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

The Trans-Carpathian Seminar on Geometry & Physics

(formerly Geometric Seminar)

2006/2007 | 2007/2008 | 2008/2009 | 2009/2010 | 2010/2011 | 2011/2012 | 2012/2013 | 2013/2014 | 2014/2015 | 2015/2016 | 2016/2017 | 2017/2018 | 2018/2019 | 2019/2020 | 2020/2021 | 2021/2022 | 2022/2023 | 2023/2024 | 2024/2025 | Seminar homepage

RSS

join us at 14:15

(KMMF)

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.

join us at 14:15

Bartłomiej Bąk (KMMF)

Kilka konstrukcji skraju i ich zastosowanie w analizie skalarnego równania falowego

A few constructions of the scri and its applications into analysis of scalar wave equation

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.

In my talk I want to present the most popular construction of the scri on Minkowski spacetime (via conformal compactification) with associated Penrose diagram. Then I will describe the behaviour of the decreasing solution of scalar wave equation on the scri and show how to rewrite data on the scri from Cauchy data and vice versa. Moreover, making some special compactification it is possible to transform Cauchy problem to the Dirichlet problem on the compact set. Thanks to this it will be shown the one more definition of the scri where any compactification is not necessary. At the end of my presentation I will introduce how to convert some general tensor wave equation to the scalar Klein-Gordon equation.

join us at 14:15

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.

One of simplest and most important objects in Quantum Information is qubit, a 2-level quantum system. I will discuss a generalisation to 2 qubits of the standard Bloch sphere representation for a single qubit, in the framework of Hopf fibrations of high dimensional spheres by lower dimensional spheres. The single qubit Hilbert space is the 3-dimensional sphere S3. The S2 base space of a suitably oriented S3 Hopf fibration is nothing but the Bloch sphere, while the circular fibres represent the qubit overall phase degree of freedom. For the two qubits case, the Hilbert space is a 7-dimensional sphere S7, which also allows for a Hopf fibration, with S3 fibres and a S4 base. A main striking result is that suitably oriented S7 Hopf fibrations are entanglement sensitive.

join us at 14:15

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.

In general relativity, connection plays role of forces deteminig motion of free falls, but Einstein equations are expressed by curvature - quantity build of derivatives of connection coefitients. The most popular measure of curvature is Riemann tensor, which has a beautifull geometric interpretation, but its physical interpretation is not trivial. The other curvature measure in Jacobi tensor, which form is strictly given by calculus of tidal forces acting on body moving along geodesics (geodesics deviation equation). In my talk I will discuss a few approaches to curvature, and using Jacobi tensor I will show physical interpretation of Ricci tensor, which is only curvature part used in Einstein equations.

join us at 14:15

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.

room 2.23, Pasteura 5 at 14:15

(FUW)

Introduction to tractor calculus

Jarosław Kopiński

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].

join us at 14:15

Andrew Bruce (University of Luxembourg)

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.

join us at 14:15

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 Meeting ID: 840 5801 4241 Passcode: algebroid

room 2.23, Pasteura 5 at 14:15

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.

join us at 14:15

Marcin Zajac (KMMF)

Principal bundle geometry in application to electrodynamics and electroweak sector of the Standard Model

W trakcie seminarium zaprezentuję podstawy matematycznegoopisu elektordynamiki oraz Lagranżjanu oddziaływań elektrosłabych wjęzyku wiązek głównych.

room 2.23, Pasteura 5 at 14:15

Rafał R. Suszek (KMMF)

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

Kostant's approach to the theory of group objects in the category of supermanifolds and their homogeneous spaces has proven very natural and fruitful in the modelling of the geometrodynamics of extended distributions of Z/2Z-graded charge in equibalance that generalises the usual lagrangean dynamics of a charged pointlike mass in external gravitational and electromagnetic fields (and has led, rather remarkably, to far-reaching insights into the measurable physics of gauge phenomena beyond the perturbative regime). Upon reviewing the construction of the Green(-Schwarz-...)-type super-sigma-model of the said dynamics, I shall explain how the supersymmetric cohomology of the gauge field coupling to the dynamical charge through a "topological" term in the action functor secretely encodes rich information on the geometry of the critical trajectories of the dynamics. The path leads through a hierarchy of geometrisations, put forward by the Speaker, of classes in the aforementioned cohomology that resemble closely those known from the study of Beilinson(-Deligne) cohomology of gauge fields in the non-Z/2Z-graded setting and can be thought of as (first) concrete models of supersymmetric higher geometry. Their discussion shall carry us congenially through the rest of the alphabet: From Cartan-Eilenberg to Chevalley-Eilenberg and back; from Rabin et al. to Tulczyjew et al., de Azcarraga et al. and van Nieuwenhuizen et al. (judiciously circumnavigating Stasheff et al.; from Gawędzki to Hitchin and Murray and so on. We shall even stumble upon Lukierski's odd kappa (-symmetry) along the way. By the end of the long day, we are bound to get super-curved and carry on, thus disposed, towards the phenomenological asymptote.

join us at 14:15

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

join us at 14:15

Tomasz Smołka (KMMF)

Quasi-lokalny opis elektromagnetyzmu i zlinearyzowanej grawitacji ilustrowany Hopfionami

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.
Meeting ID: 840 5801 4241
Passcode: algebroid

join us at 14:15

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.

Zoom Meeting ID: 840 5801 4241, Passcode: algebroid

join us at 14:15

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.
Meeting ID: 840 5801 4241 Passcode: algebroid

join us at 14:15

(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.Zoom Meeting ID: 840 5801 4241, Passcode: algebroid

My talk will be devoted to the concept of an algebroid. I shall disscuss many different ways of encoding the same structure. I hope that the younger members of our community will benefit form the talk. One can treat it as a form of introduction to the next talk by Mikołaj Rotkiewicz on the interesting generalization of the concept of an algebroid.Zoom Meeting ID: 840 5801 4241, Passcode: algebroidWróć

join us at 14:15

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.
Zoom Meeting ID: 840 5801 4241, Passcode: algebroid

My talk will be devoted to the concept of an algebroid. I shall disscuss many different ways of encoding the same structure. I hope that the younger members of our community will benefit form the talk. One can treat it as a form of introduction to the next talk by Mikołaj Rotkiewicz on the interesting generalization of the concept of an algebroid.
Zoom Meeting ID: 840 5801 4241, Passcode: algebroid

join us at 14:15

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

Meeting ID: 840 5801 4241
Passcode: algebroid