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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.

Natural differential operators from Lie algebra homology

Vit Tucek
Wednesday, 10. April 2019 - 11:30 to 12:30
in IM building, ground floor
 I will present a generalization of construction of Calderbank and Diemer of the so called Bernstein-Gelfand-Gelfand sequences of differential operators which act on sections of bundles associated to any Cartan geometry of parabolic type. These operators were originally constructed by Čap, Slovák and Souček and include many important and interesting operators, e.g. operators whose kernels consists of various Killing fields or operators whose kernels provide Einstein metrics. Classically these operators are all strongly overdetermined but the generalization produces also other types of operators such as the conformally invariant modification of the Laplace-Beltrami operator or the Dirac operator. The construction of Calderbank and Diemer proceeds by explicitely constructing homotopy transfer data for a twisted deRham sequence... more

Legendrian Isotopy from a Quantitative Viewpoint

Georgios Dimitroglou Rizell
Uppsala University
Wednesday, 3. April 2019 - 11:15 to 12:45
in IM building, ground floor

We use techniques from persistence homology applied to the Chekanov-Eliashberg algebra in order to obtain a restriction on the oscillatory norm of a contact Hamiltonian that displaces a Legendrian in the contact vector space from its image under the Reeb flow. These techniques are also used to show that a Legendrian which admits an augmentation cannot C0-approximate a loose Legendrian, and to obstruct the existence of small positive Legendrian loops. This is joint work with M. Sullivan.


Combinatorial homotopy theory for operads

Jovana Obradovic
Institute of Mathematics, CAS
Wednesday, 27. March 2019 - 11:30 to 12:30
in IM building, ground floor
We introduce an explicit combinatorial characterization of the minimal model of the
coloured operad encoding non-symmetric operads, introduced in [3]. The polytopes
of our characterization are hypergraph polytopes [1, 2] whose hypergraphs arise in a
certain way from rooted trees – we refer to them as operadic polytopes. In particular,
each operadic polytope displays the homotopy relating different ways of composing
the nodes of the corresponding rooted tree. In this way, our operad structure gener-
alizes the structure of Stasheff’s topological A ∞ -operad: the family of associahedra
corresponds to the suboperad determined by linear rooted trees. We then further
generalize this construction into a combinatorial resolution of the coloured operad
encoding non-symmetric cyclic operads.
[1] K. Došen, Z. Petrić, Hypergraph polytopes, Topology and its Applications 158,1405–1444, 2011.
[2] P.-L. Curien, J.... more

Segal-Shale-Weil representation and connected global analysis

Svatopluk Krysl
Wednesday, 20. March 2019 - 11:30 to 12:30
in IM building, ground floor

The Segal-Shale-Weil representation, which is a unitary representation of the double cover of a finite dimensional symplectic Lie group, was introduced by D. Shale a A. Weil in the beginning of sixties. This representation is unitary and splits into two irreducible (i.e., simple closed) modules. Its complex version can be induced to the principle bundle of the double cover of (complexified) symplectic frames defined over a symplectic manifold, similarly as spinor bundles are associated to spin structures on Riemannian manifolds. Resulting bundles are called symplectic spinor bundles.

For the induced bundle structures, one can define several operators of Dirac type (K. Habermann around 1995). However since the symplectic spin bundles are of infinite rank, analysis for these operators is rather difficult. Especially till now, just spectral problems over homogeneous symplectic manifolds can be treated and no qualitative results for more general cases are... more