Past

Both sides of the geometric Langlands correspondence have natural Hecke
symmetries. I will explain an identification between the Hecke
symmetries on both sides via the geometric Satake equivalence. On the
abelian level it relates the topology of a variety associated to a group
and the representation category of its Langlands dual group.
 

Incomplete Cholesky factorizations are commonly used as black-box preconditioners for the iterative solution of large sparse symmetric positive definite linear systems. Traditionally, incomplete 
factorizations are obtained by dropping (i.e., replacing by zero) some entries of the factors during the factorization process. Here we consider a less common way to approximate the factors : through low-rank approximations of some off-diagonal blocks. We focus more specifically on approximation schemes that satisfy the orthogonality condition: the approximation should be orthogonal to the corresponding approximation error.

The resulting incomplete Cholesky factorizations have attractive theoretical properties. First, the underlying factorization process can be shown breakdown-free. Further, the condition number of the 
preconditioned system, that characterizes the convergence rate of standard iterative schemes, can be shown bounded as a function of the accuracy of individual approximations. Hence, such a bound can benefit from better approximations, but also from some algorithmic peculiarities. Eventually, the above results can be shown to hold for any symmetric positive definite system matrix.

On the practical side, we consider a particular variant of the preconditioner. It relies on a nested dissection ordering of unknowns to  insure an attractive memory usage and operations count. Further, it exploits in an algebraic way the low-rank structure present in system matrices that arise from PDE discretizations. A preliminary implementation of the method is compared with similar Cholesky and 
incomplete Cholesky factorizations based on dropping of individual entries.

we discuss various conjectures about the absolute Galois group G_Q  of the field Q of rational numbers and to what extent it encodes the elementary theory of Q.

We saw earlier that a subquadratic isoperimetric inequality implies a linear one. I will give examples of groups, due to Brady and Bridson, which prove that this is the only gap in the isoperimetric spectrum. 

Given a commutative ring A with ideal m, we consider the formal completion of A at m, and we ask when algebraic structures over the completion can be approximated by algebraic structures over the ring A itself. As we will see, Artin's approximation theorem tells us for which types of algebraic structures and which pairs (A,m) we can expect an affirmative answer. We will introduce some local notions from algebraic geometry, including formal and etale neighbourhoods. Then we will discuss some algebraic structures and rings arising in algebraic geometry and satisfying the conditions of the theorem, and show as a corollary how we can lift isomorphisms from formal neighbourhoods to etale neighbourhoods of varieties.

http://hottformath.wikispaces.com

The goal of this talk is to discuss a link between the Homogeneous Monge Ampere Equation in complex geometry, and a certain flow in the plane motivated by some fluid mechanics.   After discussing and motivating the Dirichlet problem for this equation I will focus to what is probably the first non-trivial case that one can consider, and prove that it is possible to understand regularity of the solution in terms of what is known as the Hele-Shaw flow in the plane. As such we get, essentially explicit, examples of boundary data for which there is no regular solution, contrary to previous expectation.  All of this is joint work with David Witt Nystrom.

Topological phases of matter exhibit Bott-like periodicity with respect to
time-reversal, charge conjugation, and spatial dimension. I will explain how
the non-commutative topology in topological phases originates very generally
from symmetry data, and how operator K-theory provides a powerful and
natural framework for studying them.

Full details are available at: http://www.stcatz.ox.ac.uk/alantayler

Cubature is the business of describing a probability measure in terms of an empirical measure sharing its support with the original measure, of small support, and with identical integrals for a class of functions (eg polynomials with degree less than k). 

Applying cubature to already discrete sets of scenarios provides a powerful tool for scenario management and summarising data.  We refer to this process as recombination. It is a feasible operation in real time and has lead to high accuracy pde solvers.

The practical complexity of this operation has changed! By a factor corresponding to the dimension of the space of polynomials. 

