GEOMETRICAL METHODS IN PHYSICS METODY GEOMETRYCZNE FIZYKI
Wednesday at 2:15 pm, niadeckich 8, room 106 roda, godzina 14:15, niadeckich 8, sala 106

Centrum Fizyki Teoretycznej PAN, Instytut Matematyczny PAN i Katedra Metod Matematycznych Fizyki

zapraszaj studentw, doktorantw i nie tylko modych pracownikw nauki na seminarium

Metody Geometryczne Fizyki

Informacje o tematyce seminarium w jzyku angielskim znajduj si tutaj. Information about the subject area of the seminar (in English) can be found here

 Rok akademicki 2017/2018 Semestr letni 16 maja 2018/ May 16th, 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 9th, 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 2nd, 2018 There will be no seminar on May 2nd. 25 kwietnia 2018/ April 25th, 2018 Pawe URBASKI (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 Urbaski. 18 kwietnia 2018/ April 18th, 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 11th, 2018 Ewa GIREJKO (Politechnika Biaostocka) On consensus in opinion formation models based on multiagents systems Multiagents 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 multiagents systems under the consensus problem will be provided. Since the most representa- tive class of agentbased models explaining phenomena in social sciences are the opinion formation models, we focus on the HegselmannKrause (H-K) and CuckerSmale (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 28th, 2018 Pawe NUROWSKI (CFT) Examples of nonholonomic systems: ants on a table. ? 21 marca 2018/ March 21st, 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 14th, 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 7th, 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 28th, 2018 Kamil NIEDZIAOMSKI (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. Semestr zimowy 24 stycznia 2018/ January 24th, 2018 Tomasz MACIEK (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 17th, 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 Brions 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 10th, 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 20th, 2017 On December 20th there will be no seminar. We meet again on January 10th. 13 grudnia 2017/ December 13th, 2017 Marcin ZAJC (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 6th, 2017 Mikoaj 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 29th, 2017 Aneta SLIEWSKA (Universytet w Biaymstoku) 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 22nd, 2017 Andrzej TRAUTMAN (FUW) Optical structures in realitivity 15 listopada 2017/ November 15th, 2017 Seminarium odwoane / Session cancelled 8 listopada 2017/ November 8th, 2017 Marek DEMIASKI (FUW) Fale grawitacyjne - nowe okno na Wszechwiat 18 i 25 padziernika 2017/ October 18th and 25th, 2017 Micha JӏWIKOWSKI (IMPAN), Mikoaj 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 padziernika 2017/ October 4th and 11th, 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. Rok akademicki 2016/2017 Semestr letni 24 maja 2017/ May 24th, 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 16th 2017, IMPAN, room 403 Frdric 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 Souriaus 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 (KostantKirillovSouriau) 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. Slides (PDF) 10 maja 2017/ May 10th, 2017 Pawe URBASKI (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*. Using techniques of double vector bundles we analyse properties of linear distributions essential for Lagrangian and Hamiltonian reductions (i.e. for E=TQ). It generalizes the framework for the Routh reduction. 26 kwietnia 2017/ April 26th, 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 19th, 2017 The seminar is cancelled due to organizational reasons 12 kwietnia 2017/ April 12th, 2017 Anatol ODZIJEWICZ (Uniwersytet w Biaymstoku) 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 5th, 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 29th, 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 22nd, 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 15th, 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 8th, 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 1st, 2017 Mikoaj 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 TkM 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 Semestr zimowy 25 stycznia 2017/ January 25th, 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 18th, 2017 Tomasz SMOKA (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 11th, 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. This is a joint work with Katarzyna Karnas. 7 grudnia 2016/ December 7th, 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 30th, 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 23rd, 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 (1) Kiselev A. V., van de Leur J. W., Variational Lie algebroids and homological evolutionary vector fields, Theor. Math. Phys. 167 (2011) n.3 Nonlinear Physics: Theory & Experiment VI. P. 772784. arXiv:1006.4227 [math.DG] (2) Kiselev A. V., Krutov A. O., Non-Abelian Lie algebroids over jet spaces, J. Nonlin.Math. Phys. 21 (2014) n.2. P. 188213. arXiv:1305.4598 [math.DG] 16 listopada 2016/ November 16th, 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 aij= 0 if |i-j|>1. It turned out that other "tridiagonal matrices", for example those for which aij=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 wiatosaw Gal (Wrocaw University). 9 listopada 2016/ November 9th, 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 odbdzie si/ on November 2nd, 2016 there will be no seminar 26 padziernika 2016/ October 26th, 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 padziernika 2016/ October 19th, 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 padziernika 2016/ October 12ve, 2016 Katarzyna GRABOWSKA, Pawe URBASKI (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 padziernika 2016/ October 5st, 2016 Katarzyna GRABOWSKA, Pawe URBASKI (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.

 Rok akademicki 2013/2014, semestr letni 4 czerwca 2014/June, 4th 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, 21st and 28th 2014 Piotr WALUK (KMMF) SYMMETRIES OF DISTRIBUTIONS AFTER KRUGLIKOV 14 maja 2014/May, 14th 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, 7th 2014 Jerzy KIJOWSKI (CFT) O PEWNEJ METODZIE ,DYSKRETYZACJI' RWNA CZSTKOWYCH POCHODZCYCH Z PROBLEMW WARIACYJNYCH Jest kilka sposobw ,,dyskretyzacji'' rwna Einsteina uywanych do oblicze w tzw. numerical gravity. Wszystkie te sposoby gwac ,,in flagranti'' struktur geometryczn tych rwna. Prelegent jest gboko przekonany, e trudnoci z obliczeniami numerycznymi w oglnej teorii wzgldnoci std wanie wynikaj i proponuje now metod jej dyskretyzacji, wynikajc z analizy tej wanie struktury. 30 kwietnia 2014/April, 30th 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, 23rd 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, 9th 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, 2nd 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, 19th and 26th 2014 Micha JӯWIKOWSKI INTEGRATION OF LIE ALGEBROIDS 12 marca 2014/ March, 12th 2014 Zbigniew JELONEK DYFEOMORFIZMY ZACHOWUJACE SYMPLEKTYCZNE LUB IZOTROPOWE PODROZMAITOSCI 5 marca 2014/ March, 5th 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, 19th and 26th 2014 Janusz GRABOWSKI (IMPAN) GRADED DIFFERENTIAL GEOMETRY

