Past

The Ramsey number $R(K_s, Q_n)$ is the smallest integer $N$ such that every red-blue colouring of the edges of the complete graph $K_N$ contains either a red $n$-dimensional hypercube, or a blue clique on $s$ vertices. Note that $N=(s-1)(2^n -1)$ is not large enough, since we may colour in red $(s-1)$ disjoint cliques of cardinality $2^N -1$ and colour the remaining edges blue. In 1983, Burr and Erdos conjectured that this example was the best possible, i.e., that $R(K_s, Q_n) = (s-1)(2^n -1) + 1$ for every positive integer $s$ and sufficiently large $n$. In a recent breakthrough, Conlon, Fox, Lee and Sudakov proved the conjecture up to a multiplicative constant for each $s$. In this talk we shall sketch the proof of the conjecture and discuss some related problems.

(Based on joint work with Gonzalo Fiz Pontiveros, Robert Morris, David Saxton and Jozef Skokan)

In this talk we explore continuous analogues of matrix factorizations.  The analogues we develop involve bivariate functions, quasimatrices (a matrix whose columns are 1D functions), and a definition of triangular in the continuous setting.  Also, we describe why direct matrix algorithms must become iterative algorithms with pivoting for functions. New applications arise for function factorizations because of the underlying assumption of continuity. One application is central to Chebfun2. 

We prove existence of solution for evolutionary variational and quasivariational inequalities defined by a first order quasilinear operator and a variable convex set, characterized by a constraint on the absolute value of the gradient (which, in the quasi-variational case, depends on the solution itself). The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable a priori estimates.

Uniqueness of solution is proved for the variational inequality. We also obtain existence of stationary solutions, by studying the asymptotic behaviour in time. We shall illustrate a simple “sand pile” example in the variational case for the transport operator were the problem is equivalent to a two-obstacles problem and the solution stabilizes in finite time. Further remarks about these properties of the solution will be presented.This is a joint work with Lisa Santos.

If times allows, using similar techniques, I shall also present the existence, uniqueness and continuous dependence of solutions of a new class of evolution variational inequalities for incompressible thick fluids. These non-Newtonian fluids with a maximum admissible shear rate may be considered as a limit class of shear-thickening or dilatant fluids, in particular, as the power limit of Ostwald-deWaele fluids.

We discuss kinematic algebras associated to the scattering equations that arise in the description of the scattering of massless particles.  We describe their role in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex and identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.

We discuss kinematic algebras associated to the scattering equations that arise in the description of the scattering of massless particles.  We describe their role in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex and identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.

I will describe how a sieve method can be used to establish the Hasse principle for the variety

$$f(t)=N(x_1,\ldots,x_k),$$

where $f$ is an irreducible cubic and $N$ is a norm form for a number field satisfying certain hypotheses.

The vanishing mean curvature flow in Minkowski space is the

natural evolutionary generalisation of the minimal surface equation,

and has applications in cosmology as a model equation for cosmic

strings and membranes. The equation clearly admits initial data which

leads to singularity formation in finite time; Nguyen and Tian have

even shown stability of the singularity formation in low dimension. On

the other hand, Brendle and Lindblad separately have shown that all

"nearly flat" initial data leads to global existence of solutions. In

this talk, I describe an intermediate regime where global existence

of solutions can be proven on a codimension 1 set of initial data; and

where the codimension 1 condition is optimal --- The

catenoid, being a minimal surface in R^3, is a static solution to the

vanishing mean curvature flow. Its variational instability as a

minimal surface leads to a linear instability under the flow. By

appropriately "modding out" this unstable mode we can show the

existence of a stable manifold of initial data that gives rise to

solutions which scatters toward to the

catenoid. This is joint work with Roland Donninger, Joachim Krieger,

and Jeremy Szeftel. The preprint is available at http://arxiv.org/abs/1310.5606v1

Abstract: The expected signature is often viewed as a direct analogue of the Laplace transform, and as such it has been asked whether, under certain conditions, it may determine the law of a random signature. In this talk we first introduce a meaningful topology on the space of (geometric) rough paths which allows us to study it as a well-defined probability space. With the help of compact symplectic Lie groups, we then define a set of characteristic functions and show that two random variables in this space are equal in law if and only if they agree on each characteristic function. We finally show that under very general boundedness conditions, the value of each characteristic function is completely determined by the expected signature, giving an affirmative answer to the aforementioned question in many cases. In particular, we demonstrate that the Stratonovich signature is completely determined in law by its expected signature, and show how a similar technique can be used to demonstrate convergence in law of random signatures.

