Past

This talk will give a description of the period ring B_dR of Fontaine, which uses de Rham algebra computations. 

This talk is part of the workshop on Beilinson's approach to p-adic Hodge  theory.

Markit is a leading global provider of financial information services. We provide products that enhance transparency, reduce risk and improve operational efficiency.

We wish to find ways to automatically detect and label ‘extreme’ occurrences in a time series such as structural breaks, nonlinearities, and spikes (i.e. outliers). We hope to detect these occurrences in the levels, returns and volatility of a time series or any other transformation of it (e.g. moving average).

We also want to look for the same types of occurrences in the multivariate case in a set of time series through measures such as e.g. correlations, eigenvalues of the covariance matrix etc. The number of time series involved is of the order 3x10^6.

We wish to explain the appearance of an ‘extreme’ occurrence or a cluster of occurrences endogenously, as an event conditional on the values of the time series in the set, both contemporaneously and/or as conditional on their time lags.

Furthermore, we would like to classify the events that caused the occurrence in some major categories, if found e.g. shock to oil supply, general risk aversion, migrations etc. both algorithmically and by allowing human corrective judgement (which could become the basis for supervised learning).

*/ /*-->*/ A {\em monoid} is a semigroup with identity. A {\em finitary property for monoids} is a property guaranteed to be satisfied by any finite monoid. A good example is the maximal condition on the lattice of right ideals: if a monoid satisfies this condition we say it is {\em weakly right noetherian}. A monoid $S$ may be represented via mappings of sets or, equivalently and more concretely, by {\em (right) $S$-acts}. Here an $S$-act is a set $A$ together with a map $A\times S\rightarrow A$ where $(a,s)\mapsto as$, such that

for all $a\in A$ and $s,t\in S$ we have $a1=a$ and $(as)t=a(st)$. I will be speaking about finitary properties for monoids arising from model theoretic considerations for $S$-acts.

Let $S$ be a monoid and let $L_S$ be the first-order language of $S$-acts, so that $L_S$ has no constant or relational symbols (other than $=$) and a unary function symbol $\rho_s$ for each $s\in S$. Clearly $\Sigma_S$ axiomatises the class of $S$-acts, where

\[\Sigma_S=\big\{ (\forall x)(x\rho_s \rho_t=x\rho_{st}):s,t\in S\big\}\cup\{ (\forall x)(x\rho_1=x)

\}.\]

Model theory tells us that $\Sigma_S$

has a model companion $\Sigma_S^*$ precisely when the class

${\mathcal E}$ of existentially closed $S$-acts is axiomatisable and

in this case, $\Sigma_S^*$ axiomatises ${\mathcal E}$. An old result of Wheeler tells us that $\Sigma_S^*$ exists if and only if for every finitely generated right congruence $\mu$ on $S$, every finitely generated $S$-subact of $S/\mu$ is finitely presented, that is, $S$ is {\em right coherent}. Interest in right coherency also arises from other considerations such as {\em purity} for $S$-acts.

Until recently, little was known about right coherent monoids and, in particular, whether free monoids are (right) coherent.

I will present some work of Gould, Hartmann and Ru\v{s}kuc in this direction: specifically we answer positively the question for free monoids.

Where $\Sigma_S^*$ exists, it is known to be

stable, and is superstable if and only if $S$ is weakly right noetherian.

By using an algebraic description of types over $\Sigma_S^*$ developed in the 1980s by Fountain and Gould,

we can show that $\Sigma_S^*$ is totally

transcendental if and only if $S$ is weakly right noetherian and $S$ is {\em ranked}. The latter condition says that every right congruence possesses a finite Cantor-Bendixon rank with respect to the {\em finite type topology}.

Our results show that there is a totally transcendental theory of $S$-acts for which Morley rank of types does not coincide with $U$-rank, contrasting with the corresponding situation for modules over a ring.

Donaldson-Thomas theory was born as a mean to attach to Calabi-Yau 3-manifolds integers, invariant under small deformation of the complex structure. Subsequent evolutions have replaced integers with cohomological invariants, more flexible and with a broader range of applicable cases.

