Past

The classical Gehring lemma for elliptic equations with measurable coefficients states that an energy solution, which is initially assumed to be $H^1$ - Sobolev regular, is actually in a better Sobolev space space $W^{1,q}$ for some $q>2$. This a consequence of a self-improving property that so-called reverse Hölder inequality implies. In the case of nonlocal equations a self-improving effect appears: Energy solutions are also more differentiable. This is a new, purely nonlocal phenomenon, which is not present in the local case. The proof relies on a nonlocal version of the Gehring lemma involving new exit time and dyadic decomposition arguments. This is a joint work with G. Mingione and Y. Sire. 

In dimension three, convex surface theory implies that every tight contact structure on a connected sum M # N can be constructed as a connected sum of tight contact structures on M and N. I will explain some examples showing that this is not true in any dimension greater than three.  The proof is based on a recent higher-dimensional version of a classic result of Eliashberg about the symplectic fillings of contact manifolds obtained by subcritical surgery. This is joint work with Paolo Ghiggini and Klaus Niederkrüger.

We consider several classes of sequences of random variables whose Laplace transform presents the same type of \textit{splitting phenomenon} when suitably rescaled. Answering a question of Kowalski-Nikeghbali, we explain the apparition of a universal term, the \textit{Gamma factor}, by a common feature of each model, the existence of an auxiliary randomisation that reveals an independence structure.
The class of examples that belong to this framework includes random uniform permutations, random polynomials or random matrices with values in a finite field and the classical Sathe-Selberg theorems in probabilistic number theory. We moreover speculate on potential similarities in the Gaussian setting of the celebrated Keating and Snaith's moments conjecture. (Joint work with R. Chhaibi)
 

Motivated by a theorem of Barbour, we revisit some of the classical limit theorems in probability from the viewpoint of the Stein method. We setup the framework to bound Wasserstein distances between some distributions on infinite dimensional spaces. We show that the convergence rate for
the Poisson approximation of the Brownian motion is as expected proportional to λ −1/2 where λ is the intensity of the Poisson process. We also exhibit the speed of convergence for the Donsker Theorem and extend this result to enhanced Brownian motion.

 

I will give an introductory account of the zeta-functions for one-parameter families of CY manifolds. The aim of the talk is to point out that the zeta-functions corresponding to singular manifolds of the family correspond to modular forms. In order to give this introductory account I will give a lightning review of finite fields and of the p-adic numbers.

Our Christmas Public Lecture this year will be presented by Marcus du Sautoy who will be examining an aspect of Christmas not often considered: the mathematics.

To register please email: @email

The Oxford Mathematics Christmas Lecture is generously sponsored by G-Research - Researching investment ideas to predict financial markets

Due to its use in cryptographic protocols such as the Diffie--Hellman

key exchange, the discrete logarithm problem attracted a considerable

amount of attention in the past 40 years. In this talk, we summarize

the key technical ideas and their evolution for the case of discrete

logarithms in small characteristic finite fields. This road leads from

the original belief that this problem was hard enough for

cryptographic purpose to the current state of the art where the

algorithms are so efficient and practical that the problem can no

longer be considered for cryptographic use.

For full details please visit:

http://blogs.bodleian.ox.ac.uk/adalovelace/files/2015/10/Ada-Lovelace-S…

We study the low-temperature limit in the Landau-de Gennes theory for liquid crystals. We prove that for minimizers for orientable Dirichlet data tend to be almost uniaxial but necessarily contain some biaxiality around the singularities of a limiting harmonic map. In particular we prove that around each defect there must necessarily exist a maximal biaxiality point, a point with a purely uniaxial configuration with a positive order parameter, and a point with a purely uniaxial configuration with a negative order parameter. Estimates for the size of the biaxial cores are also given.

This is joint work with Apala Majumdar and Adriano Pisante.

It has been conjectured that marine ice sheets (those that

flow into the ocean) are unconditionally unstable when the underlying

bed-slope runs uphill in the direction of flow, as is typical in many

regions underneath the West Antarctic Ice Sheet. This conjecture is

supported by theoretical studies that assume a two-dimensional flow

idealization. However, if the floating section (the ice shelf) is

subject to three-dimensional stresses from the edges of the embayments

into which they flow, as is typical of many ice shelves in Antarctica,

then the ice shelf creates a buttress that supports the ice sheet.

This allows the ice sheet to remain stable under conditions that may

otherwise result in collapse of the ice sheet. This talk presents new

theoretical and experimental results relating to the effects of

three-dimensional stresses on the flow and structure of ice shelves,

and their potential to stabilize marine ice sheets.

