Past

Oxford Mathematics Public Lectures - Andrew Wiles, 28th November, 6.30pm, Science Museum, London SW7 2DD

Oxford Mathematics in partnership with the Science Museum is delighted to announce its first Public Lecture in London. World-renowned mathematician Andrew Wiles will be our speaker. Andrew will be talking about his current work and will also be 'in conversation' with mathematician and broadcaster Hannah Fry after the lecture.

This lecture is now sold out, but it will be streamed live and recorded. https://livestream.com/oxuni/wiles
 

Roelcke precompact groups are exactly the topological groups that can be realized as automorphism groups of omega-categorical structures (in continuous logic). In this talk, I will discuss a model-theoretic framework for the study of those groups and their dynamical systems as well as two concrete applications. The talk is based on joint work with Itaï Ben Yaacov and Tomás Ibarlucía.

The specialization question for rationality is the following one: assume that very general fibers of a flat proper morphism are rational, does it imply that all fibers are rational? I will talk about recent solution of this question in characteristic zero due to myself and Nicaise, and Kontsevich-Tschinkel. The method relies on a construction of various specialization morphisms for the Grothendieck ring of varieties (stable rationality) and the Burnside ring of varieties (rationality), which in turn rely on the Weak Factorization and Semi-stable Reduction Theorems.

The purpose of this work is to introduce a new constraint functional for shape optimization problems, which enforces the constructibility by means of additive manufacturing processes, and helps in preventing the appearance of overhang features - large regions hanging over void which are notoriously difficult to assemble using such technologies. The proposed constraint relies on a simplified model for the construction process: it involves a continuum of shapes, namely the intermediate shapes corresponding to the stages of the construction process where the total, final shape is erected only up to a certain level. The shape differentiability of this constraint functional is analyzed - which is not a standard issue because of its peculiar structure. Several numerical strategies and examples are then presented. This is a joint work with G. Allaire, R. Estevez, A. Faure and G. Michailidis.

In this joint work with D. Ciubotaru, we introduce the notion of local and global indices of Dirac operators for a rational Cherednik algebra H, with underlying reflection group G. In the local theory, I will report on some relations between the (local) Dirac index of a simple module in category O, the graded G-character and the composition series polynomials for standard modules. In the global theory, we introduce an "integral-reflection" module over which we define and compute the index of a (global) Dirac operator and show that the index is independent of the parameters. If time permits, I will discuss some local-global relations.

A new generation of low cost 3D tomography systems is based on multiple emitters and sensors that partially convolve measurements. A successful approach to deconvolve the measurements is to use nonlinear compressed sensing models. We discuss such models, as well as algorithms for their solution. 

This will be a discussion of the paper https://arxiv.org/abs/1607.06978.

Snap-through buckling is a type of instability in which an elastic object rapidly jumps from one state to another, just as an umbrella flips upwards in a gust of wind. While snap-through under dry, mechanical loads has already been harnessed in engineering to generate fast motions between two states, the mechanisms underlying snapping in bulk fluid flows remain relatively unexplored. In this talk we demonstrate how elastic snap-through may be used to passively control fluid flows at low Reynolds number, in contrast to some pre-existing valves that rely on active control. We study viscous flow through a channel in which one of the bounding walls is an elastic arch. By performing experiments at the macroscopic scale, we show that snap-through of the arch rapidly changes the channel from a constricted to an unconstricted state, increasing the hydraulic conductivity by up to an order of magnitude. We also observe nonlinear pressure-flux characteristics away from snapping due to the coupling between the driving flow and elasticity. This behaviour is confirmed by a mathematical model that also shows the device may readily be scaled down for microfluidic applications. Finally, we demonstrate that such a device may be used to create a fluidic analogue of a fuse: the fluid flux through a channel may not rise above a given value. 

Current technological progress has raised concerns about automation of tasks performed by workers resulting in job losses. Previous studies have used machine learning techniques to compute the automation probability of occupations and thus, studied the impact of automation on employment. However, such studies do not consider second-order effects, for example, an occupation with low automation probability can have a  surplus of labor supply due to similar occupations being automated. In this work, we study such second-order effects of automation using a network approach.  In our network – the Job Space – occupations are nodes and edges link occupations which share a significant amount of work activities. By mapping employment, automation probabilities into the network, and considering the movement of workers, we show that an occupation’s position in the network may be crucial to determining its employment future.