This talk is meant to be a gentle induction to the topic. We start with an introduction on virtual fundamental classes, and how they relate to deformation and obstruction spaces of a moduli space; then we pass on to the Calabi-Yau 3-dimensional case, stressing how some homological conditions are essential and can lead to generalisation. First we describe the global construction using virtual fundamental classes, then the local approach via the Behrend function and the virtual Euler characteristic.
We introduce quivers with potential, which provide a profitable framework in which to build DT-theory, as they are a source of moduli spaces locally presented as degeneracy loci. Finally, we overview the problem of categorification, introducing the DT-sheaf and showing how it relates to the numerical invariants.

When considering $E_k$ numbers (products of exactly $k$ primes), it is natural to ask, how they are distributed in short intervals. One can show much stronger results when one restricts to almost all intervals. In this context,  we seek the smallest value of c such that the intervals $[x,x+(\log x)^c]$ contain an $E_k$ number almost always. Harman showed that $c=7+\varepsilon$ is admissible for $E_2$ numbers, and this was the best known result also for $E_k$ numbers with $k>2$.

We show that for $E_3$ numbers one can take $c=1+\varepsilon$, which is optimal up to $\varepsilon$. We also obtain the value $c=3.51$ for $E_2$ numbers. The proof uses pointwise, large values and mean value results for Dirichlet polynomials as well as sieve methods.

Dave Hewett: Canonical solutions in wave scattering

By a "canonical solution" I have in mind a closed-form exact solution of the scalar wave equation in a simple geometry, for example the exterior of a circular cylinder, or the exterior of an infinite wedge. In this talk I hope to convince you that the study of such problems is (a) interesting; (b) important; and (c) a rich source of (difficult) open problems involving eigenfunction expansions, special functions, the asymptotic evaluation of integrals, and matched asymptotic expansions.

Enhanced Langlands parameters for a p-adic group G are pairs formed by a Langlands parameter for G and an irreducible character of a certain component group attached to the parameter. We will first introduce a notion
of cuspidality for these pairs. The cuspidal pairs are expected to correspond to the supercuspidal irreducible representations of G via the local Langlands correspondence.
We will next describe a construction of  a cuspidal support map for enhanced Langlands parameters, the key tool of which is an extension to disconnected complex Lie groups of the generalized Springer correspondence due to Lusztig.
Finally, we will use this map to decompose  the set of enhanced Langlands parameters into Bernstein series.
This is joint work with Ahmed Moussaoui and Maarten Solleveld.

To predict the behaviour of a dynamical system using a mathematical model, an accurate estimate of the current state of the system is needed in order to initialize the model. Complete information on the current state is, however, seldom available. The aim of optimal state estimation, known in the geophysical sciences as ‘data assimilation’, is to determine a best estimate of the current state using measured observations of the real system over time, together with the model equations. The problem is commonly formulated in variational terms as a very large nonlinear least-squares optimization problem. The lack of complete data, coupled with errors in the observations and in the model, leads to a highly ill-conditioned inverse problem that is difficult to solve.

To understand the nature of the inverse problem, we examine how different components of the assimilation system influence the conditioning of the optimization problem. First we consider the case where the dynamical equations are assumed to model the real system exactly. We show, against intuition, that with increasingly dense and precise observations, the problem becomes harder to solve accurately. We then extend these results to a 'weak-constraint' form of the problem, where the model equations are assumed not to be exact, but to contain random errors. Two different, but mathematically equivalent, forms of the problem are derived. We investigate the conditioning of these two forms and find, surprisingly, that these have quite different behaviour.

We will explain a result of Bridson, Howie, Miller and Short on the finiteness properties of subgroups of direct products of surface groups. More precisely, we will show that a subgroup of a direct product of n surface groups is of finiteness type $FP_n$ if and only if there is virtually a direct product of at most n finitely generated surface groups. All relevant notions will be explained in the talk.

 

Partial metric spaces generalise metric spaces by allowing self-distance
to be a non-negative number. Originally motivated by the goal to
reconcile metric space topology with the logic of computable functions
and Dana Scott's innovative theory of topological domains they are now
too rigid a form of mathematics to be of use in modelling contemporary
applications software (aka 'Apps') which is increasingly concurrent,
pragmatic, interactive, rapidly changing, and inconsistent in nature.
This talks aims to further develop partial metric spaces in order to
catch up with the modern computer science of 'Apps'. Our illustrative
working example is that of the 'Lucid' programming language,and it's
temporal generalisation using Wadge's 'hiaton'.

