Veranstaltungen

After recalling what is an ASIP and how it can appear in dynamics, we will discuss and motivate the following joint result with Magnus Aspenberg and Tomas Persson: Consider the quadratic family \(T_a(x) = a x (1 - x)\), for \(x\) in \([0, 1]\) and parameters a in \((2,4)\). For any transversal Misiurewicz parameter b, we find a positive measure subset Omega of mixing Collet-Eckmann parameters such that for any Holder function f with nonvanishing autocorrelation for b, the functions \(f_a(T_a^{k}(1/2))\) (where \(f_a\) is a suitable normalisation of \(f\)) for the normalised Lebesgue measure on a positive measure subset of Omega (depending on \(f\)) satisfy an ASIP.

An important problem with sensor networks is that they do not provide information about the regions that are not covered by their sensors. If the sensors in a network are static, then the Alexander Duality Theorem from classic algebraic topology is sufficient to determine the coverage of a network. However, in many networks the sensors change position with time. In the case of dynamic sensor networks, one considers the covered and uncovered regions as parametrized spaces with respect to time. In this talk, I will define parametrized homology, a variant of zigzag persistent homology that measures how the homology of the level sets of the space changes as we vary the parameter. I will also present a version of the Alexander Duality Theorem in the setting of parametrized homology. This approach sheds light on the practical problem of loss of coverage within dynamic sensor networks.

n-Laplace systems with antisymmetric potential are known to govern geometric equations such as n-harmonic maps between manifolds and generalized prescribed H-surface equations. Due to the nonlinearity of the leading order n-Laplace and the criticality of the equation they are very difficult to treat. I will discuss some progress we obtained, combining stability methods by Iwaniec and nonlinear potential theory for vectorial equations by Kuusi-Mingione. Joint work with Dorian Martino

The DKW inequality is a non-asymptotic, high probability estimate on the L_\infty distance between the distribution function of a real-valued random variable and its empirical (random) counterpart. Little was known on generalisations of that inequality to high dimensions, where instead of a single random variable one is interested in the behaviour of empirical distribution functions of a family of marginals of a random vector. Based on chaining methods, we show that the behaviour of various notions of distance (including the L_\infty one) between the empirical and actual distributions of marginals in the given family can be fully characterised in terms of some (rather surprising) intrinsic complexity parameter of the family. Based on joint work with Shahar Mendelson.

We present a new approach to the gluing problem in General Relativity, that is, the problem of matching two solutions of the Einstein equations along a spacelike or characteristic (null) hypersurface. In contrast to previous constructions, the new perspective actively utilizes the nonlinearity of the constraint equations. As a result, we are able to remove the 10-dimensional spaces of obstructions to gluing present in the literature. As application, we show that any asymptotically flat spacelike initial data set can be glued to Schwarzschild initial data of sufficiently large mass. This is joint work with I. Rodnianski.

Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability. We will first review these motivations, then present the ''mean-field'' derivation of effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order behavior, giving information on the configurations at the microscopic level and connecting with crystallization questions,and finish with the description of the effect of temperature.

The talk will first exemplify Hybrid Modeling, that is combining first-principle based with data-driven models, on a toy example. Next, an approach for formalizing hybrid modeling will be presented, in terms of architectural design patterns. Afterwards, the benefits of Hybrid Modeling will be demonstrated in two applications: (i) data-driven electromagnetic field simulation, where the constitutive law will be directly inferred from data, and (ii) irregular time series, where mathematical structures such as Kálmán filter and stochastic ODEs are integrated within deep neural networks. The talk concludes with some suggestions for research questions.

The Lie algebra of vector fields on a manifold acts on differential forms by Lie derivatives and contractions, and these operations are related by the Cartan relations. We will explain an interpretation of these relations from the point of view of Lie theory, and describe how this leads to a categorification of the Chern-Weil homomorphism. For a Lie group G, we consider the space of smooth singular chains C(G), which is a differential graded Hopf algebra. We show that the category of sufficiently local modules over C(G) can be described infinitesimally, as the category of representations of a dg-Lie algebra which is universal for the Cartan relations. If G is compact and simply connected, the equivalence of categories can be promoted to an A-infinity equivalence of dg-categories, which are also A-infinity equivalent to the category of infinity local systems on the classifying space BG. The equivalence can be realized explicitly to provide a categorification of the Chern-Weil homomorphism. The talk is based on joint works with A. Quintero and S. Pineda, and work in progress with M. Rivera and F. Bischoff.