Riccardo Benedetti's research activity

Main research themes: Until the 90s: mostly real (semi) algebraic geometry and the real algebraic models for smooth or polyhedral structures (main collaborators: A. Tognoli, M. Dedo', J-J. Risler, M. Shiota, F. Acquistapace, F. Broglia). Later: mostly the geometry and topology of low dimensional manifolds; classical and quantum 3-dimensional hyperbolic geometry; classical and quantum 3-dimensional gravity (main collaborators: C. Petronio, S. Baseilhac, F. Bonsante, A. Marin, R. Silhol, R. Frigerio, E. Guadagnini, J-M. Gambaudo).

Current research topics::

"Volumes conjectures": analytic families of quantum hyperbolic invariants and their asymptotic when N tends to infinity.
Fine study of the phase ambiguity and of the symmetrization procedure for the quantum hyperbolic invariants (and other quantum state sum invariants based on some 6j-symbols theory); relations with different notions of "structured triangulations" (branched, taut, veering,..) considered up to the corresponding ``transit equivalence''. Relations between quantum hyperbolic invariants and the finite dimensional representation theory of the quantum Teichmuller space. Relations between quantum hyperbolic and quantum Teichmuller TQFT.

I have directed the PhD theses of Daniele Alessandrini, Silvia Benvenuti, Francesco Bonsante, Francois Costantino, Rosa Gini, Domenico Luminati, Carlo Petronio who are all active researchers in mathematics.

Some of my papers

Connectivity results for surface branched triangulations ,
Rend. Istit. Mat. Univ. Trieste Volume 52 (2020), 1–34 qui
Lectures on Differential Topology
Book, preliminary version arXiv:1907.10297
IDEAL TRIANGULATIONS OF 3-MANIFOLDS UP TO DECORATED TRANSIT EQUIVALENCES ,

