Past

Quiver varieties, as introduced by Nakaijma, play a key role in representation theory. They give a very large class of symplectic singularities and, in many cases, their symplectic resolutions too. However, there seems to be no general criterion in the literature for when a quiver variety admits a symplectic resolution. In this talk I will give necessary and sufficient conditions for a quiver variety to admit a symplectic resolution.  This result is based on work of Crawley-Bouvey and of Kaledin, Lehn and Sorger. The talk is based on joint work with T. Schedler.
 

I will introduce classical results on finiteness theorem with a way of connecting them to idea of covering spaces. I will talk about the proof of FLT under this connection.

There is a simple way to “glue together” a coupled pair of continuum random trees to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which describes the “interface” between the trees). We present an explicit and canonical way to embed the sphere into the Riemann sphere. In this embedding, the measure is Liouville quantum gravity with parameter gamma in (0,2), and the curve is space-filling version of SLE with kappa=16/gamma^2. Based on joint work with Bertrand Duplantier and Scott Sheffield

TBC

I will describe a “cubical flat torus theorem” for a group G acting properly and cocompactly on a CAT(0) cube complex.
This states that every “highest” free abelian subgroup of G acts properly and cocompactly on a convex subcomplex that is quasi-isometric to a Euclidean space.
I will describe some simple consequences, as well as the original motivation which was to prove the “bounded packing property” for cyclic subgroups of G.
This is joint work with Daniel Woodhouse.

We consider a controlled system, in which an input $X: [0, T] \rightarrow E:= \mathbb{R}^{d}$ is a continuous but potentially highly oscillatory path and the corresponding output $Y$ is the line integral along $X$, for some unknown function $f: E \rightarrow E$. The rough paths theory provides a general framework to answer the question on which mild condition of $X$ and $f$, the integral $I(X)$ is well defined. It is robust enough to allow to treat stochastic integrals in a deterministic way. In this paper we are interested in identification of controlled systems of this type. The difficulty comes from the high dimensionality caused by the input of a function type. We propose novel adaptive and non-parametric algorithms to learn the functional relationship between the  input and the output from the data by carefully choosing the feature set of paths based on the rough paths theory and applying linear regression techniques. The algorithms is demonstrated on a financial application where the task is to predict the P$\&$L of the unknown trading strategy.

From a complex analytic perspective, both Teichmüller spaces and
symmetric spaces can be realised as contractible bounded domains, that
have several features in common but also exhibit many differences. In
this talk we will study isometric maps between these two important
classes of bounded domains equipped with their intrinsic Kobayashi metric.

Calabi-Yau manifolds without flux are perhaps the best-known
supergravity backgrounds that leave some supersymmetry unbroken. The
supersymmetry conditions on such spaces can be rephrased as the
existence and integrability of a particular geometric structure. When
fluxes are allowed, the conditions are more complicated and the
analogue of the geometric structure is not well understood.

In this talk, I will define the analogue of Calabi-Yau geometry for
generic D=4, N=2 backgrounds with flux in both type II and
eleven-dimensional supergravity. The geometry is characterised by a
pair of G-structures in 'exceptional generalised geometry' that
interpolate between complex, symplectic and hyper-Kahler geometry.
Supersymmetry is then equivalent to integrability of the structures,
which appears as moment maps for diffeomorphisms and gauge
transformations. Similar structures also appear in D=5 and D=6
backgrounds with eight supercharges.

As a simple application, I will discuss the case of AdS5 backgrounds
in type IIB, where deformations of these geometric structures give
exactly marginal deformations of the dual field theories.

In the course of work with Jamshid Derakhshan on definability in adele rings, we came upon various problems about definability and model completeness for possibly infinite dimensional algebraic extensions of p-adic fields (sometimes involving uniformity across p). In some cases these problems had been closely approached in the literature but never  explicitly considered.I will explain what we have proved, and try to bring out many big gaps in our understanding of these matters. This  seems appropriate just over 50 years after the breakthroughs of Ax-Kochen and Ershov.