15, 22 stycznia 2014/ January, 15th and 22nd, 2014

Mikoaj ROTKIEWICZ

CAKOWANIE NA SUPER-ROZMAITOCIACH

18 grudnia 2013/ December, 18th, 2013

Jose MOURAO (U Lisboa)

QUANTIZATION IN KHLER AND REAL POLARIZATIONS

11 grudnia 2013/December, 11th, 2013

WIZY W STATYCE I DYNAMICE

4 grudnia 2013/ November, 4th, 2013

Tomasz RYBICKI (AGH)

A LOOK AT THE HAMILTONIAN GROUP OF A SYMPLECTIC MANIFOLD THROUGH ITS QUANTOMORPHISM GROUP

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

27 listopada 2013/ November, 27th, 2013

Jacek JEZIERSKI

GEOMETRIA POWIERZCHNI ZEROWYCH A HORYZONTY EKSTREMALNE

20 listopada 2013/ November, 20th, 2013

Krzysztof DRACHAL (PW)

ON SOME APPLICATIONS OF DIFFERENTIAL SPACES IN A SENSE OF SIKORSKI (cd)

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

13 listopada 2013/ November, 13th, 2013

Andrew BRUCE

ON CURVES AND JETS OF CURVES ON SUPERMANIFOLDS (cd)

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

6 listopada 2013/ November, 6th, 2013

Tiffany COVOLO (U Luxembourg)

TRACE AND BEREZINIAN OF MATRICES OVER A CLIFFORD ALGEBRA

30 padziernika 2013/ October, 30th, 2013

Micha JӏWIKOWSKI

O WIZACH WAKONOMICZNYCH I NIEHOLONOMICZNYCH

23 padziernika 2013/ October, 23rd , 2013

N-PARTICLE QUANTUM STATISTICS ON GRAPHS

16 padziernika 2013/ October, 16th , 2013

Katarzyna GRABOWSKA

CO O RWNANIU HAMILTONA-JACOBIEGO

9 padziernika 2013/ October, 9th , 2013

Andrew BRUCE

ON CURVES AND JETS OF CURVES ON SUPERMANIFOLDS

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

2 padziernika 2013/ October, 2nd , 2013

Krzysztof DRACHAL (PW)

ON SOME APPLICATIONS OF DIFFERENTIAL SPACES IN A SENSE OF SIKORSKI

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

5 czerwca 2013/ Juner, 5th , 2013

Maria SOROKINA (U Saint Petersburg)

HAMILTONIAN FORMALISM ON MANIFOLDS WITH SINGULARITIES

The configuration spaces of many real mechanical systems appear to be manifolds with singularity. A singularity often indicates that the geometry of motion may change at the point. In such cases we face the conceptual problem describing mechanics even for ideal models: since the configuration space is not a smooth manifold, thus, the fully developed machinery of Hamiltonian mechanics cannot be applied. I will present a way of conquering the aforementioned conceptual problem by considering a certain algebra instead of the configuration space. Configuration space is the real spectrum of the algebra. The structure of this algebra is completely determined by the geometry of the singularity. For a broad class of singularities the desired algebra can be described directly since it is the pullback of two already known algebras which can be easily described. Availability of the algebra enables to use Differential operator theory. The elementary examples of mechanical systems to which this algorithm is applicable, are flat linkages. In the frames of the presented approach a Poisson structure is built on a manifold with a one-dimensional singularity. The same result can be obtained for some other kinds of singularities. At the end of the talk I will present the specific results for the case of a configuration space consisting of two curves on the plane having random order of contact.

29 maja 2013/ May, 29th , 2013

22 maja 2013/ May, 22th , 2013

Prof. Jos F. CARINENA (Zaragoza)

RECENT RESULTS IN THE THEORY OF LIE SYSTEMS AND ITS GENERALIZATIONS

A generalization of the concept of Lie systems (systems of first-order differential equations possessing a superposition rule) will be presented and the theory will be illustrated with several examples. Moreover, other recent results on applications in physics of Lie systems compatible with some geometric structures, as symplectic, presymplectic, or more generally Dirac structures, will also be discussed.

15 maja 2013/ May, 15th , 2013

Wojciech DOMITRZ (IFT)

THE WIGNER CAUSTIC ON SHELL AND SINGULARITIES OF ODD FUNCTIONS

24 kwietnia, 8 maja 2013/ April, 24th and May, 8th , 2013

Mikoaj ROTKIEWICZ i Micha JӏWIKOWSKI (IFT)

VARIATIONAL CALCULUS AND HIGHER LIE ALGEBROIDS

17 kwietnia 2013/ April, 17th , 2013

METROLOGIA KWANTOWA, CZYLI CO?

Co geometria kanaw kwantowych moe powiedzie o interferometrach fal grawitacyjnych i zegarach atomowych.

10 kwietnia 2013/ April, 10th , 2013

Piotr SOTAN

ALGEBRY HOPFA

27 marca 2013/ March, 27th , 2013

Marek KU

LOKALNA RWNOWANO STANW KWANTOWYCH

20 marca 2013/ March, 20th , 2013

Gabriel PIETRZKOWSKI

ROZWIZANIE EXPLICITE UKADU STEROWANIA TYPU sl(2)

6, 13 marca 2013/ March, 6, 13th , 2013

Andrew James BRUCE

ODD JACOBI MANIFOLDS AND JACOBI ALGEBROIDS

In this talk I will present some of the basic ideas of my approach to odd Jacobi manifolds and focus on applications in the theory of Jacobi algebroids and odd contact geometry. The talk will be based on "Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids" Extracta Math. 27(1) (2012), 91-123.

27 lutego 2013/ February, 27th , 2013

Micha JӏWIKOWSKI

TEORIA DETW w/g EHRESMANNA

20 lutego 2013/ February, 20th , 2013

Mikoaj ROTKIEWICZ

FUNKTORY WEILA W SUPERGEOMETRII

23 stycznia 2013/ January, 23rd , 2013

MIkoaj ROTKIEWICZ

FUNKTORY WEILA W GEOMETRII I SUPERGEOMETRII

16 stycznia 2013/ January, 16th , 2013

Pawe NUROWSKI

LOCAL CONFORMAL INVARIANTS VIA FEFFERMAN-GRAHAM METRICS

12 grudnia 2012, 9 stycznia 2013/ December, 12th , 2012, and January, 9th , 2013

Piotr FRCKIEWICZ

KORELACJE KLASYCZNE, KWANTOWE I SUPERKWANTOWE

Podczas wykadu omwi rnice pomidzy klasycznymi, kwantowymi i superkwantowymi. W tym celu wykorzystam pojcie nierwnoci Bella, w szczeglnoci nierwno CHSH oraz problem opisany w pracy M. L. Almeida, J.-D. Bancal, N. Brunner, A. Acin, N. Gisin, S. Pironio: "Guess your neighbour's input: a multipartite non-local game with no quantum advantage" Phys. Rev. Lett. 104, 230404 (2010).

