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 2.23, Pasteura 5 at 14:15

Marian Wiatr (KMMF)

Energy of field as a generating function of dynamics

Every physics student knows that the energy of an electromagnetic field is equal to 1/2(E^2+B^2). However, there are a lot of ways to obtain this formula. Some lecturers use heuristic arguments about transferring charged particles from infinity. Another idea is to derive it from the energy-momentum tensor - then the canonical energy-momentum tensor must be artificially improved by Belinfante-Rosenfeld procedure to obtain the desired formula. However, these derivations are not so satisfying from the formal and physical point of view. I will show how to obtain this formula as a generating function of dynamics using symplectic formalism proposed by W. M. Tulczyjew in a paper "A Symplectic Framework of Linear Field Theories". First, I will define the energy of a field in every linear field theory using symplectic space associated with current given by the three-dimensional volume V and transversal time vector field on it. Next, I will show how to apply this construction to Maxwell electrodynamics, and how it can be useful in proving the existence of a solution of the initial-boundary problem by evolutionary methods.

room 2.23, Pasteura 5 at 14:15

Adam Maskalaniec (FUW)

Introduction to supergeometry

In this talk I will introduce the fundamentals of supergeometry. At the beginning the main notions and examples of vector superspaces and superalgebras will be presented. Next I will explain the basics of differential calculus on supermanifolds. Finally the super Poincare lemma will be proved.

room 2.23, Pasteura 5 at 14:15

Janusz Grabowski (IMPAN)

The geometry of quantum states (part II)

A differential geometric picture for some basic issues of Quantum Mechanics on finite-dimensional Hilbert spaces will be introduced. The true infinite dimensional case will be considered in a separate complementary seminar.

join us at 14:15

Luca Schiavone (Universita di Napoli)

Poisson brackets on the space of solutions of Hamiltonian Field Theories: Yang-Mills theories

The aim of this talk is to show whether and how the space of solutions of the equations of motion of Hamiltonian field theories can be equipped with a Poisson structure. In particular, we will show how, in the multisympletic formalism, a canonical pre-symplectic structure on the space of solutions naturally emerges out of the principle of least action and, then, we ask whether and how such a pre-symplectic structure may give rise to a Poisson one. We will use the coisotropic embedding theorem to provide an answer to this question and we argue that in some cases it is possible to construct a Poisson structure directly on the space of solutions whereas in some situations one is forced to define it on an enlarged manifold. The guiding example we will consider along the talk, namely Yang-Mills theories, lies in the latter case and we will interpret the additional degrees of freedom emerging from the above mentioned enlargement as the ghost fields introduced in Quantum Field Theory by Fadeev and Popov.

If you want to join the seminar write to konieczn at fuw.edu.pl

room 2.23, Pasteura 5 at 14:15

Manuel de León (ICMAT-CSIC and Real Academia de Ciencias)

Brackets, nonholonomic mechanics and quantization

The nonholonomic dynamics can be described by the so-called nonholonomic bracket on the constrained submanifold, which is a non-integrable modification of the Poisson bracket of the ambient space, in this case, of the canonical bracket on the cotangent bundle of the configuration manifold. This bracket was defined using a convenient symplectic decomposition of the tangent bundle of the cotangent bundle over the constraint submanifold. On the other hand, another bracket, also called nonholonomic bracket, was defined using the description of the problem in terms of skew-symmetric algebroids. Recently, reviewing two older papers by R. J. Eden, we have defined a new bracket which we call Eden’s bracket. In this lecture, we prove that these three brackets coincide. Moreover, the description of the nonholonomic bracket à la Eden has allowed us to make important advances in the study of Hamilton Jacobi theory and the quantization of nonholonomic systems.

room 2.23, Pasteura 5 at 14:15

Janusz Grabowski (IMPAN)

The geometry of quantum states

room 2.23, Pasteura 5 at 14:15

Arman Taghavi-Chabert (FUW)

Introduction to CR geometry and applications (Part II)

