Veranstaltungen

Intermittent dynamics, where systems irregularly alternate between long periods of different types of dynamical behaviour, has been studied since the work of Pomeau and Manneville in 1980. In random dynamical systems this phenomenon has only been well understood in a few specific cases. A random dynamical system consists of a family of deterministic systems, one of which is chosen to be applied at each time step according to some probabilistic rule. In this talk we will describe the intermittency of some families of random systems with a particular emphasis on how the intermittency of the random system depends on the intermittency of the underlying deterministic systems. This talk is based on joint works with Ale Jan Homburg, Tom Kempton, Valentin Matache, Marks Ruziboev, Masato Tsujii, Evgeny Verbitskiy and Benthen Zeegers.

Filtered Lagrangian Floer homology gives rise to a barcode associated to a pair of Lagrangians. It is well-known that the lengths of the finite bars and the spectral distance are lower bounds of the Lagrangian Hofer metric. In this talk we will discuss a reverse inequality: We will show an upper bound of the Lagrangian Hofer distance between equators in the cylinder in terms of a weighted sum of the lengths of the finite bars and the spectral distance.

We will consider the Cauchy problem for Nonlinear Diffusion equations of porous medium type $u_t=-\mathcal{L} u^m$, with $m>1$ and investigate whether or not integrable data produce bounded solutions. The diffusion operator belongs to a quite general class of nonlocal operators, and we will see how different assumption on the operator imply (or not) smoothing properties. We will briefly compare the approach based on Moser iteration and the approach through Green functions. On the one hand, we show that if the linear case ($m=1$) enjoys smoothing properties, also the nonlinear will do. On the other hand, we see that in some cases the nonlinear diffusion enjoys the smoothing properties also when the linear counterpart does not, thanks to the convex nonlinearity. Following Nash' ideas, we see how smoothing properties are often equivalent to the validity of Gagliardo-Nirenberg-Sobolev (and Nash) inequalities: we explore these implications also in the nonlinear and nonlocal context and the connection with dual inequalities (Hardy-Littlewood-Sobolev) and Green function estimates. This is a work with J. Endal (NTNU, Trondheim). Finally, I shall present smoothing and higher regularity estimates for a class of zero-order p-Laplacian evolution equations, showing another case in which the regularization is true in the nonlinear setting while it fails for the linear counterpart. This is a work in progress with A. Salort (Univ. Buenos Aires).

Tree graphical models are common statistical models for data in a wide variety of applications. Tree models are particularly popular in phylogenetics, where an important task is to infer the evolutionary history of current species. Given observations at the leaves of the tree, a common problem is to reconstruct the tree's latent structure. We present two simple spectral-based methods for tree recovery: <br> (i) A bottom up spectral neighbor joining method (SNJ); and <br>(ii) STDR - a spectral based top down method. <br> We prove that under suitable assumptions, both methods are consistent and derive finite sample recovery guarantees. We illustrate the competitive performance of our algorithms in comparison with popular tree recovery methods.

The study of abstract reflection groups is due to Tits, who defines Coxeter systems (W,S) through their presentation. There are many different combinatorial and geometric objects associated to such Coxeter systems (W,S). One of them is the Davis complex. The goal of this talk is to introduce abstract Coxeter groups and describe their associated Davis complex.

This cycle of talks wants to highlight how ideas from tropical geometry have contributed not only to the solution, but also to the development of enumerative geometric problems regarding moduli spaces of curves, and maps from curves to curves. We will spend a little of time reviewing the origins of this story, i.e. the development of tropical Hurwitz numbers as combinatorial analogues for the classical Hurwitz numbers. We will discuss a more recent interpretation that views tropical Hurwitz numbers as the natural computation for the intersection number of the double ramification cycle with an element of the log Chow ring of the moduli space of curves (called in this case the branch polynomial, as it is presented as a piecewise polynomial function on the moduli spaces of tropical curves) which is determined by the tropical moduli space of covers of the projective line. We will see that from the tropical perspective analogous piecewise polynomial functions may be associated to $k$-DR cycles (cycles arising from spaces of twisted pluri-differentials), thus giving rise to $k$-analogues of Hurwitz numbers (called leaky Hurwitz numbers) that enjoy many of the algebro-combinatorial properties of Hurwitz numbers - such as piecewise polynomiality and wall crossings. We will present some work in progress which intends to incorporate descendants into these pictures. Tropical algorithms are developed that give rise to some intruiguingly simple formulas in the case when one point is fully ramified. The material presented is based on many years of joint work with several people, including Paul Johnson, Hannah Markwig, Dhruv Ranganathan and Johannes Schmitt.

