Laboratoire de Mathématiques
U.F.R. Sciences Exactes et Naturelles
Moulin de la Housse - BP 1039
51687 REIMS cedex 2
France 

Office: +33 (0)3 26 91 83 93

e-mail: firstname.lastname(at)univ-reims.fr

I am a graduate of Moscow School for Mathematics No 2 (web site in Russian), class of 1981

PhD received in 1989 from Moscow State University

Areas of Interest:

q-deformed rationals and irrationals, a theory developed with Sophie Morier-Genoud.

-projective differential geometry (Schwarzian derivative,...);

-discrete integrable systems (Pentagram map,...);

-algebraic combinatorics (Coxeter friezes, square identities,...);

-cohomology of Lie algebras of vector fields and diffeomorphism groups (Virasoro algebra, modules of differential operators on smooth manifolds,...);

-graded commutative algebras (quaternions, octonions, Lie antialgebras,...).

Together with Christian Duval and Pierre Lecomte we developed equivariant quantization.

See my publications at Google scholar:

Google scholar,

I am on the Editorial Board of:

Journal of Nonlinear Mathematical Physics,

You can download (a preprint version of) my book with Sergei Tabachnikov:

Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups

Cambridge University Press, Cambridge Tracts in Mathematics, 165, 2005.

Videos of some of my talks:

The Pentagram map: a discrete integrable system talk .

Talks on the Hurwitz-Radon function and its applications two , three , four (one is missing).

Frieze patterns and linear difference operators one , two , three .

Frieze patterns, Pentagram map and combinatorics: more open questions than solutions here .

Projective differential geometry old and new here .

"Whas is" papers:

• Well, Papa, can you multiply triplets? (with S. Morier-Genoud) 

• What is... the Schwarzian derivative? (with S. Tabachnikov) 

• Oddtown and Eventown I and Oddtown and Eventown II 

• Hopf fibrations and Hurwitz-Radon numbers (with S. Tabachnikov) 

• Real Clifford algebras and quadratic forms over ${\mathbb F}_2^n$: two old problems become one 

Hurwitz square identities & applications:

• On fibrations with flat fibers. (with S. Tabachnikov) 

• Extremal set theory, cubic forms on ${\mathbb F}_2^n$ and Hurwitz square identities. (with S. Morier-Genoud) 

• Orthogonal Designs and a Cubic Binary Function. (with S. Morier-Genoud) 

• A series of algebras generalizing the octonions and Hurwitz-Radon identity. (with S. Morier-Genoud) 

• New solutions to the Hurwitz problem on square identities. (with A. Lenzhen and S. Morier-Genoud) 

Graded-commutative algebra:

• Higher Trace and Berezinian of Matrices over a Clifford Algebra. (with T. Covolo,N. Poncin) 

• Simple graded commutative algebras. (with S. Morier-Genoud) 

• Lie antialgebras: prémices. 

• Alternated Hochschild Cohomology. (with P. Lecomte) 

• Supertransvectants and symplectic geometry. (with H. Gargoubi) 

Pentagram map, discrete integrable systems:

• Liouville-Arnold integrability of the pentagram map on closed polygons. (with R. Schwartz, S. Tabachnikov) 

• The Pentagram map: a discrete integrable system. (with R. Schwartz, S. Tabachnikov) 

• Quasiperiodic motion for the Pentagram map (with R. Schwartz, S. Tabachnikov) 

Combinatorics:

• Partitions of unity in SL(2,Z), negative continued fractions, and dissections of polygons. 

• Rotundus: triangulations, Chebyshev polynomials, and Pfaffians. (with C. Conley) 

• Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences. (with S. Tabachnikov) 

• On tangent cones of Schubert varieties. (with D. Fuchs, A. Kirillov, S. Morier-Genoud) 

• A step towards cluster superalgebras 

• Linear difference equations, frieze patterns and combinatorial Gale transform. (with S. Morier-Genoud, R. Schwartz and S. Tabachnikov) 

• 2-frieze patterns and the cluster structure of the space of polygons. (with S. Morier-Genoud and S. Tabachnikov) 

• Introducing supersymmetric frieze patterns and linear difference operators. (with S. Morier-Genoud and S. Tabachnikov) 

• $\SL_2(\Z)$-tiling of the torus, Coxeter-Conway friezes and Farey triangulations. (with S. Morier-Genoud and S. Tabachnikov) 

• Coxeter's frieze patterns and discretization of the Virasoro orbits. (with S. Tabachnikov) 

Equivariant quantization and symbol calculus:

• Projectively Equivariant Symbol Calculus. (with P. Lecomte) 

• Conformally equivariant quantization: existence and uniqueness. (with C. Duval and P. Lecomte) 

