Past

For Logic Seminar: Note change of time and place.

In this talk, I will discuss the effect of boundary conditions on the solvability of PDEs that have formally an integrable structure, in the

sense of possessing a Lax pair. Many of these PDEs arise in wave propagation phenomena, and boundary value problems for these models are very important in applications. I will discuss the extent to which general approaches that are successful for solving the initial value problem extend to the solution of boundary value problem.

I will survey the solution of specific examples of integrable PDE, linear and nonlinear. The linear theory is joint work with David Smith. For the nonlinear case, I will discuss boundary conditions that yield boundary value problems that are fully integrable, in particular recent joint results with Thanasis Fokas and Jonatan Lenells on the solution of boundary value problems for the elliptic sine-Gordon equation.

We propose a model where an algorithmic trader takes a view on the distribution of prices at a future date and then decides how to trade in the direction of her predictions using the optimal mix of market and limit orders. As time goes by, the trader learns from changes in prices and updates her predictions to tweak her strategy. Compared to a trader that cannot learn from market dynamics or form a view of the market, the algorithmic trader's profits are higher and more certain. Even though the trader executes a strategy based on a directional view, the sources of profits are both from making the spread as well as capital appreciation of inventories. Higher volatility of prices considerably impairs the trader's ability to learn from price innovations, but this adverse effect can be circumvented by learning from a collection of assets that co-move.

Many problems in the physical sciences

require the determination of an unknown

function from a finite set of indirect measurements.

Examples include oceanography, oil recovery,

water resource management and weather forecasting.

The Bayesian approach to these problems

is natural for many reasons, including the

under-determined and ill-posed nature of the inversion,

the noise in the data and the uncertainty in

the differential equation models used to describe

complex mutiscale physics. The object of interest

in the Bayesian approach is the posterior

probability distribution on the unknown field [1].

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However the Bayesian approach presents a

computationally formidable task as it

results in the need to probe a probability

measure on separable Banach space. Monte

Carlo Markov Chain methods (MCMC) may be

used to achieve this [2], but can be

prohibitively expensive. In this talk I

will discuss approximation of probability measures

by a Gaussian measure, looking for the closest

approximation with respect to the Kullback-Leibler

divergence. This methodology is widely

used in machine-learning [3]. In the context of

target measures on separable Banach space

which themselves have density with respect to

a Gaussian, I will show how to make sense of the

resulting problem in the calculus of variations [4].

Furthermore I will show how the approximate

Gaussians can be used to speed-up MCMC

sampling of the posterior distribution [5].

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[1] A.M. Stuart. "Inverse problems: a Bayesian

perspective." Acta Numerica 19(2010) and

http://arxiv.org/abs/1302.6989

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[2] S.L.Cotter, G.O.Roberts, A.M. Stuart and D. White,

"MCMC methods for functions: modifying old algorithms

to make them faster". Statistical Science 28(2013).

http://arxiv.org/abs/1202.0709

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[3] C.M. Bishop, "Pattern recognition and machine learning".

Springer, 2006.

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[4] F.J. Pinski G. Simpson A.M. Stuart H. Weber, "Kullback-Leibler

Approximations for measures on infinite dimensional spaces."

http://arxiv.org/abs/1310.7845

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[5] F.J. Pinski G. Simpson A.M. Stuart H. Weber, "Algorithms

for Kullback-Leibler approximation of probability measures in

infinite dimensions." In preparation.

(Joint work with Jochen Koenigsmann) Admitting a p-henselian
valuation is a weaker assumption on a field than admitting a henselian
valuation. Unlike henselianity, p-henselianity is an elementary property
in the language of rings. We are interested in the question when a field
admits a non-trivial 0-definable p-henselian valuation (in the language
of rings). They often then give rise to 0-definable henselian
valuations. In this talk, we will give a classification of elementary
classes of fields in which the canonical p-henselian valuation is
uniformly 0-definable. This leads to the new phenomenon of p-adically
(pre-)Euclidean fields.

 The theory of derivators is an approach to homotopical algebra
that focuses on the existence of homotopy Kan extensions. Homotopy
theories (e.g. model categories) typically give rise to derivators by
considering the homotopy categories of all diagrams categories
simultaneously. A general problem is to understand how faithfully the
derivator actually represents the homotopy theory. In this talk, I will
discuss this problem in connection with algebraic K-theory, and give a
survey of the results around the problem of recovering the K-theory of a
good Waldhausen category from the structure of the associated derivator.