Background material: http://arxiv.org/abs/1307.3580

Factorization homology is an invariant of an n-manifold M together with an n-disk algebra A. Should M be

a circle, this recovers the Hochschild complex of A; should A be a commutative algebra, this recovers the

homology of M with coefficients in A. In general, factorization homology retains more information about

a manifold than its underlying homotopy type.

In this talk we will lift Poincare' duality to factorization homology as it intertwines with Koszul

duality for n-disk algebras -- all terms will be explained. We will point out a number of consequences

of this duality, which concern manifold invariants as well as algebra invariants.

This is a report on joint work with John Francis.

A review of a valuation strategy to price American-style option contracts in a “limited information” framework is discussed where sequential Monte Carlo (SMC) techniques, as presented in Doucet, de Freitas, and Gordon’s text Sequential Monte Carlo Methods in Practice, and the least–squares Monte Carlo (LSM) approach of Longstaff and Schwartz (Review of Financial Studies 14:113-147, 2001), are used as part of the valuation methodology. We utilize a risk–neutralized version of a mean-reverting model to model the volatility process. We assume that volatility is a latent stochastic process, and we capture information about it using “summary vectors” based on sequential Monte Carlo posterior filtering distributions. Of primary interest in this work is an empirical assessment of American options governed by a stochastic volatility model where the focus is on the market price of volatility risk (or the volatility risk premium). We discuss statistical modeling of the market price of volatility risk as our current evidence reveals interesting nuances about the volatility risk premium, and we hypothesize that switching models or more sophisticated time-series models could be of value to understand the empirical observations we found on the market price of volatility risk. Prior studies have shown that the magnitude of the volatility risk premium changes markedly when an American index option (NYSE Arca Oil Index Options) is in its expiration month relative to prior months, or that the magnitude varies across equities. Our objective is to study if useful information can be extracted from the volatility risk premium process, and how this information can better inform holders of American options when making decisions under uncertainty.

Key words: American options, stochastic volatility, volatility risk, sequential, Monte Carlo, risk premium, decisions, uncertainty

Disclaimer: The views expressed in this abstract (and the paper that will accompany it) are solely those of the authors and do not, in any way, reflect the opinions of the Office of the Comptroller of the Currency (OCC).

Raushan Buzyakova asked if a space is hereditarily D provided 
that the extent and Lindelöf numbers coincide for every subspace. We 
will introduce interval topologies on trees and present Nyikos' 
counterexample to this conjecture.

We consider over-the-counter (OTC) transactions with bilateral default risk, and study the optimal design of the Credit Support Annex (CSA). In a setting where agents have access to a trading technology, default penalties and collateral costs arise endogenously as a result of foregone investment opportunities. We show how the optimal CSA trades off the costs of the collateralization procedure against the reduction in exposure to counterparty risk and expected default losses. The results are used to provide insights on the drivers of different collateral rules, including hedging motives, re-hypothecation of collateral, and close-out conventions. We show that standardized collateral rules can have a detrimental impact on risk sharing, which should be taken into account when assessing the merits of standardized vs. bespoke CSAs in non-centrally cleared OTC instruments. This is joint work with D. Bauer and L.R. Sotomayor (GSU).

In {\it Was sind und was sollen die Zahlen?} (1888), Dedekind proves the Recursion Theorem (Theorem 126), and applies it to establish the categoricity of his axioms for arithmetic (Theorem 132). It is essential to these results that mathematical induction is formulated using second-order quantification, and if the second-order quantifier ranges over all subsets of the first-order domain (full second-order quantification), the categoricity result shows that, to within isomorphism, only one structure satisfies these axioms. However, the proof of categoricity is correct for a wide class of non-full Henkin models of second-order quantification. In light of this fact, can the proof of second-order categoricity be taken to establish that the second-order axioms of arithmetic characterize a unique structure?

In this talk, I will describe how the eigenvalues of the Atkin operator on overconvergent modular forms might be related to the classical study of the Laplacian on certain manifolds. The goal is to phrase everything geometrically, so as to maximally engage the audience in discussion on possible approaches to study the spectral flow of this operator.

The ocean is populated by an intense geostrophic eddy field with a dominant energy-containing scale on the order of 100 km at midlatitudes. Ocean climate models are unlikely routinely to resolve geostrophic eddies for the foreseeable future and thus development and validation of improved parameterisations is a vital task. Moreover, development and validation of improved eddy parameterizations is an excellent strategy for testing and advancing our understanding of how geostrophic ocean eddies impact the large-scale circulation.

