Past

This is the first talk of the workshop organised by F. Brown, M. Kim and D. Rössler on Beilinson's approach to p-adic Hodge theory. 

In this talk, we shall give the definition and recall various properties of the cotangent complex, which was originally defined by L. Illusie in his monograph "Complexe cotangent et déformations" (Springer LNM 239, 1971).

Over the years, nonlinear and nonparametric models have attracted a great deal of attention. This is mainly due to the fact that most time series arising from the real-world exhibit nonlinear behavior, whereas nonparametric models, in principle, do not make strong prior assumptions about the true functional form of the underlying data generating process.

In this workshop, we will focus on the use of nonlinear and nonparametric modelling approaches for time series forecasting, and discuss the need and implications of accurate forecasts for informed policy and decision-making. Crucially, we will discuss some of the major challenges (and potential solutions) in probabilistic time series forecasting, with emphasis on: (1) Modelling in the presence of regime shifts, (2) Effect of model over-fitting on out-of-sample forecast accuracy, and, (3) Importance of using naïve benchmarks and different performance scores for model comparison. We will discuss the applications of different modelling approaches for: Macroeconomics (US GNP), Energy (electricity consumption recorded via smart meters), and Healthcare (remote detection of disease symptoms).

Given a collection F of holomorphic functions, we consider how to describe all the holomorphic functions locally definable from F. The notion of local definability of holomorphic functions was introduced by Wilkie, who gave a complete description of all functions locally definable from F in the neighbourhood of a generic point. We prove that this description is not complete anymore in the neighbourhood of non-generic points. More precisely, we produce three examples of holomorphic functions which each suggest that at least three new definable operations need to be added to Wilkie's description in order to capture local definability in its entirety. The construction illustrates the interaction between resolution of singularities and definability in the o-minimal setting. Joint work with O. Le Gal, G. Jones, J. Kirby.

Let X be a complex manifold with divisor D. I will describe a construction, which is due to Deligne, whereby given a choice of a branch of the logarithm one can canonically extend a holomorphic flat connection on the complement of the divisor X\D to a flat logarithmic connection on X.

In a recent work Andreata, Iovita, and Pilloni constructed the eigenvariety for cuspidal Siegel modular forms. This eigenvariety has the expected dimension (the genus of the Siegel forms) but it parametrizes only cuspidal forms. We explain how to generalize the construction to the non-cuspidal case. To be precise, we introduce the notion of "degree of cuspidality" and we construct an eigenvariety that parametrizes forms of a given degree of cuspidability. The dimension of these eigenvarieties depends on the degree of cuspidality we want to consider: the more non-cuspidal the forms, the smaller the dimension. This is a joint work with Riccardo Brasca.

In this talk, we will establish existence and uniqueness for a wide class of Markovian systems of backward stochastic differential equations (BSDE) with quadratic nonlinearities. This class is characterized by an abstract structural assumption on the generator, an a-priori local-boundedness property, and a locally-H\"older-continuous terminal condition. We present easily verifiable sufficient conditions for these assumptions and treat several applications, including stochastic equilibria in incomplete financial markets, stochastic differential games, and martingales on Riemannian manifolds. This is a joint work with Gordan Zitkovic.

Surface plasmons are collective electron-density oscillations at a metal-dielectric interface. In particular, highly localised surface-plasmon modes of nanometallic structures with narrow nonmetallic gaps, which enable a tuneable resonance frequency and a giant near-field enhancement, are at the heart of numerous nanophotonics applications. In this work, we elucidate the singular near-contact asymptotics of the plasmonic eigenvalue problem governing the resonant frequencies and modes of such structures. In the classical regime, valid for gap widths > 1nm, we find a generic scaling describing the redshift of the resonance frequency as the gap width is reduced, and in several prototypical dimer configurations derive explicit expressions for the plasmonic eigenvalues and eigenmodes using matched asymptotic expansions; we also derive expressions describing the resonant excitation of such modes by light based on a weak-dissipation limit. In the subnanometric ``nonlocal’’ regime, we show intuitively and by systematic analysis of the hydrodynamic Drude model that nonlocality manifests itself as a potential discontinuity, and in the near-contact limit equivalently as a widening of the gap. We thereby find the near-contact asymptotics as a renormalisation of the local asymptotics, and in particular a lower bound on plasmon frequency, scaling with the 1/4 power of the Fermi wavelength. Joint work with Vincenzo Giannini, Richard V. Craster and Stefan A. Maier. 