28 listopada 2012 / November, 28th , 2012

Andriy PANASYUK (UWM)

O KLASYFIKACJI PAR ZGODNYCH NAWIASW LIEGO, Z KTRYCH JEDEN JEST PӣPROSTY

21 listopada 2012 / November, 21st , 2012

NIEINWAZYJNY POMIAR KWANTOWY

Standardowy, rzutowy pomiar kwantowy jest inwazyjny (ingeruje w ukad) prowadzi do paradoksw: efektu Zenona i amania nierwnoci Bella. Pomiar nieinwazyjny jest przeciwiestwem rzutowego, unika powyszych paradoksw, ale prowadzi do zaskakujcych nowych, np. amania symetrii czasu.

14 listopada 2012 / November, 14th , 2012

Elisa GUZMAN (U La Laguna)

REDUCTION FOR PRINCIPAL BUNDLES AND TULCZYJEW'S TRIPLE

Abstract: Here some ideas about reduction for principal bundles are presented. It is known that Classical Field theories of first order can be formulated as Lagrangian submanifolds of premultisymplectic manifolds through a particular Tulczyjew's Triple. In the case, the Classical Field theories is a G-principal bundle a new triple and new Lagrangian submanifolds related with the previous ones are obtained to formulated the Euler-Poincare and the Hamilton-Poincare equations when the Lagrangian and the Hamiltonian density are G-invariant.

7 listopada 2012 / November, 7th , 2012

Micha JӏWIKOWSKI

GEOMETRIA WIZEK NIESKOCZONYCH DETW

Abstrakt: Podczas wykadu omwi podstawowe pojcia i konstrukcje zwizane z wizk nieskoczonych detw.

24, 31 padziernika 2012 / October, 24th and 31st , 2012

Micha JӏWIKOWSKI

GEOMETRIA WIZEK DETW

Podczas dwch wykadw przedstawi wprowadzenie do przestrzeni detw. W szczeglnoci omwi pojcie dystrybucji Cartana i scharakteryzuje jej maksymalne cakowe podrozmaitoci. Ten wynik posuy do dowodu tw. Lie-Backlunda charakteryzujcego wszystkie symetrie przestrzeni detw. Opowiem te o geometrycznym podejciu do czstkowych rwna rniczkowych i o wizkach nieskoczonych detw i formalnej cakowalnoci.

17 padziernika 2012 / October, 17th , 2012

Pawe NUROWSKI

GEOMETRIA TOCZCYCH SI POWIERZCHNI

3, 10 padziernika 2012 / October, 3rd and 10th , 2012

Micha JӏWIKOWSKI

6 czerwca 2012 / June, 6th 2012

Ryszard KERNER (UMC Paris)

SPACETIMES SYMMETRIES FROM Z_3-GRADED QUARK ALGEBRA

We investigate certain Z_3-graded associative algebras with cubic Z_3-invariant constitutive relations. The invariant forms on finite algebras of this type are given in the low dimensional cases with two or three generators. We show how the Lorentz symmetry represented by the SL(2,C) group emerges naturally without any notion of Minkowskian metric, just as the invariance group of the Z_3-graded cubic algebra and its constitutive relations. Its representation is found in terms of Pauli matrices. The relationship of this construction with the operators defining quark states is also considered, and a third-order analogue of the Klein-Gordon equation is introduced. Cubic products of its solutions may provide the basis for the familiar wave functions satisfying Dirac and Klein-Gordon equations. Pdf

30 maja 2012 / May, 30th 2012

Bronisaw JAKUBCZYK

O REGULARNOCI ABNORMALNYCH EKSTREMAL MINIMALIZUJACYCH

23 maja 2012 / May, 23rd 2012

16 maja 2012 / May, 16th 2012

Giovanni MORENO (SU Opava)

INITIAL DATA OF A NONLINEAR PDE AND THEIR NATURAL STRUCTURES

9 maja 2012 / May, 9th 2012

Simon BRAIN (U Luxembourg)

THE DIFFERENTIAL AND TWISTOR GEOMETRY OF SELF-DUAL YANG-MILLS GAUGE FIELDS

25 kwietnia 2012 / April, 25th 2012

Mikoaj ROTKIEWICZ (IM UW)

FORMY WIELOSYMPLEKTYCZNE

4 kwietnia 2012 / April, 4th 2012

ukasz SKOWRONEK (UJ)

ELEMENTY GEOMETRII ALGEBRAICZNEJ W TEORII SPLTANIA KWANTOWEGO

Poka, jak rozstrzygnem, z wykorzystaniem twierdzenia Bezout, kwesti charakteryzacji pewnej rodziny stanw, wanych z punktu widzenia teorii spltania kwantowego. S to stany spltane z dodatni czciow transpozycj, o rzdzie cztery. Okazuje si, e wszystkie je mona uzyska poprzez elementarn, dobrze znan konstrukcj, poczon z dowoln lokaln transformacj stanu. Postaram si rwnie, w miar moliwoci, nakreli inne problemy, dajce si traktowa metodami geometrii algebraicznej, a bdce w krgu zainteresowa osb zajmujcych si teori spltania kwantowego i pokrewnymi zagadnieniami. W tym wypadku bdzie chodzio o rozwizywanie rwna wielomianowych.

28 marca 2012 / March, 28th 2012

Maciej BASZAK (UAM)

KWANTYZACJA DEFORMACYJNA MECHANIKI HAMILTONOWSKIEJ

W wykadzie prezentuj w systematyczny sposb alternatywne sformuowanie mechaniki kwantowej, zwane mechanik kwantow na przestrzeni fazowej lub kwantowaniem deformacyjnym. Rozpatrywana klasa deformacji zawiera jako szczeglne przypadki wszystkie znane w literaturze deformacje klasycznej algebry obserwabli. Ponadto zdefiniowana jest przestrze zawierajca dopuszczalne stany kwantowe i posiadajca struktur algebry Hilberta ze wzgldu na odpowiednie *-mnoenie. Podczas wykadu postaram si uzasadni tez, i prezentowany formalizm jest bardziej fundamentalny od standardowej aksjomatycznej mechaniki kwantowej. Standardowa mechanika kwantowa pojawia si w prezentowanym formalizmie jako naturalna reprezentacja mechaniki kwantowej na przestrzeni fazowej. Ta uyteczna i prosta reprezentacja, istniejca przynajmniej dla sformuowania kanonicznego, wynika z istnienia odpowiedniej klasy transformacji Wignera-Moyal'a speniajcych wszystkie wasnosci iloczynu tensorowego odpowiednich przestrzeni Hilberta.