Additive combinatorics enable one to characterise subsets S of elements in a group such that S+S has

small cardinality. In particular a theorem of Vosper says that subsets of integers modulo a prime p

with minimal sumsets can only be arithmetic progressions, apart from some degenerate cases. We are

interested in q-analogues of these results, namely characterising subspaces S in some algebras such

that the linear span of its square S^2 has small dimension.

Analogues of Vosper's theorem will imply that such spaces will have bases consisting of elements in

geometric progression.

We derive such analogues in extensions of finite fields, where bounds on codes in the space of

quadratic forms play a crucial role. We also obtain that under appropriately formulated conditions,

linear codes with small squares for the component-wise product can only be generalized Reed-Solomon

codes.



Based on joint works with Christine Bachoc and Oriol Serra, and with Diego Mirandola.

For a holomorphic function (“superpotential”)  W: X —> C on a complex manifold X, one defines the (2-periodic) matrix factorisation category MF(X;W), which is supported on the critical locus Crit(W) of W. At a Morse singularity, MF(X;W) is equivalent to the category of modules over the Clifford algebra on the tangent space TX. It had been suggested by Kapustin and Rozansky that, for Morse-Bott W, MF(X;W) should be equivalent to the (2-periodicised) derived category of Crit(W), twisted by the Clifford algebra of the normal bundle. I will discuss why this holds when the first neighbourhood of Crit(W) splits, why it fails in general, and will explain the correct general statement.

Many of the fastest known algorithms for factoring large integers rely on finding subsequences of randomly generated sequences of integers whose product is a perfect square. Motivated by this, in 1994 Pomerance posed the problem of determining the threshold of the event that a random sequence of N integers, each chosen uniformly from the set
{1,...,x}, contains a subsequence, the product of whose elements is a perfect square. In 1996, Pomerance gave good bounds on this threshold and also conjectured that it is sharp.

In a paper published in Annals of Mathematics in 2012, Croot, Granville, Pemantle and Tetali significantly improved these bounds, and stated a conjecture as to the location of this sharp threshold. In recent work, we have confirmed this conjecture. In my talk, I shall give a brief overview of some of the ideas used in the proof, which relies on techniques from number theory, combinatorics and stochastic processes. Joint work with Béla Bollobás and Robert Morris.

In this talk I'll give a general presentation of my recent work that a purely loxodromic Kleinian group of Hausdorff dimension<1 is a classical Schottky group. This gives a complete classification of all Kleinian groups of dimension<1. The proof uses my earlier result on the classification of Kleinian groups of sufficiently small Hausdorff dimension. This result in conjunction to another work (joint with Anderson) provides a resolution to Bers uniformization conjecture. No prior knowledge on the subject is assumed.

Singular stochastic control problems ae largely studied in literature.The standard approach is to study the associated Hamilton-Jacobi-Bellman equation (with gradient constraint). In this work, we use a different approach (BSDE:Backward stochastic differntial equation approach) to show that the optimal value is a solution to BSDE.

The advantage of our approach is that we can study this kind of singular stochastic control with path-dependent coefficients

Question: Given a smooth compact manifold $M$ without boundary, does $M$
 admit a Riemannian metric of positive scalar curvature?

 We focus on the case of spin manifolds. The spin structure, together with a
 chosen Riemannian metric, allows to construct a specific geometric
 differential operator, called Dirac operator. If the metric has positive
 scalar curvature, then 0 is not in the spectrum of this operator; this in
 turn implies that a topological invariant, the index, vanishes.

  We use a refined version, acting on sections of a bundle of modules over a
 $C^*$-algebra; and then the index takes values in the K-theory of this
 algebra. This index is the image under the Baum-Connes assembly map of a
 topological object, the K-theoretic fundamental class.

 The talk will present results of the following type:

 If $M$ has a submanifold $N$ of codimension $k$ whose Dirac operator has
 non-trivial index, what conditions imply that $M$ does not admit a metric of
 positive scalar curvature? How is this related to the Baum-Connes assembly
 map? 

 We will present previous results of Zeidler ($k=1$), Hanke-Pape-S. ($k=2$),
 Engel and new generalizations. Moreover, we will show how these results fit
 in the context of the Baum-Connes assembly maps for the manifold and the
 submanifold. 
 

