Past

One of the simplest, and yet largely still open, questions that one can ask about special Lagrangian submanifolds is whether they exist in a given homology class. One possible approach to this problem is to evolve a given Lagrangian submanifold under mean curvature flow in the hope of reaching a special Lagrangian submanifold in the same homology class. It is known, however, that even for 'nice' initial conditions the flow will develop singularities in finite time. 

I will talk about a joint work with Tom Begley, in which we prove a short time existence result for Lagrangian mean curvature flow, where the initial condition is a Lagrangian submanifold of complex Euclidean space with a certain type of singularity. This is a first step to proving, as conjectured by Joyce, that one may 'continue' Lagrangian mean curvature flow after the occurrence of singularities.

There are several classes of random function that appear naturally in mathematical physics, probability, number theory, and other areas of mathematics. I will give a brief overview of some of these random functions and explain what they are and why they are important. Finally, I will explain how I use chebfun to study these functions.
 

The Graham-Pollak theorem states that to decompose the complete graph $K_n$ into complete bipartite subgraphs we need at least $n-1$ of them. What
happens for hypergraphs? In other words, suppose that we wish to decompose the complete $r$-graph on $n$ vertices into complete $r$-partite $r$-graphs; how many do we need?

In this talk we will report on recent progress on this problem. This is joint work with Luka Milicevic and Ta Sheng Tan.

Stochastic Hamiltonian systems with multiplicative noise are a mathematical model for many physical systems with uncertainty. For example, they can be used to describe synchrotron oscillations of a particle in a storage ring. Just like their deterministic counterparts, stochastic Hamiltonian systems possess several important geometric features; for instance, their phase flows preserve the canonical symplectic form. When simulating these systems numerically, it is therefore advisable that the numerical scheme also preserves such geometric structures. In this talk we propose a variational principle for stochastic Hamiltonian systems and use it to construct stochastic Galerkin variational integrators. We show that such integrators are indeed symplectic, preserve integrals of motion related to Lie group symmetries, demonstrate superior long-time energy behavior compared to nonsymplectic methods, and they include stochastic symplectic Runge-Kutta methods as a special case. We also analyze their convergence properties and present the results of several numerical experiments. 

In this meeting we will talk about the first two chapters of Robert Ghrist's book "Elementary Applied Topology". The book is freely available at the following link: https://www.math.upenn.edu/~ghrist/notes.html

A successful programme of personalised discounts and recommendations relies on identifying products that customers want, based both on items bought in the past and on relevant products that the customers have not yet purchased. Using basket-level grocery shopping data, we aim to use clustering ("community detection") techniques to identify groups of shoppers with similar preferences, along with the corresponding products that they purchase, in order to design better recommendation systems.

Stochastic block models (SBMs) are an increasingly popular class of methods for community detection. In this talk, I will expand on some work done by Newman and Clauset [1] that uses a modified SBM for community detection in annotated networks. In these networks, additional information in the form of node metadata is used to improve the quality of the inferred community structure. The method can be extended to bipartite networks, which contain two types of nodes and edges only between nodes of different types. I will show some results obtained from applying this method to a bipartite network of customers and products. Finally, I will discuss some desirable extensions to this method such as incorporating edge weights and assessing the relationship between metadata and network structure in a statistically robust way.

[1] Structure and inference in annotated networks, MEJ Newman and A Clauset, Nature Communications 7, 11863 (2016).

Note: This talk will cover similar topics to my presentation in the InFoMM group meeting on Friday, November 25 but it won't be exactly the same. I will focus more on the mathematical details for my JAMS talk.
 

After a brief review of holographic features of general relativity in 3 and 4 dimensions, I will show how to derive the transformation laws of the Bondi mass and angular momentum aspects under finite supertranslations, superrotations and complex Weyl rescalings.
 

I will begin the talk by reviewing the definition of commutative K-theory, a generalized cohomology theory introduced by Adem and Gomez. It is a refinement of topological K-theory, where the transition functions of a vector bundle satisfy a commutativity condition. The theory is represented by an infinite loop space which is called a “classifying space for commutativity”.  I will describe the homotopy type of this infinite loop space. Then I will discuss the graded ring structure on its homotopy groups, which corresponds to the tensor product of vector bundles.
 

We show global uniqueness in an inverse problem for the fractional Schrödinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where the measurements are taken in an arbitrary open subset of the exterior. The results apply in any dimension $\geq 2$ and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically different from the traditional Calderón problem. This is a joint work with T. Ghosh (HKUST) and G. Uhlmann (Washington).
 

