topology@ub

Research Group in Algebraic Topology

Seminar



Guest talks Spring 2025

  • Jordi Daura (UB)
    20 February 2025, 12:00, Aula IA
    Accions de grups finits grans i accions iterades en varietats asfèriques I: Accions de grups finits grans
    Abstract: Donada una varietat tancada M, podem determinar els grups finits que actuen de forma efectiva i contínua a M? Aquesta és una pregunta bàsica de la teoria de grups finits de transformació, però resoldre-la és, en general, molt difícil. Per obtenir problemes més assequibles podem fer servir el següent enfocament: en comptes d’estudiar directament les propietats d’un grup finit F que actua sobre M, ens centrem en les propietats de l’acció restringida a certs subgrups H ⊆ F d’índex acotat per una constant C que depèn únicament de M. En aquesta xerrada explicarem diversos problemes que s’alineen amb aquesta filosofia i donarem resposta a aquests problemes per a varietats tancades connexes asfèriques localment homogènies i per a varietats tancades connexes i orientables que admeten una aplicació de grau diferent de zero cap a una nilvarietat.
  • Jonas Stelzig (LMU München)
    27 February 2025, 12:00, Aula IA
    Pluripotential formality over the rationals of toric varietis and compact homogeneous Kähler manifolds
    Abstract: In recent years, higher operations on complex manifolds have emerged, akin to classical Massey products, but dependent on the complex structure. By a theorem of Deligne, Griffiths, Morgan and Sullivan, compact Kähler manifolds are rationally formal, in particular all classical Massey products vanish. In contrast to this, the holomorphic higher operations are nontrivial quite often even on compact Kähler manifolds - e.g. on all smooth curves of genus at least two, or after finitely often blowing up (in specific configurations) a given manifold of dimension at least four. These holomorphic operations can be seen as obstructions to a holomorphic, bigraded notion of formality, called 'strong formality'. Given the abundance of examples with nontrivial products, one may wonder how restrictive this notion is. We show that two natural geometric classes are indeed strongly formal: Smooth toric manifolds and compact Kähler homogeneous spaces. In fact, we even show a stronger statement: They are strongly formal over Q. This is a new notion implying rational and strong formality but vice versa conversely.
  • Nicholas Meadows (Università di Bologna)
    6 March 2025, 10:30, Aula IA
    Definable obstruction theory
    Abstract: A series of recent papers by Bergfalk, Lupini and Panagiotopoulus developed the foundations of a field known as 'definable algebraic topology', in which classical cohomological invariants are enriched by viewing them as groups with a Polish cover. This allows one to apply techniques from descriptive set theory to the study of cohomology theories. In this talk, we will establish a 'definable' version of a classical theorem from obstruction theory, and use this to study the potential complexity of the homotopy relation on the space of continuous maps C(X, |K|), where X is a locally compact Polish space, and K is a locally finite countable simplicial complex. We will also characterize the Solecki groups of the Čech cohomology of X, which are canonical approximations of a Polishable subgroup of a Polish group.
  • Lander Hermans (Universiteit Antwerpen)
    6 March 2025, 12:00, Aula IA
    A minimal model for prestacks via Koszul duality for box operads
    Abstract: Prestacks are algebro-geometric objects appearing in both commutative and non-commutative algebraic geometry. For commutative geometers, they appear as algebraic stacks and enjoy a certain notoriety as complicated objects one is forced to work with. For non-commutative geometers, prestacks appear if we aim for a non-commutative deformation theory of schemes which moreover turns out to be better behaved. In this sense, prestacks model ‘non-commutative spaces’. For me, the most instructive way to think about prestacks is to view them as the 2-categorical version of presheaves of associative algebras (such as the structure sheaf of a scheme). In the algebraic world, presheaves are often called diagrams of algebras, and they can be encoded by an operad that is notoriously nonquadratic. As a result, classical Koszul duality for operads does not apply which impedes the typical operadic machinery. For prestacks the situation is even more dire: they have besides the quadratic-cubic relations present for presheaves, also cubic-quartic relations. In this talk, I will generalize (nonsymmetric) operads to box operads and sketch key components from our main result: a Koszul duality for box operads that is able to deal with the nonquadratic relations appearing for prestacks. In particular, we obtain a deformation complex by endowing the Gerstenhaber-Schack complex for prestacks with an L-infinity structure governing its deformations. In the second part of the talk, we will apply these results to the box operad encoding prestacks in order to obtain its minimal model. These strongly homotopy prestacks should play an analogous role as A-infinity algebras for associative algebras. This opens up the way for an operadic approach to the deformation and homotopy theory of (dg) prestacks.

Past Seminars