Past

A new benchmark, High Performance Conjugate Gradient (HPCG), finally was introduced recently for the Top500 list and the Green500 list. This will draw more attention to performance of sparse iterative solvers on distributed supercomputers and energy efficiency of hardware and software. At the same time, this will more widely promote the concept that communications are the bottleneck of performance of iterative solvers on distributed supercomputers, here we will go a little deeper, discussing components of communications and discuss which part takes a dominate share. Also discussed are mathematics tricks to detect some metrics of an underlying supercomputer.

Nesetril and Ossona de Mendez introduced a new notion of convergence of graphs called FO convergence. This notion can be viewed as a unified notion of convergence of dense and sparse graphs. In particular, every FO convergent sequence of graphs is convergent in the sense of left convergence of dense graphs as studied by Borgs, Chayes, Lovasz, Sos, Szegedy, Vesztergombi and others, and every FO convergent sequence of graphs with bounded maximum degree is convergent in the Benjamini-Schramm sense.

FO convergent sequences of graphs can be associated with a limit object called modeling. Nesetril and Ossona de Mendez showed that every FO convergent sequence of trees with bounded depth has a modeling. We extend this result

to all FO convergent sequences of trees and discuss possibilities for further extensions.

The talk is based on a joint work with Martin Kupec and Vojtech Tuma.

We discuss the development of finite element techniques and solvers for magma dynamics computations. These are implemented within the FEniCS framework. This approach allows for user-friendly, expressive, high-level code development, but also provides access to powerful, scalable numerical solvers and a large family of finite element discretizations. The ability to easily scale codes to three dimensions with large meshes means that efficiency of the numerical algorithms is vital. We therefore describe our development and analysis of preconditioners designed specifically for finite element discretizations of equations governing magma dynamics. The preconditioners are based on Elman-Silvester-Wathen methods for the Stokes equation, and we extend these to flows with compaction.  This work is joint with Andrew Wathen and Richard Katz from the University of Oxford and Laura Alisic, John Rudge and Garth Wells from the University of Cambridge.

This talk will be a gentle introduction to the main ideas behind some of the obstructions to the Hasse principle. In particular, I'll focus on the Brauer-Manin obstruction and on the descent obstruction, and explain briefly how other types of obstructions could be constructed.

A new nematic phase has recently been discovered and characterized experimentally. It embodies a theoretical prediction made by Robert B. Meyer in 1973 on the basis of mere symmetry considerations to the effect that a nematic phase might also exist which in its ground state would acquire a 'heliconical' configuration, similar to the chiral molecular arrangement of cholesterics, but with the nematic director precessing around a cone about the optic axis. Experiments with newly synthetized materials have shown chiral heliconical equilibrium structures with characteristic pitch in the range of 1o nanometres and cone semi-amplitude of about 20 degrees. In 2001, Ivan Dozov proposed an elastic theory for such (then still speculative) phase which features a negative bend elastic constant along with a quartic correction to the nematic energy density that makes it positive definite. This lecture will present some thoughts about the possibility of describing the elastic response of twist-bend nematics within a purely quadratic gradient theory.

Abstract:In this talk we consider a continuous time random walk $X$ on $\mathbb{Z}^d$ in an environment of random conductances taking values in $[0, \infty)$. Assuming that the law of the conductances is ergodic with respect to space shifts, we present a quenched invariance principle for $X$ under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser's iteration scheme. Under the same conditions we also present a local limit theorem. For the proof some Hölder regularity of the transition density is needed, which follows from a parabolic Harnack inequality. This is joint work with J.-D. Deuschel and M. Slowik.

There are various Floer-theoretical invariants of links and 3-manifolds

which take the form of homology groups which are the E_infinity page of

spectral sequences starting from Khovanov homology. We shall discuss recent

work, joint with Raphael Zentner, and work in progress, joint with John

Baldwin and Matthew Hedden, in investigating and exploiting these spectral

sequences.

When cast in a Bayesian setting, the solution to an inverse problem is given as a distribution over the space where the quantity of interest lives. When the quantity of interest is in principle a field then the discretization is very high-dimensional. Formulating algorithms which are defined in function space yields dimension-independent algorithms, which overcome the so-called curse of dimensionality. These algorithms are still often too expensive to implement in practice but can be effectively used offline and on toy-models in order to benchmark the ability of inexpensive approximate alternatives to quantify uncertainty in very high-dimensional problems. Inspired by the recent development of pCN and other function-space samplers [1], and also the recent independent development of Riemann manifold methods [2] and stochastic Newton methods [3], we propose a class of algorithms [4,5] which combine the benefits of both, yielding various dimension-independent and likelihood-informed (DILI) sampling algorithms. These algorithms can be effective at sampling from very high-dimensional posterior distributions.

