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ABSTRACTS Version: July 5, 2018 LOCAL ORGANIZING COMMITEE C.ˇ Burd´Ik(Chairman), G

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ABSTRACTS Version: July 5, 2018 LOCAL ORGANIZING COMMITEE C.ˇ Burd´Ik(Chairman), G ABSTRACTS Version: July 5, 2018 LOCAL ORGANIZING COMMITEE C.ˇ Burd´ık(Chairman), G. Chadzitaskos, O. Hul´ık,J. Hrivn´ak, B. Jurˇco,P. Kerouˇs, M. Korbel´aˇr,S. Krysl, M. Schnabl, L. Snobl,ˇ V. Souˇcek,J. Tolar, O. Navr´atil,S. Poˇsta,P. Preˇsnajder, R. von Unge, P. Zusmanovich CONTACT Prof. Cestm´ırBurd´ıkˇ Department of Mathematics Faculty of Nuclear Sciences and Physical Engineering Czech Technical University Trojanova 13, 120 00 Prague Czech Republic E-mail: group32@fjfi.cvut.cz WWW: http://http://www.group32.cz/ PLENARY SPEAKERS Caron-Huot, Simon Unexpected symmetries and causality Physical systems sometimes exhibit unexpected, or `hidden' symmetries. A prominent example is the Kepler problem, whose Laplace-Runge-Lenz vector ensures that planetary orbits form closed ellipses, and accounts for the degeneracies of the hydrogen spectrum. While broken in Nature by relativistic effects, I will describe how it survives in a very special, integrable, model: supersymmetric Yang-Mills theory in the planar limit. I will emphasize the role of relativistic causality as a powerful computa- tional tool in this context. Hoˇrava,Petr Aristotelian Supersymmetry We introduce and study supersymmetric gauge and matter theories in nonrelativistic spacetimes with Aristotelian symmetries, in diverse dimensions. Under certain cirmunstances, such nonrelativistic the- ories can naturally flow at low energies to their relativistic super-Poincar´einvariant counterparts, thus providing a new mechanism for the short-distance completion of relativistic supersymmetry. Iachello, Francesco Algebraic models of quantum many-body systems I will give an overview of the method of spectrum generating algebras (SGA) and dynamical symme- tries (DS) and apply it to a class of bosonic systems (the so-called s-b boson models) with SGA ≡ U(n). I will then discuss quantum phase transitions (QPT ) in these systems by introducing the coset spaces U(n)=U(n − 1) ⊗ U(1). Finally, I will consider spectrum generating superalgebras (SGSA) and dy- namical symmetries (SUSY ) of a class of Bose-Fermi systems with SGSA ≡ U(n=m). Isaev, Alexey Bethe subalgebras in affine Birman{Murakami{Wenzl algebras and flat connections for q-KZ equations Commutative sets of Jucys-Murphyelements for affine braid groups of A(1);B(1);C(1);D(1) types were defined. Construction of R-matrix representations of the affine braid group of type C(1) and its distinguish commutative subgroup generated by the C(1)-type Jucys{Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik- Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the C(1)-type Jucys{Murphy elements. We specify our general construction to the case of the Birman{ Murakami{Wenzl algebras. As an application we suggest a baxterization of the Dunkl{Cherednik elements Y's in the affine Hecke algebra of type A. Kac, Victor Integrable Hamiltonian partial differential and difference equations and related algebraic structures The related algebraic structures are additive and multiplicative Poisson vertex algebras. This turned out to be an adequate tool for the construction and study of integrable Hamiltonian partial differential and difference equations. The most famous of the former is the KdV equation, describing water waves in narrow channel. The most famous of the latter is the Volterra lattice, describing predator-prey in- teractions. Some knowledge of Lie algebras, but no knowledge of integrable systems, will be assumed. Kobayashi, Toshiyuki Branching problems and symmetry breaking operators I begin with general results on restricting representations of reductive groups to their subgroups. Then I will focus on a concrete geometric question arising from conformal geometry, giving the complete classification of conformally covariant symmetry breaking operators for differential forms on the model space. Some of the symmetry breaking operators are differential operators and some others are ob- tained as the meromorphic continuation of integral operators. Lechtenfeld, Olaf Rational Maxwell knots via de Sitter space We set up a correspondence between solutions of the YangDMills equations on <×S3 and in Minkowski spacetime via de Sitter space. Some known Abelian and non-Abelian exact solutions are rederived. For the Maxwell case we present a straightforward algorithm to generate an infinite number of explicit solutions, with fields and potentials in Minkowski coordinates given by rational functions of increasing complexity. We illustrate our method with some nontrivial examples. Marquette, Ian Higher order superintegrability, Painlev´etranscendents