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

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

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

Metody Geometryczne Fizyki

Informacje o tematyce seminarium w języku angielskim znajdują się tutaj. Information about the subject area of the seminar (in English) can be found here






Rok akademicki 2017/2018
Semestr zimowy

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 ZAJĄC (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

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).

29 listopada 2017/ November 29th, 2017

Aneta SLIŻEWSKA (Universytet 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

22 listopada 2017/ November 22nd, 2017

Andrzej TRAUTMAN (FUW)

Optical structures in realitivity

15 listopada 2017/ November 15th, 2017

Seminarium odwołane / Session cancelled

8 listopada 2017/ November 8th, 2017

Marek DEMIAŃSKI (FUW)

Fale grawitacyjne - nowe okno na Wszechświat

18 i 25 października 2017/ October 18th and 25th, 2017

Michał JÓŹWIKOWSKI (IMPAN), Mikołaj 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 października 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

Frédéric BARBARESCO (Thales Air Systems)

Symplectic Geometry of Heat based on Souriau Lie Groups Thermodynamics and Koszul Hessian Information Geometry.

We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. The Souriau-Fisher metric is linked to KKS (Kostant–Kirillov–Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We conclude with Higher order extension of Souriau model based on works of R..S Ingarden and W. Jaworski. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics.

Slides (PDF)

10 maja 2017/ May 10th, 2017

Paweł URBAŃSKI (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 Białymstoku)

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

Mikołaj 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 SMOŁKA (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

Michał JÓŹWIKOWSKI (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. 772–784. 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. 188–213. 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 Światosław Gal (Wrocław 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 odbędzie się/ on November 2nd, 2016 there will be no seminar

26 października 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 października 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 października 2016/ October 12ve, 2016

Katarzyna GRABOWSKA, Paweł URBAŃSKI (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 października 2016/ October 5st, 2016

Katarzyna GRABOWSKA, Paweł URBAŃSKI (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 2015/2016
Semestr letni

1 czerwca 2016/ June 1st, 2016

Stefan RAUCH-WOJCIECHOWSKI (Linköping University)

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

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

25 maja 2016/ May 25th, 2016

Artur GIŻYCKI (MiNI PW)

Representations of groups in relation to representations of transformation groupoids

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

18 maja 2016/ May 18th, 2016

Paweł CIOSMAK (MIMUW)

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

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

11 maja 2016/ May 11th, 2016

Paweł Urbański (KMMF)

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

27 kwietnia 2016/ April 27th, 2016

Paweł Urbański (KMMF)

Kaluza-Klein theory contra geometry of affine values

20 kwietnia 2016/ April 20th, 2016

Andrzej WEBER (MIMUW)

"Motivic" decomposition and C* action

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

13 kwietnia 2016/ April 13th, 2016

Jorge JOVER (Univesidad de Zaragoza))

Dynamics on the Space of States: a Geometrical Description

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

6 kwietnia 2016/ April 6th, 2016

Norbert PONCIN (UniLu)

Towards integration on Z2n supermanifolds

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

30 marca 2016/ March 30th, 2016

Mikołaj ROTKIEWICZ (MIMUW)

On smooth monoid actions on manifolds (II)

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

23 marca 2016/ March 23rd, 2016

Michał JÓŻWIKOWSKI (IMPAN)

On smooth monoid actions on manifolds

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

16 marca 2016/ March 16th, 2016

Luca VITAGLIANO (University of Salerno)

Generalized geometry in odd dimensions

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

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

Giovanni MORENO (IMPAN)

Introduction to BV-BFV theories on manifolds with boundary

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

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

Semestr zimowy

20 stycznia 2016/ January 20th, 2016

Giovanni MORENO (IMPAN)

Finite-dimensional symplectic formalism for higher-order field theories

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

13 stycznia 2016/ January 13th, 2016

Piotr Waluk (KMMF)

The Ricci flow

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

16 grudnia 2015/ December 16th, 2015

Rafał R Suszek (KMMF)

A geometrisation of the (T)QFT functor

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

9 grudnia 2015/ December 9th, 2015

Javier de Lucas (KMMF)

Lie systems and geometric phases

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

2 grudnia 2015/ December 2nd, 2015

Piotr KOPACZ (UJ, Akademia Morska w Gdyni)

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

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

18 listopada 2015/ November 18th, 2015

Michał JÓŹWIKOWSKI (IMPAN)

A contact approach to the sub-Riemannian geodesic problem

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

28 października 2015/ October 28th, 2015

Andrzej DRAGAN (FUW)

Ideal clocks - a convenient fiction

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

21 października 2015/ October 21th, 2015

Giovanni MORENO (IMPAN)

Invariant hypersurfaces in low-dimensional Lagrangian Grassmannians.

