Riccardo Benedetti

He has been visiting professor or researcher at the following institutions: University Paris-Sud Orsay, University Paris VII, Steklov Institute-Leningrad, Institut Fourier-Grenoble, Nagoya University, ULP-Strasbourg, University of Dijon, EHT Zurich, Institute Mittag-Leffler, University of Toulouse, RIMS-Kyoto, University Paris VI,Columbia University, Newton Institute (Cambridge UK); invited speaker at the second EMC - Budapest 1996; director of the Department of Mathematics of Pisa; associated investigator of the research unit of Pisa for several national research projects; "scientist in charge" for EU individual Marie Curie Fellowships. Since the 2005 he is editor of JKTR. He has been the advisor of several PhD theses, and these persons are all active researchers in mathematics.
Until the 90s he has been mostly concerned with finiteness and effectiveness questions in semi-algebraic geometry or about the real algebraic models for smooth or polyhedral structures. Among the main results of this period : the proof that Sullivan's even local Euler characteristic condition characterizes the topology of (compact) real 2D algebraic sets, the determination of an effective topological obstruction against the possibility of representing homology classes mod(2) of a given compact smooth manifold by means of real algebraic cycles in any non singular affine algebraic model of the manifold, whose existence is assured by a celebrated Nash - Tognoli theorem (with Dedo'); the solution for 3-manifolds of a famous Nash's conjecture about the existence of rational real algebraic models (with Alexis Marin). Since more than 20 years, he has been mostly interested by the geometry and topology of low dimensional manifolds. He has promoted the creation at Pisa of a research team in this field, that has now a well established international reputation. Among the main products of a long collaboration with Carlo Petronio: a monograph that has become a widespread reference on hyperbolic manifolds; the foundation and application of the theory of "branched spines" of 3-manifolds. Branched manifolds technology enters also the construction of QHFT mentioned below, and in the study of the dynamics of G-solenoids and the proof of the so called "Gap Labelling Conjecture" (with J. Bellissard and/or J-M. Gambaudo) about the (non periodic) tilings of R^n, relevant for the quantum modeling of liquid crystals. Another main and current research theme is the classical and quantum hyperbolic geometry (more generally Riemannian or Lorentzian gravity) in 3D. With Baseilhac, he has constructed and is still developing the so called "Quantum Hyperbolic Field Theory". QHFT provides a consistent non commutative counterpart of the 3D gauge theory with group PSL(2,C) and complex action "Chern-Simons + i Volume", initiated by Dupont and Sah and culminated in W. Neumann's work on the generalized Bloch group. Fundamental examples of QHFT interactions are the one among the ends of tame hyperbolic 3-manifolds. With F. Bonsante he has constructed a general theory of "canonical Wick rotations" in 3D gravity. In particular canonical Wick rotations are able to convert such ends into globally hyperbolic AdS space-times. Very recently he has turned to the knotting of handlebodies in the 3- sphere. With Frigerio he has introduced several "levels of knotting" and studied the relations between such levels by means of many different topological geometric or algebraic tools. By applying some of these tools (quandle coloring) they have also obtained new lower bounds for link genera.