**Guest talks Spring 2023**

- Vicente Muñoz (Universidad Complutense de Madrid)

22 May 2023, 12:00, IMUB

Strongly inflexible manifolds and mapping degree sets

**Abstract:**Let*M*,*N*be two oriented closed connected manifolds of dimension*n*. We define the mapping degree set deg(*M*,*N*) as the set of degrees of maps from*M*to*N*. It is relevant to construct inflexible manifolds*M*, i.e., when deg(*M*,*M*) is bounded, and strongly inflexible manifolds*M*, i.e., deg(*N*,*M*) is bounded for all*N*. They serve to produce functorial seminorms on*n*-manifolds in the vein of the simplicial volume of Gromov. Using rational homotopy methods, simply connected inflexible manifolds have been constructed. We show that the existing examples in the literature are actually not strongly inflexible. To date, no strongly inflexible simply connected manifold is known.

On the other hand, one may ask which sets of integers can appear as deg(*M*,*N*) for some*M*,*N*. By cardinality reasons, not all (infinite) sets can be mapping degree sets. We prove that any finite set of integers*A*containing 0 is a mapping degree set for some choice of*M*and*N*. We extend this question to the rational homotopy theory setting, where an affirmative answer is also given, using Sullivan models.

This is joint work with C. Costoya and A. Viruel.

- Magdalena Kędziorek (Radboud Universiteit)

10 May 2023, 12:15, IMUB

Splitting rational incomplete Mackey functors

**Abstract:**One of the important (but somehow isolated) results in rational equivariant homotopy theory is the splitting theorem of Greenlees-May/Thévenaz-Webb for rational*G*-Mackey functors, where*G*is a finite group. This theorem, together with the description of the split pieces, has interesting consequences. In particular it implies that the category of rational*G*-Mackey functors is of homological dimension zero, when*G*is a finite group.

In this talk I will present joint work with D. Barnes and M. Hill generalising the splitting to rational*incomplete**G*-Mackey functors. Here*incomplete*refers to the existence of some, but perhaps not all, additive transfers. Such a situation is modelled by an*N*_{∞}-operad of Blumberg-Hill. I will also describe the split pieces and discuss the consequences of this result for the homological dimension of the category of rational incomplete*G*-Mackey functors.

- Sergio García (Universidad Autónoma de Madrid)

20 March 2023, 15:00, IMUB

Una introducció a l'espectre de Khovanov

**Abstract:**En aquesta xerrada, s'oferirà una descripció general de l'espectre de Khovanov, un invariant important en la teoria de nusos que categorifica l'homologia de Khovanov. S'introduiran breument el polinomi de Jones i l'homologia de Khovanov, per a continuació presentar la construcció de l'espectre de Khovanov, que es donarà per a enllaços anulars. Així mateix, es mostrarà com aquesta construcció permet categoritficar l'homologia de Khovanov anular quàntica, que és una deformació de l'homologia de Khovanov anular.

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