I will discuss recent developments in the study of scattering amplitudes in Einstein-Yang-Mills theory. At tree level we find new structures at higher order collinear limits and novel connections with amplitudes in Yang-Mills theory using the CHY formalism. Finally I will comment on unitarity based observations regarding one-loop amplitudes in the theory. 

In this presentation, we investigate the spectrum of the Neumann-Poincaré operator associated to a periodic distribution of small inclusions with size ε, and its asymptotic behavior as the parameter ε vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the `trivial' eigenvalues 0 and 1, and of a subset which stays bounded away from 0 and 1 uniformly with respect to ε. This non trivial part is the reunion of the Bloch spectrum, accounting for the collective resonances between collections of inclusions, and of the boundary layer spectrum, associated to eigenfunctions which spend a not too small part of their energies near the boundary of the macroscopic device. These results shed new light about the homogenization of the voltage potential uε caused by a given source in a medium composed of a periodic distribution of small inclusions with an arbitrary (possibly negative) conductivity a surrounded by a dielectric medium, with unit conductivity.

The cyclic surgery theorem of Culler, Gordon, Luecke, and Shalen implies that any knot in S^3 other than a torus knot has at most two nontrivial cyclic surgeries. In this talk, we investigate the weaker notion of SU(2)-cyclic surgeries on a knot, meaning surgeries whose fundamental groups only admit SU(2) representations with cyclic image. By studying the image of the SU(2) character variety of a knot in the “pillowcase”, we will show that if it has infinitely many SU(2)-cyclic surgeries, then the corresponding slopes (viewed as a subset of RP^1) have a unique limit point, which is a finite, rational number, and that this limit is a boundary slope for the knot. As a corollary, it follows that for any nontrivial knot, the set of SU(2)-cyclic surgery slopes is bounded. This is joint work with Raphael Zentner.

Abstract: In this talk we describe an invariance principle for a class of non-homogeneous martingale random walks in $\RR^d$ that can be recurrent or transient for any dimension $d$. The scaling limit, which we construct, is a martingale diffusions with law determined uniquely by an SDE with discontinuous coefficients at the origin whose pathwise uniqueness may fail. The radial component of the diffusion is a Bessel process of dimension greater than 1. We characterize the law of the diffusion, which must start at the origin, via its excursions built around the Bessel process: each excursion has a generalized skew-product-type structure, in which the angular component spins at infinite speed at the start and finish of each excursion. Defining a Riemannian metric $g$ on the sphere $S^{d−1}$, different from the one induced by the ambient Euclidean space, allows us to give an explicit construction of the angular component (and hence of the entire skew-product decomposition) as a time-changed Browninan motion with drift on the Riemannian manifold $(S^{d−1}, g)$. In particular, this provides a multidimensional generalisation of the Pitman–Yor representation of the excursions of Bessel process with dimension between one and two. Furthermore, the density of the stationary law of the angular component with respect to the volume element of $g$ can be characterised by a linear PDE involving the Laplace–Beltrami operator and the divergence under the metric $g$. This is joint work with Nicholas Georgiou and Andrew Wade.

The interplay of minimum degree and 'structural properties' of large graphs with a given forbidden subgraph is a central topic in extremal graph theory. For a given graph $F$ we define the homomorphism threshold as the infimum $\alpha$ such that every $n$-vertex $F$-free graph $G$ with minimum degree $>\alpha n$ has a homomorphic image $H$ of bounded size (independent of $n$), which is $F$-free as well. Without the restriction of $H$ being $F$-free we recover the definition of the chromatic threshold, which was determined for every graph $F$ by Allen et al. The homomorphism threshold is less understood and we present recent joint work with O. Ebsen on the homomorphism threshold for odd cycles.

We propose a new splitting algorithm to solve a class of quasilinear PDEs with convex and quadratic growth gradients. 

By splitting the original equation into a linear parabolic equation and a Hamilton-Jacobi equation, we are able to solve both equations explicitly. 

In particular, we solve the associated Hamilton-Jacobi equation by the Hopf-Lax formula, 

and interpret the splitting algorithm as a stochastic Hopf-Lax approximation of the quasilinear PDE.  

We show that the numerical solution will converge to the viscosity solution of the equation.  

The upper bound of the convergence rate is proved based on Krylov's shaking coefficients technique, 

while the lower bound is proved based on Barles-Jakobsen's optimal switching approximation technique. 

Based on joint work with Shuo Huang and Thaleia Zariphopoulou.

