Katedra Metod Matematycznych Fizyki

zaprasza studentów, doktorantów i nie tylko młodych pracowników nauki na seminarium

Metody Geometryczne Fizyki

9 czerwca 2021 / June 9^th, 2021

Bartosz Zawora (FUW)

An energy-momentum method

In my talk, I will present the energy-momentum method, which allows for studying stability and bifurcation of Hamiltonian systems with symmetry. First I will introduce a symplectic action of Lie groups and momentum map. After this introduction, I will discuss the symplectic structure on reduced space (quotient space) and explain the conditions on the stability of these systems. In the last part, I will present the energy-momentum method and briefly describe how it can be applied.

The meeting will take place on Zoom. If you are interested ask for a link at katarzyna.konieczna at fuw edu pl

2 czerwca 2021 / June 2^nd, 2021

Bartłomiej Bąk (KMMF)

A few constructions of the scri and its applications into analysis of scalar wave equation / Kilka konstrukcji skraju i ich zastosowanie w analizie skalarnego równania falowego

Podczas swojej prezentacji chciałbym przedstawić najbardziej znaną konstrukcję skraju na czasoprzestrzeni Minkowskiego (poprzez konforemne uzwarcenie) oraz stowarzyszony z nią diagram Penrose'a. Następnie przejdę do omówienia zachowywania się zanikającego w nieskończoności rozwiązania skalarego równania falowego na skraju i pokazania, że znając dane Cauchy'ego można całkowicie odtworzyć dane "wypromieniowane" na skraj i vice versa. Co więcej, dokonując odpowiedniego uzwarcenia można sprowadzić problem Cauchy'ego do problemu Dirichleta na zwartym obszarze. Dzięki tym rozważaniom możliwe będzie zdefniowanie skraju w naturalny, nieodwołujący się do uzwarcania sposób. Na sam koniec opowiem w jaki sposób sprowadza się tensorowe równanie falowe do poziomu skalarnego równania Kleina-Gordona.

26 maja 2021 / May 26^th, 2021

Jan Szypulski (FUW)

Two and three qubits geometry and Hopf fibration

Jednym z najprostszych i najważniejszych obiektów w Informacji kwantowej jest kubit, dwupoziomowy system kwantowy. Omówię uogólnienie na 2 kubity standardowej reprezentacji sfery Blocha dla pojedynczego kubitu, w ramach fibracji Hopfa sfer wysokowymiarowych przez sfery niższych wymiarów. Pojedyncza kubitowa przestrzeń Hilberta to trójwymiarowa sfera S3. Przestrzeń podstawowa S2 odpowiednio zorientowanej fibracji Hopfa S3 to nic innego jak sfera Blocha, podczas gdy okrągłe włókna reprezentują ogólny stopień swobody fazy kubitu. W przypadku dwóch kubitów przestrzeń Hilberta jest 7-wymiarową sferą S7, która umożliwia również fibrację Hopfa, z włóknami S3 i podstawą S4. Głównym uderzającym wynikiem jest to, że odpowiednio zorientowane fibracje S7 Hopfa są wrażliwe na splątanie.

19 maja 2021 / May 19^th, 2021

Marian Wiatr (KMMF)

Krzywizna jako pole sił pływowych / Curvature as field of tidal forces

W szczególnej teorii względności, koneksja ma interpretację siły determinującej trajektorie spadków swobodnych, a do równań Einsteina wchodzi krzywizna - wielkość zbudowana z pochodnych współczynników koneksji. Najbardziej znaną i praktycznie jedyną użwaną miarą krzywiznjest tensor Riemanna, który ma bardzo ładną interpretację geometryczną .Nie ma on jednak bezpośredniej interpretacji fizycznej. Inną ciekawąmiarą krzywizny jest tensor Jacobiego, którego postać wynika wprost z rachunku sił pływowych działających na ciało poruszające się po geodezyjnej (równanie dewiacji geodezyjnej). W moim referacie dokładnie omówię kilka podejść do krzywizny i używając tensora Jacobiego pokażęfizyczną interpretację tensora Ricciego, który jest jedyną częścią krzywizny wchodzącą do równań Einsteina.

12 maja 2021 / May 12^th, 2021

Paweł Urbański (KMMF)

Important examples of singular mechanical systems

During the seminar some examples of singular mechanical systems that cannot be investigated with the usual textbook tools will be presented.

28 kwietnia 2021/ April 28^th, 2021

Jarosław Kopiński (CFT)

Introduction to tractor calculus

The purpose of this talk is to give an introduction to the tractor calculus, which is an efficient and powerful tool for constructing conformal invariants on conformal manifolds. I will start with a review of the (Weyl) pseudo-Riemannian and conformal invariants, discuss a connection of the tractors with the almost Einstein equation, and finish with a strategy of constructing conformal invariants from the tractor curvature.
This talk is based on "An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity" by S. Curry and R. Gover, arXiv:1412.7559 [math.DG].

7 kwietnia 2021/ April 7^th, 2021

Andrew Bruce (UniLu)

Connections adapted to graded bundles

Graded bundles are a particularly nice 'species' of graded manifolds that can be thought of as a generalisation vector bundles. We define the notion of a weighted A-connection, where A is a Lie algebroid. One should view weighted connections as respecting the graded structure just as linear connections respect the linear structure of a vector bundle. We will discuss the existence of such adapted connections and use them to define (quasi-)actions of Lie algebroids on graded bundles.

24 marca 2021/ March 24^th, 2021

Katarzyna Grabowska (KMMF)

Not a very serious seminar on tetrahedral chains

Since it will be the first spring seminar in 2021 we will try to enjoy some mathematics related to Euclidean geometry and group theory considering the question of (non) existence of closed chains composed of tetrahedra, such that the next tetrahedron is a reflection of the previous with respect to one of its sides. For other regular polyhedra closed chains exist. The seminar is a result of random leafing through Extracta Mathematicae where the title "Scottish Book" and some familiar names as prof. Stanisław Janeczko and Hugo Steinhaus caught my eye. The talk is based on the paper by Ian Steward "Tetrahedral chains and a curious semigroup", Extracta Mathematicae, vol 34, No 1, 2019

10 marca 2021/ March 10^th, 2021

Oliver Rinne (Hochschule für Technik und Wirtschaft Berlin)

Mathematical aspects of gravitational radiation

I will begin with a review of the Einstein field equations and show how their linearisation gives rise to gravitational wave solutions. The issue of gauge freedom and the analogy with electrodynamics will be discussed. I will also briefly treat gauge-invariant perturbations of black hole spacetimes. Next I will introduce the concept of conformal infinity and show how it allows for an unambiguous definition of gravitational radiation. Finally I will present some results on the numerical evolution of axisymmetric vacuum gravitational wave spacetimes.

13 stycznia 2021/ January 13^th, 2021

Rafał R. Suszek (KMMF)

The KGB (and the rest of the alphabet) for the Brane New Superworld

The abstract

16 grudnia 2020/ December 16^th, 2020

Katja Sagerschnig (CFT)

Parabolic Geometries in Dimension Five

In this talk we will discuss constructions between parabolic geometries in low dimensions, in particular in dimension five

9 grudnia 2020/ December 9^th, 2020

Tomasz Smołka (KMMF)

Quasi-lokalny opis elektromagnetyzmu i zlinearyzowanej grawitacji ilustrowany Hopfionam

Przedstawię metodę opisu elektromagnetyzmu za pomocą danych zredukowanych. Proponowany formalizm wykorzystuje foliację przestrzeni dwu-wymiarowymi sferami. Użycie rozkładu Hodge'a-Kodairy dla jednoform w T*(S^2) oraz operatora odwrotnego do laplasjanu na S^2 pozwala zredukować opis elektromagnetyzmu do jednej zespolonej funkcji skalarnej. Analogiczny formalizm z powodzeniem wykorzystuje się dla słabego pola grawitacyjnego. Teoria bedzie ilustrowana Hopfionami --- rozwiązaniami próżniowej teorii Maxwela (zlinearyzowanej grawitacji) o nietrywialnej strukturze topologicznej.

18 listopada 2020/ November 18^th, 2020

Piotr Waluk (KMMF)

How to weigh gravity

In spite of General Relativity Theory being around for over a century, we still do not have a satisfactory understanding of the concept of energy of the gravitational field. The very nature of Einstein's theory makes this supposedly basic physical concept highly non-trivial. Although one can assign a value of global energy to some gravitating systems, the concept of local energy density of gravity turns out to be meaningless.As a way around this, researchers are trying to define energy for finite, but extended regions of spacetimes. This is the still open problem of quasi-local mass. I will discuss some of the difficulties that make the search for a viable definition of quasi-local energy so challenging and show how the approximated theory of linearized gravity may offer some clues in the subject.

12 listopada 2020/ November 12^th, 2020

Mikołaj Rotkiewicz (MIMUW)

Higher order analogs of Lie algebroids

I will introduce the concept of a higher algebroid - the notion generalizing Lie algebroids and higher order tangent bundles. Our approach (joint works with Michał Jóźwikowski) is based on the description of a (Lie) algebroid as a vector bundle comorphism (Zakrzewski morphism) - a relation of special kind. In case of the Lie algebroid of a Lie groupoid G such a relation is obtained by a (natural) reduction of the canonical involution T T G -> T T G and it has a direct generalization when T G is replaced with the higher order tangent bundle of G. The discussed notion of a k-th order (Lie) algebroid (k\geq 1) has a rich algebraic structure and is very natural from the perspective of a geometric formalism of a k-th order variational calculus.
I will also discuss the other concept of higher order analogs of Lie algebroids present in the literature.

4 listopada 2020/ November 4^th, 2020

Katarzyna Grabowska (KMMF)

O Algebroidach na wiele sposobów / Algebroids in many guises

Mój referat przeznaczony będzie przede wszystkim dla młodszych członków naszej społeczności. Opowiem o pojęciu algebroidu i o tym jak można tę samą strukturę opisać na wiele różnych sposobów. Przygotujmy się w ten sposób do kolejnego referatu dr Mikołaja Rotkewicza, który będzie traktował o interesującym uogólnieniu struktury algebroidu.

28 października 2020/ October 28^th, 2020

Seminar cancelled due to strike

21 pazdziernika 2020/ October 21^st, 2020

Zohreh Ravanpak (IMPAN)

Discrete mechanics on non-associative object

Discrete Lagrangian mechanics on Lie groups and groupoids has been developed in many papers. Nevertheless, the generalization of the discrete mechanics to non-associative objectsis still lacking. I will talk about how we can generalize the discrete Lagrangian and Hamiltonian mechanics on Lie groups to non-associative objects generalizing Lie groups (smooth loops). This shows that the associativity assumption is not crucial for mechanics and opens new perspectives. I will show the process of the formulation of the discrete Lagrangian and Hamiltonian mechanics on unitary octonions as a motivating example.