We consider fully nonlinear parabolic equations of the form $du = F(t,x,u,Du,D^2 u) dt + H(x,Du) \circ dB_t,$ which can be made sense of by the Lions-Souganidis theory of stochastic viscosity solutions. I will first recall the ideas of this theory, and will discuss more recent developments (including the use of rough path theory in this context). In the second part of my talk, I will explain how in the case where $H(x,Du)=|Du|^2$, the solution $u$ may enjoy better regularity properties than the solution to the unperturbed equation, which can be measured by (a pair of) solutions to a reflected SDE. Based on joint works with P. Friz, B. Gess, P.L. Lions and P. Souganidis.

We want to discuss a collection of results around the following Question: Given a smooth compact manifold $M$ without boundary, does $M$ admit a Riemannian metric of positive scalar curvature?

We focus on the case of spin manifolds. The spin structure, together with a chosen Riemannian metric, allows to construct a specific geometric differential operator, called Dirac operator. If the metric has positive scalar curvature, then 0 is not in the spectrum of this operator; this in turn implies that a topological invariant, the index, vanishes.
 

We use a refined version, acting on sections of a bundle of modules over a $C^*$-algebra; and then the index takes values in the K-theory of this algebra. This index is the image under the Baum-Connes assembly map of a topological object, the K-theoretic fundamental class.

The talk will present results of the following type:
 
If $M$ has a submanifold $N$ of codimension $k$ whose Dirac operator has non-trivial index, what conditions imply that $M$ does not admit a metric of positive scalar curvature? How is this related to the Baum-Connes assembly map? 

We will present previous results of Zeidler ($k=1$), Hanke-Pape-S. ($k=2$), Engel and new generalizations. Moreover, we will show how these results fit in the context of the Baum-Connes assembly maps for the manifold and the submanifold. 
 

Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalization. The spectral theory of tensors was pioneered by Lim and Qi around 2005. It has numerous applications, and ties in closely with optimization and dynamical systems.  We present an introduction that emphasizes algebraic and geometric aspects

Inverse problems arise in many applications. One could solve them by adopting a Bayesian framework, to account for uncertainty which arises from our observations. The solution of an inverse problem is given by a probability distribution. Usually, efficient methods at hand to extract information from this probability distribution involves the solution of an optimization problem, where the objective function is highly nonconvex. In this talk, we explore a reformulation of inverse problems, which helps in convexifying the objective function. We also discuss a method to sample from this probability distribution.

Following the notes and an article of B. Bhatt, we shall compute the de Rham algebra of the immersion of the zero-section of affine space over Z/p^nZ.

This talk is part of the workshop on Beilinson's approach to p-adic Hodge theory.

We are entering a world where unmanned vehicles will be common. They have the potential to dramatically decrease the cost of services whilst simultaneously increasing the safety record of whole industries.

Autonomous technologies will, by their very nature, shift decision making responsibility from individual humans to technology systems. The 2010 Flash Crash showed how such systems can create rare (but not inconceivably rare) and highly destructive positive feedback loops which can severely disrupt a sector.

In the case of Unmanned Air Systems (UAS), how might similar effects obstruct the development of the Commercial UAS industry? Is it conceivable that, like the high frequency trading industry at the heart of the Flash Crash, the algorithms we provide UAS to enable autonomy could decrease the risk of small incidents whilst increasing the risk of severe accidents? And if so, what is the relationship between probability and consequence of incidents?

Quasihereditary algebras are the 'finite' version of a highest weight category, and they classically occur as blocks of the category O and as Schur algebras.

They also occur as endomorphism algebras associated to modules endowed with special filtrations. The quasihereditary algebras produced in these cases are very often strongly quasihereditary (i.e. their standard modules have projective dimension at most 1).

In this talk I will define (strongly) quasihereditary algebras, give some motivation for their study, and mention some nice strongly quasihereditary algebras found in nature.

Abstract: A hyperkähler manifold is a Riemannian manifold $(M,g)$ with three complex structures $I,J,K$ satisfying the quaternion relations, i.e. $I^2=J^2=K^2=IJK=-1$, and such that $(M,g)$ is Kähler with respect to each of them. I will describe a construction due to Kronheimer which gives such a structure on the cotangent bundle of any complex reductive group.
 

