theory of D-cap modules ``on the nose". (Work in progress).

Kaufman constructed a family of Fourier-decaying measures on the set of badly approximable numbers. Pollington and Velani used these to show that Littlewood’s conjecture holds for a full-dimensional set of pairs of badly approximable numbers. We construct analogous measures that have implications for inhomogeneous diophantine approximation. In joint work with Agamemnon Zafeiropoulos and Evgeniy Zorin, our idea is to shift the continued fraction and Ostrowski expansions simultaneously.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

Global scale ocean currents are strongly constrained by the Earth’s rotation, while this effect is generally negligible at small scales. In between, motions with scales from 1-10km are marginally affected by the Earth’s rotation. These intermediate scales, collectively termed the ocean submesoscale, have been hidden from view until recent years. Evidence from field measurements, numerical models, and satellite data have shown that submesoscales play a particularly important role in the upper ocean where they help to control the transport of material between the ocean surface and interior. In this talk I will review some recent work on submesoscale dynamics and their influence on biogeochemistry and accumulation of microplastics in the surface waters.

This talk is based on a joint work with Vincent Jinhe Ye. I will define various classes of hyperbolic varieties (Broody hyperbolic, algebraically hyperbolic, bounded, groupless) and discuss existence of model companions of classes of fields that exclude them. This is related to moduli spaces of maps to hyperbolic varieties and to the (open) question whether the above mentioned hyperbolicity notions are in fact equivalent.

For a hyperbolic knot there are two types of invariants, the hyperbolic invariants coming from the geometric structure and the classical invariants coming from the topology or combinatorics. It has been observed in many different cases that these seemingly different types of invariants are in fact related. I will give examples of these relationships and discuss in particular a link by Stoimenow between the determinant and volume.  

We model a tumor as an incompressible flow considering two antagonistic effects: repulsion of cells when the tumor grows (they push each other when they divide) and cell-cell adhesion which creates surface tension. To take into account these two effects, we use a 4th-order parabolic equation: the Cahn-Hilliard equation. The combination of these two effects creates a discontinuity at the boundary of the tumor that we call the pressure jump.  To compute this pressure jump, we include an external force and consider stationary radial solutions of the Cahn-Hilliard equation. We also characterize completely the stationary solutions in the incompressible case, prove the incompressible limit and prove convergence of the parabolic problems to stationary states.

I will give an introduction to quasidiagonality of group actions wherein an action on a C^*-algebra is approximated by actions on matrix algebras.  This has implications for crossed product C^*-algebras, especially as pertains to finite dimensional approximation.  I'll sketch the proof that all isometric actions are quasidiagonal, which we can view as a dynamical Petr-Weyl theorem.  Then I will discuss an interplay between quasidiagonal actions and semiprojectivity of C^*-algebras, a property that allows "almost representations" to be perturbed to honest ones.  

We will survey work of Birman-Craggs, Johnson, and Sato on the abelianization of the level 2 congruence group of the mapping class group of a surface, and of the corresponding Torelli group. We will then describe recent work of Lewis providing a common framework for both abelianizations, with applications including a partial answer to a question of Johnson.

Fourth-order elliptic problems arise in a variety of applications from thin plates to phase separation to liquid crystals. A conforming Galerkin discretization requires a finite dimensional subspace of $H^2$, which in turn means that conforming finite element subspaces are $C^1$-continuous. In contrast to standard $H^1$-conforming $C^0$ elements, $C^1$ elements, particularly those of high order, are less understood from a theoretical perspective and are not implemented in many existing finite element codes. In this talk, we address the implementation of the elements. In particular, we present algorithms that compute $C^1$ finite element approximations to fourth-order elliptic problems and which only require elements with at most $C^0$-continuity. We also discuss solvers for the resulting subproblems and illustrate the method on a number of representative test problems.

Given a matrix $A$ with integer entries, a subset $S$ of an abelian group and $r\in\mathbb N$, we say that $S$ is $(A,r)$-Rado if any $r$-colouring of $S$ yields a monochromatic solution to the system of equations $Ax=0$. A classical result of Rado characterises all those matrices $A$ such that $\mathbb N$ is $(A,r)$-Rado for all $r \in \mathbb N$. Rödl and Ruciński, and Friedgut, Rödl and Schacht proved a random version of Rado’s theorem where one considers a random subset of $[n]:=\{1,\dots,n\}$.

