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2020 Fiscal Year Research-status Report

Loop and double loop geometry

Research Project

Project/Area Number 19K14495
Research InstitutionThe 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)

Project Period (FY) 2019-04-01 – 2022-03-31
KeywordsCoulomb 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.

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.

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.

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.

  • Research Products

    (7 results)

All 2021 2020 Other

All Journal Article (2 results) (of which Int'l Joint Research: 2 results) Presentation (4 results) (of which Int'l Joint Research: 2 results,  Invited: 4 results) Remarks (1 results)

  • [Journal Article] On a conjecture of Pappas and Rapoport about the standard local model for GL_d d2020

    • Author(s)
      Muthiah Dinakar、Weekes Alex、Yacobi Oded
    • Journal Title

      Journal fur die reine und angewandte Mathematik (Crelles Journal)

      Volume: 2021 Pages: 175~185

    • DOI

      10.1515/crelle-2020-0030

    • Int'l Joint Research
  • [Journal Article] The equations defining affine Grassmannians in type A and a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman.2020

    • Author(s)
      Muthiah Dinakar、Weekes Alex、Yacobi Oded
    • Journal Title

      Int. Math. Res. Not. IMRN

      Volume: 2020 Pages: -

    • DOI

      10.1093/imrn/rnaa13

    • Int'l Joint Research
  • [Presentation] Affine Lusztig-Kato formula2021

    • Author(s)
      Dinakar Muthiah
    • Organizer
      University of Glasgow Algebra Seminar
    • Int'l Joint Research / Invited
  • [Presentation] Equations for affine Grassmannians and their Schubert varieties2021

    • Author(s)
      Dinakar Muthiah
    • Organizer
      GRT at Home Seminar
    • Invited
  • [Presentation] Double affine Bruhat order2020

    • Author(s)
      Dinakar Muthiah
    • Organizer
      Workshop on Combinatorial Problems of Algebraic Origin
    • Int'l Joint Research / Invited
  • [Presentation] Approaches to double loop groups2020

    • Author(s)
      Dinakar Muthiah
    • Organizer
      IPMU Postdoc Colloquium
    • Invited
  • [Remarks]

    • URL

      https://dmuthiah.github.io/

URL: 

Published: 2021-12-27  

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