In its simplest form, CR geometry can be understood as the study of real hypersurfaces in complex space. In this second lecture, I will introduce the audience to more geometrical aspects of contact CR geometry including the Webster-Tanaka calculus and possibly its tractor calculus. Time permitting, I will explain how it interacts with Lorentzian conformal geometry especially in connection with mathematical relativity.

join us at 14:15

Luca Vitagliano (University of Salerno)

Characteristics and Fold Type Singularities of Solutions of PDEs

I will discuss the precise relationship between characteristic Cauchy data for partial differential equations (PDEs) and certain singularities of solutions called fold-type singularities. If we interpret a solution as a wave, then a fold-type singularity can be interpreted as a wave front. I’ll show that every PDE defines a new equation describing the propagation of such wave fronts. I will use the geometric language of jet spaces, so all the discussion will be manifestly coordinate independent. This is a review talk and I don’t claim any originality on the subject.

For the link please write to konieczn at fuw.edu.pl

room 2.23, Pasteura 5 at 14:15

Jagna Wiśniewska (ETH)

Rabinowitz Floer homology in analysis of Hamiltonian dynamics on non-compact energy hypersurfaces

Modern symplectic geometry started as a mathematical formulation of classical mechanics. It can be used to analyse dynamical systems arising in classical mechanics, for example trajectories of charged particles in magnetic fields or orbits of satellites under the gravitational forces of planets and stars. In my research I have investigated the existence of periodic orbits for Hamiltonian systems on a fixed, non-compact energy level. Nontrivial examples of these systems are too complex to describe their evolution with a precise formula. Therefore in the analysis of Hamiltonian dynamics in order to characterise the properties of the system one has to use tools from different fields of mathematics such as differential geometry (contact and symplectic geometry), functional analysis (calculus of variations) and algebraic topology (Floer homology). These tools enable us to analyse the dynamical behaviour of the systems in question and whether their properties are preserved under small perturbations.

room 2.23, Pasteura 5 at 14:15

Iryna Yehorchenko (IMPAN)

Using contact transformations to solve partial differential equations

We demonstrate using hodograph and contact transformations to solve first-order nonlinear PDE and overdetermined systems of first- and second-order PDEs. As a result, we find parametric general solutions of these equations. We will discuss relations of the found solutions to symmetry solutions and to symmetry of the original equations.

room 2.23, Pasteura 5 at 14:15

Arman Taghavi-Chabert (FUW)

Introduction to CR geometry and applications

In its simplest form, CR geometry can be understood as the study of real hypersurfaces in complex space. In this talk, I will present the basics of the topic and introduce the audience to the Webster-Tanaka calculus in the case where the CR structure is also contact. I will explain how it interacts with Lorentzian conformal geometry especially in connection with mathematical relativity.

join us at 14:15

Alfonso Tortorella (University of Salerno)

Homogeneous G-Structures

The theory of G-structures provides us with a unified framework for a large class of geometric structures, including symplectic, complex and Riemannian structures, as well as foliations and many others. Surprisingly, contact geometry - the "odd-dimensional counterpart" of symplectic geometry - does not fit naturally into this picture. In this paper, we introduce the notion of a homogeneous G-structure, which encompasses contact structures, as well as some other interesting examples that appear in the literature. This is joint work with L. Vitagliano and O. Yudilevich.To get the link please write to konieczn@fuw.edu.pl

join us at 14:15

Alfonso Tortorella (University of Salerno)

Homogeneous G-Structures

The theory of G-structures provides us with a unified framework for a large class of geometric structures, including symplectic, complex and Riemannian structures, as well as foliations and many others. Surprisingly, contact geometry - the "odd-dimensional counterpart" of symplectic geometry - does not fit naturally into this picture. In this paper, we introduce the notion of a homogeneous G-structure, which encompasses contact structures, as well as some other interesting examples that appear in the literature. This is joint work with L. Vitagliano and O. Yudilevich.Link can be obrained by writting to konieczn@fuw.edu.pl

room 2.23, Pasteura 5 at 14:15

Bartosz Zawora (KMMF)

