Gioia M. Vago - Research

Research

Gioia M. VAGO

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My research interests are centred on topological, algebraic, combinatorial and algorithmic aspects of discrete and continuous dynamical systems.

The left-hand side picture, realised by Charles Cordier, refers to the study of the dynamical information carried by the topology of the unstable manifold of certain hyperbolic diffeomorphisms of surfaces. This is a perfect example of a question that could be algorithmically solved by algebraically enriching the combinatorics associated with the bending rules defining such dynamics (see [Va-1], [Va-2], [Va-3], but also [Ba-Jack-Va]).

The right-hand side figure naturally brings to mind another class of dynamical systems, that of Morse flows, and their underlying combinatorics, that of Morse graphs. It refers to another question, representative of my interests, this time in the continuous setting: the study of the Ogasa invariant in the case of dimension three. There again, the initial problem - namely, measuring a certain topological complexity of a manifold, by means of Morse dynamics - has been cleared up by mixing topological methods together with tools from combinatorial graph theory. Moreover, the sharp estimates we obtained algorithmically are intimately related to other algebraic invariants of the underlying manifold. This work, joint with Michel Boileau, is voluminous and the writing is in its final stage (as for the combinatorics of Lyapunov graphs, see also [Be-dR-Va], [Be-dR-Ma-Va-1] and [Be-dR-Ma-Va-2]).

Publications

	[BR-Sa-Va]	K. BEN REJEB, E. SALHI, G.M. VAGO, Nonexpansive homeomorphisms, Topology Appl, Vol 160, Issue15 (2013), 1969-1986 ;
	[J-Sa-Va]	A. JMEL, E. SALHI, G. VAGO, Homeomorphisms of the Sierpinski curve with periodic properties, Dynamical Systems, Vol 28, Issue 2 (2013), 203-213 ;
	[Be-dR-Ma-Va-2]	M.A. BERTOLIM, K.A. DE REZENDE, O. MANZOLI NETO, G.M. VAGO, On the variations of the Betti numbers of regular levels of Morse flows, Topology Appl, Vol 158, Issue 6 (2011), 761-774 ;
	[Bo-Ha-Sa-Va]	C. BONATTI, H. HATTAB, E. SALHI, G.M. VAGO, Hasse diagrams and orbit class spaces, Topology Appl, Vol 158, Issue 6 (2011), 729-740 ;
	[Bo-Cr-Va-Wi]	C. BONATTI, S. CROVISIER, G.M. VAGO, A. WILKINSON, Local density of dieomorphisms with large centralizers Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 6, 925-954 ;
	[Be-dR-Ma-Va-1]	M.A. BERTOLIM, K.A. DE REZENDE, O. MANZOLI NETO, G.M. VAGO, Isolating blocks for Morse flows, Geom. Dedicata , Vol 121, n.1 (2006), 19-41 ;
	[Be-dR-Va]	M.A. BERTOLIM, K.A. DE REZENDE, G.M. VAGO, Minimal Morse flows on compact manifolds, Topology Appl, Vol 153, Issue 18 (2006), 3450-3466 ;
	[Va-3]	G.M. VAGO, Topological and dynamical classication of the unstable manifolds of one-rectangle systems, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1563-1596 ;
	[Va-2]	G.M. VAGO, Conjugate unstable manifolds and their underlying geometrized Markov partitions, Topology Appl, Vol 104 Issue 1-3 (2000), 255-291 ;
	[Ba-Jack-Va]	M. BARGE, J. JACKLITCH, G.M. VAGO, Homeomorphisms of one-dimensional inverse limits with applications to substitution tilings, unstable manifolds and tent maps, Geometry and Topology in Dynamics, Contemp. Math. 246, Amer. Math. Soc, Providence, RI, 1999, 1-15 ;
	[Va-1]	G.M. VAGO, Variétés instables d'ensembles hyperboliques, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) no. 10, 913-918.

Preprints

[Be-Jacq-Va]	M.A. BERTOLIM, A. JACQUEMARD, G.M. VAGO, Integration of a Dirac Comb and the Bernoulli Polynomials ;
[dR-Le-Ma-Va]	K.A. DE REZENDE, G.E. LEDESMA, O. MANZOLI NETO, G.M. VAGO, Circle Valued Morse Functions on Surfaces.