3 czerwca 2020/ June 3^rd, 2020

Vladimir Salnikov (CNRS/La Rochelle University)

Graded and generalized geometry for mechanics

In this talk I will describe some objects of the generalized geometry that appear naturally in the qualitative analysis of mechanical systems. In particular we will discuss the Dirac structures within the framework of the systems with constraints and eventually of port-Hamiltonian systems. From the mathematical point of view, Dirac structures generalize simultaneously symplectic and Poisson structures. As for mechanics, the idea is to design numerical methods that preserve these structures and thus guarantee good physical behaviour in simulations. Then, I will present a framework which is even more general - the one of differential graded manifolds (also called Q-manifolds), and discuss some possible ways of using them for the ``structure preserving integrators'' in mechanics.

27 maja 2020/ May 27^th, 2020

Paweł Urbański (KMMF)

Kilka uwag o opisie hamiltonowskim w teoriach klasycznych / Few remarks on Hamiltonian approach in classical theories

Zacznę od przypomnienia czym jest opis wariacyjny (z przykładami). Układ opisywany jest podzbiorem w wiązce kostycznej (do przestrzeni konfiguracji), będącej zbiorem róniczek działań. Podam odpowiednie konstrukcje w przypadku mechaniki punktu i teorii pola. W niektórych przypadkach oprócz opisu lagranżowskiego istnieje też opis hamiltonowski. Ustosunkuję się do próby opisu układu, jak w teorii pola, jako układu dynamicznego na przestrzeni (rozmaitości) banachowskiej.

20 maja 2020/ May 20^th, 2020

Bartłomiej Bąk (FUW)

The affine variational principle in the theory of gravity

To present the affine variational principle I start from typical (metric) variational formulation of gravity for Hilbert and scalar fields lagrangians and go step by step to an affine case. Then I give a general recipe for a transformation from a metric picture to an affine one and after that I apply given procedure to a few examples.

13 maja 2020/ May 13^th, 2020

Michał Jżwikowski (MIM UW)

Geodesic problem in sub-Riemannian geometry / Geodezyjne w geometrii subriemannowskiej

Simply speaking sub-Riemannian geometry is Riemannian geometry with linear constraints put on velocities. Surprisingly this simple modification makes the problem of finding (and also existence) of length-minimizing curves (geodesics) much more difficult. In the talk I will introduce sub-Riemannian geometry and derive first-order necessary conditions for a curve to be a geodesic in sub-Riemannain geometry.

6 maja 2020/ May 6^th, 2020

Andrew Bruce (University of Luxembourg)

Supermanifolds: Motivation and Introduction

Undoubtedly Berezin is the farther of "supermathematics" who, based on methods of quantum field theory realised the importance of Grassmann algebras in physics. He realised that one should be able to treat commuting and anticommuting variables on an equal footing from both an algebraic and geometric perspective. Spurred on by the development of supersymmetry, Berezin & Leites in 1975 gave the world the notion of a supermanifold. To understand supermanifolds rigorously one needs tools from algebraic geometry, specifically sheaves and locally ringed spaces. However, a working knowledge can quickly be gained using local coordinates in a way almost identical to smooth manifolds. In this talk, I will sketch the coordinate approach to supermanifolds and present some applications of the theory in differential geometry and physics.

4 marca 2020/ March 4^th, 2020

Katarzyna Grabowska (KMMF)

O pewnych nietypowych r�niczkowaniach tranzytywnych nilpotentnych algebr p�l wektorowych

R�niczkowanie algebry, najog�lniej m�wi�c, to liniowe odwzorowanie spe�niaj�ce regu�� Leibniza. R�niczkowania algebry funkcji g�adkich na rozmaito�ci, o warto�ciach rzeczywistych, nad homomorfizmem b�d�cym ewaluacj� funkcji w punkcie to wektory styczne. R�niczkowania algebry funkcji g�adkich na rozmaito�ci o warto�ciach w tej samej algebrze nad identyczno�ci� to g�adkie pola wektorowe. Mo�emy m�wi� tak�e o r�niczkowaniach algebry Liego p�l wektorowych na rozmaito�ci. To�samo�� Jacobiego dla nawiasu Liego p�l wektorowych interpretowa� mo�na jako stwierdzenie, �e ka�de pole wektorowe zadaje r�niczkowanie algebry p�l. Referat po�wi�cony b�dzie r�niczkowaniom tranzytywnych nilpotentnych algebr p�l wektorowych. Okazuje si�, �e w szczeg�lnych sytuacjach istniej� r�niczkowania inne ni� wewn�trzne, tzn. zadane przez elementy algebry.

26 lutego 2020/ February 26^th, 2020

Jerzy Kijowski (CFT)

W jaki spos�b geometria �obaczewskiego pojawia si� w elektrodynamice Maxwella

Poka�� jak fundamentalny problem fizyczny (tzw. "back reaction") prowadzi do trudnych oblicze�, kt�re znacznie upraszczaj� si�, gdy zastosowa� geometri� �obaczewskiego. W warstwie rachunkowej referat b�dzie od� ku czci wsp�rz�dnych bisferycznych.

22 stycznia 2020/ January 22^nd, 2020

Giovanni Moreno (KMMF)

G_2 group and geometric duality / Grupa G_2 i geometryczna dualno��

G_2 group is the first exeptional simple Lie group. The first in the sense, that it has the lowest dimension in the series of exeptional groups. There exists several methods of describing this group. For the purpose of the seminar I choose the algebro-geometric version, since it shows the interesting and not obvious link between some five dimensional manifolds with contact structure and some other five dimensional manifods with the distribution of (2,3,5) type. Such a link is called "geometric duality".

15 stycznia 2020/ January 15^th, 2020

Bartosz Zawora (FUW)

Non-holonomic system of two rolling balls

In this talk, we will discuss non-holonomic system of two balls rolling on each other, with the no-slip and no-spin condition, which defines a rank 2 distribution. Depending on the ratio of the radii of two balls, the symmetry group preserving that distribution may be non-trivial. In particular, if the ratio is 1:3, the symmetry group turns out to be the exceptional Lie group G2. The main results obtained will be presented, and based on the works by G. Bor and R. Montgomery.

8 stycznia 2020/ January 8^th, 2020

Katarzyna Grabowska (KMMF)

Wst�p do geometrii multisymplektycznej (2) / An invitation to multisymplectic geometry (2)

W trakcie seminarium postaram si� opowiedzie� o kanonicznej strukturze multisymplektycznej na wi�zce wielokowektor�w i o tym jak funkcjonuje ona w klasycznej teorii pola.

18 grudnia 2019/ December 18^th, 2019

Eryk Buk (IFT)

Wst�p do geometrii multisymplektycznej / An invitation to multisymplectic geometry

Referat po�wi�cony b�dzie poj�ciu struktury multisymplektycznej. Struktura tego rodzju pojawia si� naturalnie w hamiltonowskim sformu�owaniu klasycznej teorii pola. W trakcie referatu om�wi� kanoniczn� struktur� multisymplektyczn� przestrzeni wielokowektor�w naturalnie pojawiaj�cej si� w teorii pola. Zaproponuj� tak�e og�ln� definicj� struktury multisymplektycznej. Referat oparty jest na przegl�dowej pracy L. Ryvkina i T. Wurzbachera.

11 grudnia 2019/ December 11^th, 2019

Konrad Topolski (FUW)

Metody konforemne w Og�lnej Teorii Wzgl�dno�ci - wprowadzenie / Conformal methods in General Relativity - a short exposition

W trakcie wyst�pienia om�wi� transformacje konforemne g�adkiej rozmaito�ci z metryk� o sygnaturze lorentzowskiej i pochodz�ce od nich transformacje wielko�ci zwi�zanych z krzywizn�. Przedstawi� przyk�ady rachunk�w w lokalnym uk�adzie wsp�rz�dnych. Wprowadz� poj�cie czasoprzestrzeni asymptotycznie prostych i s�abo asymptotycznie prostych. Wyprowadz� pewne wnioski na temat charakteru brzegu rozmaito�ci przy za�o�eniu r�nych w�asno�ci wielko�ci geometrycznych.

4 grudnia 2019/ December 4^th, 2019

Jacek Wojtkiewicz (KMMF)

O spinowych stanach koherentnych w kwantowym modelu Heisenberga / On spin coherent states in quantum Heisenberg model

Stan koherentny spinowy otrzymamy
je�li Exponensem od operator�w momentu p�du zadzia�amy
Na jaki� stan wybrany;
kt�ry bierzemy jako najbardziej namagnesowany.
Jedynk� przez stany koherentne wyrazimy
i do sumy statystycznej wstawimy;
Nieliniowy Model Sigma t� drog� dostaniemy;
obecno�� cz�onu topologicznego (WZ) przy tym wykryjemy.

Alternatywne podej�cie kr�tko wzmiankuj�,
gdzie Botta-Borela-Weila tw. si� wykorzystuje;
geometrii wok� tego mrowie a mrowie;
je�li kto� to lubi - b�dzie mia� u�ywania, co si� zowie.
Wszelako u�yteczno�� tego na razie nieudokumentowana;
je�li chcesz to, s�uchaczu, zmieni� - to aktywno�� twoja mile widziana.

27 listopada 2019/ November 27^th, 2019

Zohreh Ravanpak (IMPAN)

Brief introduction to Poisson geometry

Poisson bracket on a smooth manifold M is a bracket on the algebra of smooth functions satisfying the familiar properties of the Lie bracket, additionally, a Leibniz identity. I will talk about the history about how study of classical Hamiltonian mechanics naturally arised a Poisson bracket of two observable quantities. Basic definitions, relation between Poisson brackets and Poisson bivectors and Hamilton's equations of motion and some relevant examples will be included.

13 listopada 2019/ November 13^th, 2019

Jacek Wojtkiewicz (KMMF)

O stanach koherentnych w zastosowaniu do modelu Heisenberga i geometrii tam�e. 0. Prolog: O FunCa�kach s��w kilka /

Heisenberg model in the language of coherent states and its geometric aspects. 0. Prologue: Few words about functional integrals

Wypisuj�c sum� statystyczn� Z dla kwantowego modelu Heisenberga, cz�sto korzysta si� z wzoru Trottera i wstawia du�o (oo wiele) jedynek,kt�re mo�na napisa� na r�ne sposoby, a jednym z nich jest wyra�enie jej przez stany koherentne. W ten spos�b Z wyra�a si� przez ca�k� funkcjonaln�. Najsampierw zamierzam powiedzie� 'jak to robi� fizycy', a o tym, jak to robi� 'lege artis', by� mo�e p�niej. W ka�dym przypadku b�dzie to poprzedzone wst�pem nt. ca�ek funkcjonalnych.

