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Cohomology in algebra, geometry, physics and statistics

usually takes place every Wednesday Institute of Mathematics of ASCR, Žitná 25, Praha 1
Chair: Anton Galaev, Roman Golovko, Igor Khavkine, Alexei Kotov, Hong Van Le and Petr Somberg

In this seminar we shall discuss topics concerning constructions and applications of cohomology theory in algebra, geometry, physics and statistics. In particular we shall discuss in first four seminars the relations between vertex algebras and foliations on manifolds, Gelfand-Fuks cohomology on singular spaces, cohomology of homotopy Lie algebras. The expositions should be accessible for all participants.

The Topological Half of the Grothendieck-Hirzebruch-Riemann-Roch Theorem

Eugenio Landi
Pennsylvania State University
Wednesday, 9. November 2022 - 13:30 to 14:30
ZOOM meeting
The HRR theorem famously states that the holomorphic Euler characteristic of X with coefficients in a holomorphic vector bundle V equals $\int_X ch(V)td(X)$. This can be rewritten as two theorems: the first one, analytical, identifying $\chi(X,V)$ with the K-theoretic pushforward of V to the point, while the second, purely topological, identifying the pushforward with the integral. The same can be said for the GHRR theorem and pushforwards along proper holomorphic maps between holomorphic manifolds. I will focus on the second half, introducing orientations and pushforwards in cohomology and explaining how the presence of the Todd class is natural and expected.
We shall open the seminar room+ZOOM meeting at 13.15 for coffee

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Steenrod Algebra and Equivariant Algebraic Topology

Foling Zou
University of Michigan
Wednesday, 2. November 2022 - 13:30 to 14:30
ZOOM meeting
Steenrod algebra give stable cohomology operations.
Non-equivariantly, the dual Steenrod algebra spectrum is a wedge of
suspensions of HZ/p. It is explicitly computed and fundamental in a lot of
computations in algebraic topology. Consider the equivariant
Eilenberg–Maclane spectra H = HZ/p for the cyclic group of order p. I will
talk about the computation of the dual Steenrod algebra of H. It turns out
that when p is odd, H ∧ H is a wedge of suspensions of H and another
spectrum, which we call HM. This is joint work with Po Hu, Igor Kriz, and
Petr Somberg
We shall open the seminar room+ZOOM meeting at 13.15 for coffee
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On some diffeologies on spaces of probabilities, spaces of measures, and spaces of means

Jean-Pierre Magnot
University d'Angers, France
Wednesday, 26. October 2022 - 13:30 to 14:30
ZOOM meeting,
Passing from probabilities to finite measures, from finite measures to measures, and from measures to infinite dimensional integrals,we develop examples of diffeologies on each of these classes of spaces, partially from works of the author, and partially from other approaches in the existing literature. The highlighted spaces include finite and infinite configurations, Monte-carlo sequences, Radon measures, Haar and Lebesgue integrals in the space of connections, and an infinite dimensional Lebesgue mean. The highlighted diffeologies include functional diffeology, vague diffeology, the Cauchy diffeology and pro-finite diffeologies. The exposition intends to give a rigorous differential geometric setting for  some actual differential geometry related to probability and integration theory.... more

Floer-Novikov (co)homology associated with non-abelian coverings and symplectic fixed points

Hong Van Le
Institute of Mathematics of the Czech Academy of Sciences
Wednesday, 19. October 2022 - 13:30 to 14:30
in IM rear building, blue lecture room, ground floor + ZOOM meeting
In my talk   I shall explain   our    with Kaoru Ono   construction   of  Floer-Novikov  cohomology  groups $HFN^* (M^{\Gamma_\xi \times H},\xi, Q)$ defined on a regular covering $M^{\Gamma_\xi \times H}$ of a  compact   symplectic  manifold   $(M, \omega)$ with  transformation group  $\Gamma_\xi \times  H$  and associated  to  a    locally symplectic isotopy ${\{\varphi_t\}}$ of $(M, \omega)$ with  flux $\xi \in H ^1 (M, R)$. Then  $H$ acts naturally on $HFN^* (M^{\Gamma_\xi \times H},\xi, Q)$.  For a subgroup $G \subset H$  denote  by $(HFN^* (M^{\Gamma_\xi \times H},\xi,  Q))^G$  the   subgroup of $HFN^* (M^{\Gamma_\xi \times H}, \xi, Q)$  consisting   of the fixed  points of the $G$-action.  We  prove that  the   rank   of  $(HFN^* (M ^{\Gamma_\... more