[1] S.L. Cotter, G.O. Roberts, A.M. Stuart, D. White. "MCMC methods for functions: modifying old algorithms to make them faster," Statistical Science (2013).

[2] M. Girolami, B. Calderhead. "Riemann manifold Langevin and Hamiltonian Monte Carlo methods," Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73 (2), 123–214 (2011).

[3] J. Martin, L. Wilcox, C. Burstedde, O. Ghattas. "A stochastic newton mcmc method for large-scale statistical inverse problems with application to seismic inversion," SIAM Journal on Scientific Computing 34(3), 1460–1487 (2012).

[4] K. J. H. Law. "Proposals Which Speed Up Function-Space MCMC," Journal of Computational and Applied Mathematics, in press (2013). http://dx.doi.org/10.1016/j.cam.2013.07.026

[5] T. Cui, K.J.H. Law, Y. Marzouk. Dimension-independent, likelihood- informed samplers for Bayesian inverse problems. In preparation.

I will define and describe in some details a large class of gauge theories in four dimensions. These theories admit a variational principle with the action a functional of only the gauge field. In particular, no metric appears in the Lagrangian or is used in the construction of the theory. The Euler-Lagrange equations are second order PDE's on the gauge field. When the gauge group is taken to be SO(3), a particular theory from this class can be seen to be (classically) equivalent to Einstein's General Relativity. All other points in the SO(3) theory space can be seen to describe "deformations" of General Relativity. These keep many of GR's properties intact, and may be important for quantum gravity. For larger gauge groups containing SO(3) as a subgroup, these theories can be seen to describe gravity plus Yang-Mills gauge fields, even though the associated geometry is much less understood in this case.

In this talk, we discuss an analogue of the Hermitian-Einstein equations for generalized Kaehler manifolds proposed by N. Hitchin. We explain in particular how these equations are equivalent to a notion of stability, and that there is a Kobayahsi-Hitchin-type of correspondence between solutions of these equations and stable objects. The correspondence holds even for non-Kaehler manifolds, as long as they are endowed with Gauduchon metrics (which is always the case for generalized Kaehler structures on 4-manifolds).

This is joint work with Shengda Hu and Reza Seyyedali.

We extend Kyle's (1985) model of insider trading to the case where liquidity provided

by noise traders follows a general stochastic process. Even though the level of noise

trading volatility is observable, in equilibrium, measured price impact is stochastic.

If noise trading volatility is mean-reverting, then the equilibrium price follows a

multivariate stochastic volatility `bridge' process. More private information is revealed

when volatility is higher. This is because insiders choose to optimally wait to trade

more aggressively when noise trading volatility is higher. In equilibrium, market makers

anticipate this, and adjust prices accordingly. In time series, insiders trade more

aggressively, when measured price impact is lower. Therefore, aggregate execution costs

to uninformed traders can be higher when price impact is lower

Clouds play a key role in the climate system. Driven by radiation, clouds power the hydrological cycle and global atmospheric dynamics. In addition, clouds fundamentally affect the global radiation balance by reflecting solar radiation back to space and trapping longwave radiation. The response of clouds to global warming remains poorly understood and is strongly regime dependent. In addition, anthropogenic aerosols influence clouds, altering cloud microphysics, dynamics and radiative properties. In this presentation I will review progress and limitations of our current understanding of the role of clouds in climate change and discuss the state of the art of the representation of clouds and aerosol-cloud interactions in global climate models, from (slightly) better constrained stratiform clouds to new frontiers: the investigation of anthropogenic effects on convective clouds.

I am interested in integer solutions to equations of the form $f(x)=0$ where $f$ is a transcendental, globally analytic function defined in a neighbourhood of $\infty$ in $\mathbb{R}^n \cup \{\infty\}$. These notions will be defined precisely, and clarified in the wider context of globally semi-analytic and globally subanalytic sets.

The case $n=1$ is trivial (the global assumption forces there to be only finitely many (real) zeros of $f$) and the case $n=2$, which I shall briefly discuss, is completely understood: the number of such integer zeros of modulus at most $H$ is of order $\log\log H$. I shall then go on to consider the situation in higher dimensions.