6 listopada 2019/ November 6^th, 2019

Giovanni Moreno (KMMF)

Lagran�owskie Grassmaniany, nieliniowe r�wnania r�niczkowe drugiego rz�du i ich charakterystyki (II) / Lagrangian Grassmanians, nonlinear second order differential equations and chracteristics (II)

Wprowadz� struktury geometryczne odpowiednie do badania nieliniowych r�wna� r�niczkowych drugiego rz�du, oparte na poj�ciu Grassmanianu Lagran�owskiego, tzn. rozmaito�ci wszystkich n-wymiarowych liniowych podprzestrzeni (2n)-wymiarowej przestrzeni symplektycznej, na kt�rych forma symplektyczna znika. Skoncentruj� si� na przypadku n=2 i om�wi� wyniki zwi�zane z r�wnaniem Monge-Ampere'a oraz jego charakterystykami.

30 pa�dziernika 2019/ October 30^rd, 2019

Giovanni Moreno (KMMF)

Lagran�owskie Grassmaniany, nieliniowe r�wnania r�niczkowe drugiego rz�du i ich charakterystyki (I) / Lagrangian Grassmanians, nonlinear second order differential equations and chracteristics (I)

Wprowadz� struktury geometryczne odpowiednie do badania nieliniowych r�wna� r�niczkowych drugiego rz�du, oparte na poj�ciu Grassmanianu Lagran�owskiego, tzn. rozmaito�ci wszystkich n-wymiarowych liniowych podprzestrzeni (2n)-wymiarowej przestrzeni symplektycznej, na kt�rych forma symplektyczna znika. Skoncentruj� si� na przypadku n=2 i om�wi� wyniki zwi�zane z r�wnaniem Monge-Ampere'a oraz jego charakterystykami.

23 pa�dziernika 2019/ October 23^rd, 2019

Pawe� Urba�ski (KMMF)

Obiekty tworz�ce relacji symplektycznych / Generating objects of symplectic relations

O obiektach tworz�cych podrozmaito�ci m�wimy, gdy rozmaito�� symplektyczna jest izomorficzna wi�zce kostycznej. Mowa b�dzie o obiektach tworz�cych relacji symplektycznych i ich sk�adaniu w kontek�cie transformacji Legendre'a.

16 pa�dziernika 2019/ October 16^th, 2019

Tomasz Smo�ka (KMMF)

Symetrie czasoprzestrzeni i wielko�ci zachowane / Symmetries of spacetimes and conserved quantities

W trakcie seminarium om�wi� konforemne pola Killinga i ich uog�lnienia do wy�szych tensor�w Killinga i Yano-Killinga oraz wielko�ci zachowane zwi�zane z symetriami.

9 pa�dziernika 2019/ October 9^nd, 2019

Katarzyna Grabowska (KMMF)

Lagran�jan, Hamiltonian i transformacja Legendre'a / Lagrangian, Hamiltonian and Legendre transformation

W trakcie seminarium inauguruj�cego nasz spotkania w nowym roku akademickim opowiem o podstawowych strukturach geometrycznych zwi�zanych z mechanik� lagran�owsk�, mechanik� hamiltonowsk� i transformacj� Legendre'a. Seminarium b�dzie dostosowane do potrzeb nowych s�uchaczy.

22 maja 2019/ May 22^nd, 2019

Katja Sagerschnig (CFT)

Introduction to projective differential geometry

A projective structure on a manifold can be viewed as the collection of all unparametrized geodesics of an affine connection. The aim of this talk is to give a quick introduction to projective structures, their invariants, symmetries, and to problems in the field.

15 maja 2019/ May 15^th, 2019

The seminar is cancelled

8 maja 2019/ May 8th^th, 2019

Tomasz SMO�KA (KMMF)

Elektrodynamika na formach r�niczkowych c.d./ Electrodynamics in the language of differential forms (continued)

17 kwietnia 2019/ April 17^th, 2019

Piotr T. Chru�ciel (University of Vienna)

Roads to topological censorship

I will review the various theorems describing the restrictions on the topology of solutions of Einstein equations, known under the name of "topological censorship"

10 kwietnia 2019/ April 10^th, 2019

Tomasz SMO�KA (KMMF)

Elektrodynamika na formach r�niczkowych / Electrodynamics in the language of differential forms

3 kwietnia 2019/ April 3^rd, 2019

Marcin ZAJ�C (KMMF)

Wst�p do teorii wi�zek g��wnych II / Introduction to principal bundles II

Struktura wi�zki g��wnej stanowi naturalny j�zyk do opisu teorii pola z cechowaniem. W ramach mojego referatu przedstawi� podstawowe w�asno�ci wi�zek g��wnych oraz ich zastosowania w fizyce.

27 marca 2019/ March 27^th, 2019

Marcin ZAJ�C (KMMF)

Wst�p do teorii wi�zek g��wnych / Introduction to principal bundles

Struktura wi�zki g��wnej stanowi naturalny j�zyk do opisu teorii pola z cechowaniem. W ramach mojego referatu przedstawi� podstawowe w�asno�ci wi�zek g��wnych oraz ich zastosowania w fizyce.

20 marca 2019/ March 20^th, 2019

Jacek WOJTKIEWICZ (KMMF)

O genezie struktury kontaktowej w termodynamice / On the genesis of contact structure in thermodynamics

Termodynamika sta�a si� teori� logicznie zamkni�t� zasadniczo pod koniec XIX wieku. Znacznie p�niej dostrze�ono, �e uk�ad termodynamiczny posiada naturaln� struktur� kontaktow�. Zamierzam om�wi�, jak si� ta struktura pojawia.

13 marca 2019/ March 13^th, 2019

Maciej NIESZPORSKI (KMMF)

Wst�p do powierzchni solitonowych / An introduction to soliton surfaces

Teoria uk�ad�w ca�kowalnych r�wna� r�niczkowych ma swoje korzenie w pracach geometr�w prze�omu XIX i XX wieku. Pokr�tce przedstawi� motywacj� autor�w takich jak Bianchi, Darboux, Tziteica, Eisenhart, Wilczy�ski czy Blaschke

6 marca 2019/ March 6th^th, 2019

Jacek JEZIERSKI (KMMF)

Geometria powierzchni zerowych / Geometry of null hypersurfaces

27 lutego 2019/ February 27^th, 2019

Janusz GRABOWSKI (IMPAN)

Wst�p do algebroid�w Liego / Introduction to Lie algebroids

23 stycznia 2019/ January 23^rd, 2019

Javier de LUCAS (KMMF)

An introduction to Poisson-Lie groups and dressing transformations

In a nutshell, a Poisson-Lie group is a Lie group with a certain type of compatible Poisson structure. From a physical viewpoint, the study of Poisson-Lie groups was initially motivated by the analysis of the Hamiltonian structure of the group of so-called dressing transformations arising in the inspection of relevant classes of integrable systems. The aim of this talk is to introduce the theory of Poisson-Lie groups and their dressing transformations. After recalling several basic results on Poisson manifolds, I will characterise Poisson-Lie groups in different equivalent ways, I will study the properties of their related Lie algebras (which leads to the theory of Lie bialgebras), and I will analyse the main properties of dressing transformations.

16 stycznia 2019/ January 16^th, 2019

Katarzyna GRABOWSKA (KMMF)

Geometria warto�ci afinicznych / Geometry of affine values

W geometrii r�niczkowej pracujemy zazwyczaj z obiektami o charakterze liniowym. Wektory styczne, kowektory i inne tensory s� elementami odpowiednich przestrzeni wektorowych. Z drugiej strony w teoriach fizycznych pojawiaj� si� wielko�ci, kt�re maj� charakter afiniczny. Na przyk�ad prawo transformacji p�du w ramach mechaniki nierelatywistycznej mi�dzy dwoma uk�adami odniesienia poruszaj�cymi si� wzgl�dem siebie jest odwzorowaniem afinicznym. Odpowiednim narz�dziem do opisu tego rodzaju sytuacji jest geometria warto�ci afinicznych (AV-geometry). W geometrii tej algebra funkcji g�adkich na rozmaito�ci zast�piona jest przez zbi�r g�adkich ci�� jednowymiarowej wi�zki afinicznej. Znajomo�� geometrii warto�ci afinicznych przydaje si� do opisu dynamiki cz�stki na�adowanej, do niezale�nego od uk�adu odniesienia sformu�owania mechaniki Newtona oraz w wielu miejsach w klasycznej teorii pola. W czasie referatu przedstawi� podstawowe konstrukcje zwi�zane z geometri� warto�ci afinicznych i wspomn� o jej zastosowaniach.

9 stycznia 2019/ January 9^th, 2019

Piotr WALUK (KMMF)

Symetrie jednorodnych wszech�wiat�w - cz�� II / Symmetries of homogeneous universes - part II

Jednorodne przestrzennie modele kosmologiczne s� niemal w pe�ni scharakteryzowane przez swoj� grup� symetrii przestrzennych. Podczas wyst�pienia om�wi� kilka przyk�ad�w jednorodnych przestrzennie czasoprzestrzeni, kt�re pomimo swej prostoty wykazuj� intryguj�ce w�asno�ci fizyczne.

19 grudnia 2018/ December 19^th, 2018

Pawe� URBA�SKI (KMMF)

R�niczkowania i koneksje: kilka bis�w / Derivations and connections: few pieces for an encore

W trakcie referatu dyskutowa� b�dziemy zagadnienia, kt�re pojawia�y si� w tej czy innej formie w trakcie kilku wcze�niejszych seminari�w. Elementem ��cz�cym te zagadnienia b�dzie u�ycie form o warto�ciach wektorowych. W miar� dost�pnego czasu om�wimy (1) wz�r na r�niczkowanie d_T w kontek�cie ca�kowania przez cz�ci w rachunku wariacyjnym, (2) nawiasy Liego r�niczkowa� w reprezentacji przez formy o warto�ciach wektorowych, tzn nawiasy Richardsona-Nijenhuisa i Froelichera-Nijenhuisa, (3) jednoforma r�niczkowa jako powi�zanie (koneksja), (3) krzywizna powi�zania wyra�ona przez nawias Froelichera-Nijenhuisa, (4) torsja powi�zania w wi�zce stycznej.

