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

[[{"fid":"23604","view_mode":"media_square","fields":{"format":"media_square","field_file_image_alt_text[und][0][value]":"Jon Keating","field_file_image_title_text[und][0][value]":"Jon Keating"},"type":"media","attributes":{"alt":"Jon Keating","title":"Jon Keating","height":"258","width":"258","class":"media-element file-media-square"}}]]

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)