We motivate and dicuss the Beilinson Theorem for sheaves on projective spaces. Hopefully we see some examples along the way.

After an MISG there is time to reflect. I will report briefly on the follow up to two problems that we have worked on.

Crack Repair:

It has been found that thin elastically weak spray on liners stabilise walls and reduce rock blast in mining tunnels. Why? The explanation seems to be that the stress field singularity at a crack tip is strongly altered by a weak elastic filler, so cracks in the walls are less likely to extend.

Boundary Tracing:

Using known exact solutions to partial differential equations new domains can be constructed along which prescribed boundary conditions are satisfied. Most notably this technique has been used to extract a large class of new exact solutions to the non-linear Laplace Young equation (of importance in capillarity) including domains with corners and rough boundaries. The technique has also been used on Poisson's, Helmholtz, and constant curvature equation examples. The technique is one that may be useful for handling modelling problems with awkward/interesting geometry.

A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical successes in various fields ranging from image to audio processing, there have only been a few theoretical arguments supporting these evidences. In particular, sparse coding, or sparse dictionary learning, relies on a non-convex procedure whose local minima have not been fully analyzed yet. Considering a probabilistic model of sparse signals, we show that, with high probability, sparse coding admits a local minimum around the reference dictionary generating the signals. Our study takes into account the case of over-complete dictionaries and noisy signals, thus extending previous work limited to noiseless settings and/or under-complete dictionaries. The analysis we conduct is non-asymptotic and makes it possible to understand how the key quantities of the problem, such as the coherence or the level of noise, can scale with respect to the dimension of the signals, the number of atoms, the sparsity and the number of observations.

This is joint work with Rodolphe Jenatton & Francis Bach.

We will discuss TQFTs (at a basic level), then higher categorical extensions, and see how these lead naturally to the notion of Segal spaces.

Given a residually finite group, we analyse a growth function measuring the minimal index of a normal subgroup in a group which does not contain a given element. This growth (called residual finiteness growth) attempts to measure how ``efficient'' of a group is at being residually finite. We review known results about this growth, such as the existence of a Gromov-like theorem in a particular case, and explain how it naturally leads to the study of a second related growth (called intersection growth). Intersection growth measures asymptotic behaviour of the index of the intersection of all subgroups of a group that have index at most n. In this talk I will introduce these growths and give an overview of some cases and properties.

This is joint work with Ian Biringer, Khalid Bou-Rabee and Martin Kassabov.

Rabinowitz Floer homology (RFH) is the Floer theory associated to the Rabinowitz action functional. One can think of this functional as a Lagrange multiplier functional of the unperturbed action functional of classical mechanics. Its critical points are closed orbits of arbitrary period but with fixed energy.

This fixed energy problem can be transformed into a fixed period problem on an enlarged phase space. This provides a way to see RFH as a "standard" Hamiltonian Floer theory, and allows one to treat RFH on an equal footing to other related Floer theories. In this talk we explain how this is done and discuss several applications.

Joint work with Alberto Abbondandolo and Alexandru Oancea.

The antitriangular factorisation of real symmetric indefinite matrices recently proposed by Mastronardi and van Dooren has several pleasing properties. It is backward stable, preserves eigenvalues and reveals the inertia, that is, the number of positive, zero and negative eigenvalues. 

In this talk we show that the antitriangular factorization simplifies for saddle point matrices, and that solving a saddle point system in antitriangular form is equivalent to applying the well-known nullspace method. We obtain eigenvalue bounds for the saddle point matrix and discuss the role of the factorisation in preconditioning. 

We prove a generalization of Helly's theorem concerning intersections of convex sets that has an interesting voting theory interpretation. We then
consider various extensions in which compelling mathematical problems are motivated from very natural questions in the voting context.

From cryptography to the proof of Fermat's Last Theorem, elliptic curves (those curves of the form y^2 = x^3 + ax+b) are ubiquitous in modern number theory.  In particular, much activity is focused on developing techniques to discover rational points on these curves. It turns out that finding a rational point on an elliptic curve is very much like finding the proverbial needle in the haystack -- in fact, there is currently no algorithm known to completely determine the group of rational points on an arbitrary elliptic curve.

 I'll introduce the ''real'' picture of elliptic curves and discuss why the ambient real points of these curves seem to tell us little about finding rational points. I'll summarize some of the story of elliptic curves over finite and p-adic fields and tell you about how I study integral points on (hyper)elliptic curves via p-adic integration, which relies on doing a bit of p-adic linear algebra.  Time permitting, I'll also give a short demo of some code we have to carry out these algorithms in the Sage Math Cloud.

