Seminar

The homotopy groups of CW complexes and of the mapping spaces between them are notoriously difficult to compute. However, if one disregards torsion, rational homotopy theory is very effective and can easily solve such problems. Moreover, it produces efficient invariants…
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In 1990, V. Drinfeld introduced the Grothendieck-Teichm¨uller group GT. This group receives the homomorphism from the absolute Galois group GQ of rational numbers and this homomorphism is injective due to Belyi’s theorem. Many challenging questions about GT are motivated by…
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A fundamental class of problems in algebraic topology are the realization problems: when is a given algebraic object realizable as an algebraic invariant of a space? For example, one might ask when a module over the Steenrod algebra arises as…
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The configuration space Confn(X) consists of ordered n-tuples of pairwise distinct points in a topological space X. This notion can be generalized by allowing certain points to collide while requiring others to remain distinct. More precisely, given a simple connected…
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Cobordisms, a fundamental concept of differential topology, play a deep role in both homotopy theory and higher category theory. In homotopy theory, (framed) n-manifolds up to cobordism are isomorphic to the n-th stable homotopy group of spheres. This isomorphism preserves…
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Many categories of algebraic structures, like abelian groups and commutative rings, can be presented in terms of “algebraic theories”, a precise term which was given a categorical formulation by Lawvere. Quillen, using the simplex category, developed a construction of “nonabelian…
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Hochschild homology was originally introduced as an invariant of associative algebras that generalizes the commutator. In subsequent decades, it has been extended both in categorical directions, such as to dg-categories, and in topological ones, such as to ring spectra. These…
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Classically, (∞, 1)-categories are modeled by complete Segal spaces. However, many examples first arise as non-complete Segal spaces, which are then completed abstractly to obtain an (∞, 1)-category. With Nuiten, we construct a model of (∞, 1)-categories in which completeness…
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The goal of this talk is to introduce the notion of a Grothendieck (∞, 2)-topos as a presentable (∞, 2)-category satisfying a categorified version of the descent axiom for (∞, 1)- topoi of Rezk-Lurie, which we call fibrational descent.As the…
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