| Project/Area Number |
19K14495
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | The University of Tokyo |
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Principal Investigator |
MUTHIAH DINAKAR 東京大学, カブリ数物連携宇宙研究機構, 客員准科学研究員 (50835410)
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Project Status |
Completed (Fiscal Year 2023)
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| Budget Amount *help |
¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2019: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
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| Keywords | Coulomb branches / KM Affine Grassmannians / KM Affine Hecke Algebras / Iwahori-Hecke algebras / Monopole Operators / Coulomb branch / affine Grassmannian / (double) loop groups / affine Grassmannians / Geometric Satake / p-adic Kac-Moody groups / bow varieties |
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| Outline of Research at the Start |
A group captures the mathematical essence of the concept of symmetry. Closely related is Representation Theory, which is about the ways a group can be linearized. I study both singly infinite-dimensional and doubly infinite-dimensional groups. The Geometric Satake Equivalence establishes a connection between singly infinite-dimensional geometry and finite-dimensional representation theory. The objective of this research is to investigate the generalization to infinite-dimensional Kac-Moody groups. This involves studying the subtle doubly infinite-dimensional geometry of loop Kac-Moody groups.
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| Outline of Final Research Achievements |
Alex Weekes and I have published two papers about the geometry of Coulomb branches. First, we classified symplectic leaves in finite-type, and second, we constructed many symplectic leaves in general using fundamental monopole operators. I have posted a paper defining double affine Kazhdan-Lusztig R-polynomials generalizing techniques of masures and point-counting for p-adic Kac-Moody groups. Alex Weekes, Oded Yacobi and I have published a paper solving a twenty-year-old conjecture of Pappas and Rapoport using techniques we developed for studying affine Grassmannians.
Additionally, I have begun and progressed several projects during the grant. Hiraku Nakajima and I are working on understanding Intersection Cohomology stalks for affine type A Coulomb branches. August Hebert and I have constructed a completion of Kac-Moody affine Hecke algebras in a work in-progress. Finally, Anna Puskas and I are about to post a preprint about pursuing Coxeter theory of Kac-Moody affine Hecke algebras.
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| Academic Significance and Societal Importance of the Research Achievements |
This is about Langlands duality for loop groops via the geometry of double loop groups. This involves Coulomb branches, which come from quantum physics, and p-adic Kac-Moody groups, which come from arithmetic and number theory. The goal is to advance and connect these rich and different areas.
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