Marino Gran (UCLouvain)
Abstract:
In recent years, numerous new applications of categorical Galois theory have emerged in various interesting non-abelian algebraic contexts. In particular, within semi-abelian categories, this approach has led to some new calculations of higher fundamental groups in terms of generalized commutators in categories such as that of compact groups, crossed modules, and skew braces. These categories share some structural properties with the categories of groups and of Lie algebras, and also with the category of cocommutative Hopf algebras over a field, which is also semi-abelian.This raises the natural question of whether similar homological methods can be applied to study cocommutative Hopf algebras as well.In this talk, after reviewing some fundamental properties of semi-abelian categories and some motivating examples, I will explain that the answer to the above question is affirmative. By using the exactness properties of cocommutative Hopf algebras and the free functor universally associating a Hopf algebra with any coalgebra it is possible to establish some new Hopf-type formulae for the homology of cocommutative Hopf algebras. An important role is played by cleft extensions, namely those surjective morphisms of Hopf algebras that are split as coalgebra morphisms.With any cleft extension, one can associate a 5-term exact sequence in homology that can be seen as a Hopf-theoretic analogue of the classical Stallings-Stammbach exact sequence in group theory. This new approach can also be applied to investigate the homology of cocommutative Hopf braces, which are interesting structures that naturally occur in the study of solutions to the so-called quantum Yang-Baxter equation. The category of cocommutative Hopf braces turns out to be both semi-abelian and monadic on the category of coalgebras, so that it is possible to investigate it from the perspective of non-abelian homological algebra.This talk is based on a joint work with Andrea Sciandra.
Sylvain Crovisier (Université Paris-Saclay)
Ergodic theory of surface diffeomorphisms
10:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Shunhua Jiang (ETH Zürich)
Generalized Flow in Nearly-linear Time on Moderately Dense Graphs abstract
Abstract:
In this talk, we consider generalized flow problems where there is an $m$-edge $n$-node directed graph $G = (V,E)$ and each edge $e \\in E$ has a loss factor $\\gamma_e >0$ governing whether the flow is increased or decreased as it crosses edge $e$. We provide a randomized $\\tilde{O}( (m + n^{1.5}) \\cdot \\mathrm{poly} \\log(\\frac{W}{\\delta}))$ time algorithm for generalized maximum flow and generalized minimum cost flow in this setting where $\\delta$ is the target accuracy and $W$ is the maximum of all costs, capacities, and loss factors and their inverses.This improves upon the previous state-of-the-art $\\tilde{O}(m \\sqrt{n} \\cdot \\log^2(\\frac{W}{\\delta}) )$ time algorithm, obtained by combining the algorithm of [Daitch-Spielman 2008] with techniques from [Lee-Sidford 2014].To obtain this result we provide new dynamic data structures and spectral results regarding the matrices associated to generalized flows and apply them through the interior point method framework of [Brand-Lee-Liu-Saranurak-Sidford-Song-Wang 2021]. Based on joint work with Michael Kapralov, Lawrence Li, and Aaron Sidford.
11:15 • EPF Lausanne, INJ114
Lucio Rosi (Universität Basel)
Abstract:
Abstract The Complex Ginzburg-Landau (CGL) equation is a fundamental nonlinear partial differential equation (PDE) frequently used to model a wide variety of evolution phenomena in a wide range of physical systems. It is also being studied as a model equation for other different nonlinear PDEs, like the incompressible Navier-Stokes equation in fluid dynamics. In this talk, we will present a numerical investigation into the existence, stability, and uniqueness of self-similar solutions to the CGL equation in the supercritical regime. The primary focus is on forward-in-time solutions in self-similar coordinates whose initial condition correspond to the blow-up profiles of backward-in-time self-similar solutions. The investigation seeks to determine whether the evolution of this forward profile from the singularity is unique.
12:15 • Universität Basel, Seminarraum 05.001, Spiegelgasse 5
Killian Davis (Ohio State University)
Nonadditivity of the Seifert Genus for Virtual Knots abstract
Abstract:
Classically, the Seifert genus for knots respects knot addition, leading to useful structural theorems, such as the fact that genus 1 knots are prime. For virtual knots, however, this is false. In this talk, we will demonstrate nonadditivity of the Seifert genus by showing two unknots which add to a genus 1 virtual knot. We will also introduce the main tool used to prove this: the homotopy Zh construction.
14:15 • Université de Genève, Conseil Général 7-9, Room 1-15
Lucas Kaufmann (University of Orléans)
Abstract:
Title: Equidistribution of periodic points for endomorphisms of P^k Abstract: Equidistribution phenomena are naturally present in several branches of mathemematics. The usual picture is that a sequence of points on a space X defined in some natural way always converge to a given limit distribution (a probability measure on X). In the case of dynamical systems, two natural choices are given by the iterated pre-images of a given point or periodic points of period tending to infinity. In the holomorphic world, it is known since the works of Lyubich in dimension 1 and Briend-Duval in any dimension that the periodic points of a holomorphic endomorphism of P^k equidisitribute towards its equilibrium measure. Arithmetic versions also exist (Ullmo-Zhang, Baker Rumely, Favre- Rivera-Letelier, Chamber-Loir, Yuan-Zhang, etc). In this talk, I will discuss results concerning the speed of convergence in the above theorems in the holomorphic category. This is a joint work with H. de Thélin and T.-C. Dinh. Seminarraum 05.002, Spiegelgasse 5
14:15 • Universität Basel
Emmanuel Kowalski (ETHZ)
Bilinear forms with galant kernels abstract
Abstract:
Bilinear form structures occur frequently in analytic number theory, for instance in describing the primes. Estimating the norms of these bilinear forms is of crucial importance for applications.The talk will explain this general philosophy and present recent progress in the case of certain bilinear forms defined using kernels associated with "gallant" $\\ell$-adic sheaves.(Joint work with É. Fouvry, Ph. Michel and W. Sawin)
14:15 • EPF Lausanne
Jeffrey Näf (Université de Genève )
Imputation under Missing at Random: How to Impute and How to Evaluate Imputations abstract
Abstract:
In this talk, we take an in-depth look at the topic of missing value imputation. Focusing on the \'missing at random\' (MAR) case, we discuss the qualities that constitute an effective imputation method and how to evaluate them in practice. Crucially, we review common pitfalls in the literature that can bias subsequent analysis and explain how to avoid them, while also discussing some of the most promising imputation methods currently available. If time permits, we also briefly explore state-of-the-art research on evaluating an imputation method for a given dataset based on imputation scores (I-Scores).
16:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Tom Hutchcroft (Caltech)
Critical long-range percolation abstract
Abstract:
It is conjectured that many models of statistical mechanics have a rich, fractal-like behaviour at and near their points of phase transition, with power-law scaling governed by critical exponents that are expected to depend on the dimension but not on the small-scale details of the model such as the choice of lattice. This is now reasonably well understood in two dimensions and in high dimensions, but remains poorly understood in intermediate dimensions (e.g. d=3). I will overview the conjectures around this area and describe recent progress on related problems for models with long-range interactions.
16:15 • Université de Genève, Conseil Général 7-9, Room 1-15
Prof. Dr. Hajer Bahouri (Laboratoire Jacques-Louis Lions, Sorbonne Université)
On the global well-posedness the derivative nonlinear Schrödinger equation on the torus abstract
Abstract:
<p>In this talk, I will present a recent joint work with Galina Perelman concerning the derivative nonlinear Schrödinger (DNLS) equation on the torus. The DNLS equation which is a canonical dispersive equation arising in a variety of physical contexts is known to be completely integrable (and then it admits a spectral formulation, an infinite number of conservation laws and explicit families of conservation laws). The main difficulty of this equation is the lack of coercivity of its conservation laws in the regime above the algebraic soliton threshold, and this makes large data global well-posedness a challenging issue. This equation was solved a short time ago on the real line, while the case of the torus is still less understood. In this work, we prove global well-posedness for DNLS equation on the torus.</p> <p>The first part of my presentation will be devoted to a general overview of DNLS and completely integrable equations, then I will on the case of the torus by providing the new arguments that enabled us to achieve our goal. </p>
16:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 35/36
Prof. Dr. Christian Bayer (WIAS Berlin)
Markovian approximations of rough volatility models abstract
Abstract:
The rough Heston model is a very popular recent model in mathematical finance; however, the lack of Markov and semimartingale properties poses significant challenges in both theory and practice. A way to resolve this problem is to use Markovian approximations of the model. Several previous works have shown that these approximations can be very accurate even when the number of additional factors is very low. Existing error analysis is largely based on the strong error, corresponding to the L2 distance between the kernels. Extending earlier results by [Abi Jaber and El Euch, SIAM Journal on Financial Mathematics 10(2):309--349, 2019], we show that the weak error of the Markovian approximations can be bounded using the L1-error in the kernel approximation for general classes of payoff functions for European style options. Moreover, we give specific Markovian approximations which converge super-polynomially in the number of dimensions, and illustrate their numerical superiority in option pricing compared to previously existing approximations. The new approximations also work for the hyper-rough case H>−1/2. In addition, we provide explicit characterisations of the state space of the Markovian approximations. Joint work with Eduardo Ani Jaber and Simon Breneis.
17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43