Willet’s contact reduction

A contact manifold is a pair (M, ξ) consisting of an odd-dimensional manifold M endowed with a one-codimensional completely non-integrable distribution ξ on M. If ξ = ker η for a differential one-form η on M, which is not unique, the pair (M, η) is called a co-orientable contact manifold. In this talk, I will present a Marsden–Weinstein reduction for co-oriented contact manifolds devised in C. Willett, Contact reduction, Trans. Amer. Math. Soc. 354 (2002) 4245–4260. Roughly speaking, this reduction uses a Lie group action of symmetries of η to obtain from it a new co-oriented manifold on a quotient of a submanifold of M in a manner that does not depend on the choice of η corresponding to a fixed ξ. More in detail, I will first introduce some basic notions on co-oriented contact manifolds and briefly explain how they can be viewed as symplectic R^×-principal bundles. In particular, I will recall the definition of a contact Lie group action and its associated momentum map. Next, I will introduce the notion of an orbifold and a contact quotient. Finally, I will present a co-oriented contact Marsden–Weinstein reduction theorem that ensures that the contact quotient is a contact orbifold. To understand the more complicated case of a co-oriented contact orbifold, I will examine a symplectic orbifold obtained through symplectic reduction. If time permits, I will discuss removing the singularity and the integrability assumptions in favour of the existence of a slice.

room 2.23, Pasteura 5 at 14:15

Xavier Rivas (KMMF)

Multicontact formulation for non-conservative field theories

A new geometric structure inspired by multisymplectic and contact geometries, which we call multicontact structure, is developed to describe non-conservative classical field theories. Using the differential forms that define this multicontact structure as well as other geometric elements that are derived from them while assuming certain conditions, we can introduce, on the multicontact manifolds, the variational field equations which are stated using sections, multivector fields, and Ehresmann connections on the adequate fiber bundles. Furthermore, it is shown how this multicontact framework can be adapted to the jet bundle description of classical field theories. The field equations are stated in the Lagrangian and the Hamiltonian formalisms and some examples are provided.

room 2.23, Pasteura 5 at 14:15

Tomasz Smołka (KMMF)

Hopfions and topological invariants in electrodynamics

Hopfions are a family of solutions of Maxwell equations which have non-trivial topological structure. Their connections with Hopf fibration will be presented. I will discuss topological invariants associated with duality symmetry. If you want to join the seminar please ask for link: konieczn at fuw edu pl

room 2.23, Pasteura 5 at 14:15

Jacek Jezierski (KMMF)

Geometry of null hypersurfaces

We will review basic geometric structures on null hypersurfaces which are useful for the description of black hole horizons

room 2.23, Pasteura 5 at 14:15

Paweł Urbański (KMMF)

Variational principles for dissipative systems

The principle of virtual work well known in statics of mechanical systems is a master model for all variational principles of classical physics. I will state a simple version of this principle. The analogue of the principle in Lagrangian formulation of the dynamics will be presented. I will discuss also the Legendre transformation for systems with viscosity and friction forces. Convex analysis will be strongly involved. Contact geometry will not be involved.

room 2.23, Pasteura 5 at 14:15

Giovanni Moreno (KMMF)

A Proof of the Lie-Bäcklund Theorem

At the end of the nineteen century Norwegian mathematician Sophus Lie (1842-1499), together with Swedish mathematician Albert Bäcklund (1845-1922), established a theorem that in modern terms reads as follows: any Lie transformation (resp., field) of some domain of the jet space J^k(n,m) is either the lift of a contact transformation (resp., field), if m=1, or the lift of a point transformation (resp., field), if m>1. The proof of this pivotal result in the geometric theory of PDEs, which I will carry out in this seminar, gives us the opportunity to examine in depth some features of the jet spaces and their natural structures.

room 2.23, Pasteura 5 at 14:15

Michał Jóźwikowski (MIMUW)