Abstract: Four manifolds are some of the most intriguing objects in topology. So far, they have eluded any attempt of classification and their behaviour is very different from what one encounters in other dimensions. On the other hand, Einstein metrics are among the canonical types of metrics one can find on a manifold. In this talk I will discuss many of the peculiarities that make dimension four so special and see how Einstein metrics could potentially help us understand more about four manifolds.

Solving systems of linear equations $Ax=b$ is easy, but how can we solve such a system when given a "noisy" version of $b$? Over the reals one can use the least squares method, but the problem is harder when working over a finite field. Recently this subject has become very important in cryptography, due to the introduction of new cryptosystems with interesting properties.

The talk will survey work in this area. I will discuss connections with coding theory and cryptography. I will also explain how Fourier analysis in finite groups can be used to solve variants of this problem, and will briefly describe some other applications of Fourier analysis in cryptography. The talk will be accessible to a general mathematical audience.

Feedforward layers are integral step in processing and transmitting sensory information across different regions the brain. Yet experiments reveal the difficulty of stable propagation through layers without causing neurons to synchronize their activity. We study the limits of stable propagation in a discrete feedforward model of binary neurons. By analyzing the spectral properties of a mean-field Markov chain model, we show when such information transmission persists. Addition of inhibitory neurons and synaptic noise increases the robustness of asynchronous rate transmission. We close with an example of feedforward processing in the input layer to cerebellum. 

We propose the use of a constraint preconditioner for the iterative solution of the linear system arising from the finite element discretization of the coupled Stokes-Darcy system. The Stokes-Darcy system is a set of coupled PDEs that can be used to model a freely flowing fluid over porous media flow. The fully coupled system matrix is large, sparse, non-symmetric, and of saddle point form. We provide for exact versions of the constraint preconditioner spectral and field-of-values bounds that are independent of the underlying mesh width. We present several numerical experiments, using the deal.II finite element library, that illustrate our results in both two and three dimensions. We compare exact and inexact versions of the constraint preconditioner against standard block diagonal and block lower triangular preconditioners to illustrate its favorable properties.

I will explain how algebraic spaces can be presented as Zariski geometries and prove some classical facts about algebraic spaces using the theory of Zariski geometries.

For a finitely generated group $G$ with subgroup $H$ we define $e(G,H)$, the relative ends of the pair $(G,H)$, to be the number of ends of the Cayley graph of G quotiented out by the left action of H. We will examine some basic properties of relative ends and will outline the theorem of Sageev showing that $e(G,H)>1$ if and only if $G$ acts essentially on a simply connected CAT(0) cube complex. If time permits, we will outline Niblo's proof of Stallings' theorem using Sageev's construction.

I will give a historical overview of some of the theorems proved by the
Ancient Greeks, which are now taken for granted but were, and are,
landmarks in the history of mathematics. Particular attention will be
given to the calculation of areas, including theorems of Hippocrates,
Euclid and Archimedes.

Let $G$ be a compact Lie group and $k$ be a field of characteristic $p\ge 0$ such that $H^*(G)$ does not have $p$-torsion. We show that a free Lagrangian orbit of a Hamiltonian $G$-action on a compact, monotone, symplectic manifold $X$ split-generates an idempotent summand of the monotone Fukaya category over $k$ if and only if it represents a non-zero object of that summand. Our result is based on: an explicit understanding of the wrapped Fukaya category through Koszul twisted complexes involving the zero-section and a cotangent fibre; and a functor canonically associated to the Hamiltonian $G$-action on $X$. Several examples can be studied in a uniform manner including toric Fano varieties and certain Grassmannians. 

A graph is quasirandom if its edge distribution is similar (in a well defined quantitative way) to that of a random graph with the same edge density. Classical results of Thomason and Chung-Graham-Wilson show that a variety of graph properties are equivalent to quasirandomness. On the other hand, in some known proofs the error terms which measure quasirandomness can change quite dramatically when going from one property to another which might be problematic in some applications.

