2020 Fiscal Year Research-status Report
Loop and double loop geometry
Project/Area Number |
19K14495
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Research Institution | The University of Tokyo |
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 – 2022-03-31
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Keywords | Coulomb branch / affine Grassmannian |
Outline of Annual Research Achievements |
My paper "On a conjecture of Pappas and Rapoport about the standard local model for $GL_d$." joint with Alex Weekes and Oded Yacobi has been published in J. Reine Angew. Math. (Crelle's Journal). My paper "Symplectic leaves for generalized affine Grassmannian slices." has been accepted to Ann. Sci. Ec. Norm. Super. My work with Hiraku Nakajima on intersection cohomology of Coulomb branches has been progressing well. We essentially have a proof of our result, and we are currently in the process of writing up our work carefully. In particular, our work will answer the Braverman-Finkelberg conjecture from 2007 in the case of affine type A. Progress on this conjecture was one of the main aims of my initial grant proposal. Furthermore, Alex Weekes and I have been working on understanding minuscule monopole operators and how they behave under restriction maps of Coulomb branches. We are able to show that these operators restrict nicely, which is quite surprising because the restriction maps are not easy to describe in purely Coulomb branch terms. With Auguste Hebert, we are making further progress on our work on completions of Iwahori-Hecke algebras. We are able to prove that the completions are correct, and we are currently working on writing up clear proofs of our results.
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Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
My work has been steadily being accepted to good journals. The work underway is progressing at a satisfactory rate. I would say that things are progressing rather smoothly.
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Strategy for Future Research Activity |
As things are progressing rather smoothly, I would firstly continue to complete the projects under way. For future work, I would aim to continue work with Hiraku Nakajima to understand intersection cohomology of Coulomb branches for all affine types. Additionally, I would aim, with Alex Weekes, to better understand how the relations between minuscule monopole operators for Coulomb branches. This would be very useful because it would give a nice presentation for Coulomb branches in all types.
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Causes of Carryover |
Research travel has been restricted to essentially zero because of COVID. For that reason, I have not been able to use my grant to facilitate research collaboration. I am hoping that these restrictions will soon be lifted, and I will once again be able to travel for research purposes.
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