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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.

Condensed mathematics and set theory

Chris Lambie-Hanson
Institute of Mathematics, Czech Academy of Sciences
Wednesday, 16. November 2022 - 13:30 to 14:30
Institute of Mathematics, the blue lecture room in the rear building, + ZOOM meeting
Recently, Clausen and Scholze initiated the study of condensed mathematics, providing a framework in which to do algebra in situations in which the algebraic structures under consideration also carry topological information. The fundamental idea is to replace the categories of topological spaces or topological abelian groups, which are poorly behaved algebraically, with the more algebraically robust categories of condensed sets or condensed abelian groups. In this talk, we will give a very brief introduction to condensed mathematics and sketch a couple of very basic applications of set theoretic techniques to foundational questions therein. Time permitting, we will also briefly touch on related applications of these set theoretic ideas to other topics in homological algebra
We shall open the seminar room+ZOOM meeting at 13.15 for coffee  and close at 15.00
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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