A new mathematical framework for parameterising ocean eddy fluxes is developed that is consistent with conservation of energy and momentum while retaining the symmetries of the original eddy fluxes. The framework involves rewriting the residual-mean eddy force, or equivalently the eddy potential vorticity flux, as the divergence of an eddy stress tensor. A norm of this tensor is bounded by the eddy energy, allowing the components of the stress tensor to be rewritten in terms of the eddy energy and non-dimensional parameters describing the mean "shape" of the eddies. If a prognostic equation is solved for the eddy energy, the remaining unknowns are non-dimensional and bounded in magnitude by unity. Moreover, these non-dimensional geometric parameters have strong connections with classical stability theory. For example, it is shown that the new framework preserves the functional form of the Eady growth rate for linear instability, as well as an analogue of Arnold's first stability theorem. Future work to develop a full parameterisation of ocean eddies will be discussed.

Sparse matrix factorization involves a mix of regular and irregular computation, which is a particular challenge when trying to obtain high-performance on the highly parallel general-purpose computing cores available on graphics processing units (GPUs). We present a sparse multifrontal QR factorization method that meets this challenge, and is up to ten times faster than a highly optimized method on a multicore CPU. Our method is unique compared with prior methods, since it factorizes many frontal matrices in parallel, and keeps all the data transmitted between frontal matrices on the GPU. A novel bucket scheduler algorithm extends the communication-avoiding QR factorization for dense matrices, by exploiting more parallelism and by exploiting the staircase form present in the frontal matrices of a sparse multifrontal method.

This is joint work with Nuri Yeralan and Sanjay Ranka.

Dislocations are line defects in crystals, and were first posited as the carriers of plastic flow in crystals in the 1934 papers of Orowan, Polanyi and Taylor. Their hypothesis has since been experimentally verified, but many details of their behaviour remain unknown. In this talk, I present joint work with Christoph Ortner on an infinite lattice model in which screw dislocations are free to be created and annihilated. We show that configurations containing single geometrically necessary dislocations exist as global minimisers of a variational problem, and hence are globally stable equilibria amongst all finite energy perturbations.

A bit more than ten years ago, Peter Oszváth and Zoltán Szabó defined Heegaard-Floer homology, a gauge theory inspired invariant of three-manifolds that is designed to be more computable than its cousins, the Donaldson and Seiberg-Witten invariants for four-manifolds. This invariant is defined in terms of a Heegaard splitting of the three-manifold. In this talk I will show how Heegaard-Floer homology is defined (modulo the analysis that goes into it) and explain some of the directions in which people have taken this theory, such as knot theory and fitting Heegaard-Floer homology into the scheme of topological field theories.

A large class of links in $S^3$ has the property that the complement admits a complete hyperbolic metric of finite volume. But is this volume understandable from the link itself, or maybe from some nice diagram of it? Marc Lackenby in the early 2000s gave a positive answer for a class of diagrams, the alternating ones. The proof of this result involves an analysis of the JSJ decomposition of the link complement: in particular of how does it appear on the link diagram. I will tell you an outline of this proof, forgetting its most technical aspects and explaining the underlying ideas in an accessible way.

I will review several known problems on the automorphism group of finite $p$-groups and present a sketch of the proof of the the following result obtained jointly with Jon Gonz\'alez-S\'anchez:

For each prime $p$ we construct a family $\{G_i\}$ of finite $p$-groups such that $|Aut (G_i)|/|G_i|$ goes to $0$, as $i$ goes to infinity. This disproves a well-known conjecture that $|G|$ divides $|Aut(G)|$ for every non-abelian finite $p$-group $G$.

The Contou-Carrère symbol has been introduced in the 90's in the study of local analogues of autoduality of Jacobians of smooth projective curves. It is closely related to the tame symbol, the residue pairing, and the canonical central extension of loop groups. In this talk we will a discuss a K-theoretic interpretation of the Contou-Carrère symbol, which allows us to generalize this one-dimensional picture to higher dimensions. This will be achieved by studying the K-theory of Tate objects, giving rise to natural central extensions of higher loop groups by spectra. Using the K-theoretic viewpoint, we then go on to prove a reciprocity law for higher-dimensional Contou-Carrère symbols. This is joint work with O. Braunling and J. Wolfson.

Crouzeix's conjecture is an exasperating problem of linear algebra that has been open since 2004: the norm of p(A) is bounded by twice the maximum value of p on the field of values of A, where A is a square matrix and p is a polynomial (or more generally an analytic function).  I'll say a few words about the conjecture and
show the beautiful proof of Pearcy in 1966 of a special case, based on a vector-valued barycentric interpolation formula.

