?

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 2023-02-21
일정종료 2023-02-23
배경색상 #77CC00
반복주기 2
반복단위 1.일(단위)
  • 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
공지 QSMS Mailing List Registration qsms 2021.09.09 13288
130 [QSMS-BK21 symplectic semiar] Part 4 (23' Aug 18 ~) qsms 2023.12.27 3372
129 [QSMS Seminar 2022-10-25] Rational liftings and gRSK qsms 2022.10.08 3374
128 [QSMS-BK21 Toric Geometry Seminar] Moduli Space of Special Lagrangians in the Complement of an Anticanonical Divisor qsms 2022.04.12 3377
127 [SNU Number Theory Seminar 2022-05-20] Distribution of Hecke eigenvalues of GL_2 automorphic representations qsms 2022.05.18 3382
126 [QSMS Geometry seminar 2023-05-25] Symplectic field theory and codimension-2 stable Hamiltonian submanifolds qsms 2023.05.25 3382
125 [QSMS Seminar 2022-09-27] Categories of Whittaker Modules over Lie superalgebras and Categorification of Fock Spaces qsms 2022.09.20 3387
124 [QSMS-BK21 Symplectic Geometry Seminar 2022-09-30] Introduction to Floer Homotopy: Bordism and Flow Categories qsms 2022.09.26 3391
123 [QSMS-BK21 Symplectic Geometry Seminar 2022-12-09] Introduction to Floer Homotopy: Formal Groups and Complex Cobordism qsms 2022.11.28 3391
122 [QSMS Monthly Seminar] Quantization of integrable differential difference equations qsms 2022.04.22 3392
121 [QSMS-BK21 Seminar] Past seminars qsms 2022.08.26 3392
120 [QSMS-BK21 Symplectic Geometry Seminar 2022-11-04] Introduction to Floer Homotopy: Kuranishi Structures qsms 2022.10.26 3392
119 [SNU Number Theory Seminar 2022-04-15] A note on non-ordinary primes for some genus-zero arithmetic groups qsms 2022.04.12 3395
118 [QSMS Monthly Seminar 2022-10-28] Oscillator representations of quantum affine algebras   qsms 2022.10.30 3395
117 [QSMS Seminar 2022.04.05] Trialities of W-algebras qsms 2022.02.15 3398
116 [QSMS-BK21 Seminar 2022-08-29] Symplectic Cohomology, String Topology, and Deformations qsms 2022.08.29 3399
115 [QSMS Monthly Seminar 2023-04-21] Lagrangian submanifolds and their Legendrian lifts qsms 2023.04.21 3399
114 [QSMS Geometry Seminar 20221230] Morphism spaces between coisotropic A-branes qsms 2022.12.27 3413
113 [QSMS Monthly Seminar] Yokonuma Hecke algebras and invariants of framed links qsms 2022.05.27 3419
» [Series of lectures 2/21, 2/23, 2/28] An introduction to geometric representation theory and 3d mirror symmetry qsms 2023.02.02 3424
111 [QSMS-BK21 Symplectic Seminar 12 May] Introduction to Floer Homotopy: Global Kuranishi Charts via Abouzaid-McLean-Smith qsms 2023.05.12 3425
Board Pagination Prev 1 2 3 4 5 6 7 8 Next
/ 8