Oscar Garcia-Prada (Madrid)
mini-course: Higher rank Teichmüller spaces and Higgs bundles abstract
Abstract:
It is well-known that the Teichmüller space of a compact surface can be identified with a connected component of the moduli space of representations of the fundamental group of the surface in PSL(2,R). Higher rank Teichmüller spaces are generalizations of this, which exist in the moduli space of representations of the fundamental group into certain real simple non-compact Lie groups of higher rank. As for the usual Teichmüller space, these spaces consist entirely of discrete and faithful representations. In this series of lectures, I will give a full classification of the groups for which higher rank Teichmüller spaces can exist, as well as a complex algebraic-geometric parametrization of these spaces in terms of Higgs bundles. This involves, in particular, the non-abelian Hodge correspondence, and the more recent Cayley correspondence developed in collaboration with Steve Bradlow, Brian Collier, Peter Gothen and André Oliveira (to appear in Annals of Mathematics, 2024, arXiv:2101.09377).
10:00 • Université de Genève, Conseil Général 7-9, Room 6-13
Neel Patel
Limitations of Stochastic Selection with Pairwise Independent Priors abstract
Abstract:
Motivated by the growing interest in correlation-robust stochastic optimization, we investigate stochastic selection problems beyond independence.Specifically, we consider the instructive case of pairwise-independent priors and matroid constraints. We obtain essentially-optimal bounds for contention resolution and prophet inequalities.The impetus for our work comes from the recent work of Caragiannis et al., who derived a constant-approximation for the single-choice prophet inequality with pairwise-independent priors.For general matroids, our results are tight and largely negative. For both contention resolution and prophet inequalities, our impossibility results hold for the full linear matroid over a finite field. We explicitly construct pairwise-independent distributions which rule out an $\\omega(1/rank)$-balanced offline CRS and an $\\omega(1/\\log rank)$-competitive prophet inequality against the (usual) oblivious adversary.For both results, we employ a generic approach for constructing pairwise-independent random vectors — one which unifies and generalizes existing pairwise independence constructions from the literature on universal hash functions and pseudorandomness. Specifically, our approach is based on our observation that random linear maps turn linear independence into stochastic independence.We then examine the class of matroids which satisfy the so-called partition property — these include most common matroids encountered in optimization. We obtain positive results for both online contention resolution and prophet inequalities with pairwise-independent priors on such matroids, approximately matching the corresponding guarantees for fully independent priors. These algorithmic results hold against the almighty adversary for both problems.
11:00 • EPF Lausanne, INJ114
Dr. Katy Loyd (University of Maryland)
Pointwise Ergodic Averages along Sequences of Slow Growth abstract
Abstract:
Following Birkhoff's proof of the Pointwise Ergodic Theorem, it is natural to consider whether convergence still holds along various subsequences of the integers. In 2020, Bergelson and Richter showed that in uniquely ergodic systems, pointwise convergence holds along the number theoretic sequence $\\Omega(n)$, where $\\Omega(n)$ denotes the number of prime factors of $n$, with multiplicities. In this talk, we will see that by removing this assumption, a pointwise ergodic theorem does not hold along $\\Omega(n)$. In fact, $\\Omega(n)$ satisfies a notion of non-convergence called the strong sweeping out property. We then further classify the strength of this non-convergence behavior by considering weaker notions of averaging. Time permitting, we will introduce a more general criterion for identifying slow growing sequences with the strong sweeping out property (based on joint work with S. Mondal).
13:30 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 28
Dr. Sam Canning (ETH Zürich)
Complex abelian varieties and their moduli IV abstract
Abstract:
Degenerating abelian varieties
13:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Dan Isaksen (Wayne State University)
Abstract:
HF2-synthetic stable homotopy theory is a deformation of classical stable homotopy theory. I will describe what this means in concrete terms, and I will demonstrate in practical terms how it helps us compute stable homotopy groups.
Umberto Pappalettera (Universität Bielefeld)
Abstract:
In this talk I will present a new "anomalous regularisation" result for solutions of the stochastic transport equation \\partial_t \\rho + rc \\partial_t W ot \\nabla \\rho = 0, where W is a Gaussian, homogeneous, isotropic noise with lpha-H\\"older space regularity and compressibility ratio \\wp < rac{d}{4lpha^2}. The proof is obtained by studying the local behaviour around the origin of solutions to a degenerate parabolic PDE in non-divergence form, which is of independent interest. Based on joint work with Theodore Drivas and Lucio Galeati.
14:15 • Universität Basel, Spiegelgasse 5, Seminarraum 05.002
Prof. Dr. Thomas Mettler (UniDistance Suisse)
Flat extensions of principal connections and the Chern—Simons 3-form abstract
Abstract:
I will introduce the notion of a flat extension of a connection on a principal bundle. Roughly speaking, a connection admits a flat extension if it arises as the pull-back of a component of a Maurer–Cartan form. For trivial bundles over closed oriented 3-manifolds, I will relate the existence of certain flat extensions to the vanishing of the Chern–Simons invariant associated to the connection. Joint work with Andreas Cap & Keegan Flood.
15:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Prof. Dr. Gigliola Staffilani (Massachusetts Institute of Technology)
A curious phenomenon in wave turbulence theory abstract
Abstract:
In this talk we will use the periodic cubic nonlinear Schrödinger equation to present some estimates of the long time dynamics of the energy spectrum, a fundamental object in the study of wave turbulence theory. Going back to Bourgain, one possible way to conduct the analysis is to look at the growth of high Sobolev norms. It turns out that this growth is sensitive to the nature of the space periodicity of the system. I will present a combination of old and very recent results in this direction.
16:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2
Dr. Alejandro Rosales Ortiz (Universität Zürich, Switzerland)
Excursion theory for Brownian motion indexed by the Brownian tree abstract
Abstract:
We begin by introducing the notion of Brownian motion indexed by the Brownian tree. We will then present the main aspects of a theory, developed in two recent works with Armand Riera, that describes the evolution of this tree-indexed process between visits to 0. The theory applies to fairly general continuous Markov processes indexed by Lévy trees. Despite the radically different setting, we will see that our results share strong similarities with the celebrated Itô excursion theory for linear Brownian motion. If time permits, we will also discuss some applications to Brownian geometry.
17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43