Struktury kontaktowe w teorii optymalnego sterowania

Contact structures in Optimal Control Theory

Teoria optymalnego sterowania to ciekawy dział matematyki leżący na styku teorii równań różniczkowych zwyczajnych, analizy funkcjonalnej i geometrii różniczkowej, a jednocześnie bliski praktycznym zastosowaniom w inżynierii, czy robotyce. Centralny rezultat tej teorii - Zasada Maksimum Pontriagina - zwykle formułowana jest w języku równań hamiltonowskich. Równania te są jednak jednorodne, co pozwala na ich interpretację w oparciu o naturalną strukturę kontaktową na projektywizacji wiązki kostycznej. W moim referacie omówię geometrię Zasady Maksimum Pontriagina i spróbuję wyjaśnić znaczenie obu struktur: symplektycznej i kontaktowej.

Optimal Control Theory is a fascinating part of mathematics, lying at the crossover of ordinary differential equations, functional analysis and differential geometry, yet at the same time near to practical applications in robotics or engineering, Its central result - the Pontryagin Maximum Principle - is usually stated in the language of Hamilton equations. These equations are, however, homogeneous, which opens a possibility to interpret them using a natural contact structure on the projectivisation of the cotangent bundle. In my talk I will discuss the geometry behind the Pontryagin Maximum Principle, trying to illuminate the role of both symplectic and contact structures involved.

room 2.23, Pasteura 5 at 14:15

Katarzyna Grabowska (KMMF)

Hamiltonian an Lagrangian systems in the contact setting

In the current literature about contact structures many authors express the opinion that contact structures can be used in Hamiltonian and possibly also Lagrangian description of mechanical nonconservative sytems. In this context I will present a contact version of the Tulczyjew triple using the idea that a contact structure is encoded in a certain homogeneous symplectic structure. The theory will be illustrated by examples.

room 2.23, Pasteura 5 at 14:15

(KMMF)

There will be no seminar on November 2nd

room 2.23, Pasteura 5 at 14:15

Janusz Grabowski (IMPAN)

Contact geometry as a chapter in symplectic geometry - part II

This will be the continuation of the last week talk: I will present an approach to the concept of a contact manifold which, in contrast to the one dominating in the physics literature, serves also for non-trivial contact structures. In this approach contact geometry is not an ‘odd-dimensional cousin’ of symplectic geometry, but rather a part of the latter, namely ‘homogeneous symplectic geometry’. This understanding of contact structures is much simpler than the traditional one and very effective in applications, reducing, for instance, the contact Hamiltonian formalism to the standard symplectic picture.

room 2.23, Pasteura 5 at 14:15

Janusz Grabowski (IMPAN)

Contact geometry as a chapter in symplectic geometry

room 2.23, Pasteura 5 at 14:15

Bartłomiej Bąk (KMMF)

Opis klasycznej termodynamiki w języku geometrii kontaktowej

Classical thermodynamics in the language of contact geometry

W trakcie mojego wystąpienia chciałbym przedstawić opis termodynamiki klasycznej w języku geometrii kontaktowej. Geometria kontaktowa umożliwia reprezentowanie fenomenologicznych praw natury (jak np. I i II zasada termodynamiki, reguły Maxwella) jako własności pewnych obiektów geometrycznych. Następnie przejdę do konstrukcji wiązki głównej nad rozmaitością wyposażoną w powyższą strukturę kontaktową. Wynikają z niej tzw. transfomacje kontaktowe ,,zachowujące'' formę kontaktową (z dokładnością do czynnika multiplikatywnego). Transformacje kontaktowe są reprezentowane przez jednoparametrowe grupy dyfeomorfizmów, zaś pola wektorowe będące generatorami tych transformacji posiadają niezwykle interesujące własności fizyczne.

In my talk I want to introduce the contact geometry description of classical thermodynamics. In contact language phenomenological laws of nature e.g. I and II law of thermodynamics, Maxwell relations) could be represented by properties of some geometrical objects. Next, I will construct the principal bundle over the manifold with the contact structure presented above. This construction provides the so-called contact contact transformations, which preserve the contact 1-form up to a multiplicative factor and are represented by one-parameter groups of diffeomorphisms. Vector fields which are generators of these transformations have very interesting physical properties.