Simonovits and Sós proved that the property that all induced subgraphs have about the expected number of copies of a fixed graph $H$ is quasirandom. However, their proof relies on the regularity lemma and gives a very weak estimate. They asked to find a new proof for this result with a better estimate. The purpose of this talk is to accomplish this.

Joint work with D. Conlon and J. Fox

We provide a simple convergence analysis unified for TR and a new ARC algorithms, which we name ARCq and which is very close in spirit to trust region methods, closer than the original ARC is. We prove global convergence to second order points. We also obtain as a corollary the convergence of the original ARC method. Since one of our aims is to achieve a simple presentation, we sacrifice some generality which we discuss at the end of our developments. In this simplified setting, we prove the optimal complexity property for the ARCq and identify the key elements which allow it. We then propose efficient implementations using a Cholesky like factorization as well as a scalable version based on conjugate gradients.

We provide a result on prospect theory decision makers who are naïve about the time inconsistency induced by probability weighting. If a market offers a sufficiently rich set of investment strategies, investors postpone their trading decisions indefinitely due to a strong preference for skewness. We conclude that probability weighting in combination with naïveté leads to unrealistic predictions for a wide range of dynamic setups. Finally, I discuss recent work on the topic that invokes different assumptions on the dynamic modeling of prospect theory.

New ambitwistor string models are presented for a variety of theories and older models are shown to work at 1 loop and perhaps higher using a simpler formulation on the Riemann sphere.

Abstract:In the first part of the talk, using classical hypergeometric identities, I will compute the harmonic measure of finite intervals and their complementaries for the Lévy stable process on the line. This gives a simple and unified proof of several results by Blumenthal-Getoor-Ray, Rogozin, and Kyprianou-Pardo-Watson. In the second part of the talk, I will consider the two-dimensional Markov process based on the stable Lévy process and its area process. I will give two explicit formulae for the harmonic measure of the split complex plane. These formulae allow to compute the persistence exponent of the stable area process, solving a problem raised by Zhan Shi. This is based on two joint works with Christophe Profeta.

In this talk I will define algebraic automorphic forms, first defined by Gross, which are objects that are conjectured to have Galois representations attached to them. I will explain how this fits into the general picture of the Langlands program and, giving some examples, briefly describe one method of proving certain cases of the conjecture. 

In the first part of this talk, we will give a very gentle introduction to the Ising model. Then , we will explain a very simple proof of a combinatorial formula for the 2D Ising model partition function using the language of Kac-Ward matrices. This approach can be used for general weighted graphs embedded in surfaces, and extends to the study of several other observables. This is a joint work with Dima Chelkak and Adrien Kassel.
 

Abstract: It has been recently shown that rough volatility models reproduce very well the statistical properties of low frequency financial data. In such models, the volatility process is driven by a fractional Brownian motion with Hurst parameter of order 0.1. The goal of this talk is to explain how such fractional dynamics can be obtained from the behaviour of market participants at the microstructural scales.

Using limit theorems for Hawkes processes, we show that a rough volatility naturally arises in the presence of high frequency trading combined with metaorders splitting. This is joint work with Thibault Jaisson.

Mammalian lung morphology is well optimized for efficient bulk transport of gases, yet most lung morphogenesis occurs prenatally, when the lung is filled with liquid - and at birth it is immediately ready to function when filled with gas. Lung morphogenesis is regulated by numerous mechanical inputs including fluid secretion, fetal breathing movements, and peristalsis. We generally understand which of these broad mechanisms apply, and whether they increase or decrease overall size and/or branching. However, we do not generally have a clear understanding of the intermediate mechanisms actuating the morphogenetic control. We have studied this aspect of lung morphogenesis from several angles using mathematical/mechanical/transport models tailored to specific questions. How does lumen pressure interact with different locations and tissues in the lung? Is static pressure equivalent to dynamic pressure? Of the many plausible cellular mechanisms of mechanosensing in the prenatal lung, which are compatible with the actual mechanical situation? We will present our models and results which suggest that some hypothesized intermediate mechanisms are not as plausible as they at first seem.

