Past

In this joint work with Daniela Giachetti (Rome) and Pedro J. Martinez Aparicio (Cartagena, Spain) we consider the problem

$$ - div A(x) Du = F (x, u) \; {\rm in} \; \Omega,$$

$$ u = 0 \; {\rm on} \; \partial \Omega,$$

(namely an elliptic semilinear equation with homogeneous Dirichlet boundary condition),

where the non\-linearity $F (x, u)$ is singular in $u = 0$, with a singularity of the type

$$F (x, u) = {f(x) \over u^\gamma} + g(x)$$

with $\gamma > 0$ and $f$ and $g$ non negative (which implies that also $u$ is non negative).

The main difficulty is to give a convenient definition of the solution of the problem, in particular when $\gamma > 1$. We give such a definition and prove the existence and stability of a solution, as well as its uniqueness when $F(x, u)$ is non increasing en $u$.

We also consider the homogenization problem where $\Omega$ is replaced by $\Omega^\varepsilon$, with $\Omega^\varepsilon$ obtained from $\Omega$ by removing many very

small holes in such a way that passing to the limit when $\varepsilon$ tends to zero the Dirichlet boundary condition leads to an homogenized problem where a ''strange term" $\mu u$ appears.

At the end of the 70s, Gross and Deligne conjectured that periods of geometric Hodge structures with multiplication by an abelian number field are always products of values of the gamma function at rational numbers, with exponents determined by the Hodge decomposition. I will explain a proof of an alternating variant of this conjecture for the cohomology groups of smooth, projective varieties over the algebraic numbers acted upon by a finite order automorphism.

We consider three nonparametric hypothesis testing problems: (1) Given samples from distributions p and q, a homogeneity test determines whether to accept or reject p=q; (2) Given a joint distribution p_xy over random variables x and y, an independence test investigates whether p_xy = p_x p_y, (3) Given a joint distribution over several variables, we may test for whether there exist a factorization (e.g., P_xyz = P_xyP_z, or for the case of total independence, P_xyz=P_xP_yP_z).

We present nonparametric tests for the three cases above, based on distances between embeddings of probability measures to reproducing kernel Hilbert spaces (RKHS), which constitute the test statistics (eg for independence, the distance is between the embedding of the joint, and that of the product of the marginals). The tests benefit from years of machine research on kernels for various domains, and thus apply to distributions on high dimensional vectors, images, strings, graphs, groups, and semigroups, among others. The energy distance and distance covariance statistics are also shown to fall within the RKHS family, when semimetrics of negative type are used. The final test (3) is of particular interest, as it may be used in detecting cases where two independent causes individually have weak influence on a third dependent variable, but their combined effect has a strong influence, even when these variables have high dimension.

An invariant random subgroup in a (finitely generated) group is a

probability measure on the space of subgroups of the group invariant under

the inner automorphisms of the group. It is a natural generalization of the

the notion of normal subgroup. I’ll give an introduction into this actively

developing subject and then discuss in more detail examples of invariant

random subgrous in groups of intermediate growth. The last part of the talk

will be based on a recent joint work with Mustafa Benli and Rostislav

Grigorchuk.

 I will discuss joint work with Daniel Greb and Matei Toma in which we introduce a notion of Gieseker-stability that depends on several polarisations.  We use this to study the change in the moduli space of Giesker semistable sheaves on manifolds giving new results in dimensions at least three, and to study the notion of Gieseker-semistability for sheaves taken with respect to an irrational Kahler class.

We will introduce a particle system interacting through a mean-field term that models the behavior of a network of excitatory neurons. The novel feature of the system is that the it features a threshold dynamic: when a single particle reaches a threshold, it is reset while all the others receive an instantaneous kick. We show that in the limit when the size of the system becomes infinite, the resulting non-standard equation of McKean Vlasov type has a solution that may exhibit a blow-up phenomenon depending on the strength of the interaction, whereby a single particle reaching the threshold may cause a macroscopic cascade. We moreover show that the particle system does indeed exhibit propagation of chaos, and propose a new way to give sense to a solution after a blow-up.

This is based on joint research with F. Delarue (Nice), E. Tanré (INRIA) and S. Rubenthaler (Nice).

The classical separate treatments of competition and predation, and an inability to provide a sensible theoretical basis for mutualism, attests to the inability of traditional models to provide a synthesising framework to study trophic interactions, a fundamental component of ecology. Recent approaches to food web modelling have focused on consumer-resource interactions. We develop this approach to explicitly represent finite resources for each population and construct a rigorous unifying theoretical framework with Lotka-Volterra Conservative Normal (LVCN) systems. We show that mixotrophy, a ubiquitous trophic interaction in marine plankton, provides the key to developing a synthesis of the various ways of making a living. The LVCN framework also facilitates an explicit redefinition of facultative mutualism, illuminating the over-simplification of the traditional definition.

We demonstrate a continuum between trophic interactions and show that populations can continuously and smoothly evolve through most population interactions without losing stable coexistence. This provides a theoretical basis consistent with the evolution of trophic interactions from autotrophy through mixotrophy/mutualism to heterotrophy.

This seminar has been cancelled

TBA

NbTi-based superconducting wires have widespread use in engineering applications of superconductivity such as MRI and accelerator magnets. Tolerance to the effects of interactions with changing (external) magnetic fields is an important consideration in wire design, in order to make the most efficient use of the superconducting material. This project aims to develop robust analytical models of the power dissipation in real conductor geometries across a broad frequency range of external field changes, with a view to developing wire designs that minimise these effects.