14 i 21 marca 2012 / March, 14th and 21st 2012

Micha JӏWIKOWSKI

JACOBI FIELDS AND SECOND VARIATIONS

The notions of a Jacobi field and of conjugate points appear naturally while considering second order optimality conditions in variational problems. At the lecture I will study the geometric nature of Jacobi fields and their relation with Euler-Lagrange equations. In particular I will show that existence of conjugate points is equivalent with existence of the null space of the symmetric form defined by the second variation for a wide class of variational problems.

29 lutego i 7 marca 2012 / February, 29th and March, 7th 2012

Janusz GRABOWSKI

22 lutego 2012 / February, 22nd 2012

TRANSFORMACJA LEGENDRE'A W DYNAMICE STRUN

15 lutego 2012 / February, 15th 2012

AN APOLOGY FOR INFINITE JETS

I will explain the fundamental importance of infinite jets for the theory of nonlinear partial differential equations and various applications to mechanics and mathematical physics.

11 i 18 stycznia 2012 / January, 11th and 18th 2012

Wojciech KRYSKI

HYDRODYNAMIKA WEDUG ARNOLDA

4 stycznia 2012 / January, 4th 2012

Witold RESPONDEK (INSA de Rouen)

MECHANICZNE UKADY STEROWANIA I ICH WSPӣZMIENNIKI

21 grudnia 2011 / December, 21st 2011

Witold RESPONDEK (INSA de Rouen)

PASKO NIEDOOKRELONYCH RWNA RӯNICZKOWYCH POCHODZCYCH OD KANONICZNYCH DYSTRYBUCJI CARTANA

14 grudnia 2011 / December, 14th 2011

Eduardo MARTINEZ (U. of Zaragoza)

MOMENTUM MAPS FOR MECHANICAL SYSTEMS ON LIE ALGEBROIDS AND REDUCTION

7 grudnia 2011 / December, 7th 2011

Marek KU

DETEKCJA STANW QUASIKLASYCZNYCH

30 listopada 2011 / November, 30 th 2011

Janusz GRABOWSKI

TRJKI TULCZYJEWA W MECHANICE I TEORII POLA

23 listopada 2011 / November, 23 rd 2011

Jacek JEZIERSKI

GEOMETRIA CZASOPRZESTRZENI W OTOCZENIU HORYZONTU CZARNEJ DZIURY

16 listopada 2011 / November, 16 th 2011

Ben WARHURST (UNSW)

SUB-RIEMANNIAN SYMMETRIES ON NILPOTENT LIE GROUPS

9 listopada 2011 / November, 9 th 2011

Pawe STRZELECKI (MIM UW)

CAKOWA KRZYWIZNA MENGERA DLA KRZYWYCH, POWIERZCHNI I INNYCH ZBIORW: EFEKTY WYGADZANIA I BRAKU SAMOPRZECI

Krzywizna Mengera trjki punktw to odwrotno promienia okrgu, przechodzcego przez te punkty, a cakowa krzywizna Mengera krzywej prostowalnej to caka z p-tej potgi krzywizny Mengera wzgldem wszystkich trjek punktw (cakujemy wzgldem dugoci uku). Okazuje si, e dla pewnych wartoci p skoczono tak zdefiniowanej krzywizny gwarantuje, e krzywa ma hoelderowsko cigy wektor styczny (co wynika z geometrycznych namiastek nierwnosci Sobolewa-Morreya) i jest pozbawiona samoprzeci. Podobne funkcjonay i wyniki mona wskaza w oglnym przypadku, dla podrozmaitosci przestrzeni euklidesowej; postaram si opowiedzie o tym w sposb nietechniczny i przystpny.

2 listopada 2011 / November, 2 nd 2011

Jacek JEZIERSKI

CYK TENSORY W KLASYCZNEJ TEORII POLA

26 padziernika 2011 / October, 26 th 2011

Jerzy KIJOWSKI

O GEOMETRYCZNEJ KWANTYZACJI

12, 19 padziernika 2011 / October, 12th and 19th 2011

WIZY HAMILTONOWSKIE w/g DIRACA I INNYCH

5 padziernika 2011 / October, 5th 2011

Katarzyna GRABOWSKA

CO POCZ Z LAGRANJANEM  DRUGIEGO RZDU?

1 czerwca 2011 / June, 1st 2011

Wiesaw SASIN (PW)

WASNOCI GEOMETRYCZNE GRUPOIDU TRANSFORMACJI NAD CZASOPRZESTRZENI

18, 25 maja 2011 / May, 18th and 25th 2011

Maciej UKASIK  (KMMF)

RACHUNEK WARIACYJNY BEZ PARAMETRYZACJI

11 maja 2011 / May, 11th 2011

Bronisaw JAKUBCZYK  (IM PAN)

KRZYWIZNY I KONEKSJE W GEOMETRII FINSLERA I INNYCH

4 maja 2011 / May, 4th 2011

Jacek JEZIERSKI  (KMMF)

KWAZI-LOKALNA MASA W GRAWITACJI

20 kwietnia 2011 / April, 20th 2011

Frank KELLER  (IM PAN)

13 kwietnia 2011 / April, 13th 2011

Barbara OPOZDA  (UJ)

PEWNE PODROZMAITOCI MINIMALNE I ICH PRZESTRZENIE MODULI

6 kwietnia 2011 /  April , 30th 2011

Mikoaj ROTKIEWICZ (IM UW)

WIZKI  JEDNORODNE

30 marca 2011 /  March, 30th 2011

Witold RESPONDEK (INSA de ROUEN)

INTRODUCTION TO SUB-RIEMANNIAN GEOMETRY

23 marca 2011 /  March, 23rd 2011

Katarzyna GRABOWSKA

ALGEBROIDY DIRACA W MECHANICE ANALITYCZNEJ
Pdf

16 marca 2011 /  March, 16th 2011

Tomasz RYBICKI (AGH)

OGRANICZONO GRUPY DYFEOMORFIZMW
Streszczenie: Mwimy, e grupa jest ograniczona, jeeli dowolna metryka bi-niezmiennicza na niej jest ograniczona.
Nastpnie, grupa jest jednostajnie doskonaa, jeeli jest ona doskonaa i jej dugo komutatorowa jest ograniczona.
Celem referatu jest przedstawienie niedawno uzyskanych wynikw dotyczcych ograniczonoci i jednostajnej doskonaoci grup dyfeomorfizmw na rozmaitoci.
W przeciwiestwie do klasycznych twierdze o prostocie i doskonaoci grup dyfeomorfizmw, twierdzenia o ograniczonoci zale od topologii rozmaitoci.
Wskazujemy te, e za pomoc dugoci komutatorowej symplektomorfizmw interpretuje si pewne niezmienniki topologii symplektycznej.