We provide a new algorithm for approximating the law of a one-dimensional diffusion M solving a stochastic differential equation with possibly irregular coefficients.
The algorithm is based on the construction of Markov chains whose laws can be embedded into the diffusion M with a sequence of stopping times. The algorithm does not require any regularity or growth assumption; in particular it applies to SDEs with coefficients that are nowhere continuous and that grow superlinearly. We show that if the diffusion coefficient is bounded and bounded away from 0, then our algorithm has a weak convergence rate of order 1/4. Finally, we illustrate the algorithm's performance with several examples.

In 1851, Carl Jacobi made the experimental observation that all integers are sums of seven non-negative cubes, with precisely 17 exceptions, the largest of which is 454. Building on previous work by Maillet, Landau, Dickson, Linnik, Watson, Bombieri, Ramaré, Elkies and many others, we complete the proof of Jacobi's observation.

We consider the impact of spatial heterogeneities on the dynamics of 
localized patterns in systems of partial differential equations (in one 
spatial dimension). We will mostly focus on the most simple possible 
heterogeneity: a small jump-like defect that appears in models in which 
some parameters change in value as the spatial variable x crosses 
through a critical value -- which can be due to natural inhomogeneities, 
as is typically the case in ecological models, or can be imposed on the 
model for engineering purposes, as in Josephson junctions. Even such a 
small, simplified heterogeneity may have a crucial impact on the 
dynamics of the PDE. We will especially consider the effect of the 
heterogeneity on the existence of defect solutions, which boils down to 
finding heteroclinic (or homoclinic) orbits in an n-dimensional 
dynamical system in `time' x, for which the vector field for x > 0 
differs slightly from that for x < 0 (under the assumption that there is 
such an orbit in the homogeneous problem). Both the dimension of the 
problem and the nature of the linearized system near the limit points 
have a remarkably rich impact on the defect solutions. We complement the 
general approach by considering two explicit examples: a heterogeneous 
extended Fisher–Kolmogorov equation (n = 4) and a heterogeneous 
generalized FitzHugh–Nagumo system (n = 6).

Oxford Mathematics Public Lectures

MC Escher - Artist, Mathematician, Man 

Roger Penrose and Jon Chapman

This lecture has now sold out

The symbiosis between mathematics and art is personified by the relationship between Roger Penrose and the great Dutch graphic artist MC Escher. In this lecture Roger will give a personal perspective on Escher's work and his own relationship with the artist while Jon Chapman will demonstrate the mathematical imagination inherent in the work. 

The lecture will be preceded by a showing of the BBC 4 documentary on Escher presented by Sir Roger Penrose. Private Escher prints and artefacts will be on display outside the lecture theatre.

5pm

Lecture Theatre 1

Mathematical Institute

Andrew Wiles Building

Radcliffe Observatory Quarter

Woodstock Road

OX2 6GG

Roger Penrose is Emeritus Rouse Ball Professor at the Mathematical Institute in Oxford

Jon Chapman is Statutory Professor of Mathematics and Its Applications at the Mathematical Institute in Oxford

This is an exciting time to study quantum algorithms. As the technological challenges of building a quantum computer continue to be met there is still much to learn about the power of quantum computing. Understanding which problems a quantum computer could solve faster than a classical device and which problems remain hard is particularly relevant to cryptography. We would like to design schemes that are secure against an adversary with a quantum computer. I'll give an overview of the quantum computing that is accessible to a general audience and use a recently declassified project called "soliloquy" as a case study for the development (and breaking) of post-quantum cryptography.

Many people talk about properties that you would expect of a group. When they say this they are considering random groups, I will define what it means to pick a random group in one of many models and will give some properties that these groups will have with overwhelming probability. I will look at the proof of some of these results although the talk will mainly avoid proving things rigorously.

I will explain how to relate the center of a cyclotomic quiver Hecke algebras to the cohomology of Nakajima quiver varieties using a current algebra action. This is a joint work with M. Varagnolo and E. Vasserot.