In this talk we consider particular solutions of the Boltzmann equation which have the form $f (x,v,t) = g (v − M (t)x,t)$ where $M (t) = A(I + tA)^{−1}$ with the matrix $A$ describing a shear flow or a dilatation or a combination of both. These solutions are known as equidispersive solutions. We will show that, for different choices for the matrix A and for different homogeneities of the collision kernel, we obtain different long time asymptotics for the corresponding equidispersive solutions. In particular we will focus on the case of simple shear flow and prove rigorously the existence of self-similar solutions with exponentially increasing internal energy.

Generalized holomorphic bundles are the analogues of holomorphic vector bundles in the generalized geometry setting. In this talk, I will discuss the deformation theory of generalized holomorphic bundles on generalized Kaehler manifolds. I will also give explicit examples of moduli spaces of generalized holomorphic bundles on Hopf surfaces and on Inoue surfaces. This is joint work with Shengda Hu and Mohamed El Alami

In this talk I will argue that a systematic classification of 4d N=2 superconformal field theories is possible through a careful analysis of the geometry of their Coulomb branches. I will carefully describe this general framework and then carry out the classification explicitly in the rank-1, that is one complex dimensional Coulomb branch, case.  We find that the landscape of rank-1 theories is still largely unexplored and make a strong case for the existence of many new rank-1 SCFTs, almost doubling the number of theories already known in the literature. The existence of 4 of them has been recently confirmed using alternative methods and others have an enlarged N=3, supersymmetry. 

While our study focuses on Coulomb Branch geometries, we can extract much more information about these SCFTs. I will spend the last part of my talk outlining what else we can learn and the extent in which our study can be complementary to other method to study SCFTs (Conformal Bootstrap above all!).

Numbers like the special values of the gamma and the Bessel functions or the Euler-Mascheroni constant are not expected to be periods in the usual sense of algebraic geometry. However, they can be regarded as coefficients of the comparison isomorphism between two cohomology theories associated to pairs consisting of an algebraic variety and a regular function: the de Rham cohomology of a connection with irregular singularities, and the so-called “rapid decay cohomology”. Following ideas of Kontsevich and Nori, I will explain how this point of view allows one to construct a Tannakian category of exponential motives over a subfield of the complex numbers. The upshot is that one can attach to exponential periods a Galois group that conjecturally governs all algebraic relations between them. Classical results and conjectures in transcendence theory may be reinterpreted in this way. No prior knowledge of motives will be assumed, and I will focus on examples rather than on the more abstract aspects of the theory. This is a joint work with P. Jossen (ETH Zürich).

Featuring

Professor Alison Etheridge, Professor of Probability in the Mathematical Institute and Department of Statistics, Oxford

Professor Ben Green, Waynflete Professor of Pure Mathematics, Oxford

Dr Heather Harrington, Royal Society University Research Fellow in the Mathematical Institute, Oxford

Professor Jon Keating, Henry Overton Wills Professor of Mathematics, Bristol and Chair of the Heilbronn Institute for Mathematical Research

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Dr Christopher Voyce, Head of Research Facilitation in the Mathematical Institute, Oxford

By gluing copies of a deforming polytope, we describe some deformations of complete, finite-volume hyperbolic cone four-manifolds. Despite the fact that hyperbolic lattices are locally rigid in dimension greater than three (Garland-Raghunathan), we see a four-dimensional analogue of Thurston's hyperbolic Dehn filling: a path of cone-manifolds $M_t$ interpolating between two cusped hyperbolic four-manifolds $M_0$ and $M_1$.

This is a joint work with Bruno Martelli.

Roxana Pamfil
Analysis of consumer behaviour with annotated networks

Rachel Philip
Modelling droplet breakup in a turbulent jet

Asbjørn Riseth
Stochastic optimal control of a retail pricing problem
 

Abstract: We define the Hochschild complex and cohomology of a monoid in an Ab-enriched monoidal category. Then we interpret some of the lower dimensional cohomology groups and discuss when the cohomology ring happens to be graded-commutative.

