We study the one-loop corrections to correlation functions in quantum field theories in the Anti de Sitter space-time. Our calculation shows the existence of non-local counterterms which however respect the AdS isometry. Our arguments are general and applicable to general non-conformal AdS field theories. We also explain why calculations in Euclidean and Lorentzian signatures should differ even at the leading order in non globaly hyperbolic manifolds.

For an arbitrary morphism of (super)manifolds, the pull-back is a linear map of the space of functions. In 2014 Th.Voronov have introduced thick morphisms of (super)manifolds which define a generally non-linear pull-back of functions. This construction was suggested as an adequate tool to describe $L_infty$ morphisms of algebras of functions provided with the structure of a homotopy Poisson algebra. It turns out that if you go down from `heaven to earth’, and consider usual (not super!) manifolds, then we come to constructions which have natural interpretation in classical and quantum mechanics. In particular in this case the geometrical object which defines the thick diffeomorphism becomes an action of classsical mechanics, and pull-back of the thick diffeomorphism with a quadratic action give a spinor representation. I will define a thick morphism and will tell shortly about their application in homotopy Poisson algebras. Then I will discuss the relation of thick morphisms with notions such as the action in classical mechanics and spinors.

The talk is based on the work: « Thick morphisms of supermanifolds, quantum mechanics and spinor representation’, J.Geom. and Phys., 2019, article Number: 103540,
https://doi.org/10.1016/j.geomphys.2019.103540,
arXiv:1909.00290
Authors: Hovhannes Khudaverdian, Theodore Voronov

In this talk, we will explain the final outcome on the Beilinson-Bloch-Kato conjecture for motives coming from certain automorphic representations of GL(n) x GL(n+1), of our recent project with Yichao Tian, Liang Xiao, Wei Zhang, and Xinwen Zhu. In particular, we show that the nonvanishing of the central L-value of the motive implies the vanishing of the corresponding Bloch-Kato Selmer group. We will also explain the main ideas and ingredients of the proof.

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Lecture 2

Kontsevich’s oriented graphs with decorated edges and outgoing edge ordering at vertices. The Poisson complex, the Poisson cohomology.

 * Universal (with respect to all affine Poisson manifolds) symmetries of the brackets.

 * The Leibniz graphs. Examples of factorizations via the Leibniz graphs.

Lecture 1

The structure of differential graded Lie algebra (dgLa) on the vector space of graphs with a wedge ordering of edges. « Zero graphs » = minus themselves. The vertex-blow-up differential, Lie bracket, its Jacobi identity. The determining properties of the complex (identities involving zero graphs).

 * Examples of nontrivial cocycles: the tetrahedron, 5-wheel + prism. The (2k+1)-wheels as markers of nontrivial cocycles originating from the Grothendieck–Teichmueller Lie algebra.

Lecture 3

The graph orientation morphism: constructing a symmetry by using a graph cocycle.

 * Endomorphisms of the space of multivector fields. The Schouten bracket, its own Jacobi identity; the master-equation. The Richardson–Nijenhuis bracket, its own Jacobi identity; the master equation.

 * The edge (the stick in the construction of graph differential). The graph orientation morphism Or takes the stick to the Schouten bracket.

 * Verifying the property of the image of a graph cocycle under Or to be a Poisson cocycle. Canonical and incidental solutions of the problem of factorization via the Jacobiator. Topological identities in the spaces of Leibniz graphs.

 * Lie algebra homomorphism: from graph cocycles to symmetries of Poisson brackets.

Lecture 4

Open problems about the graph complex and Poisson bracket symmetries: the combinatorics of cocycles and Poisson non-triviality of the Kontsevich flows.

 * Deformation quantization: the noncommutative associative star-product, its associator, the factorization problem.

 * The action of graph complex on the space of Kontsevich’s star-products.

To sum up: We will define complementary series for hyperbolic groups and prove their irreducibility.

More precisely: The complementary series representations are a family of unitary representations that can be realized on the Gromov boundary of the hyperbolic group. They can be viewed as a one-parameter deformation of the quasi-regular representation arising on the boundary, « sometimes » approaching the trivial representation, in a certain sense. The starting point of this work is to find a suitable scalar product in order to unitarize the complementary series. Then, a spectral estimates combined with counting estimates enable us to prove an ergodic theorem à la Bader-Muchnik to achieve the irreducibility.

Joint work with Kevin Boucher and Jean-Claude Picaud.

In this course, we will discuss the use of convex integration to construct wild weak solutions in the context of the Euler and Navier-Stokes equations. In particular, we will outline the resolution of Onsager’s conjecture as well as the recent proof of non-uniqueness of weak solutions to the Navier-Stokes equations.

Onsager’s conjecture states that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy, and conversely, there exit weak solutionslying in any Hölder space with exponent less than 1/3 which dissipate energy. The conjecture itself is linked to the anomalous disspoation of energy in turbulent flows, which has been called the zeroth law of turbulence.

For initial datum of finite kinetic energy, Leray has proven that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. We prove that weak solutions of the 3D Navier-Stokes equations are not unique, within a class of weak solutions with finite kinetic energy. The non-uniqueness of Leray-Hopf solutions is the subset of a famous conjecture of Ladyzenskaja in ’69, and to date, this conjecture remains open.

In this course, we will discuss the use of convex integration to construct wild weak solutions in the context of the Euler and Navier-Stokes equations. In particular, we will outline the resolution of Onsager’s conjecture as well as the recent proof of non-uniqueness of weak solutions to the Navier-Stokes equations.

Onsager’s conjecture states that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy, and conversely, there exit weak solutionslying in any Hölder space with exponent less than 1/3 which dissipate energy. The conjecture itself is linked to the anomalous disspoation of energy in turbulent flows, which has been called the zeroth law of turbulence.

For initial datum of finite kinetic energy, Leray has proven that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. We prove that weak solutions of the 3D Navier-Stokes equations are not unique, within a class of weak solutions with finite kinetic energy. The non-uniqueness of Leray-Hopf solutions is the subset of a famous conjecture of Ladyzenskaja in ’69, and to date, this conjecture remains open.

In this course, we will discuss the use of convex integration to construct wild weak solutions in the context of the Euler and Navier-Stokes equations. In particular, we will outline the resolution of Onsager’s conjecture as well as the recent proof of non-uniqueness of weak solutions to the Navier-Stokes equations.

Onsager’s conjecture states that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy, and conversely, there exit weak solutionslying in any Hölder space with exponent less than 1/3 which dissipate energy. The conjecture itself is linked to the anomalous disspoation of energy in turbulent flows, which has been called the zeroth law of turbulence.

For initial datum of finite kinetic energy, Leray has proven that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. We prove that weak solutions of the 3D Navier-Stokes equations are not unique, within a class of weak solutions with finite kinetic energy. The non-uniqueness of Leray-Hopf solutions is the subset of a famous conjecture of Ladyzenskaja in ’69, and to date, this conjecture remains open.

We investigate properties of some spherical fonctions defined on hyperbolic groups using boundary representations on the Gromov boundary endowed with the Patterson-Sullivan measure class. We prove sharp decay estimates for spherical functions as well as spectral inequalities associated with boundary representations. This point of view on the boundary allows us to view the so-called property RD (Rapid Decay, also called Haagerup’s inequality) as a particular case of a more general behavior of spherical functions on hyperbolic groups. Then I will explain how these representations are related to the so-called « complementary series ». The problem of the unitarization of such representations will be at the heart of the discussion.
If time permits, I will try to explain the idea of a constructive proof, using a boundary unitary representation, of a result due to de la Harpe and Jolissaint asserting that hyperbolic groups satisfy property RD.