Transmural propagation is a little studied feature of mammalian electrophysiology, this talk reviews our experimental work using different optical techniques to characterise this mode
of conduction under physiological and pathophysiological conditions.

In finance, the default time of a counterparty is sometimes modeled as the
first passage time of a credit index process below a barrier. It is
therefore relevant to consider the following question:
   If we know the distribution of the default time, can we find a unique
barrier which gives this distribution? This is known as the Inverse
First Passage Time (IFPT) problem in the literature.
   We consider a more general `smoothed' version of the inverse first
passage time problem in which the first passage time is replaced by
the first instant that the time spent below the barrier exceeds an
independent exponential random variable. We show that any smooth
distribution results from some unique continuously differentiable
barrier. In current work with B. Ettinger and T. K. Wong, we use PDE
methods to show the uniqueness and existence of solutions to a
discontinuous version of the IFPT problem.

Multidimensional single molecule microscopy (MSMM) generates image time series of biomolecules in a cellular environment that have been tagged with fluorescent labels. Initial analysis steps of such images consist of image registration of multiple channels, feature detection and single particle tracking. Further analysis may involve the estimation of diffusion rates, the measurement of separations between molecules that are not optically resolved and more. The analysis is done under the condition of poor signal to noise ratios, high density of features and other adverse conditions. Pushing the boundary of what is measurable, we are facing among others the following challenges. Firstly the correct assessment of the uncertainties and the significance of the results, secondly the fast and reliable identification of those features and tracks that fulfil the assumptions of the models used. Simpler models require more rigid preconditions and therefore limiting the usable data, complexer models are theoretically and especially computationally challenging.

Abstract: (Joint work with Sylvy Anscombe) We consider four properties 
of a field K related to the existence of (definable) henselian 
valuations on K and on elementarily equivalent fields and study the 
implications between them. Surprisingly, the full pictures look very 
different in equicharacteristic and mixed characteristic.

In 1954 Thom showed that there is an isomorphism between the cobordism groups of manifolds and the homotopy groups of the Thom spectrum. I will define what these words mean and present the explicit, geometric construction of the isomorphism.

In this talk, I will present a family of forward performance processes in
discrete time. These processes are predictable with regards to the market
information. Examples from a binomial setting will be given which include
the time-monotone exponential forward process and the completely monotonic
family.

A period is a certain type of number obtained by integrating algebraic differential forms over algebraic domains. Examples include pi, algebraic numbers, values of the Riemann zeta function at integers, and other classical constants.
Difficult transcendence conjectures due to Grothendieck suggest that there should be a Galois theory of periods.
I will explain these notions in very introductory terms and show how to set up such a Galois theory in certain situations.
I will then discuss some applications, in particular to Kim's method for bounding $S$-integral solutions to the equation $u+v=1$, and possibly to high-energy physics.

We study the motion of a eukaryotic cell on a substrate and investigate the dependence of this motion on key physical parameters such as strength of protrusion by actin filaments and adhesion. This motion is modeled by a system of two PDEs consisting of the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. The two key properties of this system are (i) presence of gradients in the coupling terms and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion perturbed by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters. In the subcritical regime, the well-posedness of the problem is proved (M. Mizuhara et al., 2015). Our main focus is the supercritical regime where we established surprising features of the motion of the interface such as discontinuities of velocities and hysteresis in the 1D model, and instability of the circular shape and rise of asymmetry in the 2D model. Because of properties (i)-(ii), classical comparison principle techniques do not apply to this system. Furthermore, the system can not be written in a form of gradient flow, which is why Γ-convergence techniques also can not be used. This is joint work with V. Rybalko and M. Potomkin.

Equations of quantum mechanics in the semiclassical regime present an enduring challenge for numerical analysts, because their solution is highly oscillatory and evolves on two scales. Standard computational approaches to the semiclassical Schrödinger equation do not allow for long time integration as required, for example, in quantum control of atoms by short laser bursts. This has motivated our approach of asymptotic splittings. Combining techniques from Lie-algebra theory and numerical algebra, we present a new computational paradigm of symmetric Zassenhaus splittings, which lends itself to a very precise discretisation in long time intervals, at very little cost. We will illustrate our talk by examples of quantum phenomena – quantum tunnelling and quantum scattering – and their computation and, time allowing, discuss an extension of this methodology to time-dependent semiclassical systems using Magnus expansions

We consider the Navier-Stokes initial boundary value problem (NS-IBVP) in a smooth exterior domain. We are interested in establishing existence of weak solutions (we mean weak solutions as synonym of solutions global in time) with an initial data in L(3,∞)

following the joint paper with L.Shaheen http://people.maths.ox.ac.uk/zilber/wLb.pdf

Quasihomomorphisms (QHMs) are maps $f$ between groups such that the
homomorphic condition is boundedly satisfied. The case of QHMs with
abelian target is well studied and is useful for computing the second
bounded cohomology of groups. The case of target non-abelian has,
however, not been studied a lot.

