Livio Ferretti (UNIGE)
Framed mapping class groups and the monodromy of positive braids abstract
Abstract:
A framing on an orientable surface with boundary is given by a nowhere vanishing vector field (considered up to isotopy). The mapping class group naturally acts on such framings, and the stabilizer of a framing is called a framed mapping class group. Framed mapping class groups arise naturally in a variety of geometric settings, but their algebraic structure is to date very poorly understood. In this talk, we will introduce the basic theory of framed mapping class groups and see how they can be used to study monodromy groups of positive braids (and plane curve singularities). If time permits, we will end discussing applications of similar ideas to the study of more general fibred links.
10:30 • Université de Genève, Conseil Général 7-9, Room 1-05
Dr. Alberto Merici (University of Milan)
Log prismatic cohomology, motivic spectra and comparison theorems abstract
Abstract:
We prove that (logarithmic, Nygaard completed) prismatic and (logarithmic) syntomic cohomologies are representable in the category of logarithmic motives. As an application, we immediately obtain Gysin maps for prismatic and syntomic cohomologies, and we precisely identify their cofibers. In the second part of the talk we develop a descent technique that we call saturated descent, inspired by the work of Niziol on log K-theory. Using this, we prove crystalline comparison theorems for log prismatic cohomology, log Segal conjectures and log analogues of the Breuil-Kisin prismatic cohomology, from which we get Gysin maps for the A_{inf} cohomology.
13:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 25
Gerhard Wanner (UNIGE)
Origin of the Fourier Series
14:00 • Université de Genève, Conseil Général 7-9, Room 1-05
Prof. Dr. Yi Zhang (Academy of Mathematics and System Science, Beijing)
John domains in variational problems abstract
Abstract:
The notion of a John domain was initially introduced in 1961 by Fritz John, and later named after him by Martio and Sarvas. Typically, its study is motivated by its connections to the properties of quasiconformal and quasisymmetric mappings. Moreover, John domains find extensive applications in the theory of Sobolev functions in metric measure spaces and functional analysis, as they represent essentially the sole class of domains that uphold the Sobolev-Poincaré inequality. In this presentation, I will introduce several recent applications of John domains in the theory of the calculus of variations.
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Frank Ferrari (Université Libre de Bruxelles)
Jackiw-Teitelboim Gravity, Random Disks of Constant Curvature, Self-Overlapping Curves and Liouville CFT1 abstract
Abstract:
Jackiw-Teitelboim quantum gravity is a model of two-dimensional gravity for which the bulk curvature is fixed but the extrinsic curvature of the boundaries is free to fluctuate. The negative curvature model has been studied extensively in the recent physics literature, in a particular ``Schwarzian\'\' limit, because of its relevance in describing quantum black holes and their SYK-like duals.A first-principle approach reveals that the description used in the literature so far is an effective theory valid on distances much larger than the curvature length scale of the bulk geometry.At the microscopic level, the theory should be defined by taking the continuum limit of a new model of random polygons. The polygons, called ``self-overlapping,\'\' are constrained to bound a disk immersed in the plane. They must be counted with an appropriate multiplicity. The solution of the model could be found in principle by solving a difficult ``dually weighted’\' Hermitian matrix model.Motivated by standard heuristic path integral arguments, mimicking similar arguments used for Liouville gravity in the 80s and the 90s, we conjecture that an equivalent description is obtained in terms of a boundary log-correlated field. This yields predictions for the critical exponents of the self-overlapping polygon models and open the path to a wide range of potential applications.
15:15 • Université de Genève, Conseil Général 7-9, Room 1-15
Andrey Krutov (Prague)
Clifford algebra analogue of the Cartan theorem abstract
Abstract:
et $\\mathfrak{g}$ be a complex simple Lie algebra. The Hopf-Koszul-Samelson theorem asserts that the algebra of $\\mathfrak{g}$-invariants in the exterior algebra of $\\mathfrak{g}$ is the exterior algebra over the space of primitive invariants $P$. Kostant proved the analogous result for the Clifford algebra of $\\mathfrak{g}$. Namely, the algebra of $\\mathfrak{g}$-invariants in $Cl(\\mathfrak{g})$ is the Clifford algebra over the space of primitive invariants. Let $\\mathfrak{k}$ be a symmetric subalgebra of $\\mathfrak{g}$ and $\\mathfrak{p}$ be the corresponding isotropy representation. The Cartan theorem states that the algebra of $\\mathfrak{k}$-invariants in theexterior algebra of $\\mathfrak{p}$ is isomorphic to the tensor product of the exterior algebra of the Samelson subspace of $P$, corresponding to the pair $(\\mathfrak{g},\\mathfrak{k})$, and a certain commutative algebra $A$. We prove a Clifford algebra analogue of the Cartan theorem. Namely, we show that the algebra of $\\mathfrak{k}$-invariants in $Cl(\\mathfrak{p})$ is the tensor product of the Clifford algebra over the space of primitive invariant of $\\mathfrak{p}$ with a certain filtered deformation of $A$.
15:30 • Université de Genève, Conseil Général 7-9, Room 1-07
Konstantin Andritsch (ETHZ)
What is... an adelic torus orbit? abstract
Abstract:
\'\'As the term suggests - Adelic torus orbits - are nothing but the orbit of an algebraic torus over the ring of Adeles. They provide a powerful tool to collectively study the behavior of collections of geometric data given by arithmetic data. In this talk we will motivate the use of adelic torus orbits by looking at a concrete example: > Already in the 19th century Gauss studied integral binary quadratic forms. He observed that there are essentially only finitely many different integral binary quadratic forms with fixed discriminant. In more modern terms, these different forms arise through a natural action of the ideal class group of a quadratic number field. To study the properties of different forms at the same time it is convenient to consider the Adelic extension of the modular curve. We will see that forms who are not equivalent over the integers might be equivalent over the Adeles. After introducing the necessary concepts and motivating the idea behind Adelic torus orbits we will discuss how they can be used to prove equidistribution results on (real) homogeneous spaces.
16:30 • UZH Zentrum, Building KO2, Room F 150