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

Artur Janda (CBK)

Hyperbolic Lagrangians and the Domain of Dependence Theorem

Hyperbolicity of a system of partial differential equations refers tocertain essentially algebraic conditions, on the other hand such a systemshould have properties like the wave equation, basically it should possessthe domain of dependence property. Starting with a brief exposition ofbasic concepts related to the action principle of the theory of mapsbetween manifolds we will sketch the proof of the domain of dependencetheorem (Christodoulou) for systems of the Euler-Lagrange equationssatisfying certain regular hyperbolicity condition. This modern notionrefers strictly to Lagrangians and it is applicable to systems which cannotsatisfy classical notions of hyperbolicity (Leray, Friedrichs).

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

Piotr Waluk (KMMF)

Symmetries of distributions after Kruglikov II

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

Piotr Waluk (KMMF)

Symmetries of distributions after Kruglikov I

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

Luca Vitagliano (U Salerno)

L-infinity algebras from multicontact geometry

I define higher versions of contact structures on manifolds as maximallynon-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher versions of pre-contact structures as distributions on manifolds whose characteristic symmetries span a constant dimensional distribution. Every distribution is almost everywhere, locally, a pre-multicontact structure. After showing that the standard symplectization of contact manifolds generalizes naturally to a (pre-)multisymplectization of (pre-) multicontact manifolds, I associate a canonical L-infinity algebra to any (pre-)multicontact structure. Such L-infinity algebra is a higher version of the Jacobi brackets on contact manifolds. Since every partial differential equation (PDE) can be geometrically understood as a manifold with a distribution, then there is a (contact invariant) L-infinity algebra attached to any PDE. Finally, I describe in local coordinates the L-infinity algebra associated with the Cartan distribution on jet spaces.

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

Jerzy Kijowski (CFT)

O pewnej metodzie ,,dyskretyzacji'' równań cząstkowych pochodzących z problemów wariacyjnych

Jest kilka sposobów ,,dyskretyzacji'' równań Einsteina używanych doobliczeń w tzw. numerical gravity. Wszystkie te sposoby gwałcą,,in flagranti'' strukturę geometryczną tych równań. Prelegentjest głęboko przekonany, że trudności z obliczeniami numerycznymiw ogólnej teorii względności stąd właśnie wynikają i proponujenową metodę jej dyskretyzacji, wynikającą z analizy tej właśniestruktury.

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

Krzysztof Drachal (PW)

Comorphisms of Lie algebroids and groupoids

The aim of the talk is to remind the notions of morphisms of vectorbundles, modules, Lie algebroids and groupoids and Poisson bundles. Afterwards, corresponding notions of comorphisms (in a sense of Higgins and Mackenzie) are introduced. Comorphisms are used to show some important dualities between selected categories.

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

Giovanni Moreno (Sl U Opava)

Geometry of 3rd order Monge-Ampère equations and their characteristics

This talk incorporates three aspects: i) a review of the general theory ofcharacteristics of PDEs with a particular emphasis on its physicalramifications, ii) an introduction to a recent study of 2nd orderMonge-Ampère equations (MAEs) with the tools of contact/symplecticgeometry, and iii) a presentation of an ongoing investigation of 3rd orderMAEs, with special attention to the peculiarities of this nearlyunexplored case. These topics were freely inspired, respectively, by arXiv:1311.3477, arXiv:1003.5177, and arXiv:1403.3521.Throughout the talk, there will be a gradual transition from an initialphysical perspective on PDEs and their characteristics in general, andMAEs in particular, towards a concluding entirely geometric picture,involving contact manifolds, their prolongations, and special sub-bundlesof some Grassmannian-like bundles, improperly called here "Lagrangian".The reason behind this denomination is that they stem from a higher-orderanalog of the symplectic form, known as "meta-symplectic". Surprisinglyenough, the so-obtained framework, in spite of its abstractedness, allowsto give an immediate answer to some non-equivalence problems and, moregenerally, to begin to understand the structure of the "space of all 3rdorder MAEs".

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

Tatiana Shulman (University of Copenhagen)

Mathematical aspects of zero-error information theory

In quantum information theory, for mathematical description of quantum channels one uses the notion of completely positive maps on matrix spaces. In the first part of the talk we will discuss basic facts about these maps and their connection with quantum channels. In the second part of the talk we will focus on some mathematical problems related with superactivation of zero-error capacities of quantum channels. This is a joint work with M. Shirokov.

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

Javier de Lucas Araujo (KMMF)

Dirac-Lie systems: theory and applications

A Lie system is a nonautonomous system of first-order ordinary differential equations possessing a superposition rule, namely a mappingallowing us to describe its general solution in terms of a generic finitefamily of particular solutions and a set of constants. Equivalently, we cancharacterise Lie systems as systems of first-order ordinary differentialequations describing the integral curves of a time-dependent vector fieldtaking values in a finite-dimensional Lie algebra of vector fields: a Vessiot--Guldberg Lie algebra.We introduce a new class of Lie systems possessing a Vessiot--Guldberg Liealgebra of Hamiltonian vector fields with respect to a Dirac structure: the Dirac--Lie systems.The use of Dirac geometry enable us to develop powerful methods to studythe constants of motion, superposition rules, and other properties of thesesystems.Our results generalise previous methods to investigate integrable systemsand certain types of Lie systems.We illustrate our findings with the study of several types of Schwarzianequations and other differential equations of interest.

