Past

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.

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.