2, 9 marca 2011 /  March, 2nd, 9th 2011

16, 23 lutego 2011 /  February, 16th, 23rd 2011

O GEOMETRII RWNANIA HIROTY

Abstrakt:Rnicowa wersja rwnania Kadomtseva-Petviashvili, zaproponowana 30 lat  temu przez Ryogo Hirot, jest jednym z waniejszych
rwna wspczesnej  fizyki matematycznej (patrz np. niedawny artyku przegladowy "T-systems  and Y-systems in integrable systems",
Atsuo Kuniba, Tomoki Nakanishi,  Junji Suzuki, arXiv:1010.1344). W moim wykadzie chciabym przedstawi  relatywnie prosta geometryczn
Odwzorowania Desarguesa sieci pierwiastkowych typu A w przestrzenie  rzutowe (nad pierscieniem z dzieleniem) scharakteryzowane s pewnym
prostym warunkiem geometrycznym, prowadzcym do problemu liniowego dla  rwnania Hiroty. Rwnanie to jest w tej interpretacji zakodowane w
konfiguracji Veblena, a jego wielowymiarowa konsystencja jest rwnowana  twierdzeniu Desarguesa. Pierwsza czs wykadu chciabym zakoczy
na  przedstawieniu zwiazku symetrii konfiguracji Desarguesa z tzw. rwnaniem  piciokta oraz na omwieniu wanego przykadu kwantyzacji pewnej
naturalnej w tym kontescie struktury Poissona.    W nastpnym tygodniu chciabym skoncentrowa si na geometrycznej  interpretacji binarnej
transformacji Darboux dla rwnania Hiroty i  omwi na tym tle przykad rozwiaza wielosolitonowych. Na koniec  porusz zwizek odwzorowa
Desarguesa z alternatywnym (lecz rwnowanym)  ujciem geometrycznym dyskretnych ukadw cakowalnych poprzez tzw.  sieci czworobokw paskich.
Wykady oparte bda na moich niedawnych artykuach (Proc R. Soc. A 466  (2010) 1177, Phys. Lett. A 375 (2011) 1219) oraz na nieopublikowanej
pracy z Sergeyem Sergeevem (University of Canaberra).

12, 19 stycznia 2011 /  January, 12th, 19th 2011

Jan DEREZISKI

FUNKCJE TYPU HIPERGEOMETRYCZNEGO I ICH SYMETRIE

Abstrakt: Funkcje typu hipergeometrycznego obejmuj wikszo  najwaniejszych funkcji specjalnych (m.in funkcj Bessela, konfluentn,  hipergeometryczn
i klasyczne wielomiany ortogonalne). Speniaj one wiele  intrygujcych tosamoci. Ich wasnoci mona zrozumie i uporzdkowa przy  uyciu grup i algebr Liego.

5 stycznia 2011 /  January, 5th 2011

FORMALIZM SKINNERA-RUSKA W MECHANICE

15 grudnia 2010 /  December, 15th 2010

Micha JӏWIKOWSKI (IMPAN)

ZASADA WARIACYJNA DLA DYNAMIKI HAMILTONOWSKIEJ (Z WIZAMI)

Opis: Zaproponuj alternatywn metod wyliczania rwna Hamiltona dla  liniowej struktury Poissona na przestrzeni fazowej.
Trajektorie fazowe  otrzymamy jako ekstremale naturalnego dziaania na sumie prostej przestrzeni  fazowej i konfiguracyjnej.
Proponowane podejcie daje si take zastosowa  dla ukadw z wizami w przestrzeni konfiguracyjnej.

1, 8 grudnia 2017 /  December, 1st , 8th 2010

Gabriel PIETRZKOWSKI (IMPAN)

ALGEBRA  W  MODELU  STANDARDOWYM

24 listopada 2010 /  November, 24th2010

Rafa SUSZEK (KMMF UW)

STRINGS, GERBES, AND ALL THAT  Symmetries and generalized geometry

It has, by now, been rather well understood that the proper language in  which to give a lagrangean formulation of the two-dimensional non-linear
sigma model, regarded as a classical description of the critical bosonic  string (and of relevance in the study of certain condensed-matter systems,
as well as models of statistical physics), is that of the theory of  gerbes. These latter are higher-cohomological structures with a  differential-geometric
realisation whose rle in string theory is  analogous to that played by fibre bundles in the modelling of the dynamics  of a charged pointlike particle moving
in an external electromagnetic  field. The talk, to be regarded as a fairly general overview of the state  of art in this field of mathematical physics in two
dimensions, aspires to  outline some basic aspects of gerbe theory relevant to the classical and  quantum description of poly-phase string world-sheets and
string-theory  dualities, laying due emphasis on the higher-categorial structure that  underlies the theory of gerbes.    In the second part of the talk,
in which we take up (and briefly  recapitulate beforehand) the subject introduced at the KMMF seminar  Theory of Duality'' of 18 XI 2010 with view to
a detailed account of the  generalised geometry underlying the groupoidal symmetries of the sigma  model, the emergence, in the study of (infinitesimal)
rigid symmetries of  the two-dimensional field theory, of algebro-differential structures akin  to the gerbe-twisted Courant algebroid shall be demonstrated
and the  naturality of these structures in the context of the canonical description  of the two-dimensional field theory, and - in particular - that of the
attendant gauge principle, shall be discussed. More specifically, we shall  examine the geometry of a family of generalised tangent bundles over
the  configuration bundle of the sigma model in the presence of the full  2-category of bundle gerbes over it, establish its direct relation to the
Noether (canonical) description of the rigid symmetries of the sigma  model, and formulate a universal gauge principle for the two-dimensional  field
theories in hand based on the concept of categorial descent and the  notion of a principal bundle with a structural groupoid over the  two-dimensional spacetime.
Pdf

17 listopada 2010 /  November, 17th2010

Pawe NUROWSKI (IFT UW)

3-WYMIAROWE STRUKTURY CR I RWNANIA RӯNICZKOWE
ZWYCZAJNE 2-GO RZDU

10 listopada 2010 /  November, 10th 2010

Frank KELLER (IM PAN)

DEFORMATION OF COURANT-DORFMAN ALGEBROIDS

We will give two different constructions of a deformation complex for   Courant-Dorfman algebroids in a purely algebraic setting. The relation
between these two complexes will be discussed. Moreover, the Fedosov   construction will be adapted to our setting in order to get a star
product on the deformation complex.

