# 6th International Young Researchers Workshop on Geometry, Mechanics and Control

Coimbra, Portugal, January 10-13, 2012

### Introduction:

The 6th International Young Researchers Workshop on Geometry, Mechanics and Control will be the sixth in a series of workshops that have previously taken place in Madrid (20062007), Barcelona (2008), Ghent (2009) and La Laguna (2010). Its goal is to bring together young researchers working in geometric mechanics and control theory and to offer a platform to present the results of their research to an international audience. The main idea of this conference is to offer 3 courses of 4 hours which serve as an introduction to different topics related to geometric structures in mechanics and control theory. The courses will be at a PhD level, so the young researchers would, at the end of the Workshop, access to the recent literature on the corresponding topics. Along with the courses, there will be short talks (30 minutes) by the participants willing to present their results. Moreover, there will be a poster session including a 5-minute presentation of each poster. Attendance is, of course, open to anyone, but in particular young researchers (PhD-students, recent PhD's) are allowed to deliver a talk or present a poster at this meeting. The following is a non-exhaustive list of topics that can be covered during the workshop:

• Geometry: (multi)symplectic geometry, Poisson and Jacobi manifolds, Lie groups, Lie algebroids and Lie groupoids.
• Mechanics: Lagrangian and Hamiltonian systems, non-holonomic mechanics, calculus of variations, mechanical systems with symmetry, conservation laws and reduction, classical field theories, geometric integration of mechanical systems, geometric quantization.
• Control: optimal control theory, control of mechanical systems, geometric control.

Furthermore, this year, on Monday January 9, there will be a special "Geometry & Physics day"

### Important dates:

• REGISTRATION: no later than December 10, 2011
• COMMUNICATION/POSTER submittal: no later than November 10, 2011

### Courses:

There will be three 4-hour courses:

• Tomoki Ohsawa (University of California, San Diego): Hamilton-Jacobi theory
I will talk about Hamilton-Jacobi theory with a special emphasis on its relationship with geometric mechanics. Starting from the basic concepts in Hamilton-Jacobi theory, I will talk about the geometry associated with Hamilton-Jacobi theory and its applications. The first lecture gives some basic concepts such as Jacobi's solution as well as applications to integrable Hamiltonian systems. The second lecture is on the geometric interpretation of the Hamilton-Jacobi equation. The third lecture will focus the relationship with optimal control theory (and, time permitting, quantum mechanics). The fourth lecture gives the contact-geometric formulation of the Hamilton-Jacobi equation.
• Mario Sigalotti (INRIA, France): Quantum control
In the first part of the course (~ 1 hour) we will present a panorama of some classes of control systems which are studied in the framework of quantum control. In particular, we will introduce the main features characterizing such systems: closed vs open, finite- vs infinite-dimensional, (bi)linear vs nonlinear. We will give some basic properties of the class of systems on which the rest of the course will focus on: closed, finite-dimensional, bilinear systems. The central part of the course (~ 2 hours) will be devoted to controllability and motion planning. We will present some general controllability results of finite-dimensional systems (Krener's and Chow's theorem, compatibility of Poisson stable vector fields) and apply them to the quantum setting. We will provide some explicit conditions for controllability and discuss some algorithms for motion planning. In the final part of the course (~1 hour) we will discuss some specific optimal control problems (mainly, time-minimal problems for 2- or 3-level quantum systems).
• Ivan Struchiner (Utrecht University, The Netherlands): Lie algebroids and groupoids
These lectures are intended to be an introduction to Lie groupoids and Lie algebroids. The first lecture will be entirely dedicated to Lie groupoids. After defining the main concepts (such as Lie groupoids and Morita equivalence of Lie groupoids), several examples will be presented. The infinitesimal counterpart of a Lie groupoid, its Lie algebroid, will be introduced in the second lecture. I will show how to construct the Lie algebroid of a Lie groupoid, provide several examples and then discuss some aspects of the Lie theory for Lie groupoids/algebroids. The third lecture will be dedicated to cohomology theories naturally associated to Lie algebroids (and Lie groupoids). I will discuss the de Rham and deformation cohomology of a Lie algebroid. If time permits, I will discuss also the differentiable cohomology of a Lie groupoid. In the last lecture I will present a geometric proof of the linearization theorem for proper Lie groupoids, which is based on a deformation argument (in the spirit of Moser's trick).

