| Project/Area Number |
16K17561
|
|---|
| Research Category |
Grant-in-Aid for Young Scientists (B)
|
| Allocation Type | Multi-year Fund |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | The University of Tokyo |
|---|
Principal Investigator |
DONOVAN WILL 東京大学, カブリ数物連携宇宙研究機構, 特任研究員 (60754158)
|
|---|
| Project Period (FY) |
2016-04-01 – 2019-03-31
|
| Project Status |
Completed (Fiscal Year 2018)
|
| Budget Amount *help |
¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
|
|---|
| Keywords | Perverse sheaves / Derived symmetries / Mirror symmetry / Flops / Variation of GIT / Deformation theory / Noncommutative algebra / Contractions / Stringy Kaehler moduli / SKMS / 3-folds |
|---|
| Outline of Final Research Achievements |
My focus was algebraic geometry, the study of solution spaces to algebraic equations using geometric methods. Such spaces are not necessarily smooth: their non-smooth points are called `singularities'. An important problem is to understand certain hidden symmetries of solution spaces, extending usual symmetries by including symmetries of the `derived category', an algebraic tool which is also related to the string-theoretic study of the space in theoretical physics.
I made progress on this problem in two main ways, completing four research papers with collaborators: first,linking hidden symmetries with `perverse schobers' which give a notion of families of categories with singular behaviour; second, associating them to quite general singularities in algebraic geometry, via deformation algebras.
|
| Academic Significance and Societal Importance of the Research Achievements |
最初M.Wemyss氏と共同で、特異点に付随する非特異空間に隠れた対称性を見出す一般的な方法を与えた（Adv.Math.に出版済）。これは、以前のこの対称性の研究方法を統一し、特異点の研究への新たな道具の導入となった。次の論文(IMRNに出版済)では、幾何学的不変式論での壁越えと偏屈圏の関連を説明した。偏屈圏を構成する新しい道具を与え、壁越えに潜む隠れた対称性を理解する新しい方法を与えた。最後に桑垣樹氏と共同で、ミラー対称性への応用を与えた。ミラー対称性は、数学と物理に跨る多くの人々によって研究される弦理論的双対性である。特に偏屈圏に対し「ミラー定理」を証明し、この活発な分野の進展に貢献した。
|
