Benvinguts a la pàgina web del Seminari Teen de Geometria

Geometric structures and Lie systems: some examples

3 de des. 2015, 1:27 publicada per Xavier Gràcia

Speaker: Silvia Vilariño (Centro Universitario de la Defensa, Zaragoza)
Date and place: Thursday 10 December 2015, 12:30; FME (UPC), room 102
Title: Geometric structures and Lie systems: some examples

Abstract:
A Lie system is a system of first-order differential equations whose general solution can be expressed as a function, the superposition rule, of a generic finite collection of particular solutions and a set of constants. Thus an important questions is: how can you get a superposition rule for a given system?
In searching for the answer to this question, an important role is played by Lie systems admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields (with respect to some geometric structure). This particular type of Lie systems gives new methods to derive superposition rules.
In this seminar we comment on some of these methods using several geometric structures: Poisson, Dirac, k-symplectic... We present some examples and in particular, in one of them, we compare the method of deriving superposition rules using k-symplectic structures and other geometric structures.

References

  1. Cariñena, J. F.; de Lucas, J.;
    Lie systems: theory, generalisations, and applications.
    Dissertationes Math. (Rozprawy Mat.) 479 (2011), 162 pp.
  2. Cariñena, J. F.; Grabowski, J.; de Lucas, J.; Sardón, C.;
    "Dirac-Lie systems and Schwarzian equations".
    J. Differential Equations 257 (2014) 2303–2340.
  3. de Lucas, J.; Sardón, C.
    "On Lie systems and Kummer-Schwarz equations".
    J. Math. Phys. 54 (2013) 033505, 21 pp.
  4. de Lucas, Javier; Tobolski, Mariusz; Vilariño, Silvia
    "A new application of k-symplectic Lie systems".
    Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550071, 6 pp.
  5. de Lucas, J.; Vilariño, S.
    "k-symplectic Lie systems: theory and applications".
    J. Differential Equations 258 (2015) 2221–2255.

Lie–Hamilton systems: properties, classification, and applications

25 de nov. 2015, 2:40 publicada per Xavier Gràcia   [ actualitzat el 30 de nov. 2015, 0:45 ]

Speaker: Cristina Sardón (Dept. of Fundamental Physics, Universidad de Salamanca)
Date and place: Thursday 3 December 2015, 12:30-14:00h; FME (UPC), room 102
Title: Lie–Hamilton systems: properties, classification, and applications

Abstract:
This seminar is concerned with the definition and analysis of a new class of Lie systems on Poisson manifolds that enjoy rich geometric properties: the so called Lie–Hamilton systems.
We start by their definition, statement of their geometric fundamentals and focus on their wide applicability by exposing a number of examples found throughout the Mathematics, Physics, Biology literature and other scientific disciplines. We devise a new method to derive their superposition rules: the so called coalgebra method, and illustrate its usefulness with concrete examples. In particular, we will study Lie–Hamilton systems on the plane and will provide their complete classification on the plane. This classification will give rise to a number of diffeomorphisms between well known systems in the Physics literature that will serve us as a canonical classification of certain phenomena occurring in nature. Solutions for such phenomena will be obtained in form of superposition rules.
To finish, we will introduce the notion of structure-Lie systems, being the latter those Lie systems admitting different geometric backgrounds whose theory, applications and properties can be retrieved from the point of view of another geometric structure.

References

  1. A. Ballesteros, A. Blasco, J.F. Cariñena, F.J. Herranz, J. de Lucas, C. Sardón, "Lie-Hamilton systems on the plane: properties, classification and applications", J. Differential Equations 258, 2873–2907 (2015).
  2. A. Ballesteros, J.F. Cariñena, F.J. Herranz, J. de Lucas, C. Sardón, "From constants of motion to superposition rules of Lie–Hamilton systems", J. Phys. A: Math. Theor. 46, 285203 (2013).
  3. A. Blasco, F.J. Herranz, J. de Lucas, C. Sardón, "Lie–Hamilton systems on the plane: applications and superposition rules", J. Phys. A: Math. Theor. 48, 345202 (2015).
  4. J.F. Cariñena, J. de Lucas, C. Sardón, "Lie–Hamilton systems: theory and applications", Int. J. Geom. Methods Mod. Phys. 10, 0912982 (2013).

Introduction to Lie systems (minicourse)

29 d’oct. 2015, 3:36 publicada per Xavier Gràcia   [ actualitzat el 26 de nov. 2015, 7:26 ]

Speaker: Xavier Gràcia (UPC, Barcelona)
Date and place: Thursdays 5, 12, 19, 26 November 2015, 12:30-14:00h; FME (UPC), room 102
Title: Introduction to Lie systems

Abstract:
Lie systems are differential equations whose general solution can be described in terms of some particular solutions by means of a superposition rule; well-known examples of this are linear equations and Riccati equation. Associated with a Lie system there is a finite-dimensional Lie algebra of vector fields, the so-called Vessiot-Guldberg Lie algebra. The main theorem of the theory is Lie-Scheffers theorem, which establishes the relationship between superposition rules and these Lie algebras.
The aim of this course is to present the basic notions of the theory together with some interesting examples. The main source is the review by Cariñena and Lucas (2011).

