Faculty of Physics University of Warsaw > Events > Seminars > The Trans-Carpathian Seminar on Geometry & Physics

The Trans-Carpathian Seminar on Geometry & Physics

(formerly Geometric Seminar)

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room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Andrey Krutov (Instytut Matematyczny PAN)

On gradings modulo 2 of simple Lie algebras in characteristic 2

In characteristic 2, the classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finite dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, (see arXiv:1407.1695 S. Bouarroudj, A. Lebedev, D. Leites, and I. Shchepochkina, "Classifications of simple Lie superalgebras in characteristic 2"). No classification of gradings was known for any type of simple Lie algebras, expect restricted Zassenhaus algebras (a.k.a. Witt algebras, i.e., Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of simple Lie algebras: special linear, two inequivalent orthogonal, and projectivizations of their derived algebras, except for psl(4) for which a conjecture is given. All of the corresponding superizations are known, but a corollary provesnon-triviality of a deformation of a simple (3|2)-dimensional Liesuperalgebra (new result). For nonrestricted Zassenhaus algebras on one indeterminate of hight n, there is an (n-2)-parametric family of modulo 2 gradings; all but one of the corresponding simple Lie superalgebras are new. Joint work with Alexei Lebedev (Stockholm).

room 106 IM PAN, Śniadeckich 8, Ip at 16:30

Frédéric Barbaresco (Thales Air Systems)

Symplectic Geometry of Heat based on Souriau Lie Groups Thermodynamics and Koszul Hessian Information Geometry

We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. The Souriau-Fisher metric is linked to KKS (Kostant–Kirillov–Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We conclude with Higher order extension of Souriau model based on works of R..S Ingarden and W. Jaworski. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Paweł Urbański (KMMF)

Linear distributions on vector bundles and reductions

A linear distribution on a vector bundle E is a double vector subbundle of TE. Such distribution has its counterpart on the dual bundle E*. Using techniques of double vector bundles we analyse properties of linear distributions essential for Lagrangian and Hamiltonian reductions (i.e. for E=TQ). It generalizes the framework for the Routh reduction.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Andrew J. Bruce (University of Luxembourg)

Modular classes of Q-manifolds

Q-manifolds are supermanifolds equipped with an odd vector field that self-commute, such vector field are called homological. Such objects are found in mathematical physics behind the BV-BRST and BFV formalism of gauge theory, as well as appearing in Poisson geometry in the guise of Lie algebroids. In this talk we revisit the notion of the modular class of a Q-manifold understood as the obstruction to the existence of a Berezin volume that is invariant with respect to the Lie derivative of the homological vector field. In this way we in fact construct a characteristic class of theQ-manifold. Although these notions seem to be known to experts little has appeared in the literature. We will look at some nice examples including L_\infty-algebroids and double Lie algebroids.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Anatol Odzijewicz (Instytut Matematyki, Uniwersytet w Białymstoku)

Geometrical structures related to W*-algebras

The operator algebras theory including the theory of von Neumann algebras gives the mathematical background for quantum mechanics. A similar role is played by Poisson geometry in classical mechanics. Such notion as: Lie groupoid and Lie algebroid, symplectic manifold, fibre-wise linear Poisson structures or Lie-Poisson space are important ingredients of the mathematical framework for the contemporary geometric mechanics. In this presentation I will show that these structures are also generated in a canonical way by the structure of W*-algebra (von Neumann algebra). Some constructions and related theorems describing this structures will be presented as well.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Jacek Jezierski (KMMF)

On the existence of degenerate (or extremal) Killing horizons - special examples of GRolitons

Some special classes of Einstein metrics lead to the notion of the `near horizon geometry'. In particular, Einstein equations reduce to the so-called basic equation. This is a non-linear PDE for unknown covector field and unknown Riemannian structure on the two-dimensional manifold.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Jerzy Kijowski (CFT PAN)

Higher order curvature tensors, higher order Bianchi identities

Trying to understand cosmological anomalies (dark energy, dark matter) many physicists consider various generalizations of the General Relativity Theory. E.g.: theories derived from a Lagrangian depending not only upon the curvature tensor, but also upon its higher covariant derivatives. Personally, I do not believe in a physical relevance of such theories. But, when analizing their mathematical structure, one discovers a beautiful "Terra Nova'' of geometric constructions, which sheds also new light on the classical notion of curvature.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Andriy Panasyuk (Uniwersytet Warmińsko Mazurski)

