Events

https://www.math.uzh.ch/mat074

For each integer n \geq 1, denote by T_n the map x \mapsto nx mod 1 from the circle group T=R/Z into itself. Let p,q \geq 2 be two multiplicatively independent integers. Using Baire Category arguments, we will show that generically, a continuous T_p-invariant probability measure \mu on T is such that (T_q^{n}\mu)_{n\geq 0} does not weak-star converge to the Lebesgue measure on T. This disproves Conjecture (C3) from a 1988 paper by R. Lyons, which is a stronger version of Furstenberg's rigidity conjecture on xp and xq invariant measures on T, and complements previous results by Johnson and Rudolph. If time permits, I will also present some generalizations of this result concerning convergence to the Lebesgue measure of sequences of the form (T_{c_{n}}\mu)_{n\geq 0}, as well as some extensions to the multidimensional setting. The talk will be based on a joint work with Catalin Badea (Lille).

In this talk I will review basic predictions for the high temperature regime, the so called replica symmetric regime, of the Sherrington-Kirkpatrick mean field spin glass. I will recall the TAP equations and their derivation in connection with the decay of the two point correlation functions. For the simplified case of vanishing external field, I will present some details on recent results that characterize the susceptibility of the model as a resolvent of the interaction matrix, which predicts in a simple way the (well-known) RS-RSB transition temperature. The talk is based on joint work with Adrien Schertzer, Changji Xu and Horng-Tzer Yau.

We propose data thinning, a new approach for splitting an observation from a known distributional family with unknown parameter(s) into two or more independent parts that sum to yield the original observation, and that follow the same distribution as the original observation, up to a (known) scaling of a parameter. This proposal is very general, and can be applied to a broad class of distributions within the natural exponential family, including the Gaussian, Poisson, negative binomial, Gamma, and binomial distributions, among others. Furthermore, we generalize data thinning to enable splitting an observation into two or more parts that can be combined to yield the original observation using an operation other than addition; this enables the application of data thinning far beyond the natural exponential family. Data thinning has a number of applications to model selection, evaluation, and inference. For instance, cross-validation via data thinning provides an attractive alternative to the "usual" approach of cross-validation via sample splitting, especially in unsupervised settings in which the latter is not applicable. We will present an application of data thinning to single-cell RNA-sequencing data, in a setting where sample splitting is not applicable. This is joint work with Anna Neufeld (Fred Hutch), Ameer Dharamshi (University of Washington), Lucy Gao (University of British Columbia), and Jacob Bien (University of Southern California)

Given a group G, finding a geometric action of G on a CAT(0) cube complex can be used to say some rather strong things about G. Such actions are not always easy to find, however, which makes it useful to have sufficient conditions, both for existence and for non-existence. This talk concerns the latter: we shall see a coarse geometric obstruction to a group admitting a cocompact cubulation. Based on joint work with Zach Munro.

A frame (x_j) for a Hilbert space H allows for a linear and stable reconstruction of any vector x in H from the linear measurements (<x,x_j>). However, there are many situations where some information of the frame coefficients is lost. In applications such as signal processing, electrical engineering, and digital photography one often uses sensors with an effective range and any measurement above that range is registered as the maximum. Depending on the context, recovering a vector from such measurements is called either declipping or saturation recovery. We will discuss a frame theoretic approach to this problem in a similar way to what Balan, Casazza, and Edidin did for phase retrieval. This perspective motivates many interesting open problems. The talk is based on joint work with W. Alharbi, D. Ghoreishi, B. Johnson, and N. Randrianarivony.

In this talk we will define the conformal volume of a manifold endowed with a conformal class of Riemannian metrics. This invariant was introduced by Li and Yau in 1982 and since then has been very influential in the development of Geometric Analysis. After setting up the framework of conformal geometry and introducing the conformal volume, we will give some examples (mostly in dimension 2) and explore its connection to minimal surfaces, the Willmore functional and the Laplace eigenvalues.

Extrapolation is crucial in many statistical and machine learning applications, as it is common to encounter test data outside the training support. However, extrapolation is a considerable challenge for nonlinear models. Conventional models typically struggle in this regard: while tree ensembles provide a constant prediction beyond the support, neural network predictions tend to become uncontrollable. This work aims at providing a nonlinear regression methodology whose reliability does not break down immediately at the boundary of the training support. Our primary contribution is a new method called ‘engression’ which, at its core, is a distributional regression technique for pre-additive noise models, where the noise is added to the covariates before applying a nonlinear transformation. Our experimental results indicate that this model is typically suitable for many real data sets. We show that engression can successfully perform extrapolation under some assumptions such as a strictly monotone function class, whereas traditional regression approaches such as least-squares regression and quantile regression fall short under the same assumptions. We establish the advantages of engression over existing approaches in terms of extrapolation, showing that engression consistently provides a meaningful improvement. Our empirical results, from both simulated and real data, validate these findings, highlighting the effectiveness of the engression method. The software implementations of engression are available in both R and Python.

''In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. The simplest example of such behaviour is described by the AdS instability conjecture, put forward by Dafermos and Holzegel in 2006; this conjecture suggests that generic small perturbations of the AdS initial data lead to the formation of trapped surfaces when reflecting boundary conditions are imposed at conformal infinity. However, whether a similar scenario also holds in the more complicated case of the exterior region of an asymptotically AdS black hole spacetime has been the subject of debate. In this talk, we will show that weak turbulence does emerge in the dynamics of a quasilinear toy model for the vacuum Einstein equations on the Schwarzschild-AdS exterior spacetimes for an open and dense set of black hole mass parameters. This is joint work with Christoph Kehle.

Im August 2024 wird Informatik zum «Grundlagenfach» an Schweizer Gymnasien. In diesem Vortrag werde ich die Geschichte des Informatikunterrichts zusammenfassen (die gar nicht so jung ist, wie man meinen könnte) und eine Vision davon skizzieren, welche Rolle die Informatik in den Schulen der Zukunft spielen wird. Ferner werden einige Fehlvorstellungen diskutiert, die sich beispielsweise ergeben, wenn Begriffe wie «Variable» oder «Gleichung» aus der Mathematik auf den Programmierunterricht treffen.

Hawkes processes have proved to be a powerful probabilistic model for various applications in neurosciences or insurance. These counting processes are defined through their intensity which is stochastic and pathwise dependent on the historical values of the process itself. This implicit definition leads to important drawbacks for performing explicit calculations involving simple quantities like for instance the inter-temporal correlation for which nothing is known to our knowledge outside the stationary case and some very particular examples. In this talk based on a joint work with C. Hillairet (ENSAE Paris), we will fill this gap by exploiting a specific decomposition named pseudo-chaotic expansion mixing some pathwise calculus together with some Malliavin calculus obtained in a previous joint work.