We will see a technique for classifying QHMs $f: G \rightarrow H$ by Fujiwara and
Kapovich. We will give examples (sometimes with proofs!) for QHM in
various cases such as

• the image $H$  hyperbolic groups,
• the image $H$ discrete rank one isometries,
• the preimage $G$ cyclic / free group, etc.

Furthermore, we point out a relation between QHM and extensions by short
exact sequences.

The massless Maxwell-Klein-Gordon system describes the interaction between an electromagnetic field (Maxwell) and a charged massless scalar field (massless Klein-Gordon, or wave). In this talk, I will present a recent proof, joint with D. Tataru, of global well-posedness and scattering of this system for arbitrary finite energy data in the (4+1)-dimensional Minkowski space, in which the PDE is energy critical.

We know that the existence of a period three point for an interval map implies much about the dynamics of the map, but the restriction of the map to the periodic orbit itself is trivial. Countable invariant subsets arise naturally in many dynamical systems, for example as $\omega$-limit sets, but many of the usual notions of dynamics degenerate when restricted to countable sets. In this talk we look at what we can say about dynamics on countable compact spaces.  In particular, the theory of countable dynamical systems is the theory of the induced dynamics on countable invariant subsets of the interval and the theory of homeomorphic countable dynamics is the theory of compact countable invariant subsets of homeomorphisms of the plane.

Joint work with Columba Perez

The representation dimension of an algebra was introduced in the early 70's by M. Auslander, with the goal of measuring how far an algebra is from having finite number of finitely generated indecomposable modules (up to isomorphism). This invariant is not well understood. For instance, it was not until 2002 that O. Iyama proved that every algebra has finite representation dimension. This was done by constructing special quasihereditary algebras. In this talk I will give an introduction to this topic and I shall briefly explain Iyama's construction.

I will survey some applications of a special kind of stratification of an algebraic stack called a theta-stratification. The goal is to eventually be able to study semistability and wall-crossing 
in a large array of moduli problems beyond the well-known examples. The most general application is to studying the derived category of coherent sheaves on the stack, but one can use this to understand the topology (K-theory, Hodge-structures, etc.) of the semistable locus and how it changes as one varies the stability condition. I will also describe a ``virtual non-abelian localization theorem'' which computes the virtual index of certain classes in the K-theory of a stack with perfect obstruction theory. This generalizes the virtual localization theorem of Pandharipande-Graber and the K-theoretic localization formulas of Teleman and Woodward.

It is easy to see that if a tournament (a complete oriented graph) has a directed cycle then it has a directed 3-cycle. We investigate the analogous question for 3-tournaments, and more generally for oriented 3-graphs.

Abstract: This is a joint work with E. Breuillard.

A conjecture of Breuillard asserts that for every positive integer d, there is a positive constant c such that the following holds. Let S be a finite subset of GL(d,C) that generates a group, which is not virtually nilpotent. Then |S^n|>exp(cn) for all n.
Considering an algebraic number a that is not a root of unity and the semigroup generated by the affine transformations x-> ax+1, x-> ax+1, the above conjecture implies that the Mahler measure of a is at least 1+c' for some c'>0 depending on c. This property is known as Lehmer's conjecture.

I will talk about the converse of this implication, namely that Lehmer's conjecture implies the uniform growth conjecture of
Breuillard.

Hawking radiation and particle creation by an expanding Universe
are paradigmatic predictions of quantum field theory in curved spacetime.
Although the theory is a few decades old, it still awaits experimental
demonstration. At first sight, the effects predicted by the theory are too
small to be measured in the laboratory. Therefore, current experimental
efforts have been directed towards siumlating Hawking radiation and
studying quantum particle creation in analogue spacetimes.
In this talk, I will present a proposal to test directly effects of
quantum field theory in the Earth's spacetime using quantum technologies.
Under certain circumstances, real spacetime distortions (such as
gravitational waves) can produce observable effects in the state of
phonons of a Bose-Einstein condensate. The sensitivity of the phononic
field to the underlying spacetime can also be used to measure spacetime
parameters such as the Schwarzschild radius of the Earth.

In quite an elementary, hands-on talk, I will discuss some Ax-Lindemann type results in the setting of modular functions.  There are some very powerful results in this area due to Pila, but in nonclassical variants we have only quite weak results, for a rather silly reason to be discussed in the talk.

