調和解析定例セミナー

第54回 2026年5月16日(土)　中央大学後楽園キャンパス 5137教室およびオンライン
　　　　　　　　　　　　　　　　　　　　　　　　　　(Room 5137 at Bulding No.5, Chuo University Korakuen Campus )

1. １４：３０〜１５：３０, １６：００〜１７：００　
講演者： 臼杵 峻亮 氏 (慶応義塾大学)
　題目： An introduction to Jean Bourgain’s pointwise ergodic theorem for polynomial sequences
　 　概要： In this talk, I will give an expository overview on Jean Bourgain’s pointwise ergodic theorem for polynomial sequences, based on the contents of the seminar my co-organizer and I held a few months ago (not including my own result). Birkhoff’s pointwise ergodic theorem is one of the most foundational and essential theorems in ergodic theory, saying that, for a probability space $(X,\mathcal{B},\mu)$, a map $T:X\to X$ preserving $\mu$, and an $L^1$ function $f:X\to\mathbb{R}$, the ergodic average $N^{-1}\sum_{n=1}^N f(T^nx)$ converges for $\mu$-almost every $x$. Together with applications of ergodic theory to number-theoretic problems, other arithmetic versions of ergodic theorems have been attracting significant interest. That is, it has been studied whether the convergence theorem holds if we replace the ergodic average with the average $N^{-1}\sum_{n=1}^N f(T^{p_n}x)$ along a sequence $(p_n)$ of integers with natural arithmetic origins. Jean Bourgain’s pointwise ergodic theorem for polynomial sequences is the first groundbreaking result of that type, saying that, for any polynomial $p(n)$ with integer coefficients and any $L^p$ function $f$ for $p>1$, the average $N^{-1}\sum_{n=1}^N f(T^{p(n)}x)$ converges for $\mu$-almost every $x$. What is interesting is not only the statement itself, but also the proof. It is a wonderful collaboration of many different areas of mathematics: probability theory, harmonic analysis and analytic number theory, rather than purely ergodic-theoretic. In this talk, I will aim to explain the whole story of the proof and the essential ideas in it.
　

次々回（第55回）：2026年6月開催予定