The confirmed speakers for the Geometry & Physics day are:

• Camille Laurent-Gengoux (Paul Verlaine University-Metz, France): Global action-angle variables
Local action angle variable always exist for non-commutative integrable
system. Classes of cohomology, and, surprisingly, some non-linear equation,
obstruct their global existence.
• Juan Carlos Marrero (University of La Laguna, Spain): Kinematic reduction and the Hamilton-Jacobi equation
In this talk, I will present a close relationship between the classical Hamilton-
Jacobi theory and the kinematic reduction of control systems by decoupling vector ﬁelds.
The geometric interpretation of this relationship relies on new mathematical techniques for
mechanics deﬁned on a skew-symmetric algebroid. This geometric structure allows us to
describe in a simpliﬁed way the mechanics of nonholonomic systems with both control and
external forces.
• Edith Padrón (University of La Laguna, Spain): The contact cover of the spherical bundle of a fiberwise linear Poisson manifold
If $A$ is a Lie algebroid over $M$ of rank $n$ and $g$ is a
bundle metric on $A$, we may consider the corresponding spherical
bundle $S^{n-1}(A^*)\to M$ of the linear Poisson manifold $A^*$. In
this talk, we will describe a contact structure on the prolongation of
$A$ with respect to this spherical bundle. This fact allows to prove
that $S^{n-1}(A^*)$ is a Jacobi manifold. We will describe these
structures in several examples of Lie algebroids.
• Mathieu Stienón (Penn State University, USA): From Atiyah classes to homotopy Leibniz algebras
• Wlodzimierz M. Tulczyjew (University of Warsaw, Poland): Hamilton principal form and holonomic constraints.
Hamilton principal form is defined in general and
presented inparticular to mechanical systems with holonomic
constraints.
• Ping Xu (Pen State University, USA): Glanon groupoids
We introduce the notion of Glanon groupoids, which are Lie groupoids
equipped with multiplicative generalized complex structures.
It combines symplectic groupoids, holomorphic Lie groupoids and
holomorphic Poisson groupoids into a unified framework. Their
infinitesimal, Glanon Lie algebroids are studied. We prove that
there is a bijection between Glanon Lie algebroids and source-simply
connected and source-connected Glanon groupoids. As a consequence,
we recover various integration theorem and obtain the integration
theorem for holomorphic Poisson groupoids.
This is a joint work with Jotz, and Stienon .
• Marco Zambon (ICMAT and Universidad Autónoma de Madrid, Spain): Singular foliations and holonomy groupoids
I will recall the notion of holonomy for a regular foliation on
a smooth manifold, and the related notion of holonomy groupoid. I
will then introduce singular foliations (the simplest example is given
by a vector field, possibly with singularities) and sketch a
construction of an associated holonomy groupoid by
Skandalis-Androulidakis. The latter is in general not smooth. I will
show that, however, under suitable conditions the restriction to leaves
is smooth. The content of this talk is the first part of a project to
find a normal form theorem for singular foliations, and is joint work
with Androulidakis.

Short talks

• Letizia Brunetti: An Osserman-Type Condition In Lorentz Context
• Thiago Drummond: Right-invariant derivations on Lie groups
• Eduardo García-Toraño: Aspects of Routh Reduction
• Elisa Lavinia Guzmán Alonso: An involution map for the iterated jet of a surjective morphism of Lie algebroids
• Madeleine Jotz: Some multiplicative geometric structures on Lie groupoids and the corresponding infinitesimal data
• Derek Krepski: Verlinde-type formulas for SO(3) through geometric quantization
• Ignazio Lacirasella: On some reduced Poisson manifolds foliated by coadjoint orbits
• Hassan Najafi Alishah: A geometric approach to the KAM theorem
• Romero Solha: Almost homotopy operators in geometric quantisation of integrable systems
• Pietro Tortella: $\Lambda$-modules and holomorphic Lie algebroid connections
• Silvia Vilariño Fernández: k-symplectic manifolds and their Lagrangian submanifolds
• Goedele Waeyaert: Driven cofactor systems and Hamilton-Jacobi separability