We discuss the algorithm and give home computed examples of nested sparse grids with only positive weights in moderate dimensions (eg degree 1-8 in dimension 7).  Positive weights have significant advantage over signed ones when available.
 

Starting with seminal work by Masur-Minsky, a lot of machinery has been
developed to study the geometry of Mapping Class Groups, and this has
lead, for example, to the proof of quasi-isometric rigidity results.
Parts of this machinery include hyperbolicity of the curve complex, the
distance formula and hierarchy paths.
As it turns out, all this can be transposed to the context of CAT(0)
cube complexes. I will explain some of the key parts of the machinery
and then I will discuss results about quasi-Lipschitz maps from
Euclidean spaces and nilpotent Lie groups into "spaces with a distance
formula".
Joint with Jason Behrstock and Mark Hagen.

We consider self-organizing systems, i.e. systems consisting of a large number of interacting entities which spontaneously coordinate and achieve a collective dynamics. Sush systems are ubiquitous in nature (flocks of birds, herds of sheep, crowds, ...). Their mathematical modeling poses a number of fascinating questions such as finding the conditions for the emergence of collective motion. In this talk, we will consider a simplified model first proposed by Vicsek and co-authors and consisting of self-propelled particles interacting through local alignment.
We will rigorously study the multiplicity and stability of its equilibria through kinetic theory methods. We will illustrate our findings by numerical simulations.

   Stéphane Mallat

   Ecole Normale Superieure

Learning functionals in high dimension requires to find sources of regularity and invariants, to reduce dimensionality. Stability to actions of diffeomorphisms is a strong property satisfied by many physical functionals and most signal classification problems. We introduce a scattering operator in a path space, calculated with iterated multiscale wavelet transforms, which is invariant to rigid movements and stable to diffeomorphism actions. It provides a Euclidean embedding of geometric distances and a representation of stationary random processes. Applications will be shown for image classification and to learn quantum chemistry energy functionals.

I will give an overview of ongoing joint work with R. Rubio and C. Tipler, in which we study the moduli problem for the Strominger system of equations. Building on the work of De la Ossa and Svanes and, independently, of Anderson, Gray and Sharpe, we construct an elliptic complex whose first cohomology group is the space of infinitesimal deformations of a solution of the strominger system. I will also discuss an intriguing link between this moduli problem and a moduli problem for holomorphic Courant algebroids over Calabi-Yau threefolds. Finally, we will see how the problem for the Strominger system embeds naturally in generalized geometry, and discuss some perspectives of this approach.

Quantum Mechanics presents a radically different perspective on physical reality compared with the world of classical physics. In particular, results such as the Bell and Kochen-Specker theorems highlight the essentially non-local and contextual nature of quantum mechanics. The rapidly developing field of quantum information seeks to exploit these non-classical features of quantum physics to transcend classical bounds on information processing tasks.

In this talk, we shall explore the rich mathematical structures underlying these results. The study of non-locality and contextuality can be expressed in a unified and generalised form in the language of sheaves or bundles, in terms of obstructions to global sections. These obstructions can, in many cases, be witnessed by cohomology invariants. There are also strong connections with logic. For example, Bell inequalities, one of the major tools of quantum information and foundations, arise systematically from logical consistency conditions.

These general mathematical characterisations of non-locality and contextuality also allow precise connections to be made with a number of seemingly unrelated topics, in classical computation, logic, and natural language semantics. By varying the semiring in which distributions are valued, the same structures and results can be recognised in databases and constraint satisfaction as in probability models arising from quantum mechanics. A rich field of contextual semantics, applicable to many of the situations where the pervasive phenomenon of contextuality arises, promises to emerge.

The entire history of mathematics in one hour, as illustrated by around 300 postage stamps featuring mathematics and mathematicians from across the world.

From Euclid to Euler, from Pythagoras to Poincaré, and from Fibonacci to the Fields Medals, all are featured in attractive, charming and sometimes bizarre stamps. No knowledge of mathematics or philately required.