3 listopada 2010 /  November, 3rd 2010

Juan Carlos MARRERO (La Laguna - Spain)

TIME-DEPENDENT MECHANICS AND LAGRANGIAN SUBMANIFOLDS
OF PRESYMPLECTIC AND POISSON MANIFOLDS

A description of time-dependent Mechanics in terms of Lagrangian   submanifolds of presymplectic and Poisson manifolds is presented.
Two new   Tulczyjew triples are discussed. The first one is adapted to the   restricted Hamiltonian formalism and the second one
is adapted to the   extended Hamiltonian formalism.

27 padziernika 2010 /  October, 27th 2010

Wodzimierz M. TULCZYJEW

MECHANICS OF INCOHERENT MATTER  (DUST, PLASMA)

20 padziernika 2010 /  October, 20th 2010

TEORIA KALUZY-KLEINA

13 padziernika 2010 /  October, 13th 2010

Katarzyna GRABOWSKA

DIRAC STRUCTURES AND GEOMETRY OF NONHOLONOMIC CONSTRAINTS

Pdf

6 padziernika 2010 /  October, 6th 2010

Janusz GRABOWSKI

COURANT BRACKETS AND DIRAC STRUCTURES

26 maja, 2 czerwca 2010 /  May, 26th , June, 2nd 2010

Krzysztof KUREK (IPJ)

CZEGO BRAKUJE W MODELU STANDARDOWYM - KONCEPCJE I IDEE

5, 12 maja 2010 /  May, 5th , 12th 2010

Javier de LUCAS (IM PAN)

LIE SYSTEMS: THEORY, GENERALIZATIONS, AND APPLICATIONS

Lie systems form a special class of differential equations admitting many interesting geometric properties, e.g. their general solution can be expressed by each generic family of particular solutions

in terms of a (nonlinear) superposition rule. The main aim of this talk is to show a modern geometric approach to these systems. Such an approach has proven to be very successful not only in describing geometric

properties of these systems but also in generalising, in different ways, the Lie system notion.

As a result, many applications of these systems have arisen in Quantum Mechanics, Classical Mechanics, integrability of differential equations,

Control Theory, Financial Mathematics, etc.

Pdf

28 kwietnia 2010 /  April, 28th 2010

David SAUNDERS (U Ostrava)

SOME GEOMETRIC ASPECTS OF THE CALCULUS OF VARIATIONS IN SEVERAL INDEPENDENT VARIABLES

In this talk I shall describe some recent research on parametric problems in the calculus of variations (of which the minimal surfaces problem is perhaps the most basic example).

I shall also explain the relationship between these problems and the type of problem more usual in physics, where there is a given space of independent variables. Aspects to be covered will include an

interpretation of the first variation formula in terms of cohomology.

21 kwietnia 2010 /  April, 21st 2010

Pawe NUROWSKI

STRUKTURY PARA-CR I RWNANIA RӯNICZKOWE

14 kwietnia 2010 /  April, 14th 2010

Wodzimierz M. TULCZYJEW

A VARIATIONAL FRAMEWORK FOR ANALYTICAL MECHANICS AND FIELD THEORY

31 marca 2010 /  March, 31st 2010

###### Mikoaj ROTKIEWICZ (IM UW)

PEWNE KONSTRUKCJE SUPER-ROZMAITOCI

W literaturze spotykamy wiele rnych koncepcji super-rozmaitoci. Pomijajc szczegy, s dwa kompletnie rne podejcia. Pierwsze, "snopowe", polegajce na zastpieniu algebry funkcji gadkich na

tradycyjnej rozmaitoci $Z/2Z$-gradowan algebr przez doczenie elementw antykomutujcych. Drugie polega na zdefiniowaniu super-rozmaitoci jako zbioru z pewn dodatkow struktur

(atlasem), bardzo podobnie jak w tradycyjnej geometrii. Okazuje si, e obie koncepcje mog by w duej mierze stosowane zamiennie (rezultat z "Two approaches to supermanifolds", M.

Batchelor), a nieco dokadniej kategorie gradowanych rozmaitoci (wprowadzonych przez B.Kostanta) i DeWitta $H^infty$ super-rozmaitoci s rwnowane.

Na seminarium omwi powyszy wynik oraz podam kilka kanonicznych przykadw super-rozmaitoci.

24 marca 2010 /  March, 24th 2010

Wodzimierz JELONEK (PK)

POLA KILLINGA ZE SPECJALNYM POTENCJAEM KAEHLERA-RICCIEGO W GEOMETRII KAEHLEROWSKIEJ I ICH ZASTOSOWANIA

Podajemy klasyfikacje zwartych, jednospjnych rozmaitoci kaehlerowskich (M,g,J) z quasi-sta homolorficzn krzywizn sekcyjn przy zaoeniu dim(M)>4.

S to rozmaitoci, ktrych holomorficzna krzywizna sekcyjna R(X,JX,JX,X), gdzie X jest jednostkowym wektorem stycznym do M, zaley tylko od punktu x i dugoci rzutu ortogonalnego wektora X na

ustalon, zespolon liniow wizka D zawart w TM. Pokazujemy, ze jeli D nie jest trywialna, to M jest holomorficzna wiazk nad przestrzeni rzutow CP^n z wknem CP^1.

Wizka D okazuje si cakowaln dystrybucj styczn do wkien CP^1 wizki.

