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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.

(MODULI) SPACES OF RIEMANNIAN METRICS
Wilderich Tuschmann
Karlsruhe Institute of Technology
Wednesday, 24. February 2021 - 11:30 to 12:30
ZOOM meeting
Consider a smooth manifold with a Riemannian metric satisfying some sort of curvature constraint like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasaki, etc. A natural question to study is then what the space of all such metrics does look like. Moreover, one can also pose this question for corresponding moduli spaces of metrics, i.e., quotients of the former by (suitable subgroups of) the diffeomorphism group of the manifold, acting by pulling back metrics.

These spaces are customarily equipped with the topology of smooth convergence on compact subsets and the quotient topology, respectively, and their topological properties then provide the right means to measure 'how many'... more

Boundary conditions and edge modes in gauge theories
Alexander Schenkel
University of Nottingham
Wednesday, 13. January 2021 - 11:30 to 12:30
ZOOM meeting
The fields of a classical gauge theory form a smooth groupoid (aka stack) with morphisms given by gauge transformations. From this perspective, the concept of "equality" of two gauge fields A and A' is not a property but rather additional data given by the choice of a gauge transformation A ---> A' which witnesses that A and A' are "the same". In this talk, I will explain how this higher-categorical point of view is useful to study gauge theories on manifolds with boundaries and defects. In particular, I will show that the additional data witnessing boundary conditions are precisely the famous edge modes from physics. As examples, I will discuss 3d Abelian Chern-Simons theory on manifolds with boundary, which is physically describing the quantum Hall system, and also the 4d holomorphic Chern-Simons theory of Costello and Yamazaki where the edge modes on surface defects... more

Deformations of symplectic foliations
Alfonso Tortorella
KU Leuven
Wednesday, 6. January 2021 - 11:30 to 12:30
ZOOM
In this talk, based on joint work with Stephane Geudens and Marco Zambon, I develop the deformation theory of symplectic foliations, i.e. regular foliations equipped with a leaf-wise symplectic form. The main result is that each symplectic foliation is attached with an L_\infty algebra controlling its deformation problem. Indeed, we establish a one-to-one correspondence between the small deformations of a given symplectic foliation and the MC elements of the associated L_\infty algebra. Further, we prove that, under this one-to-one correspondence, the equivalence by isotopies of symplectic foliations agrees with the gauge equivalence of MC elements. Finally, we show that the infinitesimal deformations of symplectic foliations can be obstructed.

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Chain models of string topology coming from symplectic geometry
Pavel Hajek
University Hamburg
Wednesday, 16. December 2020 - 11:30 to 12:30
in IM back building, ground floor, blue lecture room + ZOOM meeting
I will recall loop spaces, natural structures on their
homology and the relation to symplectic geometry of the cotangent bundle
(specifically to chain level structures defined by counting holomorphic
curves). I will then zoom in on the equivariant case and a chain model
based on de Rham forms and Chern-Simons theory. I will show some
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