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日仏数学拠点FJ-LMIセミナー
過去の記録 ～09/04｜次回の予定｜今後の予定 09/05～
| 担当者 | 小林俊行, ミカエル ペブズナー |
|---|
2026年09月09日(水)
15:00-16:00 数理科学研究科棟(駒場) 126号室
Herbert GANGL 氏 (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
Herbert GANGL 氏 (Durham University)
Introduction to polylogarithms, with connections to number theory and algebraic $K$-theory (英語)
[ 講演概要 ]
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.
[ 講演参考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
