Veranstaltungen

Given two four-dimensional symplectic manifolds, together with knots in their boundaries, we define an ``anchored symplectic embedding'' to be a symplectic embedding, together with a two-dimensional symplectic cobordism between the knots (in the four-dimensional cobordism determined by the embedding). We use techniques from embedded contact homology to determine quantitative critera for when anchored symplectic embeddings exist, for many examples of toric domains. In particular we find examples where ordinarily symplectic embeddings exist, but they cannot be upgraded to anchored symplectic embeddings unless one enlarges the target domain.

Simplicial sets are an abstraction notion of some well-known spaces studied in topology. This not only allows one to look at spaces from a purely algebraic perspective, but also gives a simple model for extending the methods used to study spaces to other areas of mathematics. In this talk, I will introduce simplicial sets, where it came from and discuss a couple of examples which demonstrate the ideas mentioned above. To make the talk more accessible, I will also briefly introduce the necessary concepts from homotopy theory and category theory.

More information: https://zucmap.ethz.ch/call_made

It all began with card shuffling. Diaconis and Shahshahani studied the random transpositions shuffle; pick two cards uniformly at random and swap them. They introduced a Fourier analysis technique to prove that it takes $1/2 n \log n$ steps to shuffle a deck of $n$ cards this way. Recently, Teyssier extended this technique to study the exact shape of the total variation distance of the transition matrix at the cutoff time from the stationary measure, giving rise to the notion of a limit profile. In this talk, I am planning to discuss a joint work with Olesker-Taylor, where we extend the above technique from conjugacy invariant random walks to general, reversible Markov chains. I will also present a new technique that allows us to study the limit profile of star transpositions, which turns out to have the same limit profile as random transpositions, and I will discuss open questions and conjectures.

The full regularity of harmonic maps from a given surface into an arbitrary Riemannian manifold has been proved by Hélein in 1991. This is not true anymore when the domain has dimension strictly greater than 2, Rivière constructed an example of harmonic map from a 3-dimensional domain which is everywhere discontinuous in 1995. There are many possible generalizations of these maps to the higher dimensional case in order to recover the regularity of some "optimal" maps. For most of these generalizations, the full regularity in the general case is still open. In this talk, we will discuss some recent progress obtained for n-harmonic maps. This is a joint work with Armin Schikorra.

Mathematics in climate research is often thought to be mainly a provider of techniques for solving the continuum mechanical equations for flows of the atmosphere and oceans, for the motion and evolution of Earth’s ice masses, and the like. Three examples will elucidate that there is a much wider range of opportunities. Climate modellers often employ reduced forms of “the continuum mechanical equations” to efficiently address their research questions of interest. The first example discusses how mathematical analysis can provide systematic guidelines for the regime of applicability of such reduced model equations. Physics-based computational models are well established for weather forecasting,which aims at deterministic prediction, and for climate scenario simulations, which aim at generating weather statics utilizing the hypothesis of ergodicity. Intermediate time scale predictions for, e.g., the occurance of El Niño events, fit into neither of these categories. Accordingly, physics-based models have difficulties reaching the predictive horizon desirable for agricultural planning and the like. Modern climate research has joined forces with economy and the social sciences to generate a scientific basis for informed political decisions in the face of global climate change. One major type of problems hampering progress of the related interdisciplinary research consists of often subtle language barriers. The third example describes how mathematical formalization of the notion of “vulnerability” has helped structuring related interdisciplinary research efforts.

We survey results about statistical properties of random dynamical systems and describe a number of open questions.

Fourier-Mukai transformations on compactified Jacobians

In his 1975 paper "Statistical Analysis of Non-Lattice Data", Julian Besag proposed the pseudolikelihood method as an alternative to the standard method of maximum likelihood estimation. This method has been very influential and successful in applications like learning graphical models from data, and also inspired another related and important method called score matching. I will discuss some recent work which connects the statistical efficiency of these estimators to the computational efficiency of related sampling algorithms.

Group cohomology comes in many variations. The standard Eilenberg-MacLane group cohomology is the cohomology of the cocomplex {f:G^q+1→ ℝ | f is G-invariant} endowed with its natural homogeneous coboundary operator. Now whenever a property P of such cochains is preserved under the coboundary one can obtain the corresponding P-group cohomology. P could be: continuous, measurable, L⁰, bounded, alternating, etc. Sometimes these various cohomology groups are known to differ (eg P=empty and P=continuous for most topological groups), in other cases they are isomorphic (eg P=empty and P=alternating (easy), P=continuous and P=L⁰ (a highly nontrivial result by Austin and Moore valid for locally compact second countable groups)). In 2006, Monod conjectured that for semisimple connected, finite center, Lie groups of noncompact type, the natural forgetful functor induces an isomorphism between continuous bounded cohomology and continuous cohomology (which is typically very wrong for discrete groups). I will focus here on the injectivity and show its validity in several new cases including isometry groups of hyperbolic n-spaces in degree 4, known previously only for n=2 by a tour de force due to Hartnick and Ott. Monod recently proved that all such continuous (bounded) cohomology classes can be represented by measurable (bounded) cocycles on the Furstenberg boundary. Our main result is that these cocycles can be chosen to be continuous on a subset of full measure. In the real hyperbolic case, this subset of full measure is the set of distinct tuples of points, easily leading to the injectivity in degree 4. This is joint work with Alessio Savini.