In 2000 Eliashberg-Polterovich introduced the natural notion of orderability of contact manifolds; that is, the existence of a natural partial order on the group of contactomorphisms. I will explain how one can study orderability questions using the machinery of Rabinowitz Floer homology. We establish a link between orderable and hypertight contact manifolds, and show that the Weinstein Conjecture holds (i.e. there exists a closed Reeb orbit) whenever there exists a positive (not necessarily contractible) loop of contactomorphisms.

Joint work with Peter Albers and Urs Fuchs.

JH: Water filtration systems typically involve flow along a channel with permeable walls and suction applied across the wall. In this ``cross-flow'' arrangement, clean water leaves the channel while impurities remain within it. A limiting factor for the operation of cross-flow devices is the build-up of a high concentration of particles near the wall due to the induced flow. Termed concentration polarization (CP), this effect ultimately leads to the blocking of pores within the permeable wall and the deposition of a ``cake'' on the wall surface. Here we show that, through strategic choices in the spatial variations of the channel-wall permeability, we may reduce the effects of CP by allowing diffusion to smear out any build up of particles that may occur. We demonstrate that, for certain classes of variable permeability, there exist optimal choices that maximize the flux of clean water out of a device.

\\

IvG: TBC

We show that string theories admit chiral infinite tension analogues in which only the massless parts of the spectrum survive. Geometrically they describe holomorphic maps to spaces of complex null geodesics, known as ambitwistor spaces. They have the standard critical space–time dimensions of string theory (26 in the bosonic case and 10 for the superstring). Quantization leads to the formulae for tree– level scattering amplitudes of massless particles found recently by Cachazo, He and Yuan. These representations localize the vertex operators to solutions of the same equations found by Gross and Mende to govern the behaviour of strings in the limit of high energy, fixed angle scattering. Here, localization to the scattering equations emerges naturally as a consequence of working on ambitwistor space. The worldsheet theory suggests a way to extend these amplitudes to spinor fields and to loop level. We argue that this family of string theories is a natural extension of the existing twistor string theories. 

In the 1930's E. Artin conjectured that a form over a p-adic field of degree d has a non-trivial zero whenever n>d^2. In this talk we will discuss this relatively old conjecture, focusing on recent developments concerning quartic and quintic forms.

We show that the spatial norm in any critical homogeneous Besov

space in which local existence of strong solutions to the 3-d

Navier-Stokes equations is known must become unbounded near a singularity.

In particular, the regularity of these spaces can be arbitrarily close to

-1, which is the lowest regularity of any Navier-Stokes critical space.

This extends a well-known result of Escauriaza-Seregin-Sverak (2003)

concerning the Lebesgue space $L^3$, a critical space with regularity 0

which is continuously embedded into the spaces we consider. We follow the

``critical element'' reductio ad absurdum method of Kenig-Merle based on

profile decompositions, but due to the low regularity of the spaces

considered we rely on an iterative algorithm to improve low-regularity

bounds on solutions to bounds on a part of the solution in spaces with

positive regularity. This is joint work with I. Gallagher (Paris 7) and

F. Planchon (Nice).

Abstract : The Bayesian approach is a possible way to build estimators in statistical models. It consists in attributing a probability measure -the prior- to the unknown parameters of the model. The estimator is then the posterior distribution, which is a conditional distribution given the information contained in the data.

The Bernstein-von Mises theorem in parametric models states that under mild regularity conditions, the posterior distribution for the finite-dimensional model parameter is asymptotically Gaussian with `optimal' centering and variance.

In this talk I will discuss recent advances in the understanding of posterior distributions in nonparametric models, that is when the unknown parameter is infinite-dimensional, focusing on a concept of nonparametric Bernstein-von Mises theorem.

The HOMFLY polynomial is an invariant of knots in S^3 which can be

extended to an invariant of tangles in B^3. I'll give a geometrical

description of this invariant for rational tangles, and

explain how this description extends to a more general invariant

(the lambda^k colored HOMFLY polynomial of a rational tangle). I'll then

use this description to sketch a proof of a conjecture of Gukov and Stosic

about the colored HOMFLY homology of rational knots.

Parts of this are joint work with Paul Wedrich and Mihaljo Cevic.