Metoda dowodu polega na wykazaniu istnienia na M pola Killinga ze specjalnym potencjaem Kaehlera-Ricciego, a nastpnie na skorzystaniu z twierdzenia Derdzinskiego-Mashlera, klasyfikujcego zwarte rozmaitoci kaehlerowskie

dopuszczajce pola Killinga ze specjalnym potencjaem Kaehlera-Ricciego, i pokazaniu, ze jedynymi takimi rozmaitociami z quasi-sta krzywizn holomorficzn s wizki nad przestrzenia rzutowa CP^n z wknem CP^1,

ktre s projektywizacj potgi wizki tautologicznej nad CP^n. Podajemy rownie zastosowania pl Killinga ze specjalnym potencjaem Kaehlera-Ricciego przy czciowej klasyfikacji zwartych, hermitowskich

rozmaitoci Graya.

17 marca 2010 /  March, 17th 2010

CZY MONA USYSZEƔ KSZTAT GRAFU KWANTOWEGO?

W 1966 roku Marc Kac zada synne pytanie Can one hear the shape of a drum?. Od tego czasu czyniono prby zarwno rekonstrukcji ksztatu na podstawie widma jak i szukano metody konstrukcji obiektw izospektralnych. W kocu lat 90 Smilansky przeformuowa pytanie Kaca w kontekcie tzw. grafw kwantowych. W czasie seminarium zaprezentuje metod konstrukcji izospektralnych grafw kwantowych opart na teorii reprezentacji grup oraz przedstawi kilka prostych przykadw jej zastosowania.

10 marca 2010 /  March, 10th 2010

Marcin MARCINIAK (UG)

ODWZOROWANIA DODATNIE NA ALGEBRACH MACIERZOWYCH

Celem wykadu jest omwienie kilku problemw dotyczcych klasyfikacji odwzorowa dodatnich. W pierwszej kolejnoci pokaemy, e owa klasyfikacja moe by zredukowana do opisu punktw eksponowanych stoka odwzorowa dodatnich, nastpnie opiszemy klas znanych punktw eksponowanych. Ponadto omwimy wasnoci dodatnich odwzorowa ekstremalnych zwizane z zachowaniem rzdu. Na koniec sformuujemy czciowe rozwizania problemw Robertsona i Osaki dotyczcych szczeglnych wasnoci dodatnich odwzorowa ekstremalnych.

3 marca 2010 /  March, 3rd 2010

Andrzej DRAGAN (IFT UW)

WYZNANIE WIARY W TEORI KWANTOW WRAZ Z MATERIAEM DOWODOWYM

17, 24 lutego 2010 /  February 17th , 24th 2010

Andrzej OKOW (IFT UW)

TEORIE YANGA-MILLSA

Teorie Yanga-Millsa odgrywaj znaczc rol we wspczesnej fizyce bdc podstaw Modelu Standardowego czstek elementarnych. Z punktu widzenia geometrii s one zwizane z wizkami gwnymi: przestrze

konfiguracyjna tych teorii jest przestrzeni koneksji na wizce gwnej. W trakcie referatu zostanie przedstawiona konstrukcja tzw. dziaania czyli funkcjonau na przestrzeni koneksji okrelajcego dynamik teorii,

nastpnie z dziaania zostan wyprowadzone rwnania Yanga-Millsa. Jako przykad zastosowania tych teorii zostanie zaprezentowany tzw. mechanizm Higgsa na przykadzie modelu oddziaywa elektrosabych.

20 stycznia 2010 /  January 20th 2010

Pawe WALCZAK (U)

POTOKI GEOMETRII ZEWNETRZNEJ NA SFOLIOWANYCH ROZMAITOSCIACH RIEMANNOWSKICH

6, 13 stycznia 2010 /  January 6th, 13th 2010

Witold RESPONDEK (INSA de ROUEN)

DYSTRYBUCJE CARTANA DLA KRZYWYCH I POWIERZCHNI: CHARAKTERYZACJA, GEOMETRIA I PASKO

9 grudnia 2009 /  December 9th 2009

Alexei KOTOV (University of Luxembourg)

A BRIEF INTRODUCTION TO SIGMA-MODELS

A short introduction to nonlinear sigma-models will be given. The theory will be illustrated by some important examples which

include the Poisson sigma model and its generalizations as a part of the AKSZ (Aleksandrov-Kontsevich-Schwarz-Zaboronsky) approach.

9 grudnia 2009 /  December 9th 2009

Wojciech KRYSKI (IM PAN)

TKANINY KRONECKERA I RWNANIA RӯNICZKOWE

2 grudnia 2009 /  December 2nd 2009

Piotr MORMUL (IM UW)

GOURSAT MONSTER I JEGO WACHLARZ OSOBLIWYCH KRZYWYCH LEGENDROWSKICH

25 listopada 2009 /  November 25th 2009

REDUKCJA ROUTHA GEOMETRYCZNIE

18 listopada 2009 /  November 18th 2009

Bronisaw JAKUBCZYK (IM PAN)

KRZYWIZNY PL WEKTOROWYCH NA ROZMAITOCI Z DYSTRYBUCJ

4 listopada 2009 /  November 4th 2009

Marek KU (CFT)

KORELACJE KWANTOWE CZSTEK IDENTYCZNYCH

28 padziernika 2009 /  October 28th 2009

Halina FRANKOWSKA (CNRS and Universite Pierre et Marie Curie)

OPTIMAL CONTROL UNDER STATE CONSTRAINTS

This talk is devoted to the Bolza optimal control problem under state constraints. We shall discuss necessary optimality conditions and provide

some geometric conditions guaranteeing their normality. We also show how they can be applied to investigate regularity of optimal

trajectories and of adjoint variables, as well as existence of optimal solutions to problems with Lagrangians not satisfying the Tonelli growth

condition.

21 padziernika 2009 /  October 21st 2009

Mark GOTAY (PIMS, Vancouver)

STRESS-ENERGY-MOMENTUM TENSORS AND THE BELINFANTE-ROSENFELD FORMULA

7 padziernika 2009 /  October 7th 2009

Katarzyna GRABOWSKA (KMMF)

O PEWNYM MODELU GEOMETRYCZNYM KLASYCZNEJ TEORII POLA

20,27 maja 2009 /  May 20th, 27th 2009

Maciej  UKASIK (KMMF)

RACHUNEK WARIACYJNY NIEZALENY OD PARAMETRYZACJI

6, 13 maja 2009 /  May 6th , 13th 2009

Micha  JOWIKOWSKI (IM PAN)

Ukady sterowania na algebroidach Liego pojawiaj si w naturalny sposb wwyniku redukcji klasycznych ukadw sterowania przez grup symetrii. Na wykadzie poka jak sformuowa Zasad Maksimum Pontriagina  podstawowe twierdzenie teorii optymalnego sterowania - dla takich ukadw. Okazuje si, e za ZMP odpowiada nieco oglniejsza struktura geometryczna  algebroidu prawie-Liego. Kluczowe okazuje si zrozumienie uoglnionego pojcia homotopii krzywych. Wspautorem omawianych wynikw jest prof. Janusz Grabowski.