12 grudnia 2018/ December 12^th, 2018

Katarzyna GRABOWSKA (KMMF)

Transport r�wnoleg�y i koneksja - ci�g dalszy/ Paralel transport and connection - continuation

W ubieg�ym tygodniu om�wione zosta�o poj�cie koneksji liniowej w wi�zce wektorowej oraz poj�cie pochodnej kowariantnej zwi�zanej z t� koneksj�. W najbli�sz� �rod� przypomn� wprowadzone wcze�niej poj�cia oraz wprowadz� kolejne, tzn. transport r�wnoleg�y i krzywizn� koneksji. W dalszej cz�ci referatu skoncentruj� si� na koneksji w wi�zce stycznej.

5 grudnia 2018/ December 5th^th, 2018

Eryk BUK (FUW)

Pracuj�c na przestrzeni wektorowej lub afinicznej mo�emy por�wnywa� wektory styczne, kowektory czy inne obiekty tensorowe w r�nych punktach przestrzeni, mo�emy tak�e przesuwa� wektory wzd�u� krzywych. Na bardziej og�lnych rozmaito�ciach do przesuwania wektor�w wzd�u� krzywych potrzebna jest dodatkowa struktura - koneksja. Referat b�dzie po�wi�cony definicji i w�asno�ciom koneksji liniowej w wi�zkach wektorowych.

28 listopada 2018/ November 28th^th, 2018

Marian WIATR(KMMF)

R�niczkowania w algebrach / Derivations in algebras

W ka�dej og�lnej algebrze mo�na zdefiniowa� r�niczkowania, jako zbi�r odwzorowa� liniowych spe�niaj�cych regu�� Leibniza. M�j referat b�dzie skupia� si� na r�niczkowaniach w algebrze form na rozmaito�ci. Opowiem tak�e o interpretacji form o warto�ciach wektorowych i znaczeniu ich r�niczkowa� w geometrii i mechanice klasycznej.

21 listopada 2018/ November 21st^th, 2018

Piotr WALUK (KMMF)

Symetrie jednorodnych wszech�wiat�w / Symmetries of homogeneous universes

Jednym z naturalnych uog�lnie� znanego modelu kosmologicznego Friedmanna-Lemaitre'a jest porzucenie za�o�enia o izotropowo�ci wszech�wiata. Ten krok otwiera istn� plejad� modeli kosmologicznych o interesuj�cych w�asno�ciach. Podczas wyst�pienia przedstawi� klasyfikacj� jednorodnych nieizotoropowych wszech�wiat�w ze wzgl�du na ich symetrie przestrzenne oraz om�wi� dok�adniej kilka wybranych przyk�ad�w.

14 listopada 2018/ November 14^th, 2018

Katarzyna GRABOWSKA (KMMF)

O wi�zkach gradowanych - cz�� II / On graded bundles - part II

Referat po�wi�cony b�dzie poj�ciu wi�zki gradowanej. Szczeg�owo om�wi� struktur� wi�zki drugich d�et�w krzywych w rozmaito�ci. Pojawi� si� te� inne naturalne przyk�ady wi�zek gradowanych.

7 listopada 2018/ November 7^th, 2018

Pawe� URBA�SKI (KMMF)

O r�wnaniu Eulera-Lagrange'a nieco inaczej / On Euler-Lagrange equation - another point of view

31 pa�dziernika 2018/ October 31^st, 2018

Giovanni MORENO (KMMF)

O geometrii r�wna� r�niczkowych cz�stkowych / On geometry of Partial Differential Equations

Wst�p do podstawowych metod geometrycznych u�ywanych w teorii r�wna� r�niczkowych cz�stkowych i ich symetrii.

24 pa�dziernika 2018/ October 24^th, 2018

Katarzyna GRABOWSKA (KMMF)

O wi�zkach wektorowych i gradowanych, pojedynczych i podw�jnych / On single and double vector and graded bundles

W trakcie referatu om�wione b�d� dwie definicje i strktura wi�zek wektorowych, pojedy�czych i pow�jnych oraz wi�zek gradowanych. Szczeg�owo zostan� om�wione wi�zka styczna i kostyczna do wi�zki wektorowej i gradowanej. Om�wiony materia� mo�e stanowi� wst�p do dalszych referat�w dotycz�cych koneksji w wi�zkach wektorowych.

17 pa�dziernika 2018/ October 17^th, 2018

Marian WIATR (KMMF)

Operators of evolution and energy in Classical Field Theory

The hamiltonian equations for field theory can be interpreted as an evolution of one complex field. I will show strict, funcional-analytic description of evolution of electromagnetic field contained in a bounded area. There will be accented connection between boundary conditions for field and energy.

10 pa�dziernika 2018/ October 10^th, 2018

Daniel WYSOCKI (KMMF)

Kr�tkie wprowadzenie do algebr Hopfa i ich deformacji

3 pa�dziernika 2018/ October 3^rd, 2018

Julia LANGE (KMMF)

Redukcja Marsdena-Weinsteina na rozmaito�ciach niesko�czonego wymiaru

16 maja 2018/ May 16^th, 2018

Pawe� GORA (MIM UW)

An introduction to machine learning

Machine learning (and artificial intelligence, in general) is becoming more and more popular and is finding applications in many areas such as computer vision, natural language processing and optimal control of complex processes. It is expected that in the future, thanks to better algorithms, greater availability of high-quality data and computational power, AI programs will be still changing the world and outperforming humans in more and more complex tasks. In the seminar, I will explain basic concepts of machine learning and present popular and modern techniques related especially to deep neural networks and optimization problems. Finally, I will also mention about possible applications of geometrical methods in machine learning.

9 maja 2018/ May 9^th, 2018

Jerzy KIJOWSKI (CFT)

Energia fal grawitacyjnych wg. A. Trautmana / Energy carried by gravitational field: an approach based on ideas proposed by A. Trautman

2 maja 2018/ May 2^nd, 2018

There will be no seminar on May 2nd.

25 kwietnia 2018/ April 25^th, 2018

Pawe� URBA�SKI (KMMF)

Analytical mechanics on algebroids once more

I shall present a significantly simplified version of Martinez approach to Euler-Lagrange equations on Lie algebroids. In particular no use of the "prolonged algebroids" will be made. To avoid disambiguities I shall recall fundamental constructions concerning Hamiltonian and Lagrangian formulations of the dynamics and the Euler+Lagrange equations. The Euler-Lagrange equations obtained via "Martinez method" will be compared with Euler-Lagrange equations proposed by Grabowski, Grabowska and Urba�ski.

18 kwietnia 2018/ April 18^th, 2018

Piotr PRAGACZ (IMPAN)

Flag bundles, Segre polynomials and push-forwards

We give Gysin formulas for partial flag bundles for the classical groups. We then give Gysin formulas for Schubert varieties in Grassmann bundles, including isotropic ones. All these formulas are proved in a rather uniform way by using the step-by-step construction of flag bundles and the Gysin formula for a projective bundle. In this way we obtain a comprehensive list of new universal formulas. This is a joint work with Lionel Darondeau.

11 kwietnia 2018/ April 11^th, 2018

Ewa GIREJKO (Politechnika Bia�ostocka)

On consensus in opinion formation models based on multi�agents systems

Multi�agents systems describe the process of agreement (consensus) in a group of interacting agents. Recently, these systems are applied not only to the consensus, formation or flocking tasks of unmanned vehicles or simple robots but also are considered as a tool of modelling social in- teractions. In this presentation, a short survey of multi�agents systems under the consensus problem will be provided. Since the most representa- tive class of agent�based models explaining phenomena in social sciences are the opinion formation models, we focus on the Hegselmann�Krause (H-K) and Cucker�Smale (C-S) models. The control strategies for both cases will be considered. Some modification of H-K and C-S models will be also presented.

28 marca 2018/ March 28^th, 2018

Pawe� NUROWSKI (CFT)

Examples of nonholonomic systems: ants on a table.

?

21 marca 2018/ March 21^st, 2018

Katja SAGERSCHNIG (University of Vienna)

The Geometry of almost Einstein (2,3,5) distributions

The first part of the talk will be a brief introduction to Cartan geometries related with the exceptional Lie group G², i.e., to (2,3,5) distributions and to contact twisted cubic structures. I will introduce the associated normal Cartan connection, and an equivalent concept, the normal tractor connection. I will also recall Nurowski's construction of a conformal structure associated with a (2,3,5) distribution.
The goal of the second part is to explain joint work with Travis Willse on the geometry of Nurowski conformal structures admitting parallel standard tractors. Such a parallel tractor determines a partition of the manifold into several submanifolds, each naturally endowed with a geometric structure. This will reveal links between (2,3,5) distributions and several other geometries, including, Sasaki-Einstein structures of signature (2,3), Fefferman-conformal structures, and their para-complex analogues.I will also relate this picture to work of Gil Bor, Pawel Nurowski and Omid Makhmali.

14 marca 2018/ March 14^th, 2018

Javier de LUCAS (KMMF)

A cohomological and geometric approach to immersion formulas for soliton surface

A geometric approach to immersion formulas for soliton surfaces is provided via a generalization of the de Rham cohomology and its differential to a space of Lie algebra-valued differential forms parametrised by a spectral parameter. This leads to introducing new Poincare type lemmas for such cohomologies, which appropriately describe integrability conditions and deformations of Lax pairs. In this language, properties of soliton surfaces, e.g. immersion formulas, become very simple and generalizations of 2D-models to soliton submanifolds appear straightforwardly. Theoretical results are illustrated by physical examples.

7 marca 2018/ March 7^th, 2018

Micha� JӏWIKOWSKI (IMPAN)

Prolongations vs. Tulczyjew triples in geometric mechanics

In the scientific literature there are basically two schools of formulating Lagrangian (or Hamiltonian) mechanics in the (Lie) algebroid setting: in terms of prolongations and in terms of Tulczyjew triples. Despite the fact that in both approaches we describe the same phenomena, so far no comparison between prolongations and Tulczyjew triples was made. In this note we aim to fill this gap. More precisely, we will strip the prolongation approach to uncover the Tulczyjew triple reality hidden inside, thus proving that the latter approach is a more basic one.

28 lutego 2018/ February 28^th, 2018

Kamil NIEDZIA�OMSKI (Uniwersytet ��dzki)

G-structures and intrinsic torsion

Consider an oriented Riemannian manifold. By a G-structure we mean a reduction of the structure group of oriented orthonormal frame bundle to a subgroup G. Such reduction implies some properties on the base manifold, for example, existence of almost hermitian structure, etc. Assuming some algebraic condition of the level of Lie algebras, a component of the Levi-Civita connection defines a G-connection. The difference of these connections is called the intrinsic torsion.