Environmental risk assessments for chemicals in the EU rely heavily upon modelled estimates of potential concentrations in soil and water.  A key parameter used by these models is the degradation of the chemical in soil which is derived from a kinetic fitting of laboratory data using standard fitting routines.  Several different types of kinetic can be represented such as: Simple First Order (SFO), Double First Order in Parallel (DFOP), and First Order Multi-Compartment (FOMC). Choice of a particular kinetic and selection of a representative degradation rate can have a huge influence on the outcome of the risk assessment. This selection is made from laboratory data that are subject to experimental error.  It is known that the combination of small errors in time and concentration can in certain cases have an impact upon the goodness of fit and kinetic predicted by fitting software.  Syngenta currently spends in the region of 4m GBP per annum on laboratory studies to support registration of chemicals in the EU and the outcome of the kinetic assessment can adversely affect the potential registerability of chemicals having sales of several million pounds.  We would therefore like to understand the sensitivities involved with kinetic fitting of laboratory studies.  The aim is to provide guidelines for the conduct and fitting of laboratory data so that the correct kinetic and degradation rate of chemicals in environmental risk assessments is used.

After mentioning, by way of motivation (mine at least), some diophantine questions concerning
sets definable in the restricted analytic, exponential field $\R_{an, exp}$, I discuss the
problem of extending a given $\R_{an, exp}$-definable function $f:(a, \infty) \to \R$ to
a holomorphic function $\hat f : \{z \in \C : Re(z) > b \} \to \C$ (for some $b > a$).
In particular, I give a necessary and sufficient condition on $f$ for such an $\hat f$ to exist and be
$\R_{an, exp}$-definable.
 

Spectra provide a way of understanding cohomology theories in terms of homotopy theory. Spectra are a bit like CW-complexes, they have homotopy groups which may be used to characterize homotopy equivalences. However, a spectrum has homotopy groups in negative degrees, too, and they are abelian groups in all degrees. We will discuss spectra representing ordinary cohomology, bordism, and K-theory.

In this talk we will start by introducing the notion of Siegel-Jacobi modular form and explain its close relation to Siegel modular forms through the Fourier-Jacobi expansion. Then we will discuss how one can attach an L-function to an appropriate (i.e. eigenform) Siegel-Jacobi modular form due to Shintani, and report on joint work with Jolanta Marzec on analytic properties of this L-function, extending results of Arakawa and Murase. 

We propose a randomised version of the Heston model--a widely used stochastic volatility model in mathematical finance--assuming that the starting point of the variance process is a random variable. In such a system, we study the small- and large-time behaviours of the implied volatility, and show that the proposed randomisation generates a short-maturity smile much steeper (`with explosion') than in the standard Heston model, thereby palliating the deficiency of classical stochastic volatility models in short time. We precisely quantify the speed of explosion of the smile for short maturities in terms of the right tail of the initial distribution, and in particular show that an explosion rate of $t^\gamma$ (gamma in [0,1/2]) for the squared implied volatility--as observed on market data--can be obtained by a suitable choice of randomisation. The proofs are based on large deviations techniques and the theory of regular variations. Joint work with Fangwei Shi (Imperial College London)

In this talk, I am reflecting on the last 8 extremely enjoyable years I spent in the department (DPhil, OCIAM, 2008-2012, post-doc, WCMB, 2012-2016). My story is a little unusual: coming from an Engineering undergraduate background, spending 8 years in the Maths department, and now moving to a faculty position at the Medical School. However, I think it highlights well the enormous breadth and applicability of mathematics beyond traditional disciplinary boundaries. I will discuss different projects during my time in Oxford, focusing on time-series, signal processing, and statistical machine learning methods, with diverse applications in real-world problems.

I will present a broad family of stochastic algorithms for inverting a matrix, including specialized variants which maintain symmetry or positive definiteness of the iterates. All methods in the family converge globally and linearly, with explicit rates. In special cases, the methods obtained are stochastic block variants of several quasi-Newton updates, including bad Broyden (BB), good Broyden (GB), Powell-symmetric-Broyden (PSB), Davidon-Fletcher-Powell (DFP) and Broyden-Fletcher-Goldfarb-Shanno (BFGS). After a pause for questions, I will then present a block stochastic BFGS method based on the stochastic method for inverting positive definite matrices. In this method, the estimate of the inverse Hessian matrix that is maintained by it, is updated at each iteration using a sketch of the Hessian, i.e., a randomly generated compressed form of the Hessian. I will propose several sketching strategies, present a new quasi-Newton method that uses stochastic block BFGS updates combined with the variance reduction approach SVRG to compute batch stochastic gradients, and prove linear convergence of the resulting method. Numerical tests on large-scale logistic regression problems reveal that our method is more robust and substantially outperforms current state-of-the-art methods.