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

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

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

14 października 2015/ October 14th, 2015

Mikołaj ROTKIEWICZ (MIMUW)

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

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

7 października 2015/ October 7th, 2015

Mikołaj ROTKIEWICZ (MIMUW)

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

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

Rok akademicki 2014/2015
Semestr letni

3 czerwca 2015/ June 3rd, 2015

Marcin ZAJĄC (KMMF)

Tulczyjew triple on a Lie group

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

27 maja 2015/ May, 27th, 2015

Alfonso Giuseppe Tortorella (Universita di Firenze)

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

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

20 maja 2015/ May, 20th, 2015

Giovanni MORENO (University of Salerno)

Hypersurfaces in Lagrangian Grassmannians and geometric theory of nonlinear PDEs

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

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

13 maja 2015/ May, 13th, 2015

Session cancelled due to the conference "Geometry of Jets and Fields in Będlewo"

6 maja 2015/ May, 6th, 2015

Artur JANDA (Centrum Badań Kosmicznych)

The action principle, electrodynamics and causality

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

29 kwietnia 2015/ April, 29th, 2015

Tomasz MACIĄŻEK (CFT)

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

22 kwietnia 2015/ April, 22nd, 2015

Jerzy LEWANDOWSKI (IFT FUW)

Generally Relativistic classical and quantum particles

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

15 kwietnia 2015/ April, 15th, 2015

Piotr SUŁKOWSKI (IFT FUW)

S-duality and quantum field theory

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

1 kwietnia 2015/ April, 1st, 2015

Bogdan BALCERZAK (Politechnika Łódzka)

Dirac operators on Lie algebroids

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

25 marca 2015/ March, 25th, 2015

Antoni PIERZCHALSKI (Uniwersytet Łódzki)

Generalized gradients

Uogólnione gradienty to nieredukowalne ze względu na działającą grupę składniki pochodnej kowariantnej. Zakodowane są w nich dane o strukturze geometrycznej rozmaitości, na której są badane. Pokazana zostanie ogólna konstrukcja gradientów oraz ich przykłady. Omówione będzie zachowanie na brzegu rozmaitości. Wspomniana także będzie konstrukcja w przypadku struktur ogólniejszych - algebroidów Liego.

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

18 marca 2015/ March, 18th, 2015

Javier de LUCAS (KMMF)

Definition, properties and applications of superdifferential equations

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

11 marca 2015/ March, 11th, 2015

Andrew BRUCE (IMPAN)

On graded bundles in the category of Lie groupoids

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

4 marca 2015/ March, 4th, 2015

Alfonso Giuseppe TORTORELLA (Universita di Firenze)

Deformations of coisotropic submanifolds in abstract Jacobi manifolds

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

25 lutego 2015/ February, 25th, 2015

Tomasz ZAWADZKI (U. Łódzki)

Conformal submersions with totally umbilical fibers

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

Presentation (PDF)

Semestr zimowy

21 stycznia 2015/ January, 21st, 2015

Tomasz TYLEC (CFT PAN)

Quantum logics and generalised probability

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

7 stycznia 2015/ January, 7th, 2015

Katarzyna GRABOWSKA (KMMF)

Dirac algebroids in action

Presentation (PDF)

17 grudnia 2014/ December, 17th, 2014

Witold RESPONDEK (INSA, Rouen)

Linearization of mechanical control systems

Presentation (PDF)

10 grudnia 2014/ December, 10th, 2014

Tatiana SHULMAN (IMPAN)

Completely positive maps and zero-error quantum information theory

In quantum information theory for mathematical description of quantum channels one uses the notion of completely positive maps on matrix spaces. In the first part of the talk we will discuss general theory of completely positive maps and their connection with quantum channels. In the second part of the talk I will focus on some mathematical problems arising in zero-error quantum information theory, namely I will talk on various zero-error capacities of quantum channels and superactivation effect. This is ajoint work with M. Shirokov.

3 grudnia 2014/ December, 3rd, 2014

Michał JÓŹWIKOWSKI (IMPAN)

On symmetries of differential equations

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

19 listopada 2014/ November, 19th, 2014

Piotr STACHURA (SGGW)

Short and biased introduction to groupoids

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

Presentation (PDF)

12 listopada 2014/ November, 12th, 2014

Bronisław JAKUBCZYK (IMPAN)

REGULAR CONTROL SYSTEMS AND LAGRANGE GEOMETRY (part II)

5 listopada 2014/ November, 5th, 2014

Bronisław JAKUBCZYK (IMPAN)

REGULAR CONTROL SYSTEMS AND LAGRANGE GEOMETRY

29 października 2014/ October, 29th, 2014

Mikołaj ROTKIEWICZ (IMPAN)

On applications and generalizations of the canonical kappa-map

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

22 października 2014/ October, 22nd, 2014

Paweł URBAŃSKI (KMMF)

SYMPLECTIC GROUPOID ACTIONS AND REDUCTIONS (part II)

15 października 2014/ October, 15th, 2014

Paweł URBAŃSKI (KMMF)

SYMPLECTIC GROUPOID ACTIONS AND REDUCTIONS (part I)

