Herbert Gangl

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 the GP/PARI 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.