TBC

We will discuss the concept of $\ell^2$-stability of a group and show some of its rigidity consequences.  We provide moreover some very concrete examples of lattices in product of trees that have many interesting properties, $\ell^2$-stability being only one of them.

Abstract: First order asymptotic of scalar renewal sequences with infinite mean characterized by regular variation has been classified in the 60's (Garsia and Lamperti). In the recent years, the question of higher order asymptotic for renewal sequences with infinite mean was motivated by obtaining 'mixing rates' for dynamical systems with infinite measure. In this talk I will present the recent results we have obtained on higher order asymptotic for renewal sequences with infinite mean and their consequences for error rates in certain limit theorems (such as arcsine law for null recurrent Markov processes).

I will discuss theta-stability, a framework for analyzing moduli problems in algebraic geometry by finding a special kind of stratification called a theta-stratification, a notion which generalizes the Kempf-Ness stratification in geometric invariant theory and the Harder-Narasimhan-Shatz stratification of the moduli of vector bundles on a Riemann surface.

The talk will consider three well-defined problems which can be interpreted as mathematical tests of the physical reality of black holes: Rigidity, stability and formation of black holes.

We will discuss how the analysis of a stochastic mean-field model for
synaptic activity can be used to reconstruct some parameters about
neuronal networks.  The method is based on a non-standard analysis of the
Fokker-Planck equation and the asymptotic computation of the spectrum for
the nonself-adjoint operator. Applications concern Up- and Down- states
and bursting activity in neuronal networks.

Systemic risk in financial markets occurs when activities that are beneficial to an agent in isolation (e.g. reducing microprudential risk) cause unintended consequences due to collective interactions (usually called macroprudential risk).  I will discuss three different mechanisms through which this occurs in financial markets.   Contagion can propagate due to the market impact of trading among agents with strongly overlapping portfolios, or due to cascading failures from chains of default caused by networks of interlinked counterparty exposures.  A proper understanding of these phenomena must take both dynamics and network effects into account.  I will discuss four different examples that illustrate these points.  The first is a simple model of the market dynamics induced by Basel-style risk management, which from extremely simple assumptions shows that excessive leverage can give rise to a slowly rising price bubble followed by an abrupt crash with a time period of 10 - 15 years.  The model gives rise to a chaotic attractor whose time series closely resembles the Great Moderation and subsequent crisis.   We show that alternatives to Basel can provide a better compromise between micro and macro prudential risk.   The second example is a model of leveraged value investors that yields clustered volatility and fat-tailed returns similar to those in financial markets.  The third example is the DebtRank algorithm, which uses a similar method to PageRank to correctly quantify the way risk propagates through networks of counterparty exposures and can be used as the basis of a systemic risk tax.  The fourth example will  be work in progress to provide an early warning system for financial stress caused by overlapping portfolios.  Finally I will discuss an often neglected source of financial risk due to imbalances in market ecologies.

A basic result in Morse theory due to Reeb states that a compact manifold which admits a smooth function with only two, non-degenerate critical points is homeomorphic to the sphere. We shall apply this idea to distance function associated to a Riemannian metric to prove the diameter-sphere theorem of Grove-Shiohama: A complete Riemannian manifold with sectional curvature $\geq 1$ and diameter $> \pi / 2$ is homeomorphic to a sphere. I shall not assume any knowledge about curvature for the talk.

Following a quick introduction to derivatives markets and the classic theory of valuation, we describe the changes triggered by post 2007 events. We re-discuss the valuation theory assumptions and introduce valuation under counterparty credit risk, collateral posting, initial and variation margins, and funding costs. A number of these aspects had been investigated well before 2007. We explain model dependence induced by credit effects, hybrid features, contagion, payout uncertainty, and nonlinear effects due to replacement closeout at default and possibly asymmetric borrowing and lending rates in the margin interest and in the funding strategy for the hedge of the relevant portfolio. Nonlinearity manifests itself in the valuation equations taking the form of semi-linear PDEs or Backward SDEs. We discuss existence and uniqueness of solutions for these equations. We present an invariance theorem showing that the final valuation equations do not depend on unobservable risk free rates, that become purely instrumental variables. Valuation is thus based only on real market rates and processes. We also present a high level analysis of the consequences of nonlinearities, both from the point of view of methodology and from an operational angle, including deal/entity/aggregation dependent valuation probability measures and the role of banks treasuries. Finally, we hint at how one may connect these developments to interest rate theory under multiple discount curves, thus building a consistent valuation framework encompassing most post-2007 effects.

Damiano Brigo, Joint work with Andrea Pallavicini, Daniele Perini, Marco Francischello.