• Projectively Equivariant Quantization and Symbol Calculus: Noncommutative Hypergeometric Functions. (with C. Duval) 

• Conformally equivariant quantum Hamiltonians. (with C. Duval) 

• Differential operators on supercircle: conformally equivariant quantization and symbol calculus. (with H. Gargoubi and N. Mellouli) 

• Projectively and Conformally Invariant Star-Products. (with C. Duval and A. El Gradechi) 

Modules of of differential operators:

• Riemannian curl in contact geometry. (with S. Bouarroudj) 

• Linear differential operators on contact manifolds. (with C. Conley) 

• Space of Second-Order Linear Differential Operators as a Module over the Lie Algebra of Vector Fields. (with C. Duval) 

• Cohomology of the Vector Fields Lie Algebra and Modules of Differential Operators on a Smooth Manifold. (with P. Lecomte)

• Multi-Parameter Deformations of the Module of Symbols of Differential Operators. (with B. Agrebaoui, F. Ammar, P. Lecomte)

• Modules of Differential Operators on the Real Line. (with H. Gargoubi) 

• Space of Linear Differential Operators on the Real Line as a Module over the Lie Algebra of Vector Fields. (with H. Gargoubi) 

• Three Cocycles on Diff(S1) Generalizing the Schwarzian Derivative. (with S. Bouarroudj) 

• Symmetries of Modules of Differential Operators. (with H. Gargoubi and P. Mathonet) 

• Generalized transvectants-Rankin-Cohen brackets. (with P. Redou) 

• Vector fields in the presence of a contact structure. 

Virasoro, KdV and generalizations:

• Looped Cotangent Virasoro Algebra and Non-Linear Integrable Systems in Dimension 2 + 1. (with C. Roger) 

• Bi-Hamiltonian nature of the equation $u_{tx}=u_{xy} u_y-u_{yy} u_x$. 

• Generalizations of Virasoro group and Virasoro algebra through extensions by modules of tensor-densities on $S^1$. (with C. Roger)

• Extension of the Virasoro and Neveu–Schwarz Algebras and Generalized Sturm–Liouville Operators. (with P. Marcel and C. Roger)

• Classification of third-order linear differential equations and symplectic leaves of the Gel'fand-Dikii bracket. 

• The super Korteweg-de Vries equation as an Euler equation. (with B. Khesin)

• Coadjoint representation of Virasoro-type Lie algebras and differential operators on tensor-densities, DMV Sem., 31, Birkhäuser, Basel, 2001. 

• Symplectic leaves of the Gel'fand-Dikii bracket and homotopy classes of nondegenerated curves. (with B. Khesin)

• Exotic deformation quantization. 

• Deformations of Poisson brackets and extensions of Lie algebras of contact vector fields. (with C. Roger)

• Deforming the Lie Algebra of Vector Fields on $S^1$ Inside the Poisson Algebra on $T^∗ S^1$. (with C. Roger)

• Deforming the Lie algebra of vector fields on $S^1$ inside the Lie algebra of pseudodifferential symbols on $S^1$. (with C. Roger)

• On identities in the Lie algebra of vector fields on the real line. (an old paper in Russian of A.A. Kirillov + a paper in English with A.A. Kirillov and O. Udalova)

Projective Differential Geometry:

• Projective Structures and Contact Forms. 

• Lagrange schwarzian derivative and symplectic Sturm theory. 

• Schwarzian derivative related to modules of differential operators on a locally projective manifold.(with S. Bouarroudj) 

• Selfadjoint differential operators and curves on a Lagrangian Grassmannian... 

• Sturm theory, Ghys theorem on zeroes of the Schwarzian derivative and flattening of Legendrian curves. (with S. Tabachnikov)

• Lorentzian Worldlines and the Schwarzian Derivative. (with C. Duval)

• Theorem on six vertices of a plane curve via Sturm theory. (with L. Guieu and E. Mourre)

• Projective geometry of polygons and discrete 4-vertex and 6-vertex theorems. (with S. Tabachnikov)

• Structures symplectiques sur les espaces de courbes projectives et affines. (with L. Guieu)

• Classification of contact-projective structures on supercircles. (with Yu. Chekanov and O. Ovsienko)

• Hyperbolic Caratheodory conjecture.(with S. Tabachnikov)
  

Hichem Gargoubi (PhD 1997)
Patrick Marcel (PhD 1999)
Sofiane Bouarroudj (PhD 1999)
Sabin Loubon-Djounga (PhD 2001)
Pascal Redou (PhD 2002)
Yael Fregier (PhD 2005)
Najla Mellouli (PhD 2010)
Tiffany Covolo (PhD 2014)
Marie Kreusch (PhD 2015)