8 października 2014/ October, 8th, 2014

Andrew BRUCE (IMPAN)

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

1 października 2014/ October, 1st, 2014

Andrew BRUCE (IMPAN)

WEIGHTED ALGEBROIDS: THEORY AND OUTLOOK FOR APPLICATIONS

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




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' RÓWNAŃ CZĄSTKOWYCH POCHODZĄCYCH Z PROBLEMÓW WARIACYJNYCH

Jest kilka sposobów ,,dyskretyzacji'' równań Einsteina używanych do obliczeń w tzw. numerical gravity. Wszystkie te sposoby gwałcą ,,in flagranti'' strukturę geometryczną tych równań. Prelegent jest głęboko przekonany, że trudności z obliczeniami numerycznymi w ogólnej teorii względności stąd właśnie wynikają i proponuje nową metodę jej dyskretyzacji, wynikającą z analizy tej właśnie 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

Michał JÓŻWIKOWSKI

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

Mikołaj ROTKIEWICZ

CAŁKOWANIE NA SUPER-ROZMAITOŚCIACH


18 grudnia 2013/ December, 18th, 2013

Jose MOURAO (U Lisboa)

QUANTIZATION IN KÄHLER AND REAL POLARIZATIONS


11 grudnia 2013/December, 11th, 2013

Katarzyna GRABOWSKA i Paweł URBAŃSKI

WIĘZY 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 października 2013/ October, 30th, 2013

Michał JÓŹWIKOWSKI

O WIĘZACH WAKONOMICZNYCH I NIEHOLONOMICZNYCH


23 października 2013/ October, 23rd , 2013

Adam SAWICKI (CFT)

N-PARTICLE QUANTUM STATISTICS ON GRAPHS


16 października 2013/ October, 16th , 2013

Katarzyna GRABOWSKA

COŚ O RÓWNANIU HAMILTONA-JACOBIEGO


9 października 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 października 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

Paweł URBAŃSKI

ZASADY ZACHOWANIA BEZ SYMETRII


22 maja 2013/ May, 22th , 2013

Prof. José 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

Mikołaj ROTKIEWICZ i Michał JÓŹWIKOWSKI (IFT)

VARIATIONAL CALCULUS AND HIGHER LIE ALGEBROIDS


17 kwietnia 2013/ April, 17th , 2013

Rafał DEMKOWICZ-DOBRZAŃSKI (IFT)

METROLOGIA KWANTOWA, CZYLI CO?

Co geometria kanałów kwantowych może powiedzieć o interferometrach fal grawitacyjnych i zegarach atomowych.


10 kwietnia 2013/ April, 10th , 2013

Piotr SOŁTAN

ALGEBRY HOPFA


27 marca 2013/ March, 27th , 2013

Marek KUŚ

LOKALNA RÓWNOWAŻNOŚĆ STANÓW KWANTOWYCH


20 marca 2013/ March, 20th , 2013

Gabriel PIETRZKOWSKI

ROZWIĄZANIE EXPLICITE UKŁADU 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

Michał JÓŹWIKOWSKI

TEORIA DŻETÓW w/g EHRESMANNA


20 lutego 2013/ February, 20th , 2013

Mikołaj ROTKIEWICZ

FUNKTORY WEILA W SUPERGEOMETRII


23 stycznia 2013/ January, 23rd , 2013

MIkołaj ROTKIEWICZ

FUNKTORY WEILA W GEOMETRII I SUPERGEOMETRII


16 stycznia 2013/ January, 16th , 2013

Paweł NUROWSKI

LOCAL CONFORMAL INVARIANTS VIA FEFFERMAN-GRAHAM METRICS


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

Piotr FRĄCKIEWICZ

KORELACJE KLASYCZNE, KWANTOWE I SUPERKWANTOWE

Podczas wykładu omówię różnice pomiędzy klasycznymi, kwantowymi i superkwantowymi. W tym celu wykorzystam pojęcie nierówności Bella, w szczególności nierówność 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 NAWIASÓW LIEGO, Z KTÓRYCH JEDEN JEST PÓŁPROSTY


21 listopada 2012 / November, 21st , 2012

Adam BEDNORZ (IFT UW)

NIEINWAZYJNY POMIAR KWANTOWY

Standardowy, rzutowy pomiar kwantowy jest inwazyjny (ingeruje w układ) prowadzi do paradoksów: efektu Zenona i łamania nierówności Bella. Pomiar nieinwazyjny jest przeciwieństwem rzutowego, unika powyższych paradoksów, ale prowadzi do zaskakujących nowych, np. łamania 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

Michał JÓŹWIKOWSKI

GEOMETRIA WIĄZEK NIESKOŃCZONYCH DŻETÓW

Abstrakt: Podczas wykładu omówię podstawowe pojęcia i konstrukcje związane z wiązką nieskończonych dżetów.