The modularity theorem saying that all (semistable) elliptic curves are modular was one of the two crucial parts in the proof of Fermat's last theorem. In this talk I will explain what elliptic curves being 'modular' means and how an alternative definition can be given in terms of Galois representations. I will then state some of the conjectures of the Langlands program which in some sense generalise the modularity theorem.

The maximal subgroups of the exceptional groups of Lie type

have been studied for many years, and have many applications, for

example in permutation group theory and in generation of finite

groups. In this talk I will survey what is currently known about the

maximal subgroups of exceptional groups, and our recent work on this

topic. We explore the connection with extending morphisms from finite

groups to algebraic groups.

 “Understanding the generation and control of pattern and form is still a challenging and major problem in the biomedical sciences. I shall describe three very different problems. First I shall briefly describe the development and application of the mechanical theory of morphogenesis and the discovery of morphogenetic laws in limb development and how it was used to move evolution backwards. I shall then describe a surprisingly informative model, now used clinically, for quantifying the growth of brain tumours, enhancing imaging techniques and quantifying individual patient treatment protocols prior to their use.  Among other things, it is used to estimate patient life expectancy and explain why some patients live longer than others with the same treatment protocols. Finally I shall describe an example from the social sciences which quantifies marital interaction that is used to predict marital stability and divorce.  From a large study of newly married couples it had a 94% accuracy. I shall show how it has helped design a new scientific marital therapy which is currently used in clinical practice.”

We will start with a recollection on factorization algebras and factorization homology. We will then explain what fully extended TFTs are, after Jacob Lurie. And finally we will see how factorization homology can be turned into a fully extended TFT. This is a joint work with my student Claudia Scheimbauer.

 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 will explain how the result of Pantev-Toën-Vaquié-Vezzosi, about shifted symplectic structures on mapping stacks, can be extended to relative mapping stacks and Lagrangian structures. We will also provide applications in ordinary symplectic geometry and topological field theories.

This talk introduces a random linear model to investigate the memory bandwidth barrier effect on current shared memory computers. Based on the fact that floating-point operations can be hidden by implicit compiling techniques, the runtime for memory intensive applications can be modelled by memory reference time plus a random term. The random term due to cache conflicts, data reuse and other environmental factors is proportional to memory reference volume. Statistical techniques are used to quantify the random term and the runtime performance parameters. Numerical results based on thousands representative matrices from various applications are presented, compared, analysed and validated to confirm the proposed model. The model shows that a realistic and fair metric for performance of iterative methods and other memory intensive applications should consider the memory bandwidth capability and memory efficiency.

The Euler-Maclaurin formula is a quadrature rule based on corrections to the trapezoid rule using odd derivatives at the end-points of the function being integrated. It appears that no one has ever thought about a related function approximation that will give us the Euler-Maclaurin quadrature rule, i.e., just like we can derive Newton-Cotes quadrature by integrating polynomial approximations of the function, we investigate, what function approximation will integrate exactly to give the corresponding Euler-Maclaurin quadrature. It turns out, that the right function approximation is a combination of a trigonometric interpolant and a polynomial.

To make the method more practical, we also look at the closely related Newton-Gregory quadrature, which is very similar to the Euler-Maclaurin formula but instead of derivatives, uses finite differences. Following almost the same procedure, we find another mixed function approximation, derivative free, whose exact integration yields the Newton-Gregory quadrature rule.

Motivated by integrals of the Calculus of Variations considered in

Nonlinear Elasticity, we study mathematical models which do not fit in

the classical existence and regularity theory for elliptic and

parabolic Partial Differential Equations. We consider general

nonlinearities with non-standard p,q-growth, both in the elliptic and

in the parabolic contexts. In particular, we introduce the notion of

"variational solution/parabolic minimizer" for a class of

Cauchy-Dirichlet problems related to systems of parabolic equations.

The elliptic curve discrete logarithm problem (ECDLP) is commonly believed to be much harder than its finite field counterpart, resulting in smaller cryptography key sizes. In this talk, we review recent results suggesting that ECDLP is not as hard as previously expected in the case of composite fields.

We first recall how Semaev's summation polynomials can be used to build index calculus algorithms for elliptic curves over composite fields. These ideas due to Pierrick Gaudry and Claus Diem reduce ECDLP over composite fields to the resolution of polynomial systems of equations over the base field.

We then argue that the particular structure of these systems makes them much easier to solve than generic systems of equations. In fact, the systems involved here can be seen as natural extensions of the well-known HFE systems, and many theoretical arguments and experimental results from HFE literature can be generalized to these systems as well.

