Past

Following last week's talk on Beilinson-Bernstein localisation theorem, we give basic notions in deformation quantisation explaining how this theorem can be interpreted as a quantised version of the Springer resolution. Having attended last week's talk will be useful but not necessary.

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.