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FJ-LMI Seminar
Seminar information archive ～09/03｜Next seminar｜Future seminars 09/04～
| Organizer(s) | Toshiyuki Kobayashi, Michael Pevzner |
|---|
2026/09/09
15:00-16:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Warm Up Session of the Workshop on Polylogarithms
Herbert GANGL (Durham University)
Introduction to polylogarithms, with connections to number theory and algebraic $K$-theory (英語)
https://fj-lmi.cnrs.fr/wp-content/uploads/2026/09/Warm_up_session_schedule_and_abstracts.pdf
Warm Up Session of the Workshop on Polylogarithms
Herbert GANGL (Durham University)
Introduction to polylogarithms, with connections to number theory and algebraic $K$-theory (英語)
[ Abstract ]
In this survey talk we discuss several basic properties of classical polylogarithms
$$ \mathrm{Li}_m(z)=\sum_{k\geq1}\frac{z^k}{k^m}
$$
in the unit disc, for $m \geq 1$, leading up to a beautiful connection to the algebraic $K$-theory of number
fields, motivated by an attempt to find a suitable generalisation of Dirichlet’s celebrated class number
formula. Bloch initiated this for the dilogarithm, and Zagier extended it to higher polylogarithms. It
turns out that an arguably even more suitable class of functions to describe this connection involves
multiple polylogarithms (MPLs = nested sums of classical ones, in several variables $z_1, \ldots , z_d)$. If time
allows, we will also try to give a glimpse of Goncharov’s highly relevant “depth reduction conjecture”
for MPLs and what is known about it.
[ Reference URL ]In this survey talk we discuss several basic properties of classical polylogarithms
$$ \mathrm{Li}_m(z)=\sum_{k\geq1}\frac{z^k}{k^m}
$$
in the unit disc, for $m \geq 1$, leading up to a beautiful connection to the algebraic $K$-theory of number
fields, motivated by an attempt to find a suitable generalisation of Dirichlet’s celebrated class number
formula. Bloch initiated this for the dilogarithm, and Zagier extended it to higher polylogarithms. It
turns out that an arguably even more suitable class of functions to describe this connection involves
multiple polylogarithms (MPLs = nested sums of classical ones, in several variables $z_1, \ldots , z_d)$. If time
allows, we will also try to give a glimpse of Goncharov’s highly relevant “depth reduction conjecture”
for MPLs and what is known about it.
https://fj-lmi.cnrs.fr/wp-content/uploads/2026/09/Warm_up_session_schedule_and_abstracts.pdf