During the talk I will discus in detail properties of the intrinsic torsion, its applications and describe my recent results concerning geometry and integral formulae for G-structures with the use of the intrinsic torsion.

24 stycznia 2018/ January 24^th, 2018

Tomasz MACI��EK (CFT)

Momentum polytope at the highest weight

I will present our recent results concerning the problem of approximating the momentum polytope for irreducible unitary representations of connected compact semisimple Lie groups. I will outline our motivation, which comes from the formulation of the quantum marginal problem of describing spectra of reduced quantum states in terms of the image of a momentum map. The approximations of the momentum polytope that we consider stem from studying the structure of the momentum image around the highest weight of the representation at hand. These are mainly representations of local unitary groups in systems with a fixed number of particles.

17 stycznia 2018/ January 17^th, 2018

Adam SAWICKI (CFT)

Convexity of the momentum map and quantum entanglement

Symplectic and algebraic geometry tools have proven to be very useful for the description of quantum correlations. They not only provide a mathematically consistent way of phrasing these problems but also offer an insight which is not available with linear algebra approach. In this talk I will discuss ideas and concepts standing behind these methods. First I will review the connection between symmetries of symplectic manifolds and the momentum map. Then I will study the situation when the considered symplectic manifold is the complex projective space of the multi-particle Hilbert space. In particular I will discuss connections with the Kirwan-Ness stratification and the Brion�s convexity theorem that lead to the concept of the entanglement polytope. Entanglement polytopes have been recently proposed as a way of witnessing multipartite entanglement classes using single particle information. I will present first asymptotic results concerning feasibility of this approach for large number of qubits.

10 stycznia 2018/ January 10^th, 2018

Katarzyna GRABOWSKA (FUW)

On the concept of filtered bundle

We present the notion of a filtered bundle as a generalisation of a non-negatively graded manifold. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of the coordinates by allowing more general `filtered' transformation laws. The key examples of such bundles include affne bundles and various jet bundles, both of which play fundamental roles in geometric mechanics and field theory.

20 grudnia 2017/ December 20^th, 2017

On December 20th there will be no seminar. We meet again on January 10th.

13 grudnia 2017/ December 13^th, 2017

Marcin ZAJ�C (FUW)

Symplectic and Poisson geometry in application to thermodynamics and statistical physics

I will show some of the basic tools in symplectic and Poisson geometry with applications to thermodynamics and statistical physics. The basic concepts of thermodynamics like the notion of equilibrium, entropy or Gibbs statistical states may be expressed in terms of differential geometry in a nice and elegant way. I will show how the concept of the Gibbs state may be generalised for a Hamiltonian action of a Lie group on the symplectic manifold representing the system. The examples of the applications in physics will be given.

6 grudnia 2017/ December 6^th, 2017

Miko�aj ROTKIEWICZ (MIMUW)

Higher algebroids via differential relations (III)

In the community of people working in Geometric Mechanics it is well-known that (Lie) algebroids provide an elegant language to describe Lagrangian mechanics and variational calculus. However, it is not that clear what geometric structures should we use to describe theories involving derivatives of order higher than one. In the talk we will propose a class of such structures - higher analogs of algebroids. Our theory is based on two elegant geometric concepts: a graded bundle and a Zakrzewski morphism (a differential relation of a special kind).

29 listopada 2017/ November 29^th, 2017

Aneta SLI�EWSKA (Universytet w Bia�ymstoku)

Poisson geometry related to Atiyah sequences

We construct and investigate a short exact sequence of Poisson VB-groupoids which is canonically related to the Atiyah sequence of a G-principal bundle P. The results include a description of the structure of the symplectic leaves of the Poisson groupoid (T*P x T*P)/G

22 listopada 2017/ November 22^nd, 2017

Andrzej TRAUTMAN (FUW)

Optical structures in realitivity

15 listopada 2017/ November 15^th, 2017

Seminarium odwo�ane / Session cancelled

8 listopada 2017/ November 8^th, 2017

Marek DEMIA�SKI (FUW)

Fale grawitacyjne - nowe okno na Wszech�wiat

18 i 25 pa�dziernika 2017/ October 18^th and 25^th, 2017

Micha� JӏWIKOWSKI (IMPAN), Miko�aj ROTKIEWICZ (MIMUW)

Higher algebroids via differential relations

In the community of people working in Geometric Mechanics it is well-known that (Lie) algebroids provide an elegant language to describe Lagrangian mechanics and variational calculus. However, it is not that clear what geometric structures should we use to describe theories involving derivatives of order higher than one. In the talk we will propose a class of such structures - higher analogs of algebroids. Our theory is based on two elegant geometric concepts: a graded bundle and a Zakrzewski morphism (a differential relation of a special kind).

4 i 11 pa�dziernika 2017/ October 4^th and 11^th, 2017

Janusz GRABOWSKI (IMPAN)

Higher order Lagrangians.

We will start with presenting a geometric approach to first order Lagrangian Mechanics a la Tulczyjew on Lie algebroids. Then, we will discuss variational and geometric approaches to Mechanic based on action functionals depending on higher derivatives of paths in configuration space. In the geometric framework we will understand higher Lagrangians as constrained Lagrangian functions on higher tangent bundles. Jacobi-Ostrogradski momenta and higher Euler-Lagrange equations will be derived from a geometric formalism of the Tulczyjew triple.

24 maja 2017/ May 24^th, 2017

Andrey KRUTOV (IMPAN)

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 proves non-triviality of a deformation of a simple (3|2)-dimensional Lie superalgebra (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).

16 maja 2017/ May 16^th 2017, IMPAN, room 403

Fr�d�ric 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.

10 maja 2017/ May 10^th, 2017

Pawe� URBA�SKI (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*.

26 kwietnia 2017/ April 26^th, 2017

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 the Q-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.

19 kwietnia 2017/ April 19^th, 2017

The seminar is cancelled due to organizational reasons

12 kwietnia 2017/ April 12^th, 2017

Anatol ODZIJEWICZ (Uniwersytet w Bia�ymstoku)

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.

5 kwietnia 2017/ April 5^th, 2017

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.

29 marca 2017/ March 29^th, 2017

Jerzy KIJOWSKI (CFT)

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.

22 marca 2017/ March 22^nd, 2017

Andriy PANASYUK (UWM)

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.

15 marca 2017/ March 15^th, 2017

Tatiana SHULMAN (IMPAN)

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.

8 marca 2017/ March 8^th, 2017

Zohreh RAVANPAK (IMPAN)

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.

1 marca 2017/ March 1^st, 2017

Miko�aj ROTKIEWICZ (MIMUW)

Polarisation of graded bundles

raded 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

25 stycznia 2017/ January 25^th, 2017

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 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.

18 stycznia 2017/ January 18^th, 2017

Tomasz SMO�KA (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.

11 stycznia 2017/ January 11^th, 2017

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.

7 grudnia 2016/ December 7^th, 2016

Micha� JӏWIKOWSKI (IMPAN)

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.

30 listopada 2016/ November 30^th, 2016

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.

23 listopada 2016/ November 23^rd, 2016

Andrey KRUTOV (IMPAN)

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

16 listopada 2016/ November 16^th, 2016

Tadeusz JANUSZKIEWICZ (IMPAN)

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 �wiatos�aw Gal (Wroc�aw University).

9 listopada 2016/ November 9^th, 2016

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).

2 listopada 2016 seminarium nie odb�dzie si�/ on November 2^nd, 2016 there will be no seminar

26 pa�dziernika 2016/ October 26^th, 2016

Andrey KRUTOV (IMPAN)

Geometry of jets spaces and PDE (I)

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).

19 pa�dziernika 2016/ October 19^th, 2016

Javier de LUCAS (KMMF)

Bundle Lie systems and applications

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 generalizations 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.

12 pa�dziernika 2016/ October 12^ve, 2016

Katarzyna GRABOWSKA, Pawe� URBA�SKI (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.

5 pa�dziernika 2016/ October 5^st, 2016

Katarzyna GRABOWSKA, Pawe� URBA�SKI (KMMF)

Geometry of Routh reduction

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.

1 czerwca 2016/ June 1^st, 2016

Stefan RAUCH-WOJCIECHOWSKI (Link�ping University)

Global dynamics of a rolling and sliding disc, asymptotic solutions, stability

The problem of a disc rolling in a plane has been treated in classical works of P.Apple, D.J.Korteweg, E.J.Routh and S.A.Chaplygin. It is described by a dynamical system of 4 equations, has 3 integrals of motion and is integrable. When sliding is allowed there are 2 more dynamical variables, equations are dissipative, non-integrable and have energy as a monotonously decreasing function of time. The key for understanding the dynamics are asymptotic solutions, their stability properties and a La�Salle type theorem on asymptotic behavior of solutions. These results, together with with computer simulations of solutions starting in vicinity of the asymptotic solutions provide a basis for global understanding of what happens for different initial conditions. I shall explain formulation of the problem, present analytical results and tell how much we have learnt from numerical simulations.

25 maja 2016/ May 25^th, 2016

Artur GI�YCKI (MiNI PW)

Representations of groups in relation to representations of transformation groupoids

I present the fundamentals of representation theory of groups and groupoids. I will define the induced representation, imprimitivity systems and theorems of Mackey and Landsman that allow to move between these concepts. I narrow our discussion to transitive transformation groupoid G ✕ K/G where G is locally compact group and K is its compact subgroup. I am going to show that representation of transformation groupoid is irreducible if and only if inducing representation of subgroup K is irreducible and several other theorems.

18 maja 2016/ May 18^th, 2016

Pawe� CIOSMAK (MIMUW)

Partition function in Yang-Mills theory and the moduli space of flat connections in dimension 2

The moduli space of flat connections over a smooth manifold can be equipped in a very natural way with a symplectic structure. According to the theorem by Witten the symplectic volume of this space is given by a certain limit of the partition function in Yang-Mills theory. In dimension two this partition function can be calculated using a discrete approximation, which happen to give the exact value of the integral. I will present these topics and give examples when gauge group is SU(2) or SO(3)

11 maja 2016/ May 11^th, 2016

Pawe� Urba�ski (KMMF)

Kaluza-Klein theory contra geometry of affine values (part II)

27 kwietnia 2016/ April 27^th, 2016

Pawe� Urba�ski (KMMF)

Kaluza-Klein theory contra geometry of affine values

20 kwietnia 2016/ April 20^th, 2016

Andrzej WEBER (MIMUW)

"Motivic" decomposition and C^* action

According to the Grothendieck idea every algebraic variety is built in some sense from indecomposable pieces. These pieces are called motives.When a cohomology theory is applied to a variety (i.e. a certain type of functor from varieties to an abelian category), then the decomposition is visible as a decomposition into a direct sum. We will discuss an example of such decomposition coming from an action of C^* (for complex varieties). We confront this decomposition with the localization theorem in equivariant cohomology.