Summary
1. Introduction
2. Historical note
3. Fundamentals
4. Superposition rules and Lie systems
5. Diagonal prolongations of vector fields
6. Proof of Lie-Scheffers theorem
7. Lie systems and Lie groups
8. Second-order Lie systems

References

  1. J.F. Cariñena, J. de Lucas; "Lie systems: theory, generalisations, and applications"; Dissertationes Mathematicae 479 (2011).
  2. J.F. Cariñena, J. Grabowski, G. Marmo; Lie-Scheffers systems: a geometric approach; Bibliopolis, Naples (2000).
  3. J.F. Cariñena, J. Grabowski, G. Marmo; "Superposition rules, Lie theorem, and partial differential equations"; Rep. Math. Phys. 60 (2007) 237-258.
  4. B. Joseph; Sistemes de Lie i l'equació de Riccati matricial; master thesis, UPC (2009).

Charge quantisation without magnetic poles

22 d’oct. 2015, 0:54 publicada per Xavier Gràcia   [ actualitzat el 22 d’oct. 2015, 1:07 ]

Speaker: Romero Solha (Dept. Mathematics, Universidade Federal de Minas Gerais, Belo Horizonte)

Date and place: Monday 26 October 2015, 12:30 h; FME (UPC), room 103

Title: Charge quantisation without magnetic poles

Abstract

In this talk I intend to present how [1] provides a theoretical explanation for the quantisation of electric charges, an open problem since Millikan's oil drop experiment in 1909. This explanation is based solely on Maxwell's theory, it recasts electromagnetic theory under the language of complex line bundles; therefore, neither magnetic poles nor quantum mechanics are invoked. Essentially, the existence of magnetic poles was the only theoretical explanation for charge quantisation (e.g. Dirac's magnetic pole), and there is no experimental data supporting their existence ---on the contrary, they have never been observed.

References

  1. Romero Solha, "Charge quantisation without magnetic poles: a topological approach to electromagnetism", J. Geom. Phys. (2015), to appear; arXiv:1409.6716.

On the definition and regularity of higher-order Hamiltonian functions

7 d’oct. 2015, 4:32 publicada per Xavier Gràcia   [ actualitzat el 7 d’oct. 2015, 4:52 ]

Speaker: Pere Daniel Prieto Martínez (UPC)

Date and place: Thursday 15 October 2015, 12:30 h; FME (UPC), room 102

Title: On the definition and regularity of higher-order Hamiltonian functions

Abstract

The aim of this talk is to generalize two results from [1], namely Proposition 3.6.7 and Theorem 3.6.9, to higher-order (autonomous) dynamical systems. The attempt to generalize these results gives rise to another more interesting problem: what is a higher-order Hamiltonian function? And what does it mean for such a function to be "regular"? These concepts are clearly defined for first-order Lagrangian or Hamiltonian functions, and also for a higher-order Lagrangian function. In this talk we propose a definition for such concepts, always taking into account the particular case of Hamiltonian functions associated to a (regular) Lagrangian system.

References

  1. R. Abraham and J.E. Marsden, Foundations of mechanics (2nd ed.), Addison-Wesley, New York, 1978.
  2. M. de León and P.R. Rodrigues, Generalized classical mechanics and field theory, North-Holland Math. Studies vol. 112, Elsevier, Amsterdam, 1985.

Berezin-Toeplitz quantization and K3 surfaces

30 d’abr. 2015, 8:14 publicada per Xavier Gràcia   [ actualitzat el 30 d’abr. 2015, 8:18 ]

Speaker: Hector Castejon-Diaz (Université du Luxemburg)

Place and date: Tuesday 12 May 2015, 15:30 h; FME (UPC), room S05

Title: Berezin-Toeplitz quantization and K3 surfaces

Abstract:

I will do a small introduction to Berezin-Toeplitz quantization. Afterwards, I will introduce K3 surfaces, which (under some regularity conditions) admit three different BT quantizations, and I will explain some relations between them. As we will see, the associated quantum line bundles are not isomorphic, but their spaces of holomorphic sections have the same dimension and there exist some canonical relations between them.

Bibliography

  • Martin Schlichenmaier, "Berezin-Toeplitz quantization for compact Kähler manifolds. A review of results", 2010.
  • Daniel Huybrechts, "Lectures on K3 surfaces" (online notes).

Lie systems and geometric structures

18 de jul. 2014, 7:55 publicada per Xavier Gràcia

Speaker: Silvia Vilariño (Centro Universitario de la Defensa, Zaragoza)

Place and date: Friday 18 July 2014, 12:00 h; FME (UPC), room 101

Title: Lie systems and geometric structures

Abstract:

A Lie system is a system of first--order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot--Guldberg Lie algebra.
Some attention has lately been paid to Lie systems admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields wirh respect to several geometric structures. In this talk we review some particular types of Lie systems and we suggest the definition of a particular class of Lie systems, the k-symplectic Lie systems, admitting a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields with respect to the presymplectic forms of a k-symplectic structure. We devise new k-symplectic geometric methods to study their superposition rules, time independent constants of motion and general properties. Our results are illustrated by examples of physical and mathematical interest.