This talk will be about the moduli space M_{d, χ} of stable 1-dimensional sheaves on the projective plane. The cohomology of these moduli spaces is conjecturally related to the enumerative geometry of local P^2. This relation predicts in particular that the (intersection) Betti numbers of M_{d, χ} do not depend on χ, which has been proven by Maulik-Shen. A natural question is whether the cohomology ring is also χ-invariant. In this talk I will answer that question negatively: except for possibly finitely many exceptional cases, the ring cohomology determines χ up to some trivial symmetries. In particular this shows that those spaces are typically not homeomorphic (previously they were only known to be non-isomorphic as algebraic varieties). This is based on joint work with W. Lim and W. Pi.

An identity, or law, for a group G is a nontrivial equation in one or more variables, which is always satisfied when the variables are evaluated in G. Every finite group satisfies a law, and it is natural to ask for the length of the shortest law for G, and to investigate how this length reflects the structure of G. These questions are explored in several papers in the literature. In ongoing joint work with Jakob Schneider and Andreas Thom, we investigate how the answers to these questions may change (or not) when our equations are allowed to contain constants in G as well as variables.

The branching Brownian motion is a particle system in which each particle evolves independently of one another. Each particle moves according to a Brownian motion in dimension d, and splits into two daughter particles after an independent exponential time of parameter 1. The daughter particles then start from their positions independent copies of the same process. We take interest in the long time asymptotic behaviour of the particles reaching farthest away from the origin. We show that these particles can be found at a distance of order $\sqrt{2} t + \frac{d-4}{2\sqrt{2}} \log t$ from the origin of the process, and that they can be grouped into a Poisson point process of families of close relatives, spreading in directions sampled according to the random measure $Z(\mathrm{d} \theta)$ that plays the role of an analogue of the derivative martingale of the branching Brownian motion.

Cruising on the surface of a rotating black hole...

Coxeter polyhedra are convex polyhedra whose dihedral angles are integer submultiples of π. They are intimately related with regular polyhedra tessellating the space and enjoy nice extremal properties. Due to the work of H.S.M Coxeter, spherical and Euclidean Coxeter polyhedra are fully classified. However, in the hyperbolic case, such a classification is far from being complete. In this talk, we go into classification results in small dimensions, providing various examples, and discuss the properties of their associated reflection groups in terms of co-volumes and growth rates.

Mirror symmetry for 3d N=4 supersymmetric gauge theories has recently received much attention in geometry and representation theory. Theories within this class admit very interesting moduli spaces of vacua, whose most relevant components are called Higgs and Coulomb branches. The study of the mathematics of the Higgs branch was initiated by Nakajima in the 90s, and since then its geometry has proved to be intimately related to the representation theory of Kac-Moody Lie algebras and quantum groups. Only recently, mathematically accurate definitions of the Coulomb branch have been proposed, and their study has started. The main mathematical prediction of 3d mirror symmetry is that the Higgs and Coulomb branches of a pair of dual theories are interchanged. Hence, both pairs of homologous branches (Higgs-Higgs and Coulomb-Coulomb) are expected to share exceptional topological and geometric properties. One of the main predictions of mirror symmetry is that the elliptic stable envelopes, which are certain topological classes intimately related to elliptic quantum groups, are the same after appropriate identifications. In this talk, I will focus on Coulomb and Higgs branches of type A, which are collectively described by a class of varieties known as Cherkis bow varieties, and I will discuss the main ideas behind the proof of mirror symmetry of sable envelopes (joint work in preparation with Richard Rimanyi).

Pearson covariance of two random variables $X$ and $Y$ measures the average joint comovements around the respective means of $X$ and $Y$. We generalise this well known measure by replacing the means with other summary statistics of the marginal distributions of $X$ and $Y$ such as quantiles, expectiles, or absolute thresholds. Deviations from these quantities are defined via generalised errors, induced by identification or moment functions. As a normalised measure of dependence, a generalised correlation, is constructed. Replacing the common Cauchy--Schwartz normalisation by a novel Fréchet--Hoeffding normalisation, we obtain attainability of the entire interval $[-1,1]$ by the generalised correlation for any given marginals. After uncovering favourable properties of these new dependence measures and establishing consistent estimators, we construct function-valued distributional correlations, exhibiting the entire dependence structure. They give rise to tail correlations, which should arguably supersede the coefficients of tail dependence. The two quantities coincide for positive tail dependence, but the novel notion may distinguish between negative dependence and asymptotic independence. Finally, we construct summary covariances (correlations), which arise as (normalised) weighted averages of distributional covariances. We retrieve Pearson covariance and Spearman correlation as special cases. The talk is based on joint work with Marc-Oliver Pohle.