Finally, we consider the application of this heuristic analysis to a particular ECDLP index calculus algorithm due to Claus Diem. As a main consequence, we provide evidence that ECDLP can be solved in heuristic subexponential time over composite fields. We conclude the talk with concrete complexity estimates for binary curves and perspectives for furture works.

The talk is based on joint works with Jean-Charles Faugère, Timothy Hodges, Yung-Ju Huang, Ludovic Perret, Jean-Jacques Quisquater, Guénaël Renault, Jacob Schlatter, Naoyuki Shinohara, Tsuyoshi Takagi

The A-theory characteristic of a fibration is a

map to Waldhausen's algebraic K-theory of spaces which

can be regarded as a parametrized Euler characteristic of

the fibers. Regarding the classifying space of the cobordism

category as a moduli space of smooth manifolds, stable under

extensions by cobordisms, it is natural to ask whether the

A-theory characteristic can be extended to the cobordism

category. A candidate such extension was proposed by Bökstedt

and Madsen who defined an infinite loop map from the d-dimensional

cobordism category to the algebraic K-theory of BO(d). I will

discuss the connections between this map, the A-theory

characteristic and the smooth Riemann-Roch theorem of Dwyer,

Weiss and Williams.

In this talk I will discuss results on the geometry of constant mean curvature (H\neq 0) disks embedded in R^3. Among other

things I will prove radius and curvature estimates for such disks. It then follows from the radius estimate that the only complete, simply connected surface embedded in R^3 with constant mean curvature is the round sphere. This is joint work with Bill Meeks.

We will introduce a few class of generalised metrisable

properties; that is, properties that hold of all metrisable spaces that

can be used to generalise results and are in some sense 'close' to

metrisability. In particular, we will discuss Moore spaces and the

independence of the normal Moore space conjecture - Is every normal

Moore space metrisable?

The Plateau's problem, named after the Belgian physicist J. Plateau, is a classic in the calculus of variations and regards minimizing the area among all surfaces spanning a given contour. Although Plateau's original concern were $2$-dimensional surfaces in the $3$-dimensional space, generations of mathematicians have considered such problem in its generality. A successful existence theory, that of integral currents, was developed by De Giorgi in the case of hypersurfaces in the fifties and by Federer and Fleming in the general case in the sixties. When dealing with hypersurfaces, the minimizers found in this way are rather regular: the corresponding regularity theory has been the achievement of several mathematicians in the 60es, 70es and 80es (De Giorgi, Fleming, Almgren, Simons, Bombieri, Giusti, Simon among others).

In codimension higher than one, a phenomenon which is absent for hypersurfaces, namely that of branching, causes very serious problems: a famous theorem of Wirtinger and Federer shows that any holomorphic subvariety in $\mathbb C^n$ is indeed an area-minimizing current. A celebrated monograph of Almgren solved the issue at the beginning of the 80es, proving that the singular set of a general area-minimizing (integral) current has (real) codimension at least 2. However, his original (typewritten) manuscript was more than 1700 pages long. In a recent series of works with Emanuele Spadaro we have given a substantially shorter and simpler version of Almgren's theory, building upon large portions of his program but also bringing some new ideas from partial differential equations, metric analysis and metric geometry. In this talk I will try to give a feeling for the difficulties in the proof and how they can be overcome.

In the sixties Griffiths constructed a holomorphic map, known as the local period map, which relates the classification of smooth projective varieties to the associated Hodge structures. Fiorenza and Manetti have recently described it in terms of Schlessinger's deformation functors and, together with Martinengo, have started to look at it in the context of Derived Deformation Theory. In this talk we propose a rigorous way to lift such an extended version of Griffiths period map to a morphism of derived deformation functors and use this to construct a period morphism for global derived stacks.

A universal D-module of dimension n is a rule assigning to every family of smooth $n$-dimensional varieties a family of D-modules, in a compatible way. This seems like a huge amount of data, but it turns out to be entirely determined by its value over a single formal disc. We begin by recalling (or perhaps introducing) the notion of a D-module, and proceed to define the category $M_n$ of universal D-modules. Following Beilinson and Drinfeld we define the Gelfand-Kazhdan structure over a smooth variety (or family of varieties) of dimension $n$, and use it to build examples of universal D-modules and to exhibit a correspondence between $M_n$ and the category of modules over the group-scheme of continuous automorphisms of formal power series in $n$ variables

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.