Past

I explore some new ideas on embedding problems for Brownian motion (and other Markov processes). I show how a (forward) Skorokhod embedding problem is transformed into an optimal stopping problem for the time-reversed process (Markov process in duality). This is deduced from the PDE (Variational Inequalities) interpretation of the classical results but then shown using probabilistic techniques and extended to give an n-marginal Root embedding. I also discuss briefly how to extend the approach to other embeddings such as the Azema-Yor embedding.

Formal truth theory sits between mathematical logic and philosophy. In this talk, I will try to give a partial overview of formal truth theory, from my particular perspective and research, in connection to some areas of mathematical logic.

We will talk about the Beilinson-Bernstein localization theorem, which is a major result in geometric representation theory. We will try to explain the main ideas behind the theorem and this will lead us to some geometric constructions that are used in order to produce representations. Finally we will see how the theorem is demonstrated in the specific case of the Lie algebra sl2

Two-dimensional viscous fluid flow problems come about either because of a thin gap geometry (Hele-Shaw flow) or plane symmetry (Stokes flow). Such problems can also involve free boundaries between different fluids, and much has been achieved in this area, including by many at Oxford. In this seminar I will discuss some new results in this field.

Firstly I will talk about some of the results of my PhD on contracting inviscid bubbles in Hele-Shaw flow, in particular regarding the effects of surface tension and kinetic undercooling on the free boundary. When a bubble contracts to a point, these effects are dominant, and lead to a menagerie of possible extinction shapes. This limiting problem is a generalisation of the curve shortening flow equation from the study of geometric PDEs. We are currently exploring properties of this generalised flow rule.

Secondly I will discuss current work on applying a free boundary Stokes flow model to the evolution of subglacial water channels. These channels are maintained by the balance between inward creep of ice and melting due to the flow of water. While these channels are normally modelled as circular or semicircular in cross-section, the inward creep of a viscous fluid is unstable. We look at some simplistic viscous dissipation models and the effect they have on the stability of the channel shape. Ultimately, a more realistic turbulent flow model is needed to understand the morphology of the channel walls.

The risk of a financial position is usually summarized by a risk measure.

As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In

statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile based risk

measures such as value-at-risk are elicitable. However, the coherent risk measure expected shortfall is not elicitable. Hence, it is unclear how to perform

forecast verification or comparison. We address the question whether coherent and elicitable risk measures exist (other than minus the expected value).

We show that one positive answer are expectiles, and that they play a special role amongst all elicitable law-invariant coherent risk measures.

When high-dimensional problems are concerned, not much algorithms can break the curse of dimensionality, and solve them efficiently and reliably. Among those, tensor product algorithms, which implement the idea of separation of variables for multi-index arrays (tensors), seem to be the most general and also very promising. They originated in quantum physics and chemistry and descent broadly from the density matrix renormalization group (DMRG) and matrix product states (MPS) formalisms. The same tensor formats were recently re-discovered in the numerical linear algebra (NLA) community as the tensor train (TT) format.

Algorithms developed in the quantum physics community are based on the optimisation in tensor formats, that is performed subsequently for all components of a tensor format (i.e. all sites or modes).
The DMRG/MPS schemes are very efficient but very difficult to analyse, and at the moment only local convergence results for the simplest algorithm are available. In the NLA community, a common approach is to use a classical iterative scheme (e.g. GMRES) and enforce the compression to a tensor format at every step. The formal analysis is quite straightforward, but tensor ranks of the vectors which span the Krylov subspace grow rapidly with iterations, and the methods are struggling in practice.

The first attempt to merge classical iterative algorithms and DMRG/MPS methods was made by White (2005), where the second Krylov vector is used to expand the search space on the optimisation step.
The idea proved to be useful, but the implementation was based on the fair amount of physical intuition, and the algorithm is not completely justified.