13 kwietnia 2016/ April 13^th, 2016

Jorge JOVER (Univesidad de Zaragoza))

Dynamics on the Space of States: a Geometrical Description

I will present a picture of Quantum Mechanics based on the geometrical description of physical observables in terms of expectation value functions, generalizing thus the so called Ehrenfest picture of quantum systems. In particular, I will describe the set of pure and mixed states of a quantum system and its geometrical properties. The basic geometrical tools reproduce the algebraic structure of the set of observables on a Hilbert space. This formalism incorporate in a natural way the probabilistic description of Quantum Mechanics. The geometrical formalism allows to analyze from a new perspective the properties of quantum systems, such as dynamical equations, uncertainty relations, coherent states and the non-unitary evolution of open systems. I will focus in the application of the formalism to describe the Markovian evolution of quantum systems determined by the Kossakowski-Lindblad equation.

6 kwietnia 2016/ April 6^th, 2016

Norbert PONCIN (UniLu)

Towards integration on Z₂ⁿ supermanifolds

The aim of the talk is to describe a generalization of Superalgebra and Supergeometry to Z₂ⁿ-gradings, n>1. The corresponding sign rule is not given by the product of the parities, but by the scalar product of the involved Z₂ⁿ-degrees. This Z₂ⁿ-Supergeometry exhibits interesting differences with classical Supergeometry, provides a sharpened viewpoint, and has better categorical properties. Further, it is closely related to Clifford calculus: Clifford algebras have numerous applications in Physics, but the use of Z₂ⁿ-gradings has never been investigated. More precisely, we discuss the geometry of Z₂ⁿ-supermanifolds, give examples of such colored supermanifolds beyond graded vector bundles, and study the generalized Batchelor-Gaw�dzki theorem. However, the main focus is on the Z₂ⁿ- Berezinian and on first steps towards the corresponding integration theory, which is related to an algebraic variant of the multivariate residue theorem.

30 marca 2016/ March 30^th, 2016

Miko�aj ROTKIEWICZ (MIMUW)

On smooth monoid actions on manifolds (II)

Grabowski and Rotkiewicz showed that a smooth action of the monoid of multiplicative reals equips a manifold with the structure of a, so called, graded bundle, being a natural generalization of the notion of a vector bundle. In the light of this result it is interesting to ask what are the natural structures on manifolds related with actions of other natural monoids generalizing the multiplicative reals. We shall discuss this problem in a series of two lectures. In the first of these (M. J�wikowski, 23. March) we will recall the known results of Grabowski and Rotkiewicz and discuss actions of the monoid of second jets of real functions which preserve zero. In the second (M.Rotkiewicz, 30. March) we will discuss actions of the multiplicative complex numbers and actions of the multiplicative reals on supermanifolds.

23 marca 2016/ March 23^rd, 2016

Micha� JӯWIKOWSKI (IMPAN)

On smooth monoid actions on manifolds

Grabowski and Rotkiewicz showed that a smooth action of the monoid of multiplicative reals equips a manifold with the structure of a, so called, graded bundle, being a natural generalization of the notion of a vector bundle. In the light of this result it is interesting to ask what are the natural structures on manifolds related with actions of other natural monoids generalizing the multiplicative reals. We shall discuss this problem in a series of two lectures. In the first of these (M. J�wikowski, 23. March) we will recall the known results of Grabowski and Rotkiewicz and discuss actions of the monoid of second jets of real functions which preserve zero. In the second (M.Rotkiewicz, 30. March) we will discuss actions of the multiplicative complex numbers and actions of the multiplicative reals on supermanifolds.

16 marca 2016/ March 16^th, 2016

Luca VITAGLIANO (University of Salerno)

Generalized geometry in odd dimensions

Generalized complex structures were introduced by Hitchin and further studied by Gualtieri. They encompass symplectic structures and complex structures as extreme cases. In the general case, a generalized complex manifold can be seen as an even dimensional Poisson manifold equipped with additional structures. In the first part of the talk, we will review the definition of generalized complex structures and outline their relation with Poisson geometry, Lie algebroids and Lie groupoids. In the second part of the talk, we will propose �generalized contact bundles� as odd dimensional analogues of generalized complex manifolds. Finally, we will outline the relation between generalized contact bundles and Jacobi manifolds, Lie algebroids and Lie groupoids.

2 i 9 marca 2016/ March 2^nd and 9^th, 2016

Giovanni MORENO (IMPAN)

Introduction to BV-BFV theories on manifolds with boundary

Batalin-Vilkovisky (BV) theories are classical field theories where the target space is Z-graded. They are particularly well-suited for gauge symmetry reduction. A BV theory comprises an action functional, a symplectic form, and a homological vector field: from the action functional one obtains the Euler-Lagrange (EL) field equations, the homological vector field encodes the gauge symmetries of the theory, and the symplectic form captures their interrelationship. Under physically reasonable assumptions, these data allows for a nice cohomological description of the tangent space to the so-called EL-moduli space (the space of solutions to the EL equations, modulo gauge symmetries), at a smooth point.

In this two-parts seminar (based on the paper �Classical BV Theories on Manifolds with Boundary�, by A.S. Cattaneo et al., Commun. Math. Phys., 2014), I will review the main features of the BV formalism on closed manifolds. Then I will switch to manifolds with boundary, and show that a BV theory induces a so-called Batalin-Fradkin-Vilkovisky (BFV) theory on the boundary, in a compatible way: the result is a particular case of a BV-BFV theory. In the BV-BFV context, the symmetry reduction, and the corresponding cohomological �infinitesimal� description of the EL-moduli space at a smooth point, go along the same conceptual lines as in the BV case, but the actual procedure is more delicate, since it has to take into account the additional boundary structures. The final output will be the symplectic EL-moduli space, which fulfils the expected gluing properties, indispensable for quantisation.

20 stycznia 2016/ January 20^th, 2016

Giovanni MORENO (IMPAN)

Finite-dimensional symplectic formalism for higher-order field theories

In Mechanics, the cotangent space to the configuration space, understood as the space of initial data for (regular) Lagrangian theories, is equipped with a natural symplectic structure. Similarly, in field theory, one gets an infinite-dimensional symplectic space of boundary fields, containing the space of initial data as a Lagrangian submanifold: such spaces are at the heart of the so-called BV-BFV theories with boundary, which are enjoying a renewed interest. In this talk, I will review a recent work with J. Kijowski, where the symplectic structures behind higher-order fields theories have been studied in detail. The so-obtained framework, based on an obvious jet-theoretic analogy with Mechanics, represents probably the simplest geometric description of the dynamics of a Lagrangian field theory. The symplectic structures involved are defined �fibre-by-fibre� and, in this sense, they can be treated as finite-dimensional.

13 stycznia 2016/ January 13^th, 2016

Piotr Waluk (KMMF)

The Ricci flow

The Ricci flow is a technique for analyzing Riemannian manifolds by evolving their metric with respect to a certain differential equation. The method was first introduced by Hamilton in his paper form 1982, as means to solve a problem concerning classification of 3-dimensional compact manifolds. My talk will be a short introduction to the topic of the Ricci flow, explaining its basic ideas and illustrating them with some application examples.

16 grudnia 2015/ December 16^th, 2015

Rafa� R Suszek (KMMF)

A geometrisation of the (T)QFT functor

I shall formulate and illustrate on a simple and physically meaningful example the general principle by which (generalised Cheeger-Simons) differential characters and related transport operators defined by geometrisations of de Rham classes on the configuration bundle of a field theory with topological charges realise - in a concrete and computable manner - Segal's ambitious dream of functorial quantisation within and beyond the topological category. Time permitting, I shall also discuss certain important consequences of that principle.

9 grudnia 2015/ December 9^th, 2015

Javier de Lucas (KMMF)

Lie systems and geometric phases

In the context of the Floquet theory, using a variation of parameter argument, we show that the logarithm of the monodromy of a real periodic Lie system with appropriate properties admits a splitting into two parts called dynamic and geometric phases. The dynamic phase is intrinsic and linked to the Hamiltonian of a periodic linear Euler system on the co-algebra. The geometric phase is represented as a surface integral of the symplectic form of a co-adjoint orbit.

2 grudnia 2015/ December 2^nd, 2015

Piotr KOPACZ (UJ, Akademia Morska w Gdyni)

Finslerian versus variational approach to the solutions to Zermelo's problem with application to navigation

We consider the solutions to the Zermelo navigation problem on Riemannian manifolds, under perturbation represented by the weak vector field, in Finsler geometry with application of Randers metric. We compare it to the variational solutions via the Euler-Lagrange equations with non-restricted wind distribution. We focus on the river-type perturbation in the corresponding low-dimensional examples in the context of real applications in navigation. We also propose the geometric modification of the standard search patterns in case of acting vector field basing on the time-optimal paths.

18 listopada 2015/ November 18^th, 2015

Micha� JӏWIKOWSKI (IMPAN)

A contact approach to the sub-Riemannian geodesic problem

I will discuss the SR geodesic problem from the point of view developed by Witold Respondek and myself in arxiv:1509.01628. The most important points of this approach are: (i) the Hamiltonian formalism is obsolete (ii) geometric reasonings are elementary (iii) the emphasis is put on the flows rather than on Lie brackets. If time allows I will address issues concerning local optimality of solutions.

28 pa�dziernika 2015/ October 28^th, 2015

Andrzej DRAGAN (FUW)

Ideal clocks - a convenient fiction

We show that no device built according to the rules of quantum field theory can measure proper time along its path. Highly accelerated quantum clocks experience the Unruh effect, which inevitably influences their time rate. This contradicts the concept of an ideal clock, whose rate should only depend on the instantaneous velocity.

21 pa�dziernika 2015/ October 21^th, 2015

Giovanni MORENO (IMPAN)

Invariant hypersurfaces in low-dimensional Lagrangian Grassmannians.

It is an easy exercise to show that the two-dimensional Monge-Ampere equations are the only two-dimensional second-order PDEs that are invariant under the natural action of the affine group of the plane. In three dimensions, an analogous statement can be proved, though it requires much more computations. In four dimensions, computations are simply unendurable, and the necessity of a more conceptual approach to the problem begins to show.