### Schedule:

 Geometry and Physics Day Young Researchers Workshop on Geometry, Mechanics and Control Monday 9 Tuesday 10 Wednesday 11 Thursday 12 Friday 13 9:00-9:30 Registration 9:00-9:30 Registration 9:00-10:00 Struchiner Sigalotti Sigalotti 9:30-10:15 Tulczyjew 9:30-10:30 Struchiner 10:00-10:30 Sigalotti Lacirasella Ohsawa 10:15-11:00 Marrero 10:30-11:00 Drummond 10:30-11:00 Jotz 11:00-11:30 Coffee 11:00-11:30 Coffee 11:00-11:30 Coffee Coffee Coffee 11:30-12:15 Padrón 11:30-12:00 Waeyaert 11:30-12:00 Guzmán Najafi Alishah Solha 12:30-14:30 Lunch 12:00-13:00 Sigalotti 12:00-13:00 Ohsawa Ohsawa Struchiner 14:30-15:15 Zambon 13:00-15:00 LUNCH 15:15-16:00 Stienon 15:00-16:00 Ohsawa 15:00-16:00 Excursion Struchiner 16:00-16:30 Coffee 16:00-16:30 Vilariño 16:00-16:30 Tortella 16:30-17:15 Laurent-Gengoux 16:30-17:00 Coffee 16:30-17:00 Krepski 17:15-18:00 Xu 17:00-17:30 Brunetti 17:00-18:00 Coffee & Posters 17:30-18:00 García-Toraño

### Registration:

Please register before December 10 (November 10 if you wish to present a short talk) by sending an e-mail to yrwgmc [at] mat [dot] uc [dot] pt with the following information:

• First Name:
• Family Name:
• Institution:
• Country:
• Date of birth:

* I wish to give a 30-min talk: Yes/No. If yes, give a title and abstract
* I wish to present a poster: Yes/No. If yes, give a title and abstract

### Lodging reservation:

We have made provisional reservations of a certain number of rooms in the following Hotels which offer special rates for conference participants.
Therefore we suggest to do your reservation mentioning YrwGMC conference before 30 November 2011.

RESIDENCIAIS (Guest Houses)

Residencial Botânico *** Rua dos Combatentes da Grande Guerra, 15. 3030-207 Coimbra
Phone: (+351) 239 714 824 Fax: (+351) 239 405 124
e-mail: residencialbotanicomail [dot] telepac [dot] pt
Special rates: single room: 36,00 EUR per night double room: 46,00EUR per night (Prices include buffet breakfast and taxes). Rooms with private bathroom, central heating, air conditioning, satellite T.V., telephone, internet wireless. 10 minutes walk - through the Botanical Garden - to the University.

Residencial Alentejana *** Rua Dr. Henrique Sêco, 1 3000-145 Coimbra
Phone: (+351) 239 825 903 / 293 825 924 Fax: (+351) 239 842 478
e-mail: residencialalentejanahotmail [dot] com
Website: www.residencialalentejana.com
Rooms with private bathroom, central heating, T.V., telephone, internet wireless. Information on special rates must be requested directly through the hotel. 15 minutes walk to the University.

Pensão Residencial Antunes**
Rua Castro Matoso, nº8 - 3000-174 Coimbra
Phone: +351 239 854 720 Fax: +351 239 854 729
E-Mail: residencialantunesmail [dot] pt
Website: http://residencialantunes.planetaclix.pt/
Special rates: single room: 36,00 EUR per night double room: 46,00 per night (Prices include a small breakfast and taxes). Rooms with private bathroom, central heating, T.V., telephone, internet wireless.
5 minutes walk to the University.

---

HOTELS

Hotel Tryp Coimbra **** Avenida Armando Gonçalves, Lote 20 3000-059 Coimbra
Phone: (+351) 239 480 800 Fax: (+351) 239 484 300
e-mail:tryp [dot] coimbrasolmeliaportugal [dot] com
Website: www.trypcoimbra.com/
Special Rates: single room: 59,00 EUR per night double room: 69,00 EUR per night (Prices include buffet breakfast and taxes) Rooms with private bathroom with hairdryer, central heating, air conditioning, cable
T.V., telephone, internet wireless. 15 minutes walking distance to the University.

Hotel Ibis** Coimbra
Telephone: +351 239 852 130 Fax: +351 239 852 140.
Email:h1672accor [dot] com
Website: www.ibishotel.com/gb/hotel-1672-ibis-coimbra/
Single/Double: 49.00€ (35 € temporary special offer on the website) Prices per room/night without breakfast and without wireless internet in the room. Breakfast – 6 €/day. Online Booking.
15 minutes walking distance (uphill) to the University.

Hotel Confort Inn Almedina ***
Website: www.almedinacoimbra.com
Telephone: +351 239 855 500 Fax: +351 239 829 906.
Email: comercialalmedinacoimbra [dot] com
Single/Double – 52.00 € Prices per room/night and include breakfast (buffet) and wireless internet in the room. 20 minutes walking distance (uphill) to the University.