29 kwietnia 2009 / April, 29th 2009

Prof. Witold  RESPONDEK (INSA de ROUEN)

KIEDY UKAD STEROWANIA JEST MECHANICZNY? cz. II

22 kwietnia 2009 / April, 22nd 2009

Prof. Witold  RESPONDEK (INSA de ROUEN)

KIEDY UKAD STEROWANIA JEST MECHANICZNY? cz. I

1 kwietnia 2009 / April, 1st 2009

Andriy PANASYUK

CZʌCIOWE OPERATORY NIJENHUISA

25 marca 2009 / March, 25th 2009

Gerd RUDOLPH (University Leipzig)
GAUGE THEORIES AND SINGULAR REDUCTION
I will give an elementary introduction into symplectic reduction of Hamiltonian systems endowed with Hamiltonian Lie group actions. As an application, I will formulate gauge theory (on a finite lattice) as a Hamiltonian system with symmetry and discuss its reduction.

18 marca 2009 / March, 18th 2009

Aleksy TRALLE (UWM)

O TOPOLOGII GRUPY HAMILTONOWSKICH SYMPLEKTOMORFIZMW PRZESTRZENI JEDNORODNYCH

11 marca 2009 / March, 11th 2009

Wojciech KRYSKI (IM PAN)

RWNOWANO DYSTRYBUCJI KORZDU 2

Z MAKSYMALNYM INDEKSEM KRONECKERA

4 marca 2009 / March, 4th 2009

###### Ben WARHURST (IM PAN)

SUB-RIEMANNIAN VS EUCLIDEAN DIMENSION COMPARISON AND CARTAN GEOMETRY ON CARNOT GROUPS

A Carnot group G is naturally equipped with equivalent Euclidean and subriemannian metrics. Gromov has asked the following question for submanifolds of G: Determine all possible pairs (A,B)
of real numbers such that there exists a submanifold M of G with Euclidean Hausdorff dimension A and sub-Riemannian Hausdorff dimension B.To answer this question in general is difficult
because the structure of the underlying Lie algebra is significant. If we consider Gromov's questions for subsets of G, then a complete answer can be formulated. The solution uses elements
of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. An interesting bi-product of this work is a relatively simple method
for calculating dimensions of nonlinear iterated function systems. This is the result of joint work with Zoltan Balogh (Bern) and Jeremy Tyson (Illinois).

18 i 25 lutego 2009 / February, 18th and 25th 2009

###### Mikoaj ROTKIEWICZ (IM UW)

STRUKTURY PODWJNE W GEOMETRII WARTOCI AFINICZNYCH

21 stycznia 2009 / January, 21st 2009

###### Piotr WOJDYO(IM PAN)

REPREZENTACJE W PASZCZYNIE CZAS-CZSTOTLIWO,

ICH ZASTOSOWANIA DO  PRZETWARZANIA SYGNAW

14 stycznia 2009 / January, 14th 2009

###### Andrzej KOSSAKOWSKI (UMK)

O  STRUKTURZE  GENERATORW  NIEMARKOWSKICH RWNA  EWOLUCJI

7 stycznia 2009 / January, 7th 2009

### PROJECTIVE STRUCTURES, TWISTOR THEORY AND ODEs

10 grudnia 2008 /  December, 10th 2008

Gabriel PIETRZKOWSKI  (IM PAN}

KWANTOWA TELEPORTACJA

3 grudnia 2008 /  December, 3rd 2008

Marcin  BOBIESKI  (IM UW)

POLA  YANGA-MILLSA  I  RWNANIA  YANGA-MILLSA

26  listopada 2008 /  November, 26th 2008

Gabriel PIETRZKOWSKI  (IM PAN)

CAKOWA REPREZENTACJA  STANW SEPAROWALNYCH

Kwantowym stanem separowalnym na iloczynie tensorowym przestrzeni Hilberta $H=K\otimes L$, nazywa si dodatni operator Hermitowski o ladzie jednostkowym, ktry mona przedstawi jako dodatni kombinacj rzutw na  wektory proste w H (tj. wektory postaci $v \otimes w \in H$). Celem wystpienia jest pokazanie formuy cakowej na stany separowalne i przeanalizowanie jej wasnoci.

19  listopada 2008 /  November, 19th 2008

Javier de LUCAS ARAUJO  (IM PAN)

SUPERPOSITION PRINCIPLES

Abstract: We say that a system of first-order differential equations admits a superposition principle if its general solution can be written somehow in terms of a finite set of particular solutions and a set of constants. Differential equations admitting this property were characterized by S. Lie. Nevertheless, this characterization has some practical problems. In this talk,  we investigate superposition principles from the point of view of the modern differential geometry and we analyze some of the practical problems of Lie's characterization.

12 listopada 2008 /  November, 12th 2008

Maciej P. WOJTKOWSKI  (UWM)

GEOMETRIA FILTRW KALMANA

5 listopada 2008 /  November, 5th 2008

David MARTIN de DIEGO  (CSIC, Madrid)

DISCRETE MECHANICS: FROM DIFFERENTIAL GEOMETRY TO NUMERICAL INTEGRATION

29 padziernika 2008 /  October, 29th 2008

O RWNANIU HAMILTONA-JACOBIEGO I METODZIE JACOBIEGO

22 padziernika 2008 /  October, 22nd 2008

Alexei KOTOV  (U. of Luxembourg)

CHARACTERISTIC CLASSES ASSOCIATED WITH Q-BUNDLES

We generalize the Chern-Weil formalism to the case of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds (graded super manifolds supplied with a holomological vector field of degree 1).

15 padziernika 2008 /  October, 15th 2008

Jacek  JEZIERSKI  (KMMF UW)

O  ISTNIENIU  METRYK  KUNDTA  I  ZDEGENEROWANYCH  HORYZONTW  KILLINGA

8 padziernika 2008 /  October, 8th 2008

Katarzyna  GRABOWSKA  (KMMF UW)

NOWY  SCHEMAT  GEOMETRYCZNY  DLA  RACHUNKU  WARIACYJNEGO  Z  WIZAMI

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