Past

Abstract: Given a sequence of random variables X_n that converge toward a Gaussian distribution, by looking at the next terms in the asymptotic E[exp(zX_n)] = exp(z^2 / 2) (1+ ...), one can often state a principle of moderate deviations. This happens in particular for sums of dependent random variables, and in this setting, it becomes useful to develop techniques that allow to compute the precise asymptotics of exponential generating series. Thus, we shall present a method of cumulants, which gives new results for the deviations of certain observables in statistical mechanics:

- the number of triangles in a random Erdos-Renyi graph;

- and the magnetization of the one-dimensional Ising model.

Given a triangulated category A, equipped with a differential

Z/2-graded enhancement, and a triangulated oriented marked surface S, we

explain how to define a space X(S,A) which classifies systems of exact

triangles in A parametrized by the triangles of S. The space X(S,A) is

independent, up to essentially unique Morita equivalence, of the choice of

triangulation and is therefore acted upon by the mapping class group of the

surface. We can describe the space X(S,A) as a mapping space Map(F(S),A),

where F(S) is the universal differential Z/2-graded category of exact

triangles parametrized by S. It turns out that F(S) is a purely topological

variant of the Fukaya category of S. Our construction of F(S) can then be

regarded as implementing a 2-dimensional instance of Kontsevich's proposal

on localizing the Fukaya category along a singular Lagrangian spine. As we

will see, these results arise as applications of a general theory of cyclic

2-Segal spaces.

This talk is based on joint work with Mikhail Kapranov.

Market events such as order placement and order cancellation are examples of the complex and substantial flow of data that surrounds a modern financial engineer. New mathematical techniques, developed to describe the interactions of complex oscillatory systems (known as the theory of rough paths) provides new tools for analysing and describing these data streams and extracting the vital information. In this paper we illustrate how a very small number of coefficients obtained from the signature of financial data can be sufficient to classify this data for subtle underlying features and make useful predictions.

This paper presents financial examples in which we learn from data and then proceed to classify fresh streams. The classification is based on features of streams that are specified through the coordinates of the signature of the path. At a mathematical level the signature is a faithful transform of a multidimensional time series. (Ben Hambly and Terry Lyons \cite{uniqueSig}), Hao Ni and Terry Lyons \cite{NiLyons} introduced the possibility of its use to understand financial data and pointed to the potential this approach has for machine learning and prediction.

We evaluate and refine these theoretical suggestions against practical examples of interest and present a few motivating experiments which demonstrate information the signature can easily capture in a non-parametric way avoiding traditional statistical modelling of the data. In the first experiment we identify atypical market behaviour across standard 30-minute time buckets sampled from the WTI crude oil future market (NYMEX). The second and third experiments aim to characterise the market "impact" of and distinguish between parent orders generated by two different trade execution algorithms on the FTSE 100 Index futures market listed on NYSE Liffe.

We consider Fano threefolds on which SL(2,C) acts with a dense

open orbit. This is a finite list of threefolds whose classification

follows from the classical work of Mukai-Umemura and Nakano. Inside

these threefolds, there sits a Lagrangian space form given as an orbit

of SU(2). We prove this Lagrangian is non-displaceable by Hamiltonian

isotopies via computing its Floer cohomology over a field of non-zero

characteristic. The computation depends on certain counts of holomorphic

disks with boundary on the Lagrangian, which we explicitly identify.

This is joint work in progress with Jonny Evans.

 We take a look at diamond and use it to build interesting 
mathematical objects.

We construct a model for asset price in a limit order book, which captures on one hand main stylized facts of microstructure effects, and on the other hand is tractable for dealing with optimal high frequency trading by stochastic control methods. For this purpose, we introduce a model for describing the fluctuations of a tick-by-tick single asset price, based on Markov renewal process.

We consider a point process associated to the timestamps of the price jumps, and marks associated to price increments. By modeling the marks with a suitable Markov chain, we can reproduce the strong mean-reversion of price returns known as microstructure noise. Moreover, by using Markov renewal process, we can model the presence of spikes in intensity of market activity, i.e. the volatility clustering. We also provide simple parametric and nonparametric statistical procedures for the estimation of our model. We obtain closed-form formulae for the mean signature plot, and show the diffusive behavior of our model at large scale limit. We illustrate our results by numerical simulations, and find that our model is consistent with empirical data on futures Euribor and Eurostoxx. In a second part, we use a dynamic programming approach to our semi Markov model applied to the problem of optimal high frequency trading with a suitable modeling of market order flow correlated with the stock price, and taking into account in particular the adverse selection risk. We show a reduced-form for the value function of the associated control problem, and provide a convergent and computational scheme for solving the problem. Numerical tests display the shape of optimal policies for the market making problem.

This talk is based on joint works with Pietro Fodra.

The use of tensed language and the metaphor of set "formation" found in informal descriptions of the iterative conception of set are seldom taken at all seriously. This talk offers an axiomatisation of the iterative conception in a bimodal language and presents some reasons to thus take the tense more seriously than usual (although not literally).

I will introduce star products and formal connections and describe approaches to the problem of finding a trivialization of the formal Hitchin connection, using graph-theoretical computations.

The dynamics of networks of interacting systems depend intricately on the interaction topology. Dynamical implications of local topological properties such as the nodes' degrees and global topological properties such as the degree distribution have intensively been studied. Mesoscale properties, by contrast, have only recently come into the sharp focus of network science but have

rapidly developed into one of the hot topics in the field. Current questions are: can considering a mesoscale structure such as a single subgraph already allow conclusions on dynamical properties of the network as a whole? And: Can we extract implications that are independent of the embedding network? In this talk I will show that certain mesoscale subgraphs have precise and distinct

consequences for the system-level dynamics. In particular, they induce characteristic dynamical instabilities that are independent of the structure of the embedding network.

We present a block LU factorization with panel rank revealing

pivoting (block LU_PRRP), an algorithm based on strong

rank revealing QR for the panel factorization.

Block LU_PRRP is more stable than Gaussian elimination with partial

pivoting (GEPP), with a theoretical upper bound of the growth factor

of $(1+ \tau b)^{(n/ b)-1}$, where $b$ is the size of the panel used

during the block factorization, $\tau$ is a parameter of the strong

rank revealing QR factorization, and $n$ is the number of columns of

the matrix. For example, if the size of the panel is $b = 64$, and

$\tau = 2$, then $(1+2b)^{(n/b)-1} = (1.079)^{n-64} \ll 2^{n-1}$, where

$2^{n-1}$ is the upper bound of the growth factor of GEPP. Our

extensive numerical experiments show that the new factorization scheme

is as numerically stable as GEPP in practice, but it is more resistant

to some pathological cases where GEPP fails. We note that the block LU_PRRP

factorization does only $O(n^2 b)$ additional floating point operations

compared to GEPP.

Calculus of Variations for $L^{\infty}$ functionals has a successful history of 50 years, but until recently was restricted to the scalar case. Motivated by these developments, we have recently initiated the vector-valued case. In order to handle the complicated non-divergence PDE systems which arise as the analogue of the Euler-Lagrange equations, we have introduced a theory of "weak solutions" for general fully nonlinear PDE systems. This theory extends Viscosity Solutions of Crandall-Ishii-Lions to the general vector case. A central ingredient is the discovery of a vectorial notion of extremum for maps which is a vectorial substitute of the "Maximum Principle Calculus" and allows to "pass derivatives to test maps" in a duality-free fashion. In this talk we will discuss some rudimentary aspects of these recent developments.

This is joint work with Angus Macintyre. I will discuss new developments in 
our work on the model theory of adeles concerning model theoretic 
properties of adeles and related issues on adelic geometry and number theory.

I will show how to construct an infinite family of totally geodesic surfaces in the figure eight knot complement that do not remain totally geodesic under certain Dehn surgeries. If time permits, I will explain how this behaviour can be understood via the theory of quadratic forms.

We study the nonhomogeneous boundary value problem for Navier--Stokes equations of steady motion of a viscous incompressible fluid in  a plane or spatial exterior domain with multiply connected boundary. We prove that this problem has a solution for axially symmetric case (without any restrictions on fluxes, etc.)  No restriction on the size of fluxes are required. This is a joint result with K.Pileckas and R.Russo.

I will discuss what it means to compactify complex Lie groups and introduce the so-called "Wonderful Compactification" of groups having trivial centre. I will then show how the wonderful compactification of PGL(n) can be described in terms of complete collineations. Finally, I will discuss how the new perspective provided by complete collineations provides a way to construct compactifications of arbitrary semisimple groups.

One of the most classical questions of modern algebra is whether the group algebra of a torsion-free group can be embedded into a skew field. I will give a short survey about embeddability of group algebras into skew fields, matrix rings and, in general, continuous rings.

In this talk we aim to study periodic orbits on the energy levels of a symplectic magnetic flow on the two-sphere using methods from contact geometry. In particular we show that, if the energy is low enough, we either have two or infinitely many closed orbits. The second alternative holds if there exists a prime contractible periodic orbit. Finally we present some generalisations and work in progress for closed orientable surfaces of higher genus.

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.