This talk will introduce the notion of quasi-convex subgroups. As an application, we will prove that the intersection of two finitely generated subgroups of a free group is again finitely generated.
 

Isogenies are algebraic group morphisms of elliptic curves. Let E, E' be two (ordinary) elliptic curves defined over a finite field of characteristic p, and suppose that there exists an isogeny ψ between E and E'. The explicit isogeny problem asks to compute a rational function expression for ψ. Various specializations of this problem appear naturally in point counting and elliptic curve cryptography. There exist essentially two families of algorithms to compute isogenies. Algorithms based on Weierstraß' differential equation are very fast and well suited in the point count setting, but are clumsier in general. Algorithms based on interpolation work more generally, but have exponential complexity in log(p) (the characteristic of the finite field). We propose a new interpolation-based algorithm that solves the explicit isogeny problem in polynomial time in all the involved parameters. Our approach is inspired by a previous algorithm of Couveignes', that performs interpolation on the p-torsion on the curves. We replace the p-torsion in Couveignes' algorithm with the ℓ-torsion for some small prime ℓ; however this adaptation requires some non-trivial work on isogeny graphs in order to yield a satisfying complexity. Joint work with Cyril Hugounenq, Jérôme Plût and Éric Schost.

Suppose that we have a finite quotient singularity $\mathbb C^n/G$ admitting a crepant resolution $Y$ (i.e. a resolution with $c_1 = 0$). The cohomological McKay correspondence says that the cohomology of $Y$ has a basis given by irreducible representations of $G$ (or conjugacy classes of $G$). Such a result was proven by Batyrev when the coefficient field $\mathbb F$ of the cohomology group is $\mathbb Q$. We give an alternative proof of the cohomological McKay correspondence in some cases by computing symplectic cohomology+ of $Y$ in two different ways. This proof also extends the result to all fields $\mathbb F$ whose characteristic does not divide $|G|$ and it gives us the corresponding basis of conjugacy classes in $H^*(Y)$. We conjecture that there is an extension to certain non-crepant resolutions. This is joint work with Alex Ritter.

It follows from the ellipsoid method and results of Grotschel, Lovasz and Schrijver that one can find an optimal colouring of a perfect graph in polynomial time. But no ''combinatorial'' algorithm to do this is known.

Here we give a combinatorial algorithm to do this in an n-vertex perfect graph in time O(n^{k+1}^2) where k is the clique number; so polynomial-time for fixed k. The algorithm depends on another result, a polynomial-time algorithm to find a ''balanced skew partition'' in a perfect graph if there is one.

Joint work with Maria Chudnovsky, Aurelie Lagoutte, and Sophie Spirkl.

The Feynman diagram expansion of scattering amplitudes in perturbative superstring theory can be written (for closed strings) as a series of integrals over compactified moduli spaces of Riemann surfaces with marked points, indexed by the genus. Therefore in genus 0 it is reasonable to find, as it often happens in QFT computations, periods of M_{0,N}, which are known to be multiple zeta values. In this talk I want to report on recent advances in the genus 1 amplitude, which are related to the development of 2 different generalizations of classical multiple zeta values, namely elliptic multiple zeta values and conical sums.

Configuration spaces of points in a manifold are well studied. Giving the points thickness has obvious physical meaning: the configuration space of non-overlapping particles is equivalent to the phase space, or energy landscape, of a hard spheres gas. But despite their intrinsic appeal, relatively little is known so far about the topology of such spaces. I will overview some recent work in this area, including a theorem with Yuliy Baryshnikov and Peter Bubenik that related the topology of these spaces to mechanically balanced, or jammed, configurations. I will also discuss work in progress with Robert MacPherson on hard disks in an infinite strip, where we understand the asymptotics of the Betti numbers as the number of disks tends to infinity. In the end, we see a kind of topological analogue of a liquid-gas phase transition.

We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function in a very directional way. Using an interpretation of this energy as a large deviation rate function for SLE_k as k goes to 0, we show that the energy of a deterministic curve from one boundary point A of a simply connected domain D to another boundary point B, is equal to the energy of its time-reversal i.e. of the same curve but viewed as going from B to A in D. In particular it measures how far does the chord differ from the hyperbolic geodesic. I will also discuss the relation between the energy of the curve with its regularity, some questions are still open. If time allows, I will present the Loewner energy for loops on the Riemann sphere, and open questions related to it as well.


