My work is concentrated around polylogarithms and variants thereof,
with connections to algebraic K-theory, algebraic number theory and
arithmetic algebraic geometry. The questions that I am interested in
lead to a variety of rather disparate topics, comprising for instance
the homology of the general linear group, configuration and moduli spaces,
combinatorial Hopf algebras, multiple zeta values or (quasi-)modular forms.
A good chunk of my research has a computer-experimental flavour, using mostly
scripting language to try to detect patterns in piles of data, to
find hitherto unknown objects or to formulate conjectures. Furthermore,
I also use the machine power available to develop algorithms for attacking
harder calculational problems.
The people involved together run the
arithmetic study group.
Preprints and publications
"From polygons and symbols to polylogarithmic functions" (with C. Duhr and J. Rhodes).
"Tame kernels and second regulators
of number fields and their subfields" (with J. Browkin)
"On special elements in higher algebraic K-theory and the Lichtenbaum-Gross Conjecture" (with D. Burns and R. de Jeu).
"Functional equations and ladders for polylogarithms".
"Perfect forms, K-theory and the cohomology of modular groups" (with Ph. Elbaz-Vincent and C. Soulé), and an older version on the arXiv.
"Regulators via iterated integrals (numerical computations)", in Clay Mathematics Proceedings, 2010.
"Multiple polylogarithms, polygons, trees and algebraic cycles" (with Alexander B. Goncharov and Andrey Levin), Proc. Symp. Pure Math (2009).
"Multiple logarithms, trees and algebraic cycles"
(with Alexander B. Goncharov and Andrey Levin),
Frontiers in Number Theory, Physics, and Geometry II (eds. Cartier, Julia, Moussa, Vanhove), Springer (2007), 759--776.
"Double zeta values and modular forms" (with Masanobu Kaneko and Don Zagier), in Automorphic forms and zeta functions, Proceedings of the conference in memory of Tsuneo Arakawa, World Scientific, 2006.
"Generators and relations for K2(OF)" (with Karim Belabas) (K-theory, 31 (2004), pages 195-231)
"Functional equations of higher logarithms" Selecta Mathematica, 9 (2003), pages 361-379.
"Quelques calculs de la cohomologie de GLN(Z) et de la K-théorie de Z" (with Philippe Elbaz-Vincent and Christophe Soulé) Comptes Rendus Mathématiques, Académie des Sciences, Paris, 335 (2002), pages 321-324.
"On poly(ana)logs I" (with Philippe Elbaz-Vincent) Compositio Mathematica, 130 (2002), pages 161-210.
"Classical and elliptic polylogarithms and special values of L-series" (with Don Zagier), in The arithmetic and geometry of algebraic cycles Banff, AB, 1998), NATO Science Series C, Mathematical and Physical Sciences, volume 548, pages 561-615; Kluwer Academic Publishers, Dordrecht, 2000.
"Polylogarithmic identities in cubical higher Chow groups" (with Stefan Müller-Stach) Proceedings of Symposia in Pure Mathematics, volume 67, pages, 25-40; AMS, Providence, 1999 ( table 1, with graphical proof of 5-term relation).
Text and Table of "Table of tame and wild kernels of quadratic imaginary fields of discriminants > -5000 (conjectural values)" (with Jerzy Browkin) Mathematics of Computation, 68 (1999), pages 291-305.
"Some Computations in weight 4 motivic cohomology" , in Regulators in analysis, geometry and number theory, Progress in Mathematics, volume 171, pages 117-125; Birkhäuser, Boston, 2000.
"Werte von Dedekindschen Zetafunktionen, Dilogarithmuswerte und Pflasterungen des hyperbolischen Raumes" (Diploma thesis)
Here is a short introduction to polygonal combinatorics for algebraic cycles.