Explosive volcanic eruptions often produce large amounts of ash that is transported high into the atmosphere in a turbulent buoyant plume.  The ash can be spread widely and is hazardous to aircraft causing major disruption to air traffic.  Recent events, such as the eruption of Eyjafjallajokull, Iceland, in 2010 have demonstrated the need for forecasts of ash transport to manage airspace.  However, the ash dispersion forecasts require boundary conditions to specify the rate at which ash is delivered into the atmosphere.

Models of volcanic plumes can be used to describe the transport of ash from the vent into the atmosphere.  I will show how models of volcanic plumes can be developed, building on classical fluid mechanical descriptions of turbulent plumes developed by Morton, Taylor and Turner (1956), and how these are used to determine the volcanic source conditions.  I will demonstrate the strong atmospheric controls on the buoyant plume rise.  Typically steady models are used as solutions can be obtained rapidly, but unsteadiness in the volcanic source can be important.  I'll discuss very recent work that has developed unsteady models of volcanic plumes, highlighting the mathematical analysis required to produce a well-posed mathematical description.

Abstract:

(i) We consider the modelling of freezing of fluids which contain particulates and fibres (imagine orange juice “with bits”) flowing in channels. The objective is to design optimum geometry/temperatures to accelerate freezing.

(ii) We present the challenge of setting-up a model for lithium or sodium ion stationary energy storage cells and battery packs to calculate the gravimetric and volumetric energy density of the cells and cost. Depending upon the materials, electrode content, porosity, packing electrolyte and current collectors. There is a model existing for automotive called Batpac.

(please see

http://www.cs.ox.ac.uk/seminars/CompBioPublicSeminars/ for details)

Note: joint with Philosophy of Physics.

Venue: Lecture Room, Radcliffe Humanities, ROQ.

In 1953 Roth proved that any positive density subset of the integers contains a non-trivial three term arithmetic progression. I will present a recent quantitative improvement for this theorem, give an overview of the main ideas of the proof, and discuss its relation to other recent work in the area. I will also discuss some closely related problems. 

The Dynamic Dictionary of Mathematical Functions (or DDMF, http://ddmf.msr-inria.inria.fr/) is an interactive website on special functions inspired by reference books such as the NIST Handbook of Special Functions. The originality of the DDMF is that each of its “chapters” is automatically generated from a short mathematical description of the corresponding function.

To make this possible, the DDMF focuses on so-called D-finite (or holonomic) functions, i.e., complex analytic solutions of linear ODEs with polynomial coefficients. D-finite functions include in particular most standard elementary functions (exp, log, sin, sinh, arctan...) as well as many of the classical special functions of mathematical physics (Airy functions, Bessel functions, hypergeometric functions...). A function of this class can be represented by a finite amount of data (a differential equation along with sufficiently many initial values), 
and this representation makes it possible to develop a computer algebra framework that deals with the whole class in a unified way, instead of ad hoc algorithms and code for each particular function. The DDMF attempts to put this idea into practice.

In this talk, I will present the DDMF, some of the algorithms and software libraries behind it, and ongoing projects based on similar ideas, with an emphasis on symbolic-numeric algorithms.

In 1983 Kerckhoff settled a long standing conjecture by Nielsen proving that every finite subgroup of the mapping class group of a compact surface can be realized as a group of diffeomorphisms. An important consequence of this theorem is that one can now try to study subgroups of the mapping class group taking the quotient of the surface by these groups of diffeomorphisms. In this talk we will study quotients of surfaces under the action of a finite group to find bounds on the cardinality of such a group.

Modularity is a quality function on partitions of a network which aims to identify highly clustered components. Given a graph G, the modularity of a partition of the vertex set measures the extent to which edge density is higher within parts than between parts; and the modularity q(G) of G is the maximum modularity of a partition of V(G). Knowledge of the maximum modularity of the corresponding random graph is important to determine the statistical significance of a partition in a real network. We provide bounds for the modularity of random regular graphs. Modularity is related to the Hamiltonian of the Potts model from statistical physics. This leads to interest in the modularity of lattices, which we will discuss. This is joint work with Colin McDiarmid.