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

Michał JÓŹWIKOWSKI

GEOMETRIA WIĄZEK DŻETÓW

Podczas dwóch wykładów przedstawię wprowadzenie do przestrzeni dżetów. W szczególności omówię pojęcie dystrybucji Cartana i scharakteryzuje jej maksymalne całkowe podrozmaitości. Ten wynik posłuży do dowodu tw. Lie-Backlunda charakteryzującego wszystkie symetrie przestrzeni dżetów. Opowiem też o geometrycznym podejściu do cząstkowych równań różniczkowych i o wiązkach nieskończonych dżetów i formalnej całkowalności.


17 października 2012 / October, 17th , 2012

Paweł NUROWSKI

GEOMETRIA TOCZĄCYCH SIĘ POWIERZCHNI


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

Michał JÓŹWIKOWSKI

WIĄZKI DŻETÓW WEDŁUG VINOGRADOWA


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

Bronisław JAKUBCZYK

O REGULARNOŚCI ABNORMALNYCH EKSTREMAL MINIMALIZUJACYCH


23 maja 2012 / May, 23rd 2012

Paweł URBAŃSKI

TRANSFORMACJA LEGENDRE'A UKŁADÓW NIELAGRANŻOWSKICH


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

Mikołaj ROTKIEWICZ (IM UW)

FORMY WIELOSYMPLEKTYCZNE


4 kwietnia 2012 / April, 4th 2012

Łukasz SKOWRONEK (UJ)

ELEMENTY GEOMETRII ALGEBRAICZNEJ W TEORII SPLĄTANIA KWANTOWEGO

Pokażę, jak rozstrzygnąłem, z wykorzystaniem twierdzenia Bezout, kwestię charakteryzacji pewnej rodziny stanów, ważnych z punktu widzenia teorii splątania kwantowego. Są to stany splątane z dodatnią częściową transpozycją, o rzędzie cztery. Okazuje się, że wszystkie je można uzyskać poprzez elementarną, dobrze znaną konstrukcję, połączoną z dowolną lokalną transformacją stanu. Postaram się również, w miarę możliwości, nakreślić inne problemy, dające się traktować metodami geometrii algebraicznej, a będące w kręgu zainteresowań osób zajmujących się teorią splątania kwantowego i pokrewnymi zagadnieniami. W tym wypadku będzie chodziło o rozwiązywanie równań wielomianowych.


28 marca 2012 / March, 28th 2012

Maciej BŁASZAK (UAM)

KWANTYZACJA DEFORMACYJNA MECHANIKI HAMILTONOWSKIEJ

W wykładzie prezentuję w systematyczny sposób alternatywne sformułowanie mechaniki kwantowej, zwane mechaniką kwantową na przestrzeni fazowej lub kwantowaniem deformacyjnym. Rozpatrywana klasa deformacji zawiera jako szczególne przypadki wszystkie znane w literaturze deformacje klasycznej algebry obserwabli. Ponadto zdefiniowana jest przestrzeń zawierająca dopuszczalne stany kwantowe i posiadająca strukturę algebry Hilberta ze względu na odpowiednie *-mnożenie. Podczas wykładu postaram się uzasadnić tezę, iż prezentowany formalizm jest bardziej fundamentalny od standardowej aksjomatycznej mechaniki kwantowej. Standardowa mechanika kwantowa pojawia się w prezentowanym formalizmie jako naturalna reprezentacja mechaniki kwantowej na przestrzeni fazowej. Ta użyteczna i prosta reprezentacja, istniejąca przynajmniej dla sformułowania kanonicznego, wynika z istnienia odpowiedniej klasy transformacji Wignera-Moyal'a spełniających wszystkie własnosci iloczynu tensorowego odpowiednich przestrzeni Hilberta.


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

Michał JÓŹWIKOWSKI

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

GRADED CONTACT STRUCTURES


22 lutego 2012 / February, 22nd 2012

Paweł URBAŃSKI

TRANSFORMACJA LEGENDRE'A W DYNAMICE STRUN


15 lutego 2012 / February, 15th 2012

Alexander VINOGRADOV (University of Salerno)

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 KRYŃSKI

HYDRODYNAMIKA WEDŁUG ARNOLDA


4 stycznia 2012 / January, 4th 2012

Witold RESPONDEK (INSA de Rouen)

MECHANICZNE UKŁADY STEROWANIA I ICH WSPÓŁZMIENNIKI


21 grudnia 2011 / December, 21st 2011

Witold RESPONDEK (INSA de Rouen)

PŁASKOŚĆ NIEDOOKREŚLONYCH RÓWNAŃ RÓŻNICZKOWYCH POCHODZĄCYCH 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 KUŚ

DETEKCJA STANÓW QUASIKLASYCZNYCH


30 listopada 2011 / November, 30 th 2011

Janusz GRABOWSKI

TRÓJKI 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

Paweł STRZELECKI (MIM UW)

