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

The Trans-Carpathian Seminar on Geometry & Physics

(formerly Geometric Seminar)

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

Paweł 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.

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

Jerzy Kijowski (CFT)

Energia fal grawitacyjnych wg. A. Trautmana

Energy carried by gravitational field: an approach based on ideas proposed by A. Trautman

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

Paweł Urbański (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 Urbański.

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

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.

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

Ewa Girejko (Politechnika Białostocka)

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.

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

Paweł Nurowski (CFT)

Examples of nonholonomic systems: ants on a table

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

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 andOmid Makhmali.

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

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.

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

Michał Jóźwikowski (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.

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

Kamil Niedziałomski (Wydział Matematyki i Informatyki, Uniwersytet Łódzki)

'Geometria G-struktur poprzez skręcenie wewnętrzne

G-structures and intrinsic torsion

Rozważmy zorientowaną rozmaitość riemannowską. G-strukturą nazywamy redukcję grupy strukturalnej wiązki zorientowanych baz ortonormalnych do grupy G. Taka redukcja implikuje pewne własności na wyjściową rozmaitość, jak na przykład, istnienie struktury prawie hermitowskiej, itd. Składowa koneksji Levi-Civity, przy pewnym warunku niezmienniczości na poziomie algebr Liego grup strukturalnych, definiuje G-koneksję. Różnicę tych koneksji nazywamy skręceniem wewnętrznym.
Podczas referatu omówię dokładniej własności skręcenia wewnętrznego, jego zastosowanie i znaczenie oraz przedstawię wyniki swoich ostatnich badań dotyczących geometrii i pewnych wzorów całkowych dla G--struktur przy użyciu skręcenia wewnętrznego.

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.

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

Tomasz Maciążek (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.

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

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.

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

Katarzyna Grabowska (KMMF)

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.

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

Marcin Zając (KMMF)

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

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

Mikołaj 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).

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

Aneta Sliżewska (Uniwersytet w Białymstoku)

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

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

Andrzej Trautman (FUW)

Optical structures in relativity

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

Marek Demiański (FUW)

Fale grawitacyjne - nowe okno na Wszechświat

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

Michał Jóźwikowski, Mikołaj Rotkiewicz (IMPAN, MIMUW)

Higher algebroids via differential relations II

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

Michał Jóźwikowski, Mikołaj Rotkiewicz (IMPAN, MIMUW)

Higher algebroids via differential relations I

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

Janusz Grabowski (IMPAN)

Higher order Lagrangians II

We will start with presenting a geometric approach to first order Lagrangian Mechanics à 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.

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

Janusz Grabowski (Instytut Matematyczny PAN)