 

Let A be an affine variety inside a complex N dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of A with a very small  sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is 0 then we can assign an invariant of our singularity called the minimal
discrepancy. We relate the minimal discrepancy with indices of certain Reeb orbits on our link. As a result we show that the standard contact
5 dimensional sphere has a unique Milnor filling up to normalization. This generalizes a Theorem by Mumford.

The class of log-concave densities on $\mathbb{R}^d$ is a very natural infinite-dimensional generalisation of the class of Gaussian densities.  I will show that it also allows the statistician to have the best of both the parametric and nonparametric worlds, in that one can obtain a fully automatic density estimator in the class (via maximum likelihood), with no tuning parameters to choose.  I'll discuss its computation, methodological consequences and theoretical properties, and in particular very recent results on minimax rates of convergence and adaptation.

In this talk I will introduce methods to use 2d gauge theories as a means to describe Calabi-Yau varieties and their moduli spaces. As I review, this description furnishes a natural framework to predict derived equivalences between pairs of (sometimes even non-birational) Calabi-Yau varieties. A prominent example of this kind is realized by the Rødland non-birational pair of Calabi-Yau threefolds.
Using the 2d gauge theory description, I will propose further examples of derived equivalences among non-birational Calabi-Yau varieties.

Beilinson has given a motivic construction of the Eisenstein cohomology on modular curves. This makes it possible to define Eisenstein classes in Deligne-Beilinson, syntomic, and ´etale cohomology. These Eisenstein classes can be computed in terms of real analytic and p-adic Eisenstein series or modular units. The resulting explicit expressions allow to prove results on special values of classical and p-adic L-functions and lead to explicit reciprocity laws. Harder has more generally defined and studied the Eisenstein cohomology for Hilbert modular varieties by analytic methods. In this talk we will explain a motivic and in particular algebraic construction of Harder’s Eisenstein cohomology classes, which generalizes Beilinson’s result. This opens the way to applications, similar as for modular curves, in the case of Hilbert modular varieties.

Approximate prime numbers -- James Maynard

I will talk about the idea of an 'almost prime' number, and how this can be used to make progress on some famous problems about the primes themselves.

Mathematical biology: An early career retrospective -- Thomas Woolley

Since 2008 Thomas has focused his attention to the application of mathematical techniques to biological problems. Through numerous fruitful collaborations he has been extremely fortunate to work alongside some amazing researchers. But what has he done in the last 8 years? What lessons has he learnt? What knowledge has he produced?

This talk will encompass a brief overview of a range of applications, from animal skin patterns to cellular mechanics, via zombies and Godzilla.

The spreading of a viscous fluid in between a rigid, horizontal substrate and an overlying elastic sheet is presented as a simplified model of the hydraulic fracturing process. In particular, the talk will focus on the case of a permeable substrate for which leak-off arrests the propagation of the fluid and permits the development of a steady state. The different regimes of  gravitationally-driven and elastically-driven flow will be explored, as will the cases of a stiff and flexible sheet, before a discussion of the influence that particles included in the fluid have on the fracture propagation. 

Zhenru Wang
Title: Multi-Index Monte Carlo Estimators for a Class of Zakai SPDEs
Abstract:   
We first propose a space-time Multi-Index Monte Carlo (MIMC) estimator for a one-dimensional parabolic SPDE of Zakai type. We compare the computational cost required for a prescribed accuracy with the Multilevel Monte Carlo (MLMC) method of Giles and Reisinger (2012). Then we extend the estimator to a two-dimensional variant of SPDE. The theoretical analysis shows the benefit of using MIMC in high dimensional problems over MLMC methods. Numerical tests confirm these finding empirically.

Vadim Kaushansky
Title: An extended structural default model with jump risk
Abstact:
We consider a structural default model in an interconnected banking network as in Itkin and Lipton (2015), where there are mutual obligations between each pair of banks. We analyse the model numerically for the case of two banks with jumps in their asset value processes. Specifically, we develop a finite difference method for the resulting two-dimensional partial integro-differential equation, and study its stability and consistency. By applying this method, we compute joint and marginal survival probabilities, as well as prices of credit default swaps (CDS) and first-to-default swaps (FTD), Credit and Debt Value Adjustments (CVA and DVA).