In this talk, we consider fringe trees of random plane trees with given vertex statistics (i.e., a given number of vertices of each degree). The main results are laws of large numbers and central limit theorems for the number of fringe trees of a given type. The key tool for our proofs is an extension to the multivariate setting of a theorem by Gao and Wormald (2004), which provides a way to show asymptotic normality by analyzing the behaviour of sufficiently high factorial moments. Our results also apply to random simply generated trees (or conditioned Galton–Watson trees) by conditioning on their degree statistic. Joint work with Cecilia Holmgren and Svante Janson (Uppsala University)

In this talk, I will introduce Bratteli-Vershik diagrams, and demonstrate their role in constructing finite-area translation surfaces, often of infinite type. I will then present a key result showing that every aperiodic ergodic flow with finite entropy is isomorphic to the vertical flow of a translation surface arising from this construction.

We study the 1D quantum many-body dynamics with a screened Coulomb potential in the mean-field setting. Combining the quantum mean-field, semiclassical, and Debye length limits, we prove the global derivation of the 1D Vlasov-Poisson equation. We tackle the difficulties brought by the pure state data, whose Wigner transforms converge to Wigner measures. We find new weighted uniform estimates around which we build the proof. As a result, we obtain, globally, stronger limits, and hence the global existence of solutions to the 1D Vlasov-Poisson equation subject to such Wigner measure data, which satisfy conservation laws of mass, momentum, and energy, despite being measure solutions. This happens to solve, even with general data, the 1D measure solution case of an open problem regarding the conservation law of the Vlasov-Poisson equation raised by Diperna and Lions.

We study how to construct a stochastic process on a finite interval with given `roughness'. We first extend Ciesielski's isomorphism along a general sequence of partitions, and provide a characterization of Hölder regularity of a function in terms of its Schauder coefficients. Using this characterization we provide a better (pathwise) estimator of Hölder exponent. Furthermore, We study the concept of (generalized) p-th variation of a real-valued continuous function along a sequence of partitions. We show that the finiteness of the p-th variation of a given function is closely related to the finiteness of ℓp-norm of the coefficients along a Schauder basis. As an additional application, we construct fake (fractional) Brownian motions with some path properties and finite moments of marginal distributions same as (fractional) Brownian motions. These belong to non-Gaussian families of stochastic processes which are statistically difficult to distinguish from real (fractional) Brownian motions.

Given a function $\psi: \mathbb{N} \to [0,1/2]$, the theorems of Khintchine and Koukoulopoulos--Maynard (formerly known as Duffin--Schaeffer conjecture) provide a satisfying answer about the number of good rational approximations $\vert \alpha - \frac{p}{q} \vert < \frac{\psi(q)}{q}$ for Lebesgue almost every $\alpha \in \mathbb{R}$.\\ However, the picture is much less understood when we allow an inhomogeneous parameter $\gamma \in \mathbb{R}$ and ask for solutions to $\vert \alpha - \frac{p+ \gamma}{q} \vert < \frac{\psi(q)}{q}$, and even less if we allow the parameter $\gamma_q$ to change with $q$. In this talk, I will explain the genuine difficulties of these questions that are surprisingly deeply connected with analytic and combinatorial number theory, and discuss recent positive results both in dimension $1$ as well as and in higher-dimensional analogues. This is partially joint work with Victor Beresnevich and Sanju Velani, respectively with Felipe Ramírez.

The geometry of moduli spaces of one-dimensional sheaves on the projective plane has attracted a lot of study recently. In this talk, I will give a new calculation of the Betti numbers of the moduli spaces of one-dimensional sheaves on the projective plane using Gromov-Witten invariants of local P^2 and local curves. The new calculation is based on the refined sheaves/GW correspondence established by Bousseau and all genus local/relative correspondence given by Bousseau-Fan-Guo-Wu. It can be used to prove the divisibility property of Poincaré polynomials of moduli spaces of one-dimensional sheaves on projective plane conjectured by Choi-van Garrel-Katz-Takahashi, and can also be used to recover the asymptotic behaviour of the Betti numbers established by Yuan. This is a work in progress with Shuai Guo.

Survival analysis has been a task of significant interest within the statistics community throughout the years. More recently, the machine learning and bioinformatics communities have also increasingly become interested in survival analysis. In this seminar, we survey recent developments focusing especially on high-dimensional multi-omics survival analysis. We concentrate particularly on empirical evaluations highlighting the difficulty of this task, the need for more data, and standardized evaluation. In the second part of the seminar, we discuss recent work on knowledge distillation for sparse survival models and methods for structure selection in sparse partially linear survival models.