In this talk two different methods for constructing complete steady and expanding Ricci solitons of cohomogeneity one will be discussed. The first is based on an estimate on the growth of the soliton potential and holds for large classes of cohomogeneity one manifolds. The second approach is specific to the two summands case and uses a Lyapunov function. This method also carries over to the Einstein case and as an application, a simplified construction of B\"ohm's Einstein metrics of positive scalar curvature on spheres will be explained.

We consider the one parameter mirror families W of the Calabi-Yau 3-folds with Picard-Fuchs  equations of hypergeometric type. By mirror symmetry the  even D-brane masses of orginial Calabi-Yau manifolds M can be identified with four periods with respect to an integral symplectic basis of $H_3(W,\mathbb{Z})$ at the point of maximal unipotent monodromy. We establish that the masses of the D4 and D2 branes at the conifold are given by the two algebraically independent values of the L-function of the weight four holomorphic Hecke eigenform with eigenvalue one of $\Gamma_0(N)$. For the quintic in  $\mathbb{P}^4$ it this Hecke eigenform of $\Gamma_0(25)$ was as found by Chad Schoen.  It was discovered  by de la Ossa, Candelas and Villegas that  its  coefficients $a_p$ count the number of  solutions of  the mirror quinitic at the conifold over the finite number field $\mathbb{F}_p$ . Using the theory of periods and quasi-periods of $\Gamma_0(N)$ and the special geometry pairing on Calabi-Yau 3 folds we can fix further values in the connection matrix between the maximal unipotent monodromy point and the conifold point.  

Richard Wade:   Classifying spaces, automorphisms, and right-angled Artin groups 

Right-angled Artin groups (otherwise known as partially commutative groups, or graph groups), interpolate between free abelian groups and free groups. These groups have seen a lot of attention recently, much of this due to some surprising links to the world of hyperbolic 3-manifolds.We will look at classifying spaces for such groups and their associated automorphism groups. These spaces are useful as they give a topological way to understand algebraic invariants of groups. This leads us to study some beautiful mathematical objects: deformation spaces of tori and trees. We will look at some recent results that aim to bridge the gap between these two families of spaces.
 
Andrey Kormilitzin:   Learning from electronic health records using the theory of rough paths

In this talk, we bring the theory of rough paths to the study of non-parametric statistics on streamed data and particularly to the problem of regression and classification, where the input variable is a stream of information, and the dependent response is also (potentially) a path or a stream.  We informally explain how a certain graded feature set of a stream, known in the rough path literature as the signature of the path, has a universality that allows one to characterise the functional relationship summarising the conditional distribution of the dependent response. At the same time this feature set allows explicit computational approaches through machine learning algorithms.

Finally, the signature-based modelling can be applied to some real-world problems in medicine, in particular in mental health and gastro-enterology.

This will be a whistle-stop tour of a few topics on infectious disease modelling, mainly influenza. Topics to include:

• challenges in capturing dynamics of pathogens with multiple co-circulating strains
• untangling the 2009 influenza pandemic from medical insurance claims data from the US
• bioinformatic methods to detect viral packaging signals
• and a big science project (top secret until the talk!)

Julia will be visiting the Mathematical Institute on sabbatical this term, and hopes this talk will help us find areas of overlapping interests.

It is thought that the hairy legs of water walking arthropods are able to remain clean and dry because the flexibility of the hairs spontaneously moves drops off the hairs. We present a mathematical model of this bending-induced motion, or bendotaxis, and study how it performs for wetting and non-wetting drops. Crucially, we show that both wetting and non-wetting droplets move in the same direction (using physical arguments and numerical solutions). This suggests that a surface covered in elastic filaments (such as the hairy leg of insects) may be able to universally self-clean. To quantify the efficiency of this effect, we explore the conditions under which drops leave the structure by ‘spreading’ rather than translating and also how long it takes to do so.

In a recent preprint, Basterra, Bobkova, Ponto, Tillmann and Yeakel
defined operads with homological stability (OHS) and showed that after
group-completion, algebras over an OHS group-complete to infinite loop
spaces. This can in particular be used to put a new infinite loop space
structure on stable moduli spaces of high-dimensional manifolds in the
sense of Galatius and Randal-Williams, which are known to be infinite
loop spaces by a different method.

To complicate matters further, I shall introduce a mild strengthening of
the OHS condition and construct yet another infinite loop space
structure on these stable moduli spaces. This structure turns out to be
equivalent to that constructed by Basterra et al. It is believed that
the infinite loop space structure due to Galatius--Randal-Williams is
also equivalent to these two structures.