In this paper, we investigate the analogous random Ramsey problem in the more general setting of abelian groups. Given a sequence $(S_n)_{n\in\mathbb N}$ of finite subsets of abelian groups, let $S_{n,p}$ be a random subset of $S_n$ obtained by including each element of $S_n$ independently with probability $p$. We are interested in determining the probability threshold for $S_{n,p}$ being $(A,r)$-Rado.

Our main result is a general black box for hypergraphs which we use to tackle problems of this type. Using this tool in conjunction with a series of supersaturation results, we determine the probability threshold for a number of different cases. A consequence of the Green-Tao theorem is the van der Waerden theorem for the primes: every finite colouring of the primes contains arbitrarily long monochromatic arithmetic progressions. Using our machinery, we obtain a random version of this result. We also prove a novel supersaturation result for $[n]^d$ and use it to prove an integer lattice generalisation of the random version of Rado's theorem.

This is joint work with Andrea Freschi and Andrew Treglown (both University of Birmingham).

We present new high-order finite elements discretizing the $L^2$ de Rham complex on triangular and tetrahedral meshes. The finite elements discretize the same spaces as usual, but with different basis functions. They allow for fast linear solvers based on static condensation and space decomposition methods.

The new elements build upon the definition of degrees of freedom given by (Demkowicz et al., De Rham diagram for $hp$ finite element spaces. Comput.~Math.~Appl., 39(7-8):29--38, 2000.), and consist of integral moments on a symmetric reference simplex with respect to a numerically computed polynomial basis that is orthogonal in both the $L^2$- and $H(\mathrm{d})$-inner products ($\mathrm{d} \in \{\mathrm{grad}, \mathrm{curl}, \mathrm{div}\}$).

On the reference symmetric simplex, the resulting stiffness matrix has diagonal interior block, and does not couple together the interior and interface degrees of freedom. Thus, on the reference simplex, the Schur complement resulting from elimination of interior degrees of freedom is simply the interface block itself.

This sparsity is not preserved on arbitrary cells mapped from the reference cell. Nevertheless, the interior-interface coupling is weak because it is only induced by the geometric transformation. We devise a preconditioning strategy by neglecting the interior-interface coupling. We precondition the interface Schur complement with the interface block, and simply apply point-Jacobi to precondition the interior block.

The combination of this approach with a space decomposition method on small subdomains constructed around vertices, edges, and faces allows us to efficiently solve the canonical Riesz maps in $H^1$, $H(\mathrm{curl})$, and $H(\mathrm{div})$, at very high order. We empirically demonstrate iteration counts that are robust with respect to the polynomial degree.

I will then explain that hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage. These have the potential to solve real-world problems---including optimisation or ground-state search---but they suffer from a large number of circuit repetitions required to extract information from the quantum state. I will explain some of our recent results as hybrid quantum algorithms that exploit so-called classical shadows (random unitary protocols) in order to extract and post-process a large amount of information from the quantum computer [PRX 12, 041022 (2022)] and [arXiv:2212.11036]. I will finally identify the most likely areas where quantum computers may deliver a true advantage in the near term.

This talk will present a new characterisation of rectifiable subsets of a complete metric space in terms of local approximation, with respect to the Gromov-Hausdorff distance, by finite dimensional Banach spaces. Time permitting, we will discuss recent joint work with Hyde and Schul that provides quantitative analogues of this statement.
 

TBC

Bicommutant categories, initially invented for the purposes of Chern-Simons theory and 2d CFT, seem to also appear in other domains of math with examples related to group theory, and dynamical systems.

One way to capture both the elastic and stochastic reaction of purchases to price is through a model where sellers control the intensity of a counting process, representing the number of sales thus far. The intensity describes the probabilistic likelihood of a sale, and is a decreasing function of the price a seller sets. A classical model for ticket pricing, which assumes a single seller and infinite time horizon, is by Gallego and van Ryzin (1994) and it has been widely utilized by airlines, for instance. Extending to more realistic settings where there are multiple sellers, with finite inventories, in competition over a finite time horizon is more complicated both mathematically and computationally. We discuss some dynamic games of this type, from static to two player to the associated mean field game, with some numerical and existence-uniqueness results.

Based on works with Andrew Ledvina and with Emre Parmaksiz.