On local bisymplectic realizations of compatible Poisson brackets

In a seminal paper "The local structure of Poisson manifold" (1983) A. Weinstein proved that for any Poisson manifold (M,P) there exists a local symplectic realization, i.e. nondegenerate Poisson manifold (M',P') and a local surjective submersion f:M'->M with f_*P'=P. Global aspects of this problem were afterwards intensively studied as they are related to the theory of symplectic and Poisson grupoids, to the integration problem of Lie algebroids, and to different quantization schemes. In this talk I will discuss a problem of local simultaneous realization of two compatible Poisson structures by means of two nondegenerate ones. Note the following essential difference between the two realization problems: there is only one local model of the nondegenerate Poisson bivector P' given by the Darboux theorem and there are many local models of bisymplectic bihamiltonian structures. So besides the problem of existence it is important to understand how many nonequivalent realizations there are in the second case.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Tatiana Shulman (Instytut Matematyczny PAN)

On almost commuting matrices

We will start with classical questions from 70's which ask whether almost commuting matrices have to be close to exactly commuting ones. Of course the notions of "almost" and "close" need clarification and generally can vary with the specific problem in question. There is also a quantifying aspect and an algorithmic issue in searching for the commuting matrices whenever they do exist. Besides being popular in Linear Algebra and Operator Theory, questions of this kind arise also in Quantum Information Theory. A further development of such questions comes from Group Theory and asks whether almost representations of a group have to be close to actual representations. We will present results on that from joint work with Don Hadwin.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Zohreh Ravanpak (Instytut Matematyczny PAN)

Study on the Lie group and Lie groupoid approach to Poisson-Nijenhuis structures

The talk focusses on multiplicative Poisson-Nijenhuis structures on a Lie group, its dual Lie group and the corresponding double groups. As an application, we study completely integrable bi-Hamiltonian systems with respect to two linear Poisson structures on a vector bundle and integrable deformations of bi-Hamiltonian systems on Poisson groupoids

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Mikołaj Rotkiewicz (MIMUW)

Polarisation of graded bundles

Abstract: Graded bundles can be viewed as a natural generalization of vector bundles. In short, they are locally trivial fibered bundles with fibers possessing a structure of a graded space, i.e. a manifold diffeomorphic to Rⁿ with a distinguished class of global coordinates with positive integer weights assigned. In a special case when these weights are all equal to 1, a graded space becomes a standard vector space and a graded bundle - a vector bundle. The fundamental example of a graded bundle isthe k-th order tangent bundle of a manifold M. Can we turn graded bundles in the realm of vector bundles? We shall construct a functor which takes a graded bundle of degree k and produces a k-fold vector bundle, mimicking the the canonical embedding T^kM into TT...T M. Can we linearise other graded structures in similar way? What is the image of the linearisation functor? Similar question will be discussed. Based on a joint paper with A. J. Bruce and J. Grabowski

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Katarzyna Karnas (CFT PAN)

Product of finite order rotations and generating infinite groups of unitary gates

I will consider the product of two rotations in three-dimensional space for which the rotation axes are perpendicular and the rotational angle is a rational multiple of pi and ask when if the obtained rotation angle is also a rational multiple of pi. The problem is reduced to finding the trigonometric minimal polynomial. I will show that this problem is related to quantum information theory. Joint work with Adam Sawicki.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Tomasz Smołka (KMMF)

Electromagnetic and gravitational Hopfions

Hopfions are a family of field solutions which have non-trivial topological structure. Their connections with Hopf fibration will be presented. I will focus on two physical applications of Hopfions: electromagnetism and linear gravitation. Using Hopfion solution, I will discuss problem of energy in linear gravitation.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Adam Sawicki (CFT PAN)

Universal quantum gates

I will consider the problem of deciding if a finite set of quantum one-qudit gates is universal, i.e if the generated group is either the special unitary or the special orthogonal group. To every gate I will assign its image under the adjoint representation. The necessary condition for the universality is that the only matrices that commute with all the adjoint representation matrices are proportional to the identity. If in addition there is an element in the considered group whose Hilbert-Schmidt distance from the centre is smaller than 1/\sqrt{2}, then the set of gates is universal. Using these I will present a simple algorithm that allows deciding the universality of any set of d-dimensional gates in a finite number of steps. Moreover, I will formulate the general classification theorem.This is a joint work with Katarzyna Karnas.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Michał Jóźwikowski (Instytut Matematyczny PAN)