Topics:

1) Marine Acoustics;

2) Air and water quality discharge and emission modelling;

3) Geospatial mapping, remote sensing and ecosystem services.

http://hottformath.wikispaces.com

The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Not all rationals between 0 and 1 occur as commuting probabilities. In fact Keith Joseph conjectured in 1977 that all limit points of the set of commuting probabilities are rational, and moreover that these limit points can only be approached from above. In this talk we'll discuss a structure theorem for commuting probabilities which roughly asserts that commuting probabilities are nearly Egyptian fractions of bounded complexity. Joseph's conjectures are corollaries.

It is well known that piecewise smooth signals are approximately sparse in a wavelet basis. However, other sparse representations are possible, such as the discrete gradient basis. It turns out that signals drawn from a random piecewise constant model have sparser representations in the discrete gradient basis than in Haar wavelets (with high probability). I will talk about this result and its implications, and also show some numerical experiments in which the use of the gradient basis improves compressive signal reconstruction.

Donaldson-Thomas invariants are fundamental deformation invariants of Calabi-Yau threefolds. We describe a recent conjecture of Oberdieck and Pandharipande which predicts that the (three variable) generating function for the Donaldson-Thomas invariants of K3xE is given by the reciprocal of the Igusa cusp form of weight 10. For each fixed K3 surface of genus g, the conjecture predicts that the corresponding (two variable) generating function is given by a particular meromorphic Jacobi form. We prove the conjecture for K3 surfaces of genus 0 and genus 1. Our computation uses a new technique which mixes motivic and toric methods.

A recurring theme in attempts to understand the quantum theory of gravity is the idea of "Gravity as the square of Yang-Mills". In recent years this idea has been met with renewed energy, principally driven by a string of discoveries uncovering intriguing and powerful identities relating gravity and gauge scattering amplitudes. In an effort to develop this program further, we explore the relationship between both the global and local symmetries of (super)gravity and those of (super) Yang-Mills theories squared. 



In the context of global symmetries we begin by giving a unified description of D=3 super-Yang-Mills theory with N=1, 2, 4, 8 supersymmeties in terms of the four division algebras: reals, complex, quaternions and octonions. On taking the product of these multiplets we obtain a set of D=3 supergravity theories with global symmetries (U-dualities) belonging to the Freudenthal magic square: “division algebras squared” = “Yang-Mills squared”! By generalising to D=3,4,6,10 we uncover a magic pyramid of Lie algebras.



We then turn our attention to local symmetries. Regarding gravity as the convolution of left and right Yang-Mills theories together with a spectator scalar field in the bi-adjoint representation, we derive in linearised approximation the gravitational symmetries of general covariance, p-form gauge invariance, local Lorentz invariance and local supersymmetry from the flat space Yang-Mills symmetries of local gauge invariance and global super-Poincaré. As a concrete example we focus on the new-minimal (12+12, N=1) off-shell version four-dimensional supergravity obtained by tensoring the off-shell (super) Yang-Mills multiplets (4+4, N =1) and (3+0, N =0).

When solving Einstein's equations with negative cosmological constant, the natural setting is that of an initial-boundary value problem. Data is specified on the timelike conformal boundary as well as on some initial spacelike (or null) hypersurface. At the PDE level, one finds that the boundary data is typically prescribed on a surface at which the equations become singular and standard energy estimates break down. I will discuss how to handle this singularity by introducing a renormalisation procedure. I will also talk about the consequences of different choices of boundary conditions for solutions of Einstein’s equations with negative cosmological constant.

We will introduce both the class of right-angled Artin groups (RAAG) and
the Nielsen realisation problem. Then we will discuss some recent progress
towards solving the problem.