We purify and generalize some techniques which were successful in the limit theory of graphs and other discrete structures. We demonstrate how this technique can be used for solving different combinatorial problems, by defining the limit problems of the Manickam--Miklós--Singhi Conjecture, the Kikuta–Ruckle Conjecture and Alpern's Caching Game.
we study a new mechanism of reaction-diffusion involving a line with fast diffusion, proposed to model the influence of transportation networks on biological invasions.
We will be interested in the existence and uniqueness of traveling waves solutions, and especially focus on their velocity. We will show that it grows as the square root of the diffusivity on the line, generalizing and showing the robustness of a result by Berestycki, Roquejoffre and Rossi (2013), and provide a characterization of the growth ratio thanks to an hypoelliptic (a priori) degenerate system.
Finally we will take a look at the dynamics and show that the waves attract a large class of initial data. In particular, we will shed light on a new mechanism of attraction which enables the waves to attract initial data with size independent of the diffusion on the line : this is a new result, in the sense than usually, enhancement of propagation has to be paid by strengthening the assumptions on the size of the initial data for invasion to happen.
We consider families of fast-slow skew product maps of the form \begin{align*}x_{n+1} = x_n+\eps^2 a_\eps(x_n,y_n)+\eps b_\eps(x_n)v_\eps(y_n), \quad
y_{n+1} = T_\eps y_n, \end{align*} where $T_\eps$ is a family of nonuniformly expanding maps, $v_\eps$ is of mean zero and the slow variables $x_n$ lie in $\R^d$. Under an exactness assumption on $b_\eps$ (automatically satisfied in the cases $d=1$ and $b_\eps\equiv I_d$), we prove convergence of the slow variables to a limiting stochastic differential equation (SDE) as $\eps\to0$. Our results include cases where the family of fast dynamical systems
$T_\eps$ consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.Similar results are obtained also for continuous time systems \begin{align*} \dot x = \eps^2 a_\eps(x,y,\eps)+\eps b_\eps(x)v_\eps(y), \quad \dot y = g_\eps(y). \end{align*}
Here, as in classical Wong-Zakai approximation, the limiting SDE is of Stratonovich type $dX=\bar a(X)\,dt+b_0(X)\circ\,dW$ where $\bar a$ is the average of $a_0$
and $W$ is a $d$-dimensional Brownian motion.
I am going to discuss Rips' conjecture that all finitely presented groups with quadratic Dehn functions have decidable conjugacy problem.
This is a joint work with A.Yu. Olshanskii.
This talk will be about Lévy processes on compact groups - discrete or continuous - and two-dimensional analogues called pure Yang-Mills fields. The latter are indexed by reduced loops of finite length in the plane and satisfy properties analogue to independence and stationarity of increments. There is a one-to-one correspondance between Lévy processes invariant by adjunction and pure Yang-Mills fields. For Brownian motions, Yang-Mills fields stand for a rigorous version of the Euclidean Yang-Mills measure in two dimension. I shall first sketch this correspondance for Lévy processes with large jumps. Then, I will discuss two applications of an extension theorem, due to Thierry Lévy, similar to Kolmogorov extension theorem. On the one hand, it allows to construct pure Yang-Mills fields for any invariant Lévy process. On the other hand, when the group acts on vector spaces of large dimension, this theorem also allows to study the asymptotic behavior of traces. The limiting objects yield a natural family of states on the group algebra of reduced loops. We characterize among them the master field defined by Thierry Lévy by a continuity property. This is a joint work with Guillaume Cébron and Franck Gabriel.
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'.