**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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