Works

HDR

Habilitation à Diriger des Recherches - Sciences et Techniques, Spécialité Mathématiques, 13th December 2013
Manuscript entitled: Sur des aspects géométriques et combinatoires en systèmes dynamiques - Geometric and combinatorial aspects of dynamical systems
Referees : J.S.W. LAMB (Imperial College), G. LEVITT (U. Caen), J. LOS (CNRS, U. Aix-Marseille) ;
Jury : M.-C. ARNAUD (U. Avignon and IUF), T. BARBOT (U. Avignon and ENS Lyon), C. BONATTI (CNRS, U. Bourgogne), X. BRESSAUD (U. Paul Sabatier Toulouse), B. HASSELBLATT (Tufts U. and Chaire Jean Morlet CIRM, U. Aix-Marseille), G. LEVITT (U. Caen) ;

Ph.D.

Doctorat de l'Université de Bourgogne Spécialité Mathématiques, 2nd December 1998
Thesis entitled: Variétés instables d'ensembles hyperboliques - Unstable manifolds of hyperbolic systems
Appreciation cum laude: mention très honorable et les félicitations du jury
Jury : C. BONATTI (Thesis Supervisor, U. Dijon), P. ARNOUX (Referee, U. Marseille), M. BARGE (Referee, Montana State U.), L. GUILLOU (U. Grenoble), L. PARIS (U. Dijon), B. PERRON (U. Dijon), M. VIANA (I.M.P.A, Rio de Janeiro).

Computer programs

Programs (VBA Applications for Excel and implementation in Maple) which optimise some families of hallosteric models relatively to some experimental data (2004).
The algorithms used by these programs were created by A. Jacquemard and myself to answer a question asked by B. Maume et J.P. Blein, Laboratoire de Phytopharmacie, INRA-Universite de Bourgogne, UMR 692. For more details on the biochemical motivation and results of this collaboration, see Modulation of the Biological Activity of a Tobacco LTP1 by Lipid Complexation N. Buhot, E. Gomes, M.L. Milat, M. Ponchet, D. Marion, J. Lequeu, S. Delrot, P. Coutos-Thevenot, and J.P. Blein, Mol Biol Cell. 2004 November; 15(11) 5047-5052.
Program (implemented in Maple) deciding, via their coding, whether two instable manifolds are conjugate, whether they are homeomorphic (1999).

Students

Post-doc student

Maria Alice Bertolim (Unicamp, Campinas - SP, Brazil).
Financial support: Ministère Français de l'Enseignement Supérieur et de la Recherche (national selection), post-doctoral research stay for young foreign students, one-year contract (2004-2005).
After that, Maria Alice Bertolim was selected as Associated Professor at Unicamp in Campinas, then worked at Regensburg University (Germany) and at Salzburg University (Austria).
Codirection of a Ph.D. Thesis

Abdelrazak Jmel (ISSAT Gabés and University of Sfax - Tunisia).
Thesis coadvised by Ezzeddine Salhi, Professeur à l'Université de Sfax (Tunisie), and myself, defended at the University of Sfax on the 5th of July 2013, entitled Dynamics in one-dimensional spaces.
A. Jmel worked on the relationships between dyniamical properties of periodic type and topological properties of the orbit space associated with such dynamics (1 published paper, 2 preprints submitted for publication).
Financial support: during its thesis, A. Jmel has kept working according to its permanent teaching position as PRAG at ISSAT (Institut Supérieur des Sciences Appliquées et de Technologie) in Gabés.
Stages École Normale Supérieure - Lyon (Mathematics)

The aim of the stages proposed in 2001 to the ENS-Lyon students Nicolas Gourmelon and François Salignat was the study of expansive homeomorphisms. Afterwards, Nicolas Gourmelon prepared his Ph.D. Thesis in Dijon under the supervision of Christian Bonatti, on the subjet Instabilité de la dynamique en l'absence de décomposition dominée. He is currently Maître de Conférences (Associate Professor) at the University of Bordeaux I.
Stage École Normale Supérieure - Lyon (Computer Science)

In 2010, Francesca Fiorenzi (MCF in Computer Science, Univ. Paris XI-Orsay) and myself have proposed to Bruno Guillon (ENS-Lyon student) a subject for a stage which was interesting both from theComputer Science point of view and from the Mathematical one. The aim of the stage was a good understanding of J. Kari's proof of the fact that the reversibility if a cellular automaton is undecidable in dimension 2, while it is decidable in dimension 1. Besides the algorithmic aspect of the subjet, my interest comes from the fact that cellular automata are very reach dicrete dynamical systems and moreover the proofs under analysis are based on a reduction of the initial problem to a problem of tiling of a plane. The stage turned out to be particularly interesting because in its last part we discussed with Bruno Guillon about some open questions that were not dealt with in the reference paper.