topology@ub

Research Group in Algebraic Topology

Seminar



Guest Talks Fall 2024

  • Leopold Zoller (LMU München)
    30 October 2024, 12:00, Aula B6
    On the Toral Rank Conjecture
    Abstract: The Toral Rank Conjecture states that the total Betti number of a compact space with a free torus action is at least as big as the total Betti number of the acting torus. This conjecture, originally due to Steve Halperin, has been open for over 40 years and has become a central open question in rational homotopy theory. In this talk, we will give an introduction to the conjecture and its current state. Furthermore, we will discuss connections to related conjectures in commutative algebra and recent advances in the field. This talk is partly based on joint work with Manuel Amann.
  • Jeffrey Bergfalk (UB)
    26 November 2024, 12:00, IMUB
    An introduction to condensed mathematics
    Abstract: In recent years, Dustin Clausen and Peter Scholze have systematically developed a far-reaching framework for amalgamating topological and algebraic constructions which they term "condensed mathematics". Loosely speaking, the material of this framework is categories of sheaves on a site with underlying category that of compact Hausdorff spaces or, equivalently, that of "profinite sets" or Stone spaces. The centrality, via Stone duality, of the latter objects to set theory is one locus of broader interactions between condensed mathematics and set theory; others involve the infinitary combinatorics of derived condensed categories. The second of our two-part series of talks will focus on the latter. This first one will focus simply on introducing the condensed landscape; no previous knowledge of it will be assumed.
  • Geoffroy Horel (Université Sorbonne Paris Nord)
    3 December 2024, 12:00, IMUB
    Motivic structure on unstable homotopy types
    Abstract: I will explain some algebraic models for rational or integral homotopy types with the motivation of formalizing what it means for an affine algebraic group to act on a homotopy type. As an application, I will sketch a construction of a motivic structure on the homotopy type of algebraic varieties.
  • Jeffrey Bergfalk (UB)
    10 December 2024, 12:00, IMUB
    Derived functors and anima in condensed mathematics
    Abstract: : In this talk, the second in a two-part series, we discuss derived categories and functors in condensed mathematics and their sensitivities to infinitary combinatorics. For example: after briefly reviewing the nonabelian derived category, or animation, operation Ani( - ) (yielding the infinity category of "spaces" or "anima" when applied to the category of sets, and the derived infinity category when applied to the category of abelian groups, for example), we show that the product of two compact anima is not, in general, compact. We then describe a number of Ext computations in the field which ultimately depend upon set theoretic assumptions beyond the ZFC axioms.

Past Seminars