I'll sketch a construction which associates a canonical p-adic L-function with a 'non-critically refined' cohomological cuspidal automorphic representation of GL(2) over an arbitrary number field F, generalizing and unifying previous results of many authors. These p-adic L-functions have good interpolation and growth properties, and they vary analytically over eigenvarieties. When F=Q this reduces to a construction of Pollack and Stevens. I'll also explain where this fits in the general picture of Iwasawa theory, and I'll point towards the iceberg of which this construction is the tip.

A sharp increase in the popularity of commodity investing in the past decade has triggered an unprecedented inflow of institutional funds into commodity futures markets. Such financialization of commodities coincided with significant booms and busts in commodity markets, raising concerns of policymakers. In this paper, we explore the effects of financialization in a model that features institutional investors alongside traditional futures markets participants. The institutional investors care about their performance relative to a commodity index. We find that if a commodity futures is included in the index, supply and demand shocks specific to that commodity spill over to all other commodity futures markets. In contrast, supply and demand shocks to a nonindex commodity affect just that commodity market alone. Moreover, prices and volatilities of all commodity futures go up, but more so for the index futures than for nonindex ones. Furthermore, financialization — the presence of institutional investors — leads to an increase in correlations amongst commodity futures as well as in equity-commodity correlations. Consistent with empirical evidence, the increases in the correlations between index commodities exceed those for nonindex ones. We model explicitly demand shocks which allows us to disentangle the effects of financialization from the effects of demand and supply (fundamentals). We perform a simple calibration and find that financialization accounts for 11% to 17% of commodity futures prices and the rest is attributable to fundamentals.

Several experimental studies have shown that the abundance distributions of proteins in a population of isogenic cells may display multiple distinct maxima. Each of these maxima may be associated with a subpopulation of a particular phenotype, the quantification of which is important for understanding cellular decision-making. I will present a novel methodology which allows us to quantify multi-modal gene expression distributions and single cell power spectra in gene regulatory networks. The method is based on an extension of the linear noise approximation; in particular we rigorously show that, in the limit of slow promoter dynamics, these distributions can be systematically approximated as a mixture of Gaussian components. The resulting closed-form approximation provides a practical tool for studying complex nonlinear gene regulatory networks that have thus far been amenable only to stochastic simulation. I will demonstrate the applicability of our approach to several examples and discuss some new dynamical characteristics e.g., how the interplay of transcriptional and translational regulation can be exploited to control the multimodality of gene expression distributions in two-promoter networks and how genetic oscillators can display concerted noise-induced bimodality and noise-induced oscillations.

This talk will be an introduction to the notion of D-modules on

prestacks. We will begin by discussing Grothendieck's definition of

crystals of quasi-coherent sheaves on a smooth scheme X, and briefly

indicate how the category of such objects is equivalent to that of

modules over the sheaf of differential operators on X. We will then

explain what we mean by a prestack and define the category of

quasi-coherent sheaves on them. Finally, we consider how the

crystalline approach may be used to give a suitable generalization

of D-modules to this derived setting.

Computer simulation of the propagation and interaction of linear waves
is a core task in computational science and engineering.
The finite element method represents one of the most common
discretisation techniques for Helmholtz and Maxwell's equations, which
model time-harmonic acoustic and electromagnetic wave scattering.
At medium and high frequencies, resolution requirements and the
so-called pollution effect entail an excessive computational effort
and prevent standard finite element schemes from an effective use.
The wave-based Trefftz methods offer a possible way to deal with this
problem: trial and test functions are special solutions of the
underlying PDE inside each element, thus the information about the
frequency is directly incorporated in the discrete spaces.

This talk is concerned with a family of those methods: the so-called
Trefftz-discontinuous Galerkin (TDG) methods, which include the
well-known ultraweak variational formulation (UWVF).
We derive a general formulation of the TDG method for Helmholtz
impedance boundary value problems and we discuss its well-posedness
and quasi-optimality.
A complete theory for the (a priori) h- and p-convergence for plane
and circular/spherical wave finite element spaces has been developed,
relying on new best approximation estimates for the considered
discrete spaces.
In the two-dimensional case, on meshes with very general element
shapes geometrically graded towards domain corners, we prove
exponential convergence of the discrete solution in terms of number of
unknowns.

This is a joint work with Ralf Hiptmair, Christoph Schwab (ETH Zurich,
Switzerland) and Ilaria Perugia (Vienna, Austria).

As the title says, in this talk I will be giving a casual introduction to higher categories. I will begin by introducing strict n-categories and look closely at the resulting structure for n=2. After discussing why this turns out to be an unsatisfying definition I will discuss in what ways it can be weakened. Broadly there are two main classes of models for weak n-categories: algebraic and geometric. The differences between these two classes will be demonstrated by looking at bicategories on the algebraic side and quasicategories on the geometric.

This is a report on joint work (still in progress) with Ellen Eischen, Jian-Shu Li,
and Chris Skinner.  I will describe the general structure of our construction of p-adic L-functions
attached to families of ordinary holomorphic modular forms on Shimura varieties attached to
unitary groups.  The complex L-function is studied by means of the doubling method;
its p-adic interpolation applies adelic representation theory to Ellen Eischen's Eisenstein 
measure.

In this talk we aim to introduce the key ideas of homotopy type theory and show how it draws on and has applications to the areas of logic, higher category theory, and homotopy theory. We will discuss how types can be viewed both as propositions (statements about mathematics) as well as spaces (mathematical objects themselves). In particular we will define identity types and explore their groupoid-like structure; we will also discuss the notion of equivalence of types, introduce the Univalence Axiom, and consider some of its implications. Time permitting, we will discuss inductive types and show how they can be used to define types corresponding to specific topological spaces (e.g. spheres or more generally CW complexes).\\

This talk will assume no prior knowledge of type theory; however, some very basic background in category theory (e.g. the definition of a category) and homotopy theory (e.g. the definition of a homotopy) will be assumed.