CAŁKOWA KRZYWIZNA MENGERA DLA KRZYWYCH, POWIERZCHNI I INNYCH ZBIORÓW: EFEKTY WYGŁADZANIA I BRAKU SAMOPRZECIĘĆ

Krzywizna Mengera trójki punktów to odwrotność promienia okręgu, przechodzącego przez te punkty, a całkowa krzywizna Mengera krzywej prostowalnej to całka z p-tej potęgi krzywizny Mengera względem wszystkich trójek punktów (całkujemy względem długości łuku). Okazuje się, że dla pewnych wartości p skończoność tak zdefiniowanej krzywizny gwarantuje, że krzywa ma hoelderowsko ciągły wektor styczny (co wynika z geometrycznych namiastek nierównosci Sobolewa-Morreya) i jest pozbawiona samoprzecięć. Podobne funkcjonały i wyniki można wskazać w ogólnym przypadku, dla podrozmaitosci przestrzeni euklidesowej; postaram się opowiedzieć o tym w sposób nietechniczny i przystępny.


2 listopada 2011 / November, 2 nd 2011

  

Jacek JEZIERSKI 
  
CYK TENSORY W KLASYCZNEJ TEORII POLA

26 października 2011 / October, 26 th 2011

  

Jerzy KIJOWSKI 
  
O GEOMETRYCZNEJ KWANTYZACJI

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

  

Paweł URBAŃSKI 
  
WIĘZY HAMILTONOWSKIE w/g DIRACA I INNYCH

5 października 2011 / October, 5th 2011

  

Katarzyna GRABOWSKA 
  
CO POCZĄĆ Z LAGRANŻJANEM  DRUGIEGO RZĘDU?

1 czerwca 2011 / June, 1st 2011  

Wiesław SASIN (PW)   

WŁASNOŚCI GEOMETRYCZNE GRUPOIDU TRANSFORMACJI NAD CZASOPRZESTRZENIĄ 

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

Maciej ŁUKASIK  (KMMF)   

RACHUNEK WARIACYJNY BEZ PARAMETRYZACJI

11 maja 2011 / May, 11th 2011  

Bronisław 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)   

GRADED GEOMETRY AND POISSON REDUCTION  
 

13 kwietnia 2011 / April, 13th 2011  

Barbara OPOZDA  (UJ)   

PEWNE PODROZMAITOŚCI MINIMALNE I ICH PRZESTRZENIE MODULI  
 

6 kwietnia 2011 /  April , 30th 2011  

Mikołaj ROTKIEWICZ (IM UW)   

WIĄZKI  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)   

 OGRANICZONOŚĆ GRUPY DYFEOMORFIZMÓW
Streszczenie: Mówimy, że grupa jest ograniczona, jeżeli dowolna metryka bi-niezmiennicza na niej jest ograniczona.
Następnie, grupa jest jednostajnie doskonała, jeżeli jest ona doskonała i jej długość komutatorowa jest ograniczona.
Celem referatu jest przedstawienie niedawno uzyskanych wyników dotyczących ograniczoności i jednostajnej doskonałości grup dyfeomorfizmów na rozmaitości.
W przeciwieństwie do klasycznych twierdzeń o prostocie i doskonałości grup dyfeomorfizmów, twierdzenia o ograniczoności zależą od topologii rozmaitości.
Wskazujemy też, że za pomocą długości komutatorowej symplektomorfizmów interpretuje się pewne niezmienniki topologii symplektycznej.  
 

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

Tadeusz MIŁOSZ (UWMCS)   

 ZASADA  MAKSIMUM PONTRIAGINA

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

Adam DOLIWA (UWM)   

 O GEOMETRII RÓWNANIA HIROTY  

 

Abstrakt:Różnicowa wersja równania Kadomtseva-Petviashvili, zaproponowana 30 lat  temu przez Ryogo Hirotę, jest jednym z ważniejszych
równań współczesnej  fizyki matematycznej (patrz np. niedawny artykuł przegladowy "T-systems  and Y-systems in integrable systems",
Atsuo Kuniba, Tomoki Nakanishi,  Junji Suzuki, arXiv:1010.1344). W moim wykładzie chciałbym przedstawić  relatywnie prosta geometrycznš
interpretację układu równań Hiroty.
Odwzorowania Desarguesa sieci pierwiastkowych typu A w przestrzenie  rzutowe (nad pierscieniem z dzieleniem) scharakteryzowane sš pewnym
prostym warunkiem geometrycznym, prowadzšcym do problemu liniowego dla  równania Hiroty. Równanie to jest w tej interpretacji zakodowane w
konfiguracji Veblena, a jego wielowymiarowa konsystencja jest równoważna  twierdzeniu Desarguesa. Pierwsza częsć wykładu chciałbym zakończyć
na  przedstawieniu zwiazku symetrii konfiguracji Desarguesa z tzw. równaniem  pięciokšta oraz na omówieniu ważnego przykładu kwantyzacji pewnej
naturalnej w tym kontescie struktury Poissona.    W następnym tygodniu chciałbym skoncentrować się na geometrycznej  interpretacji binarnej
transformacji Darboux dla równania Hiroty i  omówić na tym tle przykład rozwiazań wielosolitonowych. Na koniec  poruszę zwišzek odwzorowań
Desarguesa z alternatywnym (lecz równoważnym)  ujęciem geometrycznym dyskretnych układów całkowalnych poprzez tzw.  sieci czworoboków płaskich.
Wykłady oparte będa na moich niedawnych artykułach (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 DEREZIŃSKI   

