?

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 4491
45 [SNU Number Theory Seminar 2022-12-02] Torsion points and concurrent exceptional curves on del Pezzo surfaces of degree one qsms 2022.11.28 1276
44 [QSMS Monthly Seminar 2022-12-02] Extended crystal structures of Hernandez-Leclerc categories qsms 2022.12.02 1175
43 [QSMS Geometry Seminar 20221230] Morphism spaces between coisotropic A-branes qsms 2022.12.27 1074
42 [QSMS 집중강연 Representation theory 2023-02-07] Mini-course on quantum symmetric pairs I qsms 2023.01.03 1188
41 [QSMS 집중강연 Representation theory 2023-02-13] Mini-course on quantum symmetric pairs II qsms 2023.01.05 1266
40 [2022 Global KMS International Conference 10/19 ~ 10/21] Related talks qsms 2023.01.05 1143
» [Series of lectures 2/21, 2/23, 2/28] An introduction to geometric representation theory and 3d mirror symmetry qsms 2023.02.02 1172
38 [SNU Number Theory Seminar 6 ~ 7 Feb] Prismatic F-crystals and applications to p-adic Galois representations qsms 2023.02.13 1295
37 [SNU Number Theory Seminar 31 Mar] Height gap theorem and almost law qsms 2023.03.02 1197
36 [QSMS-BK21 Symplectic Seminar 10 Mar] Mirror Symmetry for Log Calabi-Yau Surfaces qsms 2023.03.09 1149
35 [QSMS Monthly Seminar 2023-03-24] Rational points on abelian varieties qsms 2023.03.24 1083
34 [SNU Number Theory Seminar 14 Apr] Homological interpretation of Hecke $L$-values over real quadratic fields qsms 2023.03.31 1073
33 [SNU Number Theory Seminar 21 Apr] Selberg's central limit theorem of L -functions near the critical line qsms 2023.04.20 1160
32 [SNU Number Theory Seminar 2 June] A refined Tamagawa number conjecture for modular forms qsms 2023.04.20 1114
31 [QSMS Monthly Seminar 2023-04-21] Lagrangian submanifolds and their Legendrian lifts qsms 2023.04.21 1065
30 [QSMS Seminar 2023-05-17] Affinizations, R-matrices and reflection functors qsms 2023.05.04 1040
29 [QSMS-BK21 Symplectic Seminar 28 Apr] PGL(2,Q_p)-ORBIT CLOSURES ON A p-ADIC HOMOGENEOUS SPACE OF INFINITE VOLUME qsms 2023.05.12 1079
28 [QSMS-BK21 Symplectic Seminar 12 May] Introduction to Floer Homotopy: Global Kuranishi Charts via Abouzaid-McLean-Smith qsms 2023.05.12 1080
27 [QSMS-BK21 Symplectic Seminar 26 May] 1. Product structures on the relative symplectic cohomology 2. Symