We are pleased to organize the sixth edition of this short meeting. We will welcome anyone interested in attending, in an informal and relaxed atmosphere, the talks of a selected group of topologists.
This year we welcome (title and abstracts below):
Organizers: Urtzi Buijs, Antonio Garvín, Vicente Muñoz and Antonio Viruel
Title: On the geometry of left-orders
Abstract: Given a finitely generated group with a left invariant order we investigate the geometry of the subset of the Cayley graph consisting on elements greater than the identity (the positive cone of the order). We will give examples where all the positive cones are coarsely connected (e.g. Braid groups) and examples where no positive cone is coarsely connected (e. g. hyperbolic surface groups and free groups). This is based on a joint work with J. Alonso, J. Brum and C. Rivas.
Title: Cellular Sheaf Laplacians
Abstract: Cellular sheaves are functors from face posets of cell complexes to an appropriate algebraic category (say, inner product spaces), of recent use in data science. This talk will cover several recent applications of the Hodge Laplacian for cellular sheaves to opnion dynamics, consensus over networks, and more. This is joint work with Jakob Hansen and Hans Riess.
Title: Real homotopy of configuration spaces
Abstract: I will present several algebraic models for the real/rational homotopy types of (ordered) configuration spaces of points and framed points in a manifold. These models can be used to establish real/rational homotopy invariance of configuration spaces under dimensionality and connectivity assumptions. Moreover, the collection of all configuration spaces of a given manifold has the structure of a right module over some version of the little disks operad, and the algebraic models are compatible with this extra structure. The proofs all use ideas from the theory of operads, namely Kontsevich’s proof of the formality of the little disks operad and – for oriented surfaces – Tamarkin’s proof of the formality of the little 2-disks operad. (Based on joint works with Campos, Ducoulombier, Lambrechts, and Willwacher.)
Title: Computational topology in intersection theory
Abstract: Given a nested pair X and Y of complex projective varieties, there is a single positive integer e which measures how X sits inside Y. This is called the Hilbert-Samuel multiplicity of Y along X, and it appears in the formulations of several standard intersection-theoretic constructions (including Segre classes, Euler obstructions, and various multiplicities). The standard method for computing e requires knowledge of the equations which define X and Y, followed by a (super-exponential) Grobner basis computation. In this talk we will see how a few hyperplane intersections and a bit of stratified Morse theory can reduce this enormous computational burden to a simple exercise in clustering point clouds. In fact, one doesn't even need the equations which define X and Y and it suffices to have dense point samples. (Those reaching furiously for their copies of Fulton can relax, because no prior knowledge of intersection theory or stratified Morse theory will be assumed.) This is joint work with Martin Helmer.
Title: On a Kahn’s group realisability problem
Abstract: In the sixties, Kahn raised the question of whether every group appears as the group of self-homotopy equivalences of a topological space. This problem has received significant attention, but initial results were focused on ad-hoc procedures for certain families of groups.
Very recently, Costoya and Viruel provided the first broad positive result by showing that every finite group is the group of self-homotopy equivalences of a (rational) space. In this talk, we improve and refine the ideas of Costoya-Viruel, both to solve more general realisability questions and to study this problem in different frameworks.
Title: On the equality of the wobbly and shaky loci
Abstract: The geometric Langlands program (GLP) generalises the fact that a rank one local system on a smooth projective curve uniquely extends to its Jacobian. According to the GLP, local systems of rank n should produce D-modules on the moduli space of GL(n,C)-bundles with the so called Hecke eigenproperty. Higgs bundles, through their Hitchin system, provide a way to dominate the moduli space of bundles by the Jacobian of a curve. A program by Donagi-Pantev aims at using this fact to deduce geometric Langlands from the case of rank one bundles, which is well understood. This requires, as a first step, to understand the resolution of a rational map. I will talk about a joint result with Christian Pauly, and the applicability of the ideas therein towards the proof of a conjecture by Donagi and Pantev, according to which the indeterminacy locus of the aforementioned rational map can be described in terms of wobbly bundles.