Hotel Astoria ***
Telephone: +351 239 853 020 Fax: +351 239 822 057.
Email: astoriaalmeidahotels [dot] com
Website: www.almeidahotels.com
Single – 72.00 € Double – 92.00 €
Prices per room/night and include breakfast (buffet) (may have better prices on www.booking.com). Internet is available in the hotel reception and main hall. 15 minutes walking distance (uphill) to the University.

Hotel Vila Galé Coimbra ****
Telephone: +351 239 240 000 Fax: +351 239 240 050.
E-mail: Rita Pereira (coimbra [dot] gruposvilagale [dot] pt)
Website: www.vilagale.pt/pages/hoteis/?hotel=23
Rates: Single Standard - 67.00 € Twin Standard APA – 68.00 € Prices per room/night and include breakfast (buffet) and wireless internet in the room (Best prices from www.booking.com). 25-30 minutes walking distance (uphill) to the University.

### Participants:

• Odinette Renée Abib (Univ. de Rouen (France), Univ. de Campinas (Brazil))
Lígia Abrunheiro (ISCAA da Universidade de Aveiro, Portugal)
Paulo Antunes (Universidade de Coimbra, Portugal)
Paula Balseiro (Universidade Federal Fluminense (UFF), Brazil)
Letizia Brunetti (University of Bari, Italy)
Lucía Búa Devesa (Universidade de Santiago de Compostela, Spain)
Henrique Bursztyn (Instituto de Matemática Pura e Aplicada (IMPA), Brazil)
Angelo V. Caldarella (University of Bari, Italy)
Margarida Camarinha (Universidade de Coimbra, Portugal)
Beniamino Cappelletti Montano (University of Cagliari, Italy)
Raquel Caseiro (Universidade de Coimbra, Portugal)
André Codeço Marques (Instituto Politécnico de Viseu, Portugal)
Antonio De Nicola (Universidade de Coimbra, Portugal)
Matias L. del Hoyo (IST - Lisbon, Portugal)
Viviana Alejandra Díaz  (Universidad Nacional del Sur, Argentina)
Thiago Drummond (UFRJ, Brazil)
Maria Lewtchuk Espindola (Universidade Federal da Paraíba, Brazil)
Eduardo García-Toraño (University of Ghent, Belgium)
Irina Mihaela Gheorghiu  (University of Zaragoza, Spain)
Elisa Lavinia Guzmán Alonso (Universidad de La Laguna, Spain)
David Iglesias Ponte (Universidad de La Laguna, Spain)
Fernando Jimenez (ICMAT-(CSIC-UAM-UC3M-UCM), Spain)
Michal Jóźwikowski (Institute of Mathematics, Polish Accademy of Sciences)
Noah Kieserman (Instituto de Matemática Pura e Aplicada (IMPA), Brazil)
Ignazio Lacirasella (University of Bari, Italy)
Camille Laurent-Gengoux (Paul Verlaine University-Metz, France)
Rui Loja Fernandes (IST - Lisbon, Portugal)
Carlos Lopez (Univ. Alcala de Henares, Madrid, Spain)
Juan Carlos Marrero (Universidad de La Laguna, Spain)
Tom Mestdag (Ghent University, Belgium)
Abdelrahim Mousa (University of Porto, Portugal)
Hassan Najafi Alishah (IST - Lisbon, Portugal)
Citlalitl Nava-Gaxiola (University of Manchester, UK)
Joana Nunes da Costa (Universidade de Coimbra, Portugal)
Tomoki Ohsawa (University of California, San Diego, USA)
Alberto Pinto (University of Porto, Portugal)
Sandra Ricardo (UTAD and ISR, Portugal)
Rui Rodrigues (ISEC - Coimbra, Portugal)
Miguel Rodriguez Olmos (Technical University of Catalonia (UPC), Spain)
Patricia Santos (Instituto Superior de Engenharia de Coimbra, Portugal)
Mario Sigalotti (INRIA, France)
Fátima Silvia Leite (Universidade de Coimbra, Portugal)
Daniele Sepe (IST - Lisbon, Portugal)
Romero Solha (Universitat Politècnica de Catalunya, Spain)
Diana Sosa (Universidad de La Laguna, Spain)
Mathieu Stienon (Penn State University, USA)
Ivan Struchiner (Utrecht University, The Netherlands)
Joana Teles (Universidade de Coimbra, Portugal)
Pietro Tortella (Université Lille 1, France)
Wlodzimierz Tulczyjew (University of Warsaw, Poland)
Joris Vankerschaver (University of California, San Diego, USA)
Miguel Vaquero (ICMAT, Spain)
Silvia Vilariño Fernández (Centro Universitario de la Defensa. Academia Mil., Spain)
Martin Vuk (University of Ljubljana, Slovenija)
Goedele Waeyaert (University of Ghent, Belgium)
Ivan Yudin (Universidade de Coimbra, Portugal)
Ping Xu (Penn State University, USA)
Carlos Zapata Carratalá (University of Valencia, Spain / Imperial College, UK)

