Algebraic and geometric properties of discrete integrable systems

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K04893
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: The University of Tokyo
• Principal Investigator: Willox Ralph 東京大学, 大学院数理科学研究科, 教授 (20361610)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 可積分系 / 離散可積分系 / 超離散可積分系

Outline of Final Research Achievements

I mainly studied discrete dynamical systems defined on low-dimensional lattices and developed methods for deciding on the integrability or non-integrability of such discrete systems. In particular, I classified the types of singularities that may arise in mappings of the complex projective plane and proposed several new methods for deciding on the integrability of such mappings, based on the structure of their singularities. I also developed a systematic method for constructing discrete Painleve equations ― the prototypical discrete integrable system ― based on the symmetries and geometric structure of integrable birational maps.
Furthermore, using certain algebraic and combinatorial properties of ultradiscrete systems, I discovered new solution methods for two types of soliton cellular automata.

Free Research Field

数物科学・数学