El efecto túnel como responsable de la evaporación de agujeros negros

29 d’abr. 2014, 5:51 publicada per Xavier Gràcia   [ actualitzat el 30 d’abr. 2014, 4:48 ]

Speaker: Ramon Torres (Dept. of Applied Phyisics, UPC)

Place and date: Thursday 8 May 2014, 12:00 h; FME (UPC), room 101

Title: El efecto túnel como responsable de la evaporación de agujeros negros

Abstract:

En 1975 Stephen Hawking realizó un descubrimiento que cogió a la comunidad científica por sorpresa: los agujeros negros no sólo atrapan todo lo que atraviesa su horizonte (como afirma la teoría de la relatividad general) sino que, además, son capaces de emitir radiación (una vez la teoría cuántica de campos es tomada en consideración). Como consecuencia se abría la puerta a la posible eventual completa evaporación de los agujeros negros. El propio Hawking explicaría físicamente este fenómeno como debido a la creación de pares de partículas virtuales en las que una de ellas consigue escapar del horizonte gracias al efecto túnel. Sólo en los últimos años se han desarrollado métodos que permiten el tratamiento matemático de la radiación de Hawking de acuerdo a esta interpretación física y teniendo en cuenta la conservación de la energía en el proceso. Esto ha permitido el estudio más detallado de la evaporación de agujeros negros incluso añadiendo correcciones cuánticas a la solución clásica de Schwarzschild.

References:
  • Torres, R.; Fayos, F.; Lorente-Espín, O.
    "Evaporation of (quantum) black holes and energy conservation"
    Physics Letters B 720 (2013) 198-204
  • Torres, R.
    "On the interior of (quantum) black holes"
    Physics Letters B 724 (2013) 338-345

Estructures simplèctiques exòtiques a R^4

17 de febr. 2014, 5:12 publicada per Xavier Gràcia   [ actualitzat el 3 de març 2014, 6:06 ]

Xerrada a càrrec de: Roger Casals (ICMAT-CSIC)

Data i lloc: divendres 21 de febrer del 2014, 12:00 h; FME (UPC), aula 102

Títol: Estructures simplèctiques exòtiques a R4

Resum:
Una estructura simplèctica ω a R2n es diu exòtica si no existeix cap embedding simplèctic dins de (R2n0), amb ω0 l’estructura simplèctica estàndard de R2n. L’existència d’aquestes estructures en R2n és coneguda des del treball de M. Gromov (1985) sobre una conjectura d’Arnol’d. Les tècniques per obtenir–ne són diverses però no constructives. En particular, llevat d’un exemple ad hoc de L. Bates i G. Peschke (1990), no s’han trobat fórmules explícites senzilles. En aquesta xerrada explicarem com construir estructures simplèctiques exòtiques en R4 utilitzant distribucions de plans en R3. Això permet una descripció simple i alhora geomètrica de formes simplèctiques exòtiques.

Referències:

  • Roger Casals, "Overtwisted disks and exotic symplectic structures", preprint arXiv:1402.7099 (2014).

Normal forms and adiabatic invariants (course by J.A. Vallejo)

20 de set. 2013, 3:31 publicada per Xavier Gràcia   [ actualitzat el 10 d’oct. 2013, 3:28 ]

Speaker: José Antonio Vallejo (Universidad Autónoma de San Luis Potosí, Mexico)
Dates: 7-9 October 2013
   Mon 7 and Tue 8: 12:30-14:00h; Wed 9: 11:00-13:00
Place: FME (UPC), room 100
Title: Normal forms and adiabatic invariants

Contents:

  1. Secular perturbation theory
  2. The method of Lindstedt-Poincaré-VonZippel
  3. Canonical perturbation theory through Lie-Deprit series
  4. Solving the homological equations: averaging techniques and normal forms
  5. Geometric interpretation. Perturbations of vector fields
  6. Global normal forms for systems admitting a circle action
  7. Computation of adiabatic invariants in slow-fast systems

References

  1. A. Deprit: "Canonical transformations depending on a small parameter". Celestial Mechanics 1 (1969) 12-30.
  2. J. Sanders, F. Verhulst and J. Murdock: Averaging Methods in Nonlinear Dynamical Systems. Springer, New York (2007).
  3. M. Avendaño-Camacho, J.A. Vallejo and Yu. Vorobjev: "A simple global representation for second-order normal forms of Hamiltonian systems relative to periodic flows". J. Phys. A: Math. Theor. 46 (2013) 395201. arXiv:1301.3244.
  4. M. Avendaño-Camacho, J.A. Vallejo and Yu. Vorobjev: "Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems". J. Math. Phys. 54 (2013) 082704. arXiv:1305.3974.

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