FUNKCJE  TYPU HIPERGEOMETRYCZNEGO I ICH SYMETRIE

Abstrakt: Funkcje typu hipergeometrycznego obejmują większość  najważniejszych funkcji specjalnych (m.in funkcję Bessela, konfluentną,  hipergeometryczną
i klasyczne wielomiany ortogonalne). Spełniają one wiele  intrygujących tożsamości. Ich własności można zrozumieć i uporządkować przy  użyciu grup i algebr Liego.  


5 stycznia 2011 /  January, 5th 2011  

Paweł URBAŃSKI  

FORMALIZM SKINNERA-RUSKA W MECHANICE


15 grudnia 2010 /  December, 15th 2010  

Michał JÓŹWIKOWSKI (IMPAN)  

ZASADA WARIACYJNA DLA DYNAMIKI HAMILTONOWSKIEJ (Z WIĘZAMI)

 

Opis: Zaproponuję alternatywną metodę wyliczania równań Hamiltona dla  liniowej struktury Poissona na przestrzeni fazowej.
Trajektorie fazowe  otrzymamy jako ekstremale naturalnego działania na sumie prostej przestrzeni  fazowej i konfiguracyjnej.
Proponowane podejście daje się także zastosować  dla układów z więzami w przestrzeni konfiguracyjnej.  

 


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

Gabriel PIETRZKOWSKI (IMPAN)  

 

ALGEBRA  W  MODELU  STANDARDOWYM 

 


24 listopada 2010 /  November, 24th2010  

Rafał 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 rôle 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  

Paweł NUROWSKI (IFT UW)  

 

3-WYMIAROWE STRUKTURY CR I RÓWNANIA RÓŻNICZKOWE
ZWYCZAJNE 2-GO RZĘDU
 

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 października 2010 /  October, 27th 2010  

Włodzimierz M. TULCZYJEW

 

MECHANICS OF INCOHERENT MATTER  (DUST, PLASMA)  

20 października 2010 /  October, 20th 2010  

Paweł URBAŃSKI  

TEORIA KALUZY-KLEINA


13 października 2010 /  October, 13th 2010  

Katarzyna GRABOWSKA  

DIRAC STRUCTURES AND GEOMETRY OF NONHOLONOMIC CONSTRAINTS

Pdf


6 października 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.

Pdf

 


21 kwietnia 2010 /  April, 21st  2010  

Paweł NUROWSKI

 

STRUKTURY PARA-CR I RÓWNANIA RÓŻNICZKOWE


14 kwietnia 2010 /  April, 14th  2010  

Włodzimierz M. TULCZYJEW

 

A VARIATIONAL FRAMEWORK FOR ANALYTICAL MECHANICS AND FIELD THEORY


31 marca 2010 /  March, 31st  2010  

Mikołaj ROTKIEWICZ (IM UW)

PEWNE KONSTRUKCJE SUPER-ROZMAITOŚCI

 

W literaturze spotykamy wiele różnych koncepcji super-rozmaitości. Pomijając szczegóły, są dwa kompletnie różne podejścia. Pierwsze, "snopowe", polegające na zastąpieniu algebry funkcji gładkich na

tradycyjnej rozmaitości $Z/2Z$-gradowaną algebrą przez dołączenie elementów antykomutujących. Drugie polega na zdefiniowaniu super-rozmaitości jako zbioru z pewną dodatkową strukturą

(atlasem), bardzo podobnie jak w tradycyjnej geometrii. Okazuje się, że obie koncepcje mogą być w dużej mierze stosowane zamiennie (rezultat z "Two approaches to supermanifolds", M.

Batchelor), a nieco dokładniej kategorie gradowanych rozmaitości (wprowadzonych przez B.Kostanta) i DeWitta $H^infty$ super-rozmaitości są równoważne.

 

Na seminarium omówię powyższy wynik oraz podam kilka kanonicznych przykładów super-rozmaitości.


24 marca 2010 /  March, 24th  2010  

Włodzimierz JELONEK (PK)

 

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

Podajemy klasyfikacje zwartych, jednospójnych rozmaitości kaehlerowskich (M,g,J) z quasi-stałą homolorficzną krzywizną sekcyjną przy założeniu dim(M)>4.