Invariants of pseudogroup actions

In the talk I will discuss the results of Kruglikov and Lychagin about the structure of the algebra of differential invariants for an algebraic pseudogroup action on a differential equation.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Giovanni Moreno (IMPAN)

Generic three-forms in dimension seven and symmetric second-order PDEs

In this talk I will show how it is possible to construct a second-order nonlinear PDE in two independent variables, with highly nontrivial symmetry group, by starting from a very simple datum - a generic three-form on a seven-dimensional linear space. The construction itself follows from the combination of several well-known ingredients, but it is hardly found in the literature in a concise self-contained form, which is the main goal of this talk. I will conclude by pointing out some related results recently obtained in collaboration with D. Alekseevsky, J. Gutt and G. Manno, as well as open problems.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Andrey Krutov (Instytut Matematyczny PAN)

Lie algebroids over infinite jet spaces

We define Lie algebroids over infinite jet spaces. The examples of such construction is given by Hamiltonian operators and Lie algebra-valued zero-curvature representations for partial differential equations. The talk is based on the following papers (1) Kiselev A. V., van de Leur J. W., Variational Lie algebroids and homological evolutionary vector fields, Theor. Math. Phys. 167 (2011) n.3 Nonlinear Physics: Theory & Experiment VI. — P. 772–784. arXiv:1006.4227 [math.DG] (2) Kiselev A. V., Krutov A. O., Non-Abelian Lie algebroids over jet spaces, J. Nonlin.Math. Phys. 21 (2014) n.2. — P. 188–213. arXiv:1305.4598 [math.DG]

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Tadeusz Januszkiewicz (Instytut Matematyczny PAN)

Geometry of isospectral, generalized tridiagonal Hermitian matrices

The set of isospectral hermitian matrices, i.e. a (partial) flag variety, is one of fundamental mathematical objects, playing a role in various parts of mathematics from algebraic geometry to combinatorics. One of most succesful ways to understand them is to study the action of the maximal torus of diagonal matrices in SU(n). Special hermitian matrices with fixed spectrum, i.e. the ones for which some of the off-diagonal entries are zero, have been studied in theory of integrable systems. They have interesting topology and beautiful symmetries. The classically studied case was that of tridiagnonal matrices, i.e a_{ij}= 0 if |i-j|>1. It turned out that other "tridiagonal matrices", for example those for which a_{ij}=0 for i>1, have equally interesting topology and symmetries. Again the good approach is to use the diagonal torus action. However new tools are needed to understand even so simple topological invariants like cohomology. There is also an interesting symplectic aspect to these manifolds which I will describe. This is a joint work with Światosław Gal (Wrocław University).

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Andrey Krutov (IMPAN)

Geometry of jets spaces and PDE (II)

We outline the geometric and algebraic structures associated with PDEs and study the properties of these structures and their interrelations. The talks cover the standard material about the infinite jet bundles, systems of differential equations, their symmetries and conservation laws and the construction of the nonlocalities (recursion operators, Hamiltonian structures, zero-curvature representations).

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

There is no seminar session on November 2nd

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Andrey Krutov (IMPAN)

Geometry of jets spaces and PDE (I)

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Javier de Lucas (KMMF)

Bundle Lie systems and applications

Please note that the seminar will take place in room 403 in IMPAN! A Lie system is a non-autonomous system of first-order ordinary differential equations whose general solution can be expressed as an autonomous function, the superposition rule, of a generic family of particular solutions and some constants. We show that these notions are not well defined under non-autonomous changes of variables. This suggests us to define and analyse the bundle Lie systems, which are well-defined geometric notions covering Lie systems and most of their generalisations as particular cases. Reductions of Wess--Zumino--Novikov--Witten equations, multidimensional Riccati equations and other physical examples are analysed so as to illustrate our results.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Paweł Urbański (KMMF)

Geometry of Routh reduction II

During the seminar we will discuss the geometric framework necessary to describe the so called Routh reduction of a mechanical system. In the second part of the seminar we will consider possible generalizations of this reduction.

room 106 IM PAN, Śniadeckich 8, Ip at 14:15

Katarzyna Grabowska, Paweł Urbański (KMMF)