Lesley Ward (University of South Australia)による数学・数理科学グローバル特別講義10が下記の要領で開催されます。履修されている方は下記のGoogleフォームのから申込をお願いします。
講師:Lesley Ward (University of South Australia)
講義日程:2025年 1月6日(月)、1月8日(水)、10日(金)、15日(水)、17日(金)
各日10:00–12:00
場所: 理学研究科 3号館 1階 127室
タイトル:Complex analysis meets Brownian motion and
probability:planar regions and their harmonic-measure distribution functions
アブストラクト:
This five-lecture short course focuses on a topic that offers a pleasing interplay of theoretical and computational work, bringing together ideas from complex analysis, probability, Brownian motion and partial differential equations as well as numerical calculation.
We will begin by defining the harmonic-measure distribution function (“h-function”) of a planar domain in terms of the location of the first exit from the domain of particles wandering according to Brownian motion. We will calculate examples of h-functions, both analytically using conformal mappings, especially the Riemann map, and the conformal invariance of harmonic measure, and numerically using simulation of Brownian motion. We will explain how h-functions are connected with the Dirichlet problem through Kakutani’s Theorem, as well as why h-functions solve instances of the Conformal Skorokhod Embedding Problem from probability.
We will present what is known about how the shape of the domain’s boundary is reflected in properties of the h-function; necessary and sufficient conditions for a function to be an h-function; techniques for explicitly constructing a domain that will have a given h-function; and the regularity and asymptotics of h-functions. We will show how recent work using the Schottky—Klein prime function has for the first time enabled explicit computation and visualisation of h-functions of multiply connected domains, whose boundaries may be made up of line segments, circles or polygons. Throughout the course we will highlight open research problems and current research.
This course is intended for graduate students and undergraduates with some background in complex analysis, including conformal mappings. Some experience with probability will be helpful but is not necessary. Some lecture time will be allocated for students to work together in small groups calculating examples and tackling questions arising from the lecture material, coached by the lecturer. There may be an opportunity for students to use MATLAB and for some numerical computations, such as simulation of Brownian motion and the use of MATLAB’s SKP package in finding h-functions of multiply connected domains; in this case students may like to bring their own device with MATLAB installed.
A useful resource is the paper “Harmonic measure distributions of planar domains: a survey”, Marie A. Snipes and Lesley A. Ward, The Journal of Analysis 24 (2016), No. 2, 293--330. DOI 10.1007/s41478-016-0019-0.
参加申込用URL:https://forms.gle/Px4tXLNRCCJtxL4t9
締切日:12月27日 (金)
※数学・数理科学グローバル講義Ⅱは数学・数理科学イノベーション人材育成強化コースにおける中核科目です。
※数学・数理科学グローバル講義Ⅱを履修するにはKULASISでの履修登録が必要です。
後期科目の履修登録期間は10月10日(木)・11日(金)。
※履修登録していなくても聴講(本学学生に限る)可(申込みは必要)。
数学・数理科学グローバル講義Ⅱでは4名の講師による特別講義が開講されます。
各特別講義のアブストラクト等の詳細は、当コースのホームページ
https://www.math.kyoto-u.ac.jp/ja/ktgu/coursesをご覧ください。