Past

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.