### Committees:

Organizing Committee

• Paulo Antunes (University of Coimbra, Portugal)
• Raquel Caseiro (University of Coimbra, Portugal)
• Antonio de Nicola (University of Coimbra, Portugal)
• David Iglesias (Universidad de La Laguna, Spain)
• Joana Nunes da Costa (University of Coimbra, Portugal)
• Joana Teles (University of Coimbra, Portugal)

Scientific Committee

• Paula Balseiro (Universidad Federal Fluminense, Brazil)
• María Barbero (Queen's University, Canada)
• Cédric M. Campos (Technische Universität München, Germany)
• Sebastián Ferraro (Universidad Nacional del Sur, Argentina)
• David Iglesias (Universidad de La Laguna, Spain)
• Marin Kobilarov (Caltech, USA)
• Tom Mestdag (Ghent University, Belgium)
• Miguel Rodríguez-Olmos (Universidad Politécnica de Catalunya, Spain)

### How to get here:

You will likely fly either to the Lisbon (Lisboa in portuguese) or the Oporto (Porto) airport. Both cities have good train and bus connections to Coimbra.
From Lisbon

• By train: you can take a taxi from the arrivals area of the airport to the Lisboa Oriente railway station (about 10 minutes and cost around € 10) or take the following line buses from the airport
• Aeroshuttle bus Carris 96 (http://www.carris.pt/en/bus/96/ascendente/), available everyday, every 30 minutes between 7am and 11pm. The ticket (3,50 euros) can be purchased onboard and also in the Tourism Office at arrivals area.
• Bus Carris #44 to Oriente (http://www.carris.pt/en/bus/44/ascendente/), the right direction one arrives on the opposite side of the street. The ticket (1.5 euros) may be purchased from the driver as you get on the bus.

Go to the upper ticketing level at the station and buy a ticket, preferably for an Intercidades or Alfa Pendular train to Coimbra-B station (costs between € 19 and € 22.50 in 2nd class, check www.cp.pt). Trains depart from Lisbon-Oriente station to Coimbra approximately every hour until 21:30, with a travel time of at most two hours. Get off at Coimbra-B station and take a taxi to the hotel (approximately 8 €).

• By bus: Bus connections between Lisbon and Oporto to Coimbra are very regular. Bus travel is usually cheaper than train, but quite slower. For further information (timetable and fares), please visit the Rede Expressos Bus Service (www.rede-expressos.pt). You can also buy your bus ticket online. If you arrive at Lisbon airport and you want to travel by bus, you should take a taxi (around 25 €) to the the main bus station, Rede Expressos terminal, which is located in Sete Rios. Price from Lisbon to Coimbra should about 14 €, single fare.

From Oporto

• By train: From the airport, which lies in the north of the town, you can go directly to the Campanhã train station by underground. This will take approximately 40 minutes and cost about 2 €. If you prefer, you can take a taxi to the main railway station of Oporto, Porto-Campanhã (this should cost around 25 €). Trains depart from the station hourly with a journey time of about one hour for Alfa Pendular and InterCidades trains. Rail fares cost between € 13 and € 16.50 (check www.cp.pt). Get off the train when you arrive at Coimbra B (the main railway station) and take a taxi to the hotel (approximately 8 €).
• By bus: If you arrive at Porto airport and you want to come by bus, you must take a taxi to the bus station, Rodoviaria Nacional (taxi fare around 20 €). Bus price from Porto to Coimbra is around 11 €, single fare. For further information (timetable and fares), please visit the Rede Expressos Bus Service (www.rede-expressos.pt).

Venue The Department of Mathematics is located in the main campus of the University of Coimbra (known as Pólo I). For further information please visit:

### Supported by:

 Ministerio de Ciencia e Innovación Ingenio Mathematica Geometry, Mechanics and Control Network Center for MathematicsUniversity of Coimbra