We have recently proposed the AMEn algorithm for linear systems, that also injects the gradient direction in the optimisation step, but in a way that allows to prove the global convergence of the resulted scheme. The scheme can be easily applied for the computation of the ground state --- the differences to the algorithm of S. White are emphasized in Dolgov and Savostyanov (2013).
The AMEn scheme is already acknowledged in the NLA community --- for example it was recently applied for the computation of extreme eigenstates by Kressner, Steinlechner and Uschmajew (2013), using the block-TT format proposed by in Dolgov, Khoromskij, Oseledets and Savostyanov (2014).

At the moment, AMEn algorithm was applied
 - to simulate the NMR spectra of large molecules (such as ubiquitin),
 - to solve the Fokker-Planck equation for the non-Newtonian polymeric flows,
 - to the chemical master equation describing the mesoscopic model of gene regulative networks,
 - to solve the Heisenberg model problem for a periodic spin chain.
We aim to extend this framework and the analysis to other problems of NLA: eigenproblems, time-dependent problems, high-dimensional interpolation, and matrix functions;  as well as to a wider list of high-dimensional problems.

This is a joint work with Sergey Dolgov the from Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany.

Steiner symmetrization is a very useful tool in the study of isoperimetric inequality. This is also due to the fact that the perimeter of a set is less or equal than the perimeter of its Steiner symmetral. In the same way, in the Gaussian setting,

it is well known that Ehrhard symmetrization does not increase the Gaussian perimeter. We will show characterization results for equality cases in both Steiner and Ehrhard perimeter inequalities. We will also characterize rigidity of equality cases. By rigidity, we mean the situation when all equality cases are trivially obtained by a translation of the Steiner symmetral (or, in the Gaussian setting, by a reflection of the Ehrhard symmetral). We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness, and through a fine analysis of the barycenter function

for a special class of sets. These results are obtained in collaboration with Maria Colombo, Guido De Philippis, and Francesco Maggi.

In an o-minimal expansion of the real field, while few holomorphic functions are globally definable, many may be locally definable. Wilkie conjectured that a few basic operations suffice to obtain all of them from the basic functions in the language, and proved the conjecture at generic points. However, it is false in general. Using Ax's theorem, I will explain one counterexample. However, this is not the end of the story.
This is joint work with Jones and Servi.

In some of their recent work Derbez and Wang studied volumes of representations of 3-manifold groups into the Lie groups $$Iso_e \widetilde{SL_2(\mathbb{R})} \mbox{ and }PSL(2,\mathbb{C}).$$ They computed the set of all volumes of representations for a fixed prime closed oriented 3-manifold with $$\widetilde{SL_2(\mathbb{R})}\mbox{-geometry}$$ and used this result to compute some volumes of Graph manifolds after passing to finite coverings.

In the talk I will give a brief introduction to the theory of volumes of representations and state some of Derbez' and Wang's results. Then I will prove an additivity formula for volumes of representations into $$Iso_e \widetilde{SL_2(\mathbb{R})}$$ which enables us to improve some of the results of Derbez and Wang.

The main result is to give a separable, Cech-complete, 0-dimensional Moore space that is not Scott-domain representable. This result answered questions in the literature; it is known that each complete mertrisable space is Scott-domain representable. The talk will give a history of the techniques involved.

We define the notion of orbit decidability in a general context, and descend to the case of groups to recognise it into several classical algorithmic problems. Then we shall go into the realm of free groups and shall analise this notion there, where it is related to the Whitehead problem (with many variations). After this, we shall enter the negative side finding interesting subgroups which are orbit undecidable. Finally, we shall prove a theorem connecting orbit decidability with the conjugacy problem for extensions of groups, and will derive several (positive and negative) applications to the conjugacy problem for groups.

The talk will focus on how the asymptotic behavior of the Riemann-Hilbert correspondence (and, conjecturally, the non-abelian Hodge correspondence) on a Riemann surface is controlled by certain harmonic maps from the Riemann surface to affine buildings. This is part of joint work with Katzarkov, Noll and Simpson, which revisits, from the perspective afforded by the theory of harmonic maps to buildings, the work of Gaiotto, Moore and Neitzke on spectral networks, WKB problems, BPS states and wall-crossing.

Nature and the world of human technology are full of
networks. People like to draw diagrams of networks: flow charts,
electrical circuit diagrams, signal flow diagrams, Bayesian networks,
Feynman diagrams and the like. Mathematically-minded people know that
in principle these diagrams fit into a common framework: category
theory. But we are still far from a unified theory of networks.

