?

단축키

Prev이전 문서

Next다음 문서

크게 작게 위로 아래로 댓글로 가기 인쇄
?

단축키

Prev이전 문서

Next다음 문서

크게 작게 위로 아래로 댓글로 가기 인쇄
Extra Form
일정시작 2023-02-28
일정종료 2023-02-28
배경색상 #77CC00
  • Date:  2023-02-21 (Tue) 10:30 ~ 12:00 

           2023-02-23 (Thu) 10:30 ~ 12:00 

                       2023-02-28 (Tue) 10:30 ~ 12:00 

  • Place:  129-104 (SNU)
  • Title:  An introduction to geometric representation theory and 3d mirror symmetry
  • Speaker:  Justin Hilburn (Perimeter Institute)
  • Abstract:

The Beilinson-Bernstein theorem, which identifies representations of semi-simple Lie algebra \mathfrak{g} with D-modules on the flag variety G/B, makes it possible to use powerful techniques from algebraic geometry, especially Hodge theory, to attack problems in representation theory. Some successes of this program are the proofs of the Kazhdan-Lusztig and Jantzen conjectures as well as discovery that the Bernstein-Gelfand-Gelfand categories O for Langlands dual Lie algebras are Koszul dual.

 

The modern perspective on these results places them in the context of deformation quantizations of holomorphic symplectic manifolds: The universal enveloping algebra U(\mathfrak{g}) is isomorphic to the ring of differential operators on G/B which is a non-commutative deformation of the ring of functions on the cotangent bundle T^*G/B. Thanks to work of Braden-Licata-Proudfoot-Webster it is known that an analogue of BGG category O can be defined for any associative algebra which quantizes a conical symplectic resolution. Examples include finite W-algebras, rational Cherednik algebras, and hypertoric enveloping algebras.

 

Moreover BLPW collected a list of pairs of conical symplectic resolutions whose categories O are Koszul dual. Incredibly, these “symplectic dual” pairs had already appeared in physics as Higgs and Coulomb branches of the moduli spaces of vacua in 3d N=4 gauge theories.  Moreover, there is a duality of these field theories known as 3d mirror symmetry which exchanges the Higgs and Coulomb branch. Based on this observation Bullimore-Dimofte-Gaiotto-Hilburn showed that the Koszul duality of categories O is a shadow of 3d mirror symmetry.

 

In this series of lectures I will give an introduction to these ideas assuming only representation theory of semi-simple Lie algebras and a small amount of algebraic geometry.

 

 

 


List of Articles
번호 제목 글쓴이 날짜 조회 수
36 [QSMS 2022 Winter School] Workshop on Representation Theory file qsms 2022.01.07 2017
» [Series of lectures 2/21, 2/23, 2/28] An introduction to geometric representation theory and 3d mirror symmetry qsms 2023.02.02 1257
34 [Symplectic Geometry] QSMS SUMMER SCHOOL 2023 qsms 2023.07.17 1469
33 [집중강연 2021.02.01~05] Auslander-Reiten translation and $\tau$-tilting theory file qsms 2020.11.18 3591
32 2021 Winter School on Number Theory qsms 2021.12.13 1827
31 2022 Summer School on Number Theory file qsms 2022.07.21 1733
30 2023 Summer School on Number Theory file qsms 2023.07.11 1567
29 2024 Algebra Camp file qsms 2024.01.05 1743
28 Cluster algebras and related topics qsms 2021.05.07 3183
27 Cluster algebras and related topics in East Asia file qsms 2023.05.15 2074
26 Combinatorics on flag varieties and related topics 2023 file qsms 2023.02.13 1841
25 CONFERENCE ON ALGEBRAIC REPRESENTATION THEORY 2021 qsms 2021.10.04 1828
24 East Asian Symplectic Conference 2023 file qsms 2023.09.04 1872
23 Mini-workshop on deformed W-algebras and q-characters I qsms 2022.05.15 1725
22 Mini-workshop on deformed W-algebras and q-characters II qsms 2022.06.03 1647
21 One Day Workshop on the Arithmetic Theory of Quadratic Forms qsms 2021.10.05 1609
20 QSMS 20/21 Winter mini-school on Mirror symmetry and related topics (Part 1) qsms 2021.01.13 15366
19 QSMS 20/21 Winter mini-school on Mirror symmetry and related topics (Part 2) qsms 2021.01.13 2169
18 QSMS 20/21 Winter School on Representation Theory qsms 2021.01.13 2189
17 QSMS 2021 Summer Workshop on Representation theory (Week1) file qsms 2021.08.02 1730
Board Pagination Prev 1 2 Next
/ 2