The Tutte polynomial of a graph $G$ is a two-variable polynomial $T(G;x,y)$, which encodes much information about~$G$. The number of spanning trees in~$G$, the number of acyclic orientations of~$G$, and the partition function of the $q$-state Potts model are all specialisations of the Tutte polynomial. Jackson and Sokal have studied the sign of the Tutte polynomial, and identified regions in the $(x,y)$-plane where it is ``essentially determined'', in the sense that the sign is a function of very simple characteristics of $G$, e.g., the number of vertices and connected components of~$G$. It is natural to ask whether the sign of the Tutte polynomial is hard to compute outside of the regions where it is essentially determined. We show that the answer to this question is often an emphatic ``yes'': specifically, that determining the sign is \#P-hard. In such cases, approximating the Tutte polynomial with small relative error is also \#P-hard, since in particular the sign must be determined. In the other direction, we can ask whether the Tutte polynomial is easy to approximate in regions where the sign is essentially determined. The answer is not straightforward, but there is evidence that it often ``no''. This is joint work with Leslie Ann Goldberg (Oxford).

I will explain some ongoing work on understanding algebraic D-moldules via their reduction to positive characteristic. I will define the p-cycle of an algebraic D-module, explain the general results of Bitoun and Van Den Bergh; and then discuss a new construction of a class of algebraic D-modules with prescribed p-cycle.

Data assimilation is a particular form of state estimation. That's partly the "what". We'll also look at the how's, the why's, some who's and some where's.

The talk will recall the results of three preprints, first two authored by my former student Mickael Launay, and the final coauthored by Mickael and myself. All three works are available on arXiv. At this point it is not clear that they will ever get published (or submitted for review) but hopefully this does not make their contents less interesting. This class of interacting urn processes was introduced in Launay's thesis, in an attempt to model more realistically the memory sharing that occurs in food trail pheromone marking or in similar collective learning phenomena. An interesting critical behavior occurs already in the case of exponential reinforcement. No prior knowledge of strong urns will be assumed, and I will try to explain the reason behind the phase transition.

The coalescing Brownian flow on $\R$ is a process which was introduced by Arratia (1979) and Toth and Werner (1997), and which formally corresponds to starting coalescing Brownian motions from every space-time point. We provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks. The invariance principle holds under a finite variance assumption and is thus optimal. In a series of previous works, this question was studied under a different topology, and a moment of order $3-\eps$ was necessary for the convergence to hold. Our proof relies crucially on recent work of Schramm and Smirnov on scaling limits of critical percolation in the plane. Our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work -- in particular similar results are obtained in the case of coalescing Brownian motions on the Sierpinski gasket. This is the first such result where the limiting paths do not enjoy the non-crossing property.

Joint work with Christophe Garban (Lyon) and Arnab Sen (Minnesota).

This is the first of a series of talks based on Gary 
Gruenhage's 'A survey of D-spaces' [1]. A space is D if for every 
neighbourhood assignment we can choose a closed discrete set of points 
whose assigned neighbourhoods cover the space. The mention of 
neighbourhood assignments and a topological notion of smallness (that 
is, of being closed and discrete) is peculiar among covering properties. 
Despite being introduced in the 70's, we still don't know whether a 
Lindelöf or a paracompact space must be D. In this talk, we will examine 
some elementary properties of this class via extent and Lindelöf numbers.

I will explain a new formulation of Einstein’s equations in 4-dimensions using the language of gauge theory. This was also discovered independently, and with advances, by Kirill Krasnov. I will discuss the advantages and disadvantages of this new point of view over the traditional "Einstein-Hilbert" description of Einstein manifolds. In particular, it leads to natural "sphere conjectures" and also suggests ways to find new Einstein 4-manifolds. I will describe some first steps in these directions. Time permitting, I will explain how this set-up can also be seen via 6-dimensional symplectic topology and the additional benefits that brings.

I will discuss two topics. Firstly, coupling of the circadian clock and cell cycle in mammalian cells. Together with the labs of Franck Delaunay (Nice) and Bert van der Horst (Rotterdam) we have developed a pipeline involving experimental and mathematical tools that enables us to track through time the phase of the circadian clock and cell cycle in the same single cell and to extend this to whole lineages. We show that for mouse fibroblast cell cultures under natural conditions, the clock and cell cycle phase-lock in a 1:1 fashion. We show that certain perturbations knock this coupled system onto another periodic state, phase-locked but with a different winding number. We use this understanding to explain previous results. Thus our study unravels novel phase dynamics of 2 key mammalian biological oscillators. Secondly, I present a radical revision of the Nrf2 signalling system. Stress responsive signalling coordinated by Nrf2 provides an adaptive response for protection against toxic insults, oxidative stress and metabolic dysfunction. We discover that the system is an autonomous oscillator that regulates its target genes in a novel way.