In general, my scientific interests are mainly concentrated around design and analysis of dynamics on discrete spaces. The model, at the heart of the subject, is a graph, appearing in different forms, from abstract graphs to more geometric ones as metric-measure spaces. Also, the dynamics on such spaces cover a vast area of applications, from algorithms as discrete dynamics to networks as geometric objects or microscopic physical models of real materials. These are the main motivations for my interest in this intersection of mathematics, computer science, and physics.

I cannot resist recalling the following quote (see [Geometry and the Quantum, Arxiv] by Alain Connes):

Grothendieck’s solution to the problem of treating the continuous and the discrete in a unified manner is the notion of a Topos. It does reconcile the usual idea of a topological space with that of a discrete combinatorial diagram. ...... This new idea is amazing in its simplicity, its connection with logics and the richness of the new class of spaces that it uncovers. In Grothendieck’s own words (see “Récoltes et Semailles”) one can sense his amazement:

Original text in French:

<<Le “principe nouveau” qui restait `a trouver, pour consommer les ´epousailles promises par des f´ees propices, ce n’´etait autre aussi que ce “lit” spacieux qui manquait aux futurs ´epoux, sans que personne jusque-l`a s’en soit seulement aper¸cu. ...

Ce “lit `a deux places” est apparu (comme par un coup de baguette magique...) avec l’id´ee du topos. Cette id´ee englobe, dans une intuition topologique commune, aussi bien les traditionnels espaces (topologiques), incarnant le monde de la grandeur continue, que les (soi-disant) “espaces” (ou “vari´et´es”) des g´eom`etres alg´ebristes abstraits imp´enitents, ainsi que  d’innombrables  autres  types  de  structures,  qui jusque-l`a avaient sembl´e  riv´ees  irr´em´ediablement  au “monde arithm´etique” des agr´egats “discontinus” ou “discrets”.>>