We discuss a new collocation-type method for numerically solving partial differential equations (PDEs) on the sphere.  The method uses radial basis function (RBF) approximations in a partition of unity framework for approximating spatial derivatives on the sphere.  High-orders of accuracy are achieved for smooth solutions, while the overall computational cost of the method scales linearly with the number of unknowns.  The discussion will be primarily limited to the transport equation and results will be presented for a few well-known test cases.  We conclude with a preliminary application to the non-linear shallow water wave equations on a rotating sphere.

We discuss some questions related to coloring the edge/vertices of randomgraphs. In particular we look at
(i) The game chromatic number;
(ii) Rainbow Matchings and Hamilton cycles;
(iii) Rainbow Connection;
(iv) Zebraic Colorings.

The classical Deligne conjecture (now a theorem with several published proofs) says that chains on the little disks operad act on Hochschild cohomology.  I'll describe a higher dimensional generalization of this result.  In fact, even in the dimension of the original Deligne conjecture the generalization has something new to say:  Hochschild chains and Hochschild cochains are the first two members of an infinite family of chain complexes associated to an arbitrary associative algebra, and there is a colored, higher genus operad which acts on these chain complexes.  The Connes differential and Gerstenhaber bracket are two of the simplest generators of the homology of this operad, and I'll show that there exist additional, independent generators.  These new generators are close cousins of Connes and Gerstenhaber which, so far as I can tell, have not been described in the literature.

A solid object placed at a liquid-gas interface causes the formation of a meniscus around it. In the case of a vertical circular cylinder, the final state of the static meniscus is well understood, from both experimental and theoretical viewpoints. Experimental investigations suggest the presence of two different power laws in the growth of the meniscus. In this talk I will introduce a theoretical model for the dynamics and show that the early-time growth of the meniscus is self-similar, in agreement with one of the experimental predictions. I will also discuss the use of a numerical solution to investigate the validity of the second power law.

In this talk, we will discuss low Weissenberg number

effects on mathematical properties of solutions for several PDEs

governing different viscoelastic fluids.

I present some recent work with Bruce Driver and Christian Litterer on rough paths 'constrained’ to lie in a d - dimensional submanifold of a Euclidean space E. We will present a natural definition for this class of rough paths and then describe the (second) order geometric calculus which arises out of this definition. The talk will conclude with more advanced applications, including a rough version of Cartan’s development map.

Trees are not just combinatorial structures: they are also

biological structures, both in the obvious way but also in the

study of evolution. Starting from DNA samples from living

species, biologists use increasingly sophisticated mathematical

techniques to reconstruct the most likely “phylogenetic tree”

describing how these species evolved from earlier ones. In their

work on this subject, they have encountered an interesting

example of an operad, which is obtained by applying a variant of

the Boardmann–Vogt “W construction” to the operad for

commutative monoids. The operations in this operad are labelled

trees of a certain sort, and it plays a universal role in the

study of stochastic processes that involve branching. It also

shows up in tropical algebra. This talk is based on work in

progress with Nina Otter [www.fair-fish.ch].

The splitting-up method is a powerful tool to solve (SP)DEs by dividing the equation into a set of simpler equations that are easier to handle. I will speak about how such splitting schemes can be derived and extended by insights from the theory of rough paths.

Finally, I will discuss numerics for real-world applications that appear in the management of risk and engineering applications like nonlinear filtering.

We explore the technique of elementary submodels to prove 
results in topology and set theory. We will in particular prove the 
delta system lemma, and Arhangelskii's result that a first countable 
Lindelof space has cardinality not exceeding continuum.

Nature and the world of human technology are full of
networks. People like to draw diagrams of networks: flow charts,
electrical circuit diagrams, signal flow diagrams, Bayesian networks,
Feynman diagrams and the like. Mathematically-minded people know that
in principle these diagrams fit into a common framework: category
theory. But we are still far from a unified theory of networks.