In his ICM address in 1983, Gromov proposed a program of classifying finitely generated groups up to quasi-isometry. One way of approaching this is by breaking a group down into simpler parts by means of a JSJ decomposition. I will give a survey of various JSJ theories and related quasi-isometric rigidity results, including recent work by Cashen and Martin.

Since the symmetric group is a finite group it’s representation theory is not too complex, however in this special case we can realise these representations in a particular nice combinatorial way using young tableaux and young symmetrizers. I will introduce these ideas and use them to describe the representation theory of Sn over the complex numbers.

In the theory of random graphs, the behaviour of the typical largest component was studied a lot. The initial results on G(n,m), the random graph on n vertices and m edges, are due to Erdős and Rényi. Recently, similar results for planar graphs were obtained by Kang and Łuczak.

In the first part of the talk, we will extend these results on the size of the largest component further to graphs embeddable on the orientable surface S_g of genus g>0 and see how the asymptotic number and properties of cubic graphs embeddable on S_g are used to obtain those results. Then we will go through the main steps necessary to obtain the asymptotic number of cubic graphs and point out the main differences to the corresponding results for planar graphs. In the end we will give a short outlook to graphs embeddable on surfaces with non-constant genus, especially which results generalise and which problems are still open.

Given a bipartite graph with m edges, how large is the set of sizes of its induced subgraphs? This question is a natural graph-theoretic generalisation of the 'multiplication table problem' of Erdős:  Erdős’s problem of estimating the number of distinct products a.b with a, b in [n] is precisely the problem under consideration when the graph in question is the complete bipartite graph K_{n,n}.

Based on joint work with J. Sahasrabudhe and I. Tomon.

We propose several optimal preconditioners for systems defined by some functions $g$ of Toeplitz matrices $T_n$. In this paper we are interested in solving $g(T_n)x=b$ by the preconditioned conjugate method or the preconditioned minimal residual method, namely in the cases when $g(T_n)$ are the analytic functions $e^{T_n}$, $\sin{T_n}$ and $\cos{T_n}$. Numerical results are given to show the effectiveness of the proposed preconditioners.

This talk will be a geophysicist's view on the emerging properties of a numerical model representing the Earth's climate and volcanic activity over the past million years.

The model contains a 2D ice sheet (Glen's Law solved with a semi-implicit scheme), an energy balance for the atmosphere and planet surface (explicit), and an ODE for the time evolution of CO2 (explicit).

The dependencies between these models generate behaviour similar to weakly coupled nonlinear oscillators.

We consider the motion of a thin liquid drop on a smooth substrate as the drop evaporates into an inert gas. Many experiments suggest that, at times close to the drop’s extinction, the drop radius scales as the square root of the time remaining until extinction. However, other experiments observe slightly different scaling laws. We use the method of matched asymptotic expansions to investigate whether this different behaviour is systematic or an artefact of experiment.

I will explain how to find an explicit embedding of the Kummer variety of a higher genus curve into projective space and discuss applications of such an embedding to the study of rational points on Jacobians of curves, as well as the original curves.

Part of the series 'What do historians of mathematics do?' 

" Over the last four centuries a huge amount of mathematics has been published.  Most of it has, however, had little or no influence.  By way of contrast, some mathematics, although unpublished in its time, has had great influence.  My hope is to illustrate this with discussion of manuscript sources and resources that have survived from Thomas Harriot (c.1560--1621), Isaac Newton (1642--1727) and Évariste Galois (1811--1832)."

We give a new characterization of the property that the elliptic measure
belongs to the infinity weight Muckenhoupt class
in terms of a Carleson measure property of bounded solutions.
This is joint work with C.Kenig, J.Pipher and T.Toro

 An "open de Rham space" refers to a moduli space of meromorphic connections on the projective line with underlying trivial bundle.  In the case where the connections have simple poles, it is well-known that these spaces exhibit hyperkähler metrics and can be realized as quiver varieties.  This story can in fact be extended to the case of higher order poles, at least in the "untwisted" case.  The "twisted" spaces, introduced by Bremer and Sage, refer to those which have normal forms diagonalizable only after passing to a ramified cover.  These spaces often arise as quotients by unipotent groups and in some low-dimensional examples one finds some well-known hyperkähler manifolds, such as the moduli of magnetic monopoles.  This is a report on ongoing work with Tamás Hausel and Dimitri Wyss.

The black hole information paradox comes about because of the classical no-hair theorems for black holes. I will discuss soft black hole hair in electrodynamics and in gravitation. Then some speculations on its relevance to the in formation paradox are presented.