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

Michał Jóźwikowski (IM PAN)

Integration of Lie algebroids II

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

Michał Jóźwikowski (IM PAN)

Integration of Lie algebroids

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

Zbigniew Jelonek (IM PAN)

Dyfeomorfizmy zachowujace symplektyczne lub izotropowe podrozmaitosci

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

Katarzyna Grabowska (KMMF)

Graded geometry: Lagrangian and Hamiltonian mechanics

The seminar will be dovoted to higher order mechanical systems,i.e. systems with Lagrangian function defined on T^kM. An example of suchsystem is motion of the end of a javelin. The framework of graded bundleswill be used.

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

Janusz Grabowski (IM PAN)

Graded differential geometry

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

Janusz Grabowski (IM PAN)

Graded differential geometry

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

Mikołaj Rotkiewicz (IM UW)

Całkowanie na super-rozmaitościach

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

Mikołaj Rotkiewicz (IM UW)

Całkowanie na super-rozmaitościach

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

Jose Mourao (U Lisboa)

Quantization in Kähler and real polarizations

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

Katarzyna Grabowska i Paweł Urbański (KMMF)

Więzy w statyce i dynamice

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

Tomasz Rybicki (AGH)

A look at the hamiltonian group of a symplectic manifold through its quantomorphism group

We study the Hamiltonian group of a symplectic manifold, satisfying some mild additional assumptions, by means of the associated quantomorphism group. Recall that, according to Souriau, the quantomorphism group is the strict contactomorphism group of the total space of a prequantization bundle over the manifold in question. Our investigations are concentrated on the problem of estimations (from below) of the Hofer metric on the Hamiltonian group. We establish the unboundedness of the metric and its non-degeneracy using contact geometry instead of hard symplectic methods known from the literature.

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

Jacek Jezierski (KMMF)

Geometria powierzchni zerowych a horyzonty ekstremalne

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

Krzysztof Drachal (PW)

On some applications of differential spaces in a sense on Sikorski

continuation

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

Andrew Bruce (IM PAN)

On curves and jets of curves on supermanifolds

Kontynuacja wykładu 9.10

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

Tiffany Covolo (U Luxembourg)

Trace and Berezinian of matrices over a Clifford algebra

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n-graded commutative associative algebra A.The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, we recover the classical Dieudonne determinant of quaternionic matrices, but in general our quaternionic determinant is different.

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

Michał Jóźwikowski (IM PAN)

O więzach wakonomicznych i nieholonomicznych

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

Adam Sawicki (CFT)

N-particle quantum statistics on graphs

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

Katarzyna Grabowska (KMMF)

Coś o równaniu Hamiltona-Jacobiego

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

Andrew Bruce (IM PAN)

On curves and jets of curves on supermanifolds

Following classical ideas as much as possible we define thenotion of a curve on a supermanifolds and proceed to define the jets ofsuch curves. We make extensive use of Grothendieck's functor of points andseem rather forced to take a categorical approach to supermanifolds. However, this allows us to define geometrically the k-th order tangentbundle of a supermanifold.

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

Krzysztof Drachal (PW)

O pewnych zastosowaniach przestrzeni różniczkowych Sikorskiego

On some applications of differential spaces in a sense of Sikorski

W referacie przedstawię podstawowe definicje związane z aparatem przestrzeni różniczkowych w sensie Sikorskiego. Są to obiekty bardziej ogólne aniżeli powszechnie używane gładkie rozmaitości różniczkowe. Interesujący jest również fakt, że tzw. przestrzenie spektralne (pojawiające się np. w formalizmie prof. A. Vinogradova oraz grupy Jet Nestruev) także okazują się być przestrzeniami różniczkowymi w sensie Sikorskiego. Przestrzenie spektralne mają z kolei ciekawe własności interpretacyjne z punktu widzenia fizyki klasycznej. Ponieważ dotychczas rozaważano pewne problemy związane z osobliwościami czasoprzestrzeni ogólnej teorii względności w języku przestrzeni różniczkowych i pozowoliło to na elegancką interpretację pewnych innych wyników, to połączenie metod spektralnych i Sikorskiego powinno pozwolić na użycie bardziej algebraicznych metod w badaniu osobliwości czasoprzestrzeni i być może na uzyskanie kolejnych interesujących wyników. Przedstawiony program badawczy jest w pewnym sensie kontynuacją prac prof. M. Hellera i W. Sasina z lat 90-tych XX w.

Differential spaces are objects which generalise smooth manifolds. It is interesting that for example so called "spectral spaces" used in the formalism of prof. A. Vinogradov and his Jet Nestruev group are also differential spaces. Spectral spaces have a nice interpretation in classical physics. As far as now differential spaces have been useful in dealing with a boundary of a spacetime in general relativity. They have given nice interpretations of some facts. Therefore it seems reasonable to expect that one can merge algebraic methods of spectral spaces and more geometrical methods of differential spaces and then (maybe) obtain some new intereting results. The research desing that I will describe is in some sense a continuation of papers of prof. M. Heller and prof. W. Sasin published in 1990s.