Let $L/K$ be an extension of number fields and let $J_L$ and $J_K$ be the associated groups of ideles. Using the diagonal embedding, we view $L^*$ and $K^*$ as subgroups of $J_L$ and $J_K$ respectively. The norm map $N: J_L\to  J_K$ restricts to the usual field norm $N: L^*\to K^*$ on $L^*$. Thus, if an element of $K^*$ is a norm from $L^*$, then it is a norm from $J_L$. We say that the Hasse norm principle holds for $L/K$ if the converse holds, i.e. if every element of $K^*$ which is a norm from $J_L$ is in fact a norm from $L^*$. 

The original Hasse norm theorem states that the Hasse norm principle holds for cyclic extensions. Biquadratic extensions give the smallest examples for which the Hasse norm principle can fail. One might ask, what proportion of biquadratic extensions of $K$ fail the Hasse norm principle? More generally, for an abelian group $G$, what proportion of extensions of $K$ with Galois group $G$ fail the Hasse norm principle? I will describe the finite abelian groups for which this proportion is positive. This involves counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which is achieved using tools from harmonic analysis.

This is joint work with Christopher Frei and Daniel Loughran.

CANCELLED

Surfactants are chemicals that adsorb onto the air-liquid interface and lower the surface tension there. Non-uniformities in surfactant concentration result in surface tension gradients leading to a surface shear stress, known as a Marangoni stress. This stress, if sufficiently large, can influence the flow at the interface.

Surfactants are ubiquitous in many aspects of technology and industry to control the wetting properties of liquids due to  their ability to modify surface tension. They are used in detergents, crop spraying, coating processes and oil recovery. Surfactants also occur naturally, for example in the mammalian lung. They reduce the surface tension within the liquid lining the airways, which assists in preventing the collapse of the smaller airways. In the lungs of premature infants, the quantity of surfactant produced is insufficient as the lungs are under- developed. This leads to a respiratory distress syndrome which is treated by Surfactant Replacement Therapy.

Motivated by this medical application, we theoretically investigate a model problem involving the spreading of a drop laden with an insoluble surfactant down an inclined and pre-wetted substrate.  Our focus is in understanding the mechanisms behind a “fingering” instability observed experimentally during the spreading process. High-resolution numerics reveal a multi-region asymptotic wave-like structure of the spreading droplet. Approximate solutions for each region is then derived using asymptotic analysis. In particular, a quasi-steady similarity solution is obtained for the leading edge of the droplet. A linear stability analysis of this region shows that the base state is linearly unstable to long-wavelength perturbations. The Marangoni effect is shown to be the dominant driving mechanism behind this instability at small wavenumbers. A small wavenumber stability criterion is derived and it's implication on the onset of the fingering instability will be discussed.

I will describe a method to find negatively curved structures on some groups, by manipulating metrics on piecewise hyperbolic complexes. As an example, I will prove that hyperbolic limit groups are CAT(-1).

Torsion in homology are invariants that have received increasing attention over the last twenty years, by the work of Lück, Bergeron, Venkatesh and others. While there are various vanishing results, no one has found a finitely presented group where the torsion in the first homology is exponential over a normal chain with trivial intersection. On the other hand, conjecturally, every 3-manifold group should be an example.

A group is right angled if it can be generated by a list of infinite order elements, such that every element commutes with its neighbors. Many lattices in higher rank Lie groups (like SL(n,Z), n>2) are right angled. We prove that for a right angled group, the torsion in the first homology has subexponential growth for any Farber sequence of subgroups, in particular, any chain of normal subgroups with trivial intersection. We also exhibit right angled cocompact lattices in SL(n,R) (n>2), for which the Congruence Subgroup Property is not known. This is joint work with Nik Nikolov and Tsachik Gelander.

Efficient factorization or efficient computation of class 
numbers would both suffice to break RSA.  However the talk lies more in 
computational number theory rather than in cryptography proper. We will 
address two questions: (1) How quickly can one construct a factor table 
for the numbers up to x?, and (2) How quickly can one do the same for the 
class numbers (of imaginary quadratic fields)? Somewhat surprisingly, the 
approach we describe for the second problem is motivated by the classical 
Hardy-Littlewood method.