This talk develops some aspects of stochastic calculus via regularization for processes with values in a general Banach space B.

A new concept of quadratic variation which depends on a particular subspace is introduced.

An Itô formula and stability results for processes admitting this kind of quadratic variation are presented.

Particular interest is devoted to the case when B is the space of real continuous functions defined on [-T,0], T>0 and the process is the window process X(•) associated with a continuous real process X which, at time t, it takes into account the past of the process.

If X is a finite quadratic variation process (for instance Dirichlet, weak Dirichlet), it is possible to represent a large class of path-dependent random variable h as a real number plus a real forward integral in a semiexplicite form.

This representation result of h makes use of a functional solving an infinite dimensional partial differential equation.

This decomposition generalizes, in some cases, the Clark-Ocone formula which is true when X is the standard Brownian motion W. Some stability results will be given explicitly.

This is a joint work with Francesco Russo (ENSTA ParisTech Paris)."

We will finish presenting Nyikos' counterexample to 
Bozyakova's conjecture: If e(Y) = L(Y) for every subspace Y of X, must X 
be hereditarily D?

The height of a rational number a/b (a,b integers which are coprime) is defined as max(|a|, |b|). A rational number with small (resp. big) height is a simple (resp. complicated) number. Though the notion height is so naive, height has played a fundamental role in number theory. There are important variants of this notion. In 1983, when Faltings proved the Mordell conjecture (a conjecture formulated in 1921), he first proved the Tate conjecture for abelian varieties (it was also a great conjecture) by defining heights of abelian varieties, and then deducing Mordell conjecture from this. The height of an abelian variety tells how complicated are the numbers we need to define the abelian variety. In this talk, after these initial explanations, I will explain that this height is generalized to heights of motives. (A motive is a kind of generalisation of abelian variety.) This generalisation of height is related to open problems in number theory. If we can prove finiteness of the number of motives of bounded height, we can prove important conjectures in number theory such as general Tate conjecture and Mordell-Weil type conjectures in many cases.

We study optimal portfolio strategies in a market

where the drift is driven by an unobserved Markov chain. Information on

the state of this chain is obtained from stock prices and from expert

opinions in the form of signals at random discrete time points. We use

stochastic filtering to transform the original problem into an

optimization problem under full information where the state variable is

the filter for the Markov chain. This problem is studied with dynamic

programming techniques and with regularization arguments. Finally we

discuss a number of numerical experiments

A projectile travelling supersonically in air creates a shock wave in the shape of a cone, with the projectile at the tip of the Mach cone. When the projectile travels over an array of microphones the shock wave is detected with different times of arrival at each microphone. Given measurements of the times of arrival, we are trying to calculate the azimuth direction of travel of the projectile. We have found a solution when the speed of the projectile is known. However the solution is ambiguous, and can take one of two possible values. Therefore we are seeking a new mathematical approach to resolve the ambiguity and thus find the azimuth direction of travel.

A major desideratum in transcendental number theory is a simple sufficient condition for a given real number to be irrational, or better yet transcendental. In this talk we consider various forms such a criterion might take, and prove the existence or non-existence of them in various settings.

By using the Weil-Gel'fand-Zak transform of Faddeev's quantum dilogarithm,

Andersen and Kasheav have proposed a new state-integral model for the

Andersen--Kashaev TQFT, where the circle valued state variables live on

the edges of oriented levelled shaped triangulations. I will look at a

couple of examples which give an idea of how the theories are coupled.

The problem of hydrodynamic turbulence is reformulated as a heat flow problem along a chain of mechanical systems which describe units of fluid of smaller and smaller spatial extent. These units are macroscopic but have few degrees of freedom, and can be studied by the methods of (microscopic) non-equilibrium statistical mechanics. The fluctuations predicted by statistical mechanics correspond to the intermittency observed in turbulent flows. Specically, we obtain the formula

$$ \zeta_p = \frac{p}{3} - \frac{1}{\ln \kappa} \ln \Gamma \left( \frac{p}{3} +1 \right) $$

for the exponents of the structure functions ($\left\langle \Delta_{r}v \rangle \sim r^{\zeta_p}$). The meaning of the adjustable parameter is that when an eddy of size $r$ has decayed to eddies of size $r/\kappa$ their energies have a thermal distribution. The above formula, with $(ln \kappa)^{-1} = .32 \pm .01$ is in good agreement with experimental data. This lends support to our physical picture of turbulence, a picture which can thus also be used in related problems.