Mathematische Nachrichten, Vol. 294, 2, February 2021, 236-264
arXiv:1903.12540
(with P. Lisca) Framing 3-manifolds with bare hands ,
arXiv:1806.04991
Published in L’ENSEIGNEMENT MATHÉMATIQUE (LEM), Vol. 64, 3/4, 2018, 395-413
(with S. Baseilhac) On the quantum Teichm\"uller invariants of fibred cusped 3-manifolds ,
final version ready to be published in
arXiv:1704.05667
Geometriae Dedicata, 197(1), 1-32 , 2018
DOI 10.1007/s10711-017-0315-0
The paper is available electronically : http://link.springer.com/article/10.1007/s10711-017-0315-0
(with C. Mantegazza) La congettura di Poincare' e il flusso di Ricci ,
Matematica, Cultura e Societa', Rivista dell'Unione Matematica Italiana, Serie I, Vol. 2, N. 3, Dicembre 2017, 245-289; qui
(with C. Mantegazza, editors) Ricci flow and geometric applications ,
Lectures Notes in Mathematics 2166, CIME Foundation Subseries, (2016)
(with S. Baseilhac) Non ambiguous structures on 3-manifolds and quantum symmetry defects ,
Quantum Topology, Volume 8, Issue 4, 2017, pp. 749–846, DOI: 10.4171/QT/101;
final version ready to be published in arXiv:1506.01174v2
(with S. Baseilhac) Analytic families of quantum hyperbolic invariants ,
Algebraic & Geometric Topology 15 (2015) 1983 - 2063.
here
(with C. Petronio) Spin structures on 3-manifolds via arbitrary triangulations,
Algebraic and Geometric Topology 14 (2014), 1005-1054 here
(with R. Frigerio) Levels of knotting of spatial handlebodies,
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 365, Number 4, April 2013, Pages 2099 - 2167 S 0002-9947(2012)05707-2
Article electronically published on August 21, 2012 pdf
(with R. Frigerio) Alexander quandle lower bounds for link genera,
Journal of Knot Theory and Its Ramifications Vol.21, No. 8 (2012) 1250076 (46 pages) pdf
(with R. Frigerio and R. Ghiloni) The topology of Helmholtz domains,
arXiv:1001.4418 Expo. Math. (2012), doi:10.1016/j.exmath.2012.09.001 pdf
(with S. Baseilhac) The Kashaev and quantum hyperbolic link invariants ,
Journal of Gokova Geometry Topology, Volume 5 (2011), 31-85 PDF
Journal of GGT
(with F. Bonsante) (2+1) Einstein spacetimes of finite type ,
Chapter in: A. Papadopoulos Ed. Handbook of Teichmuller Theory, Vol II. vol. 13, European Mathematical Society (2009), 533-609 ps - almost final version.
(with F. Bonsante) Canonical Wick Rotations in 3-Dimensional Gravity ,
Memoirs of the A.M.S. 198 (2009), 1-165 (monograph) ps - almost final version.
(with L. Andersson, T. Barbot, F. Bonsante, W.M. Goldman, F. Labourie, K.P. Scannel, J-M. Schlenker) Notes on a paper of Mess ,
Geometriae Dedicata 126 (1), (2007), 47-60 PDF
(with S. Baseilhac) Quantum hyperbolic geometry ,
Algebraic and Geometric Topology 7 (2007), 845-917 PDF
(with J. Bellissard and J-M. Gambaudo) Spaces of tilings, finite telescopic approximations and gap-labelling
Communications in Mathematical Physics 261 (2006), 1-41 PDF
(with S. Baseilhac) Classical and quantum dilogarithmic invariants of flat PSL(2,C)-bundles over 3-manifolds
Geometry & Topology 9 (2005), 493-570 PDF
About a Quantum Field Theory for 3D Gravity
Milan j. math. 72 (2004) 1â€“21 PDF
(with S. Baseilhac) Quantum Hyperbolic invariants of 3-manifolds with PSL(2,C)-characters
Topology 43 (6) (2004), 1373-1423 PDF
(with J-M. Gambaudo) On the dynamics of G-Solenoids. Application to Delone sets
Ergodic Theory & Dynamical Systems 23 (2003), 673-691 PDF
(with E. Guadagnini) Cosmological time in (2+1)-gravity
Nuclear Physics B 613 (2001), 330-352 PDF file
(with G. Amendola, F. Costantino, C. Petronio) Branched spines of 3-manifolds and Reidemeister torsion of Euler structures
Rend. Istit. Mat. Univ. Trieste Suppl. 1 Vol. XXXII, 1â€“33 (2001) PDF
(with C. Petronio) Reidemeister-Turaev torsion of 3-dimensional Euler Structure with simple boundary tangency and pseudo-Legendrian knots
Manuscripta Mathematica 106 (2001), 13-61 PDF
(with C. Petronio) Combed 3-manifolds with concave boundary, framed links, and pseudo-Legendrian links
Journal of Knot Theory and its Ramifications 1,10 (2001), 1-35 ps - almost final version.
(with C. Petronio) Branched Spines and Contact Structures on 3-manifolds
Ann. Mat. Pura Appl. (4) 178 (2000), 81â€“102 PDF
(with E. Guadagnini) Geometric cone surfaces and (2+1)-gravity coupled to particles
Nuclear Physics B 588 (2000), 436-450 PDF
(with E. Guadagnini) Classical Teichmuller theory and (2+1) gravity
Physics Letters, Section B 441 (1998), 60-68 PDF
(with M. Shiota) On real algebraic links in S3
Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (3) (1998) 585â€“609 PDF
(with C. Petronio) Branched standard spines of 3-manifolds
Lecture Notes in Math. 1653, Springer (1997) (monograph) qui
(with C. Petronio) On Roberts' proof of the Turaev-Walker theorem
Journal of Knot Theory and its Ramifications 5 (1996), 427-439
(with R. Silhol) Spin and Pin- structures, immersed and embedded surfaces and a result of Segre on real cubic surfaces
Topology 34 (3) (1995), 651-678 PDF
(with C. Petronio) A Finite graphic calculus for 3-manifolds
Manuscripta Mathematica 88 (1995), 219-310 PDF
(with A. Marin) Dechirures de varietes de dimension trois et la conjecture de Nash de rationalite' en dimension trois
Comment. Math. Helv. 67 (4) (1992), 514-545 PDF
(with C. Petronio) Lectures on hyperbolic geometry
Universitext. Springer-Verlag, Berlin, (1992) (book) qui
(with M. Shiota) Finiteness of semialgebraic types of polynomial functions
Math. Z. 208 (4) (1991), 589-596 PDF
(with F. Loeser and J-J. Risler) Bounding the number of connected components of a real algebraic set
Discrete Comput. Geom. 6 (3) (1991), 191-209 PDF
(with J-J. Risler) Real algebraic and semialgebraic sets
Hermann, Paris (1990) (book)
(with F. Acquistapace e F. Broglia) Effectiveness-noneffectiveness in semialgebraic and PL geometry
Invent. Math. 102 (1) (1990), 141-156 PDF
On 3-manifolds up to blowing-up
in "Differential analysis and differential topological geometry (Kyoto, 1990)", Kyoto University Research Information Repository, Surikaisekikenkyusho Kokyuroku No. 725 (1990), 1â€“15 PDF
(with M. Dedo') A false inequality which is true for a "concrete observer"
Rendiconti Seminario Facolta' Scienze Universita' Cagliari Vol. 60 I, (1990) 23-31
(with M. Dedo') Global inequalities for curves and surfaces in three-space
Annali di Matematica Pura ed Applicata (IV), Vol. CLV (1989), 213-241 PDF
(with M. Dedo') A geometric inequality for the total curvature of plane curves
Geometriae Dedicata 22 (1987) 105-115 PDF
On a resolution theorem for homology cycles of a real algebraic variety
Boll. Un. Mat. Ital. A (6)4 (3) (1985), 459-466
(with M. Dedo') Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism
Compositio Math. 53 (2) (1984), 143-151 PDF
(with M. Dedo') Serching around H^alg_*(-)
Rocky Mountain J. Math. Volume 14, Number 4 (1984), 939-940 PDF
(with P. Cragnolini) On simultaneous diagonalization of one Hermitian form and one symmetric form
Linear Algebra and its Application 57 (1984), 215-226 PDF
(with P. Cragnolini) Versal families of matrices with respect to unitary conjugation
Adv. in Math. 54 (3) (1984), 314-335 PDF
(with M. Dedo') A cycle is the fundamental class of an Euler space
Proc. Amer. Math. Soc. 87 (1) (1983), 169-174 PDF
(with A. Tognoli) Remarks and counterexamples in the theory of real algebraic vector bundles and cycles
In: Lecture Notes in Math, Springer 959 (1982), 198-211
(with M. Dedo') The topology of two-dimensional real algebraic varieties
Ann. Mat. Pura Appl. 127 (4) (1981), 141-171 PDF
(with A. Tognoli) Theoremes d'approximation en geometrie algebrique reelle. Semin. sur la geometrie algebrique relle
Publ. Math. Univ. Paris VII, 9 (1980), 123-145 .
(with A. Tognoli) On real algebraic vector bundles
Bull. Sci. Math. 104 (2) (1980), 89-112
Density of Morse functions on a complex space
Math. Ann. 229 (2) (1977), 135-139 PDF

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