Są to rozmaitości, których holomorficzna krzywizna sekcyjna R(X,JX,JX,X), gdzie X jest jednostkowym wektorem stycznym do M,  zależy tylko od punktu x i długości rzutu ortogonalnego wektora X na

ustaloną, zespoloną liniową wiązka D zawartą w TM. Pokazujemy, ze jeśli D nie jest trywialna, to M jest holomorficzna wiazką  nad przestrzenią rzutową CP^n z włóknem CP^1.

Wiązka D okazuje się całkowalną dystrybucją styczną do włókien CP^1 wiązki.

Metoda dowodu polega na wykazaniu istnienia na M pola Killinga ze specjalnym potencjałem Kaehlera-Ricciego, a następnie na skorzystaniu z twierdzenia  Derdzinskiego-Mashlera, klasyfikującego zwarte rozmaitości kaehlerowskie

dopuszczające pola Killinga ze specjalnym potencjałem Kaehlera-Ricciego, i pokazaniu, ze jedynymi takimi rozmaitościami z quasi-stałą krzywizną  holomorficzną są wiązki nad przestrzenia rzutowa CP^n z włóknem CP^1,

które są projektywizacją potęgi wiązki tautologicznej nad CP^n. Podajemy rownież zastosowania pól Killinga ze specjalnym potencjałem  Kaehlera-Ricciego przy częściowej klasyfikacji zwartych, hermitowskich

rozmaitości Graya.

 


17 marca 2010 /  March, 17th  2010  

Adam SAWICKI (CFT PAN)

 

CZY MOŻNA „USŁYSZEĆ” KSZTAŁT GRAFU KWANTOWEGO?

W 1966 roku Marc Kac zadał słynne pytanie „ Can one hear the shape of a drum?”. Od tego czasu czyniono próby zarówno rekonstrukcji kształtu na podstawie widma jak i szukano metody konstrukcji obiektów izospektralnych. W końcu lat 90 Smilansky przeformułował pytanie Kaca w kontekście tzw. grafów kwantowych. W czasie seminarium zaprezentuje metodę konstrukcji izospektralnych grafów kwantowych opartą na teorii reprezentacji grup oraz przedstawię kilka prostych przykładów jej zastosowania.


10 marca 2010 /  March, 10th  2010  

Marcin MARCINIAK (UG)

 

ODWZOROWANIA DODATNIE NA ALGEBRACH MACIERZOWYCH
 

Celem wykładu jest omówienie kilku problemów dotyczących klasyfikacji odwzorowań dodatnich. W pierwszej kolejności pokażemy, że owa klasyfikacja może być zredukowana do opisu punktów  eksponowanych stożka odwzorowań dodatnich, następnie opiszemy klasę znanych punktów eksponowanych. Ponadto omówimy własności  dodatnich odwzorowań ekstremalnych związane z zachowaniem rzędu. Na koniec sformułujemy częściowe rozwiązania problemów Robertsona i Osaki dotyczących szczególnych własności dodatnich odwzorowań ekstremalnych.


3 marca 2010 /  March, 3rd  2010  

Andrzej DRAGAN (IFT UW)

 

WYZNANIE  WIARY  W  TEORIĘ  KWANTOWĄ  WRAZ  Z  MATERIAŁEM  DOWODOWYM


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

Andrzej OKOŁÓW (IFT UW)

 

TEORIE  YANGA-MILLSA
 

Teorie Yanga-Millsa odgrywają znaczącą rolę we współczesnej fizyce będąc podstawą Modelu Standardowego cząstek elementarnych. Z punktu  widzenia geometrii są one związane z wiązkami głównymi: przestrzeń

konfiguracyjna tych teorii jest przestrzenią koneksji na wiązce głównej. W  trakcie referatu zostanie przedstawiona konstrukcja tzw. działania czyli  funkcjonału na przestrzeni koneksji określającego dynamikę teorii, 

następnie z działania zostaną wyprowadzone równania Yanga-Millsa. Jako  przykład zastosowania tych teorii zostanie zaprezentowany tzw. mechanizm  Higgsa na przykładzie modelu oddziaływań elektrosłabych.


20  stycznia 2010 /  January  20th 2010  

Paweł WALCZAK  (UŁ)

 

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 PŁASKOŚĆ


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 KRYŃSKI (IM PAN)

TKANINY  KRONECKERA  I  RÓWNANIA  RÓŻNICZKOWE


2 grudnia 2009 /  December 2nd 2009  

Piotr  MORMUL  (IM UW)

 

GOURSAT MONSTER I JEGO WACHLARZ  OSOBLIWYCH KRZYWYCH LEGENDROWSKICH


25 listopada 2009 /  November 25th 2009  

Paweł URBAŃSKI

REDUKCJA  ROUTHA GEOMETRYCZNIE


18 listopada 2009 /  November 18th 2009  

Bronisław  JAKUBCZYK  (IM PAN)

KRZYWIZNY  PÓL  WEKTOROWYCH  NA  ROZMAITOŚCI  Z  DYSTRYBUCJĄ


4 listopada 2009 /  November 4th 2009  

Marek  KUŚ  (CFT)
 