In this talk I will recall that hypersurfaces of Lagrangian Grassmannians and second-order PDEs are basically the same thing, so that the notion of the invariancy (with respect to a given Lie group G) of a (multidimensional) second-order PDE can be formulated in terms of the G-invariancy of the corresponding hypersurface of the Lagrangian Grassmannian. Via the Plucker embedding, hypersurfaces of Lagrangian Grassmannians can be embbeded in a projective space. Such a projective space turns out to be a natural G-module, so that repesentation theory can be used for finding all the (relative) G-invariants polynomials, whose zero loci corresponds to G-invariant hypersurfaces. Up to 3 independent variables, such a method reveals nothing new, and it is just another way to show that Monge-Ampere equations corresponds precisely to GL(n)-invariant hypersurfaces. Surprisingly enough, for n=4, a new unexpected class of invariant second-order PDEs pops out, which is not made of Monge-Ampere equations.

This talk is based on a joint work with D. Alekseevsky and G. Manno.

14 pa�dziernika 2015/ October 14^th, 2015

Miko�aj ROTKIEWICZ (MIMUW)

Metric double vector bundles vs. the linearization of graded vector bundles (part II)

The full linearization functor enables studying graded bundles in a framework of multi-vector bundles. I will explain an unexpected relation between the linearization of graded bundles (of degree 2) and symplectic and metric double vector bundles discussed recently by Jotz Lean. This will be preceded by a gentle introduction to a beautiful theory of double vector bundles.

7 pa�dziernika 2015/ October 7^th, 2015

Miko�aj ROTKIEWICZ (MIMUW)

Metric double vector bundles vs. the linearization of graded vector bundles (part I)

The full linearization functor enables studying graded bundles in a framework of multi-vector bundles. I will explain an unexpected relation between the linearization of graded bundles (of degree 2) and symplectic and metric double vector bundles discussed recently by Jotz Lean. This will be preceded by a gentle introduction to a beautiful theory of double vector bundles.

3 czerwca 2015/ June 3^rd, 2015

Marcin ZAJ�C (KMMF)

Tulczyjew triple on a Lie group

In the talk I will present Tulczyjew formalism in the case when the configuration space is the tangent bundle to a Lie group. The tangent bundle to a Lie group has natural structure of a trivial bundle. The diagram of the triple can therefore be trivialized due to the group structure. I will disscuss kappa, alpha, and beta isomorphisms in the trivialization. Then I will consider Lagrangians invariant under the group action, and reduce Tulczyjew triple due to this symmetry.

27 maja 2015/ May, 27^th, 2015

Alfonso Giuseppe Tortorella (Universita di Firenze)

The BV-complex and the deformation problem of a coisotropic submanifold

In this talk, following works by Schaetz, I will review the construction of the BV-complex and its role in the coisotropic deformation problem. In a Poisson manifold, each coisotropic submanifold comes attached with two cohomological resolutions of its reduced Poisson algebra. The first one is the L-infinity-algebra introduced by Oh&Park in the symplectic case, and by Cattaneo&Felder in the Poisson case. As well-known this L-infinity-algebra controls the formal coisotropic deformation problem, and under a generically non-trivial necessary and sufficient condition the non-formal deformation as well. The second one is the BV-complex originally introduced by physicists dealing with Hamiltonian systems with symmetries. As proved by Schaetz, the BV-complex and the L-infinity-algebra are L-infinity quasi-isomorphic, and so they control equally well the formal coisotropic deformation problem. However, as proven by Schaetz, the BV-complex encodes also further information about the non-formal deformations, which generically is not captured by the L-infinity-algebra. If time permits I will start to describe how these results can be transferred into the more general framework of Jacobi manifolds.

20 maja 2015/ May, 20^th, 2015

Giovanni MORENO (University of Salerno)

Hypersurfaces in Lagrangian Grassmannians and geometric theory of nonlinear PDEs

I will begin by reviewing the simplest nontrivial case of a Lagrangian Grassmannian manifold, namely the three-dimensional L(2,4), stressing its remarkable isomorphism with the Lie quadric Q^3, mentioning also its "meta-symplectic counterpart", the four-dimensional L(2,5), which is a rather unexplored object. In spite of the low dimensionality of the objects involved, even at this level, it is possible to formulate rather tricky questions. Then I will switch to the PDE side, examining the standard framework for 2nd order 2D (nonlinear) PDEs based on the Lagrangian bundle of a 5D contact manifold (perhaps better known as "2nd order jet space"), with a particular emphasis on the geometric formulation of Cauchy problems, which, in turn, needs the notion of a characteristic Cahuchy surface. Coming back to the Lagrangian Grassimannians, I will show that the PDEs correspond, in fact, to their hypersurfaces, and that the presence of a lot of characteristics is a well-known phenomenon in Algebraic Geometry known as a "ruling". The advantage of such a bridge between the two disciplines is that is allows to recast known results in a more transparent way, and to formulate new ones as well, especially in the meta-symplectic context, i.e., that of 3rd order 2D (nonlinear) PDEs. As an example, I will demonstrate that 2D parabolic Monge-Ampere equations correspond to the so-called hyperplane sections of Q^3. To conclude with, I will bring in the dual variety of the generic Lagrangian Grassmannian L(n,2n), as well as its singular loci, which are in correspondence with remarkable classes of PDEs, like, e.g., the linearisable ones. In such a framework, a still open conjecture by E. V. Ferapontov, about the integrability of n-dimensional PDEs, may become more easy to work out.

The content of this seminar is based on the project "GEOGRAL", which I'm going to carry out at IMPAN with a Marie Sk�odowska-Curie fellowship, commencing September 1st. Due to the highly multidisciplinary character of GEOGRAL, collaborative efforts are mandatory, and I hope that this small introduction will stimulate the attention of local personnel potentially interested in joining in.

13 maja 2015/ May, 13^th, 2015

Session cancelled due to the conference "Geometry of Jets and Fields in B�dlewo"

6 maja 2015/ May, 6^th, 2015

Artur JANDA (Centrum Bada� Kosmicznych)

The action principle, electrodynamics and causality

The classical electrodynamics may be cosidered as a gauge invariant theory of sections of the cotangent bundle over the spacetime manifold. Basic notions of causality applicable to a general theory of nonlinear electrodynamics of continuous media will be presented.

29 kwietnia 2015/ April, 29^th, 2015

Tomasz MACI��EK (CFT)

Critical points of the squared norm of the momentum map for complex projective spaces

22 kwietnia 2015/ April, 22^nd, 2015

Jerzy LEWANDOWSKI (IFT FUW)

Generally Relativistic classical and quantum particles

The canonical theory of a test particle in a background curved spacetime is a fantastic example of a Hamiltonian constrained system, that is a system whose Hamiltonian is a generator of the gauge transformations. On this example we will explain definitions of the physical observables and the procedure of the canonical quantization. Taking into account that the existence of a (specially) relativistic quantum mechanics is usually questioned, the results I am going to present in this talk may even go beyond the well known quantum physics. In particular, the issue of a quantum position and quantum time operators will be revisited and solved within the geometric framework.

15 kwietnia 2015/ April, 15^th, 2015

Piotr SU�KOWSKI (IFT FUW)

S-duality and quantum field theory

I will review the notion of S-duality in quantum field theory, i.e. the equivalence of pairs of quantum field theories, respectively with weak and strong interactions. I will present a historical development of this and related ideas, as well as modern perspectives, in particular a prediction of the existence of strongly coupled quantum field theories without lagrangian description.

1 kwietnia 2015/ April, 1^st, 2015

Bogdan BALCERZAK (Politechnika ��dzka)

Dirac operators on Lie algebroids

Dirac type operators on Lie algebroids with respect to different geometric structures will be defined and discussed.

25 marca 2015/ March, 25^th, 2015

Antoni PIERZCHALSKI (Uniwersytet ��dzki)

Generalized gradients

Uog�lnione gradienty to nieredukowalne ze wzgl�du na dzia�aj�c� grup� sk�adniki pochodnej kowariantnej. Zakodowane s� w nich dane o strukturze geometrycznej rozmaito�ci, na kt�rej s� badane. Pokazana zostanie og�lna konstrukcja gradient�w oraz ich przyk�ady. Om�wione b�dzie zachowanie na brzegu rozmaito�ci. Wspomniana tak�e b�dzie konstrukcja w przypadku struktur og�lniejszych - algebroid�w Liego.

Components of the Lie derivative irreducible with respect to the group action are called generalized gradients. They encode information about the geometrical structure of the underlying manifold. General construction of gradients as well as their properties on the boundary will be presented. The analogous construction in the case of Lie algebroids will also be discussed.

18 marca 2015/ March, 18^th, 2015

Javier de LUCAS (KMMF)

Definition, properties and applications of superdifferential equations

There exist several non-equivalent meanings of the term "superdifferential equation" in the literature. In this talk, we aim to describe them and to motivate that one of these meanings amounts to a certain type of differential equations on graded bundles. Next, we present some preliminary results on the geometric and algebraic properties of such differential equations.

11 marca 2015/ March, 11^th, 2015

Andrew BRUCE (IMPAN)

On graded bundles in the category of Lie groupoids

Lie groupoids are to be found throughout differential geometry including the theory of group actions, foliations, Poisson geometry, connection theory and the study of singular spaces such as orbifolds. Over the years there has been much interest in 'categorified 'objects in the category of Lie groupoids; this has been spurred on by the original ideas of Mackenzie such as 'double Lie groupoids' etc. Very recently there has been a resurgence of interest in VB-groupoids and VB-algebroids, which are vector bundles in the category of Lie groupoids and Lie algebroids respectively. Part of this resurgence is motivated by the representation theory of Lie groupoids and algebroids. However, the original definitions of 'VB-objects' are very complicated and not obvious. Bursztyn, Cabrera and de Hoyo realised last year that VB-groupoids and VB-algebroids can be neatly described using regular actions of the multiplicative monoid of real numbers a la Grabowski and Rotkiewicz. In this talk I will highlight how, rather naturally, one can generalise and simplify the notion of 'VB-objects' by using homogeneity structures that is smooth actions of the multiplicative monoid of reals. It is known, via Grabowski and Rotkiewicz that homogeneity structures always lead to what they called graded bundles; these are manifolds with a non-negative grading on their structure sheaf and give particularly nice examples of polynomial bundles. Graded bundles can be viewed as a very natural generalisation of a vector bundle. Thus, we can pass from VB-groupoids to weighted Lie groupoids which we understand as graded bundles in the category of Lie groupoids, or indeed vice-versa. We will highlight natural examples of this very rich geometric theory and briefly describe the Lie theory relating weighted Lie groupoids and weighted Lie algebroids. Time permitting I will also describe weighted Poisson-Lie groupoids and weighted Courant algebroids. This talk is based on joint work with K. Grabowska and J. Grabowski.