and representations of polynomial algebras I will review results on classification of quantum superintegrable systems on two-dimensional Euclidean space allowing separation of variables in Cartesian coordinates and possessing an extra integral of third or fourth order. The exotic quantum potential satisfy a nonlinear ODE and have been shown to ex- hibit the Painlev´eproperty. I will also present different constructions of higher order superintegrable Hamiltonians involving Painlev´etranscendents using four types of building blocks which consist of 1D Hamiltonians allowing operators of the type Abelian, Heisenberg, Conformal or Ladder. Their inte- grals generate finitely generated polynomial algebras and representations can be exploited to calculate the energy spectrum. I will point out that for certain cases associated with exceptional orthogonal polynomials, these algebraic structures do not allow to calculate the full spectrum and degeneracies. I will describe how other sets of integrals can be build and used to provide a complete solution. Ragoucy, Eric Bethe vectors, scalar products and form factors in integrable models We review our series of works on the nested algebraic Bethe ansatz applied to integrable models based on algebras with rank higher than 2. We present some explicit representations for the Bethe vectors and their scalar products, in the framework of periodic generalized models, that encompass all inte- grable spin chain models with (twisted) periodic boundary conditions. Reshetikhin, Nicolai Hamiltonian surperintegrable systems and their quantization A superintegrable system on a 2n dimensional phase space has k ≤ n commuting integrals of motion. The case k = n corresponds to the Liouville integrability. When k < n the dimension of Liouville tori is k < n. The extreme case of k = 1 is when the system is maximally superintegrable and, in particular, all compact trajectories are periodic. The model of a hydrogen atom is an example of a superintegrable system. In the first part of the talk I will remind the definition of a superintegrable system and present multiple examples related to Poisson Lie groups. The second part will be focused on the relation between the Racah-Wigner 6j symbols for simple Lie groups and corresponding inte- grable and superintegrable systems. Tarasov, Vitaly Completeness of the Bethe ansatz for integrable models I will review results on completeness of the Bethe ansatz for the Gaudin and XXX-type models on tensor products of finite-dimensional representations of the Lie algebra glN . Those results are of ge- ometric flavour. For the Gaudin models, the completeness of the Bethe ansatz states an isomorphism of the action of the transfer-matrices with the regular representation of the algebra of functions on an intersection of Schubert cells relative to osculating flags in spaces of quasi-exponentials. In other words, Bethe eigenvectors are in bijection with ordinary differential equations whose spaces of solu- tions are spanned by quasi-exponentials. For the XXX-type models, differential equations are replaced by difference equations with spaces of solutions spanned by quasi-exponentials. Geometric models for the action of the transfer-matrices are available for the gl2 case and for tensor products of vector rep- resentations for arbitrary N. However, the whole picture is not completely worked out yet for N > 2 . Tolstoy, Valeriy All basic quantum quasitriangular Hopf deformations of o(4; C) and of its real forms: Euclidean o(4), Lorentzian o(3; 1), Kleinian o(2; 2) and quaternionic o∗(4) symmetries Complete classification of nonisomorphic (basic) quantum quasitriangular Hopf deformations of the complex orthogonal Lie algebra o(4; C) and of its real forms: Euclidean o(4), Lorentzian o(3; 1), Kleinian o(2; 2) and quaternionic o∗(4) Lie algebras is given in terms of classical r-matrices. All the r-matrices are skew-symmetric, and they satisfy homogeneous and nonhomogeneous classical Yang- Baxter equation. Using the isomorphisms o(4; C) ' o(3; C) ⊗ o(3; C), o(3; C) ' sl(2; C) we show that these classical r-matrices are multiparametric subordinated sums of standard and Jordanian classical r-matrices for sl(2; C) and its real forms su(2) and su(1; 1) ' sl(2; R) and also of Abelian classical r-matrices for sl(2; C) ⊗ sl(2; C) and its real forms. Such structure of the basic classical r-matrices makes possible explicitly to find all basic quantum quasitriangular Hopf deformations of o(4; C) and of all its real forms: o(4), o(3; 1), o(2; 2) and o∗(4). van Holten, Jan-Willem D = 1 Supergravity as a constrained system D = 1 supergravity is a framework to model systems with finite numbers of commuting and anti- commuting degrees of freedom in a supersymmetric and reparametrization-invariant way. As a gauge theory
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