KORELACJE  KWANTOWE  CZĄSTEK  IDENTYCZNYCH

28 października 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 października 2009 /  October 21st 2009  

Mark GOTAY (PIMS, Vancouver)
 

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


7 października 2009 /  October 7th 2009  

Katarzyna GRABOWSKA (KMMF)

O PEWNYM MODELU GEOMETRYCZNYM KLASYCZNEJ TEORII POLA

 


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

Maciej  ŁUKASIK (KMMF)

RACHUNEK  WARIACYJNY  NIEZALEŻNY OD PARAMETRYZACJI

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

Michał  JOŹWIKOWSKI (IM PAN)

O UOGÓLNIENIU ZASADY MAKSIMUM PONTRIAGINA
Układy sterowania na algebroidach Liego pojawiają się w naturalny sposób wwyniku redukcji klasycznych układów sterowania przez grupę symetrii. Na wykładzie pokażę jak sformułować Zasadę Maksimum Pontriagina – podstawowe twierdzenie teorii optymalnego sterowania - dla takich układów. Okazuje się, że za ZMP odpowiada nieco ogólniejsza struktura geometryczna – algebroidu prawie-Liego. Kluczowe okazuje się zrozumienie uogólnionego pojęcia homotopii krzywych. Współautorem omawianych wyników jest prof. Janusz Grabowski.

29 kwietnia 2009 / April, 29th 2009   

Prof. Witold  RESPONDEK (INSA de ROUEN)

KIEDY UKŁAD STEROWANIA JEST MECHANICZNY? cz. II

22 kwietnia 2009 / April, 22nd 2009   

Prof. Witold  RESPONDEK (INSA de ROUEN)

KIEDY UKŁAD STEROWANIA JEST MECHANICZNY? cz. I

1 kwietnia 2009 / April, 1st 2009   

Andriy PANASYUK

CZĘŚCIOWE  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 SYMPLEKTOMORFIZMÓW PRZESTRZENI JEDNORODNYCH

11 marca 2009 / March, 11th 2009   

Wojciech KRYŃSKI (IM PAN)

RÓWNOWAŻNOŚĆ  DYSTRYBUCJI  KORZĘDU  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   

Mikołaj ROTKIEWICZ (IM UW)

 STRUKTURY PODWÓJNE W GEOMETRII WARTOŚCI AFINICZNYCH

 

21 stycznia 2009 / January, 21st 2009   

Piotr WOJDYŁŁO (IM PAN)

REPREZENTACJE W PŁASZCZYŹNIE CZAS-CZĘSTOTLIWOŚĆ,

ICH ZASTOSOWANIA DO  PRZETWARZANIA SYGNAŁÓW


14 stycznia 2009 / January, 14th 2009   

Andrzej KOSSAKOWSKI  (UMK)

O  STRUKTURZE  GENERATORÓW  NIEMARKOWSKICH RÓWNAŃ  EWOLUCJI


7 stycznia 2009 / January, 7th 2009   

Maciej  DUNAJSKI  (Cambridge)

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  BOBIEŃSKI  (IM UW)

POLA  YANGA-MILLSA  I  RÓWNANIA  YANGA-MILLSA


26  listopada 2008 /  November, 26th 2008   

Gabriel PIETRZKOWSKI  (IM PAN)

CAŁKOWA REPREZENTACJA  STANÓW SEPAROWALNYCH

Kwantowym stanem separowalnym na iloczynie tensorowym przestrzeni Hilberta $H=K\otimes L$, nazywa się dodatni operator Hermitowski o śladzie jednostkowym, który można przedstawić jako dodatnią kombinację rzutów na  wektory proste w H (tj. wektory postaci $v \otimes w \in H$). Celem wystąpienia jest pokazanie formuły całkowej na stany separowalne i przeanalizowanie jej własności.


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 FILTRÓW KALMANA


5 listopada 2008 /  November, 5th 2008   

David MARTIN de DIEGO  (CSIC, Madrid)

DISCRETE MECHANICS: FROM DIFFERENTIAL GEOMETRY TO NUMERICAL INTEGRATION


29 października 2008 /  October, 29th 2008   

Paweł URBAŃSKI  (KMMF UW)

O RÓWNANIU HAMILTONA-JACOBIEGO I METODZIE JACOBIEGO


22 października 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 października 2008 /  October, 15th 2008   

Jacek  JEZIERSKI  (KMMF UW)

O  ISTNIENIU  METRYK  KUNDTA  I  ZDEGENEROWANYCH  HORYZONTÓW  KILLINGA


8 października 2008 /  October, 8th 2008   

Katarzyna  GRABOWSKA  (KMMF UW)

NOWY  SCHEMAT  GEOMETRYCZNY  DLA  RACHUNKU  WARIACYJNEGO  Z  WIĘZAMI


 


 

More about the seminar.

Dane z lat poprzednich można obejrzeć tutaj

Please send your comments to urbanski@fuw.edu.pl.