4 marca 2015/ March, 4^th, 2015

Alfonso Giuseppe TORTORELLA (Universita di Firenze)

Deformations of coisotropic submanifolds in abstract Jacobi manifolds

In this talk, using the Atiyah algebroid and first order multi-differential calculus on non-trivial line bundles, we attach an L-infinity algebra to any coisotropic submanifold S in an abstract (or Kirillov�s) Jacobi manifold. Our construction generalizes and unifies analogous constructions in the symplectic case (Oh and Park), the Poisson case (Cattaneo and Felder), locally conformal symplectic case (Le and Oh). As a new special case, we attach an L-infinity algebra to any coisotropic submanifold in a contact manifold, including Legendrian submanifolds. The L-infinity algebra of a coisotropic submanifold S governs the (formal) deformation problem of S.

25 lutego 2015/ February, 25^th, 2015

Tomasz ZAWADZKI (U. ��dzki)

Conformal submersions with totally umbilical fibers

Conformal submersions are a natural generalisation of Riemannian submersions. Under an additional assumption of fibers being totally umbilical (which is a reasonable geometric condition, due to the conformal invariance of umbilicity), we examine those mappings, obtaining their existence conditions expressed in terms of curvatures of the domain and the image. We also present a number of examples of conformal submersions with umbilical fibers and discuss their relations with Riemannian submersions.

Presentation (PDF)

21 stycznia 2015/ January, 21^st, 2015

Tomasz TYLEC (CFT PAN)

Quantum logics and generalised probability

Immediately after formulation of quantum mechanics there were attempts to �derive� it from some small set of plausible axioms. The two most successful approaches were: algebraic axiomatisation (started by Jordan, von Neumann and Wigner) and so called quantum logic approach, based on the partially ordered structure of �propositions� about physical system (started by Birkhoff and von Neumann). During many years of development the latter approach became mostly the theory of partially ordered structures with a little application in physics. Since 1990s some physicist working on quantum information theory become interested in the so-called no-signalling theories. These are very intuitive, in some sense minimal, models of compound systems. It seemed that such theories are more general than quantum mechanics. Our idea is to rigorously reformulate assumptions of non-signalling theories in the framework of quantum logics and then study its properties and true relation to quantum mechanics.

7 stycznia 2015/ January, 7^th, 2015

Katarzyna GRABOWSKA (KMMF)

Dirac algebroids in action

Presentation (PDF)

17 grudnia 2014/ December, 17^th, 2014

Witold RESPONDEK (INSA, Rouen)

Linearization of mechanical control systems

Presentation (PDF)

10 grudnia 2014/ December, 10^th, 2014

Tatiana SHULMAN (IMPAN)

Completely positive maps and zero-error quantum 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 general theory of completely positive maps and their connection with quantum channels. In the second part of the talk I will focus on some mathematical problems arising in zero-error quantum information theory, namely I will talk on various zero-error capacities of quantum channels and superactivation effect. This is ajoint work with M. Shirokov.

3 grudnia 2014/ December, 3^rd, 2014

Micha� JӏWIKOWSKI (IMPAN)

On symmetries of differential equations

In the talk I will discuss symmetries of differential equations from the geometric viewpoint in the spirit of Alexandre M. Vinogradov. As a particular application I will discuss symmetries of a k-th order ODE.

19 listopada 2014/ November, 19^th, 2014

Piotr STACHURA (SGGW)

Short and biased introduction to groupoids

The approach to groupoids started by S. Zakrzewski will be recalled. In particular various (but equivalent) definitions of morphisms and actions of groupoids will be presented with examples (mainly on algebraic level).

Presentation (PDF)

12 listopada 2014/ November, 12^th, 2014

Bronis�aw JAKUBCZYK (IMPAN)

REGULAR CONTROL SYSTEMS AND LAGRANGE GEOMETRY (part II)

5 listopada 2014/ November, 5^th, 2014

Bronis�aw JAKUBCZYK (IMPAN)

REGULAR CONTROL SYSTEMS AND LAGRANGE GEOMETRY

29 pa�dziernika 2014/ October, 29^th, 2014

Miko�aj ROTKIEWICZ (IMPAN)

On applications and generalizations of the canonical kappa-map

The canonical kappa involution for the iterated tangent bundle TTM has a surprising number of important applications and interesting generalizations. One can study the kappa map for M being a Lie groupoid. One can also look for generalizations to the higher order tangent bundles. These questions will be discussed in my talk which is partially based on my joint paper with Micha� J�wikowski "Prototypes of higher algebroids with applications to variational calculus", arXiv:1306.3379.

22 pa�dziernika 2014/ October, 22^nd, 2014

Pawe� URBA�SKI (KMMF)

SYMPLECTIC GROUPOID ACTIONS AND REDUCTIONS (part II)

15 pa�dziernika 2014/ October, 15^th, 2014

Pawe� URBA�SKI (KMMF)

SYMPLECTIC GROUPOID ACTIONS AND REDUCTIONS (part I)

8 pa�dziernika 2014/ October, 8^th, 2014

Andrew BRUCE (IMPAN)

WEIGHTED ALGEBROIDS: THEORY AND OUTLOOK FOR APPLICATIONS ( part II )

1 pa�dziernika 2014/ October, 1^st, 2014

Andrew BRUCE (IMPAN)

WEIGHTED ALGEBROIDS: THEORY AND OUTLOOK FOR APPLICATIONS

In this talk I will outline the theory of the recently discovered weighted Lie algebroids, which should be considered as a higher version of a Lie algebroid. We will then suggest how such structures can be employed in higher order Lagrangian mechanics. This is joint work with K. Grabowska and J. Grabowski.

4 czerwca 2014/June, 4^th 2014

Artur JANDA

HYPERBOLIC LAGRANGIANS AND THE DOMAIN OF DEPENDENCE THEOREM

Hyperbolicity of a system of partial differential equations refers to certain essentially algebraic conditions, on the other hand such a system should have properties like the wave equation, basically it should possess the domain of dependence property. Starting with a brief exposition of basic concepts related to the action principle of the theory of maps between manifolds we will sketch the proof of the domain of dependence theorem (Christodoulou) for systems of the Euler-Lagrange equations satisfying certain regular hyperbolicity condition. This modern notion refers strictly to Lagrangians and it is applicable to systems which cannot satisfy classical notions of hyperbolicity (Leray, Friedrichs).

21 i 28 maja 2014/May, 21^st and 28^th 2014

Piotr WALUK (KMMF)

SYMMETRIES OF DISTRIBUTIONS AFTER KRUGLIKOV

14 maja 2014/May, 14^th 2014

Luca VITAGLIANO (UNISA)

L_\infty ALGEBRAS FROM MULTICONTACT GEOMETRY

I define higher versions of contact structures on manifolds as maximally non-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.

7 maja 2014/May, 7^th 2014

Jerzy KIJOWSKI (CFT)

O PEWNEJ METODZIE ,DYSKRETYZACJI' R�WNA� CZ�STKOWYCH POCHODZ�CYCH Z PROBLEM�W WARIACYJNYCH

Jest kilka sposob�w ,,dyskretyzacji'' r�wna� Einsteina u�ywanych do oblicze� w tzw. numerical gravity. Wszystkie te sposoby gwa�c� ,,in flagranti'' struktur� geometryczn� tych r�wna�. Prelegent jest g��boko przekonany, �e trudno�ci z obliczeniami numerycznymi w og�lnej teorii wzgl�dno�ci st�d w�a�nie wynikaj� i proponuje now� metod� jej dyskretyzacji, wynikaj�c� z analizy tej w�a�nie struktury.

30 kwietnia 2014/April, 30^th 2014

Krzysztof DRACHAL (PW)

COMORPHISMS OF LIE ALGEBROIDS AND GROUPOIDS

The aim of the talk is to remind the notions of morphisms of vector bundles, 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.

23 kwietnia 2014/April, 23^rd 2014

Giovanni MORENO (Sl. U. OPAVA)

GEOMETRY OF 3RD ORDER MONGE-AMPERE EQUATIONS AND THEIR CHARACTERISTICS

his talk incorporates three aspects: i) a review of the general theory of characteristics of PDEs with a particular emphasis on its physical ramifications, ii) an introduction to a recent study of 2nd order Monge-Ampere equations (MAEs) with the tools of contact/symplectic geometry, and iii) a presentation of an ongoing investigation of 3rd order MAEs, with special attention to the peculiarities of this nearly unexplored 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 initial physical perspective on PDEs and their characteristics in general, and MAEs in particular, towards a concluding entirely geometric picture, involving contact manifolds, their prolongations, and special sub-bundles of some Grassmannian-like bundles, improperly called here "Lagrangian". The reason behind this denomination is that they stem from a higher-order analog of the symplectic form, known as "meta-symplectic". Surprisingly enough, the so-obtained framework, in spite of its abstractedness, allows to give an immediate answer to some non-equivalence problems and, more generally, to begin to understand the structure of the "space of all 3rd order MAEs".

Presentation (PDF)

9 kwietnia 2014/April, 9^th 2014

Tatiana SHULMAN (U 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.

2 kwietnia 2014/April, 2^nd 2014

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 mapping allowing us to describe its general solution in terms of a generic finite family of particular solutions and a set of constants. Equivalently, we can characterise Lie systems as systems of first-order ordinary differential equations describing the integral curves of a time-dependent vector field taking 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 Lie algebra 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 study the constants of motion, superposition rules, and other properties of these systems. Our results generalise previous methods to investigate integrable systems and certain types of Lie systems. We illustrate our findings with the study of several types of Schwarzian equations and other differential equations of interest.

19 i 26 marca 2014/ March, 19^th and 26^th 2014

Micha� JӯWIKOWSKI

INTEGRATION OF LIE ALGEBROIDS

12 marca 2014/ March, 12^th 2014

Zbigniew JELONEK

DYFEOMORFIZMY ZACHOWUJACE SYMPLEKTYCZNE LUB IZOTROPOWE PODROZMAITOSCI

5 marca 2014/ March, 5^th 2014

Katarzyna GRABOWSKA (KMMF)

GRADED GEOMETRY: LAGRANGIAN AND HAMILTONIAN MECHANICS

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

19, 26 lutego 2014/ February, 19^th and 26^th 2014

Janusz GRABOWSKI (IMPAN)

GRADED DIFFERENTIAL GEOMETRY