The aim of this lecture is to give a general introduction to

the interacting particle system and applications in finance, especially

in the pricing of American options. We survey the main techniques and

results on Snell envelope, and provide a general framework to analyse

these numerical methods. New algorithms are introduced and analysed

theoretically and numerically.

This talk will give an introduction to generalized complex geometry, where complex and symplectic structures are particular cases of the same structure, namely, a generalized complex structure. We will also talk about a sister theory, generalized complex geometry of type Bn, where generalized complex structures are defined for odd-dimensional manifolds as well as even-dimensional ones.

In this work, we want to construct the solution $(Y,Z,K)$ to the following BSDE

$$\begin{array}{l}

Y_t=\xi+\int_t^Tf(s,Y_s,Z_s)ds-\int_t^TZ_sdB_s+K_T-K_t, \quad 0\le t\le T, \\

{\mathbf E}[l(t, Y_t)]\ge 0, \quad 0\le t\le T,\\

\int_0^T{\mathbf E}[l(t, Y_t)]dK_t=0, \\

\end{array}

$$

where $x\mapsto l(t, x)$ is non-decreasing and the terminal condition $\xi$

is such that ${\mathbf E}[l(T,\xi)]\ge 0$.

This equation is different from the (classical) reflected BSDE. In particular, for a solution $(Y,Z,K)$,

we require that $K$ is deterministic. We will first study the case when $l$ is linear, and then general cases.

We also give some application to mathematical finance. This is a joint work with Philippe Briand and Romuald Elie.

In Parkinson’s disease, increased power of oscillations in firing rate has been observed throughout the cortico-basal-ganglia circuit. In

particular, the excessive oscillations in the beta range (13-30Hz) have been shown to be associated with difficulty of movement initiation. However, on the basis of experimental data alone it is difficult to determine where these oscillations are generated, due to complex and recurrent structure of the cortico-basal-ganglia-thalamic circuit. This talk will describe a mathematical model of a subset of basal-ganglia that is able to reproduce experimentally observed patterns of activity. The analysis of the model suggests where and under which conditions the beta oscillations are produced.

For Logic Seminar: Note change of time and location!

This talk will focus on extremizers for

a family of Fourier restriction inequalities on planar curves. It turns

out that, depending on whether or not a certain geometric condition

related to the curvature is satisfied, extremizing sequences of

nonnegative functions may or may not have a subsequence which converges

to an extremizer. We hope to describe the method of proof, which is of

concentration compactness flavor, in some detail. Tools include bilinear

estimates, a variational calculation, a modification of the usual

method of stationary phase and several explicit computations.

I will explain why one can symplectically embed closed symplectic manifolds (with integral symplectic form) into CPⁿ and compute the weak homotopy type of the space of all symplectic embeddings of such a symplectic manifold into CP^∞.

The classical small cancellation theory goes back to the 1950's and 1960's when the geometry of 2-complexes with a unique 0-cell was studied, i.e. the standard 2-complex of a finite presentation. D.T. Wise generalizes the Small Cancellation Theory to 2-complexes with arbitray 0-cells showing that certain classes of Small Cancellation Groups act properly discontinuously and cocompactly on CAT(0) Cube complexes and hence have codimesion 1-subgroups. To be more precise I will introduce "his" version of small Cancellation Theory and go roughly through the main ideas of his construction of the cube complex using Sageeve's famous construction. I'll try to make the ideas intuitively clear by using many pictures. The goal is to show that B(4)-T(4) and B(6)-C(7) groups act properly discontinuously and cocompactly on CAT(0) Cube complexes and if there is time to explain the difficulty of the B(6) case. The talk should be self contained. So don't worry if you have never had heard about "Small Cancellation".

In this talk, I will explain part of the programme of Gorenstein, Lyons

and Solomon (GLS) to provide a new proof of the CFSG. I will focus on

the difference between the initial notion of groups of characteristic

$2$-type (groups like Lie type groups of characteristic $2$) and the GLS

notion of groups of even type. I will then discuss work in progress

with Capdeboscq to study groups of even type and small $2$-local odd

rank. As a byproduct of the discussion, a picture of the structure of a

finite simple group of even type will emerge.