Thu, 07 Nov 2013

17:15 - 18:15
L6

What does Dedekind’s proof of the categoricity of arithmetic with second-order induction show?

Dan Isaacson
(Oxford)
Abstract

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?

Thu, 07 Nov 2013

16:00 - 17:30
C6

Quantum ergodicity and arithmetic heat kernels

Jan Vonk
Abstract

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.

Thu, 07 Nov 2013

16:00 - 17:00
L3

A geometric framework for interpreting and parameterising ocean eddy fluxes

David Marshall
(AOPP)
Abstract

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.

Thu, 07 Nov 2013

14:00 - 15:00
Rutherford Appleton Laboratory, nr Didcot

Sparse multifrontal QR factorization on the GPU

Professor Tim Davis
(University of Florida)
Abstract

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.

Thu, 07 Nov 2013

12:00 - 13:00
L6

Existence and stability of screw dislocations in an anti-plane lattice model

Thomas Hudson
(OxPDE, University of Oxford)
Abstract

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.

Wed, 06 Nov 2013

16:00 - 17:00
C6

Introduction to Heegaard-Floer Homology

Thomas Wasserman
(Oxford)
Abstract

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.

Wed, 06 Nov 2013

10:30 - 11:30
Queen's College

Link diagrams vs. hyperbolic volume of the complement: the alternating case

Antonio de Capua
Abstract

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.

Tue, 05 Nov 2013
17:00
C5

Finite p-groups with small automorphism group

Andrei Jaikin-Zapirain
(Madrid)
Abstract

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 $G$.

Tue, 05 Nov 2013

15:45 - 16:45
L4

Delooping and reciprocity

Michael Groechenig
((Imperial College, London))
Abstract

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.

Tue, 05 Nov 2013

14:30 - 15:00
L5

Pearcy's 1966 proof and Crouzeix's conjecture

L. Nick Trefethen
(University of Oxford)
Abstract

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.

Tue, 05 Nov 2013

14:30 - 15:30
L3

The Tutte polynomial: sign and approximability

Mark Jerrum
(University of London)
Abstract

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).

Tue, 05 Nov 2013

14:00 - 15:00
L4

Cycles of algebraic D-modules in positive characteristic II.

Chris Dodd
Abstract

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.

Tue, 05 Nov 2013

14:00 - 14:30
L5

Optimal domain splitting in Chebyshev collocation

Toby Driscoll
(University of Delaware)
Abstract
Chebfun uses a simple rule, essentially a binary search, to automatically split an interval when it detects that a piecewise Chebyshev polynomial representation will be more efficient than a global one. Given the complex singularity structure of the function being approximated, one can find an optimal splitting location explicitly. It turns out that Chebfun really does get the optimal location in most cases, albeit not in the most efficient manner. In cases where the function is expensive to evaluate, such as the solution to a differential equation, it can be preferable to use Chebyshev-Padé approximation to locate the complex singularities and split accordingly. 

 

Tue, 05 Nov 2013

13:15 - 14:00
C4

Introduction to Data Assimilation

Partick Raanes
(OCIAM, Oxford)
Abstract

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.

Mon, 04 Nov 2013

15:45 - 16:45
Oxford-Man Institute

On interacting strong urns

Vlada Limic
(Universite Paris Sud)
Abstract

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.

Mon, 04 Nov 2013

14:15 - 15:15
Oxford-Man Institute

Coalescing flows: a new approach

Nathanael Berestycki
(University of Cambridge)
Abstract

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).

Mon, 04 Nov 2013
14:00
C6

D-spaces: (1) Extent and Lindelöf numbers

Robert Leek
(Oxford)
Abstract

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.