Tag - Topology

Walter van Suijlekom: Non-commutative geometry and spectral triples

16th August, 2021

Our starting point is a spectral approach to geometry, starting with the simple question ’can one hear the shape of a drum?’ This was phrased by Mark Kac in the 1960s, and led to many developments in spectral geometry. For us, it is the motivation for considering spectral triples, which is the key technical device used to describe non-commutative Riemannian spin manifolds. We will give many motivating examples, and also explain how gauge symmetries naturally arise in this context. The connection to the other main theme of the workshop is found via the spectral action principle. It allows for a derivation of an action functional from any given spectral triple. This includes the Hermitian matrix model, but more interesting matrix models appear beyond. We will consider some recent developments for such models by deriving a perturbative series expansion for the spectral action.

Elba Garcia-Failde: Introduction to topological recursion

16th August, 2021

In this mini-course I will introduce the universal procedure of topological recursion, both by treating examples and by presenting the general formalism. We will study the classical case of the Hermitian matrix model in detail, which combinatorially corresponds to ribbon graphs, beginning from the loop equations, which correspond to Tutte’s recursion in the combinatorial setting. This will be the starting point to make the connection to free probability, which moreover provides a combinatorial way of exploring the variation of the topological recursion output when applying a symplectic transformation to the input. Apart from the (conjectural) property of symplectic invariance, topological recursion has many other interesting features and, together with its generalizations, has established connections to various domains of mathematics and physics, like intersection theory of the moduli space of curves and integrability. We will explain some of these properties and connections, giving several ideas why this is worth considering, and is the starting or gluing point of an active field of research, and finally hoping to instigate the search of new beautiful connections.

Yuri Berest: Spaces of quasi-invariants and homotopy Lie groups

22nd July, 2021

Quasi-invariants are natural algebraic generalizations of classical invariant polynomials of finite reflection groups. They first appeared in mathematical physics - in the work of O. Chalykh and A. Veselov on quantum integrable systems - in the early 1990s, and since then have found many interesting applications in other areas: most notably, representation theory, algebraic geometry and combinatorics. In this talk, I will explain how the algebras of quasi-invariants arise in topology: as cohomology rings of certain spaces naturally attached to compact connected Lie groups. Our main result is a generalization of a well-known theorem of A. Borel that realizes the algebra of classical invariant polynomials of a Weyl group W(G) as the cohomology ring of the classifying space BG of the corresponding Lie group G. Perhaps most interesting here is the fact that our construction of spaces of quasi-invariants is purely homotopy-theoretic. It can therefore be extended to some non-Coxeter (p-adic pseudo-reflection) groups, in which case the compact Lie groups are replaced by the so-called p-compact groups (a.k.a. homotopy Lie groups).

Giulio Lo Monaco: Vopěnka’s principle in ∞-categories

24th June, 2021

Vopěnka's principle has arisen as a model-theoretical statement, provably independent of ZFC set theory. However, there are a number of categorical ways of formulating it, preventing the existence of proper classes of objects with some conditions in presentable categories, and these are what our attention will be focused on. In particular, we will look at analogous statements in the context of ∞-categories and we will ask how these new statements interact with the older ones. Moreover, some of the consequences of Vopěnka's principle on classes of subcategories of presentable categories are investigated and to some extent generalized to ∞-categories. A parallel discussion is undertaken about the similar but weaker statement known as weak Vopěnka's principle.

Leonardo Larizza: Lax factorization systems and categories of partial maps

17th June, 2021

Factorization systems describe morphisms in a category by factorizing them into pairs of composable morphisms. Their definition depends on a kind of orthogonality relation between morphisms, which entails the existence of some diagonal morphisms for certain squares. In this seminar we present the new notion of lax weak orthogonality between morphisms, which involves lax squares and the factorization systems it generates. Then we will introduce lax versions of functorial and algebraic weak factorization systems and some of their properties. These lax factorization systems are discussed, keeping the theory of ordinary factorization systems as a blueprint and providing useful properties. An overview of the examples of such lax factorization systems is presented in the context of partial maps. We conclude with a discussion of general constructions of these examples and their description in the particular case of sets with partial maps.

Uri Bader: Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity

6th June, 2021

Compact hyperbolic manifolds are very interesting geometric objects. Maybe surprisingly, they are also interesting from an algebraic point of view: They are completely determined by their fundamental groups (this is Mostow's Theorem), which is naturally a subgroup of the rational valued invertible matrices in some dimension, GLn(ℚ). When the fundamental group essentially consists of the integer points of some algebraic subgroup of GLn we say that the manifold is arithmetic. A question arises: is there a simple geometric criterion for arithmeticity of hyperbolic manifolds? Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen. In a recent work with Fisher, Miller and Stover we proved this conjecture. Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which Alex Furman and I have developed in recent years. In my talk I will survey the subject and focus on the relation between the geometric, algebraic and arithmetic concepts

Raffael Stenzel: Proof relevance in higher topos theory

27th May, 2021

In the short course of its definition and early exploration, the theory of higher toposes (by which I specifically mean (∞,1)-toposes) has been found to exhibit various traits which appear rather odd from the perspective of ordinary topos theory. Motivated by the fact that the internal language of every higher (Grothendieck) topos is a univalent type theory - and hence is intrinsically 'proof relevant' - we reconsider the basic characteristic notions associated to a higher topos from a purely logical proof relevant point of view.

Given a small ∞-category C, this will motivate the notion of a logical structure sheaf on C whose ideals correspond exactly to the left exact localizations of the infinity-category [Cop, S] of presheaves over C. This in turn will naturally lead to a corresponding notion of generalized Grothendieck topologies on C which, first, capture all higher toposes embedded in [Cop, S], and second, correspond exactly to the classical notion of Grothendieck topologies in the monic (i.e. the proof irrelevant) context. We will see that these notions induce a Kripke-Joyal semantics valued in spaces (rather than in the classical subobject classifier) in obvious fashion as well. In the end of the talk we will take a look at a few examples of such topologies and, if time permits (which it rarely ever does, time appears to be pretty absolute when it comes to this), end with a discussion of some open questions.

Stephan Tillmann: On the space of properly convex projective structures

20th May, 2021

This talk will be in two parts. I will outline joint work with Daryl Cooper concerning the space of holonomies of properly convex real projective structures on manifolds whose fundamental group satisfies a few natural properties. This generalizes previous work by Benoist for closed manifolds. A key example, computed with Joan Porti, is used to illustrate the main results.

Tashi Walde: Higher Segal spaces via Higher Excision

20th May, 2021

Starting from the classical Segal spaces, Dyckerhoff and Kapranov introduced a hierarchy of what they call higher Segal structures. While the first new level (2-Segal spaces) has been well studied in recent years, not much is known about the higher levels and the hierarchy as a whole.

In this talk I explain how this hierarchy can be understood conceptually in close analogy to the manifold calculus of Goodwillie and Weiss. I describe a natural 'discrete manifold calculus' on the simplex category and on the cyclic category, for which the polynomial functors are precisely the higher Segal objects. Furthermore, this perspective yields intrinsic categorical characterizations of higher Segal objects in the spirit of higher excision.

Walter Tholen: Spaces vs Categories, Perfect Maps vs Discrete Fibrations

6th May, 2021

We consider perfect maps of topological spaces and discrete cofibrations of categories to guide us into Burroni's notions of T-category and T-functor. In that environment we establish a so-called comprehensive factorization system that entails the classical Street-Walters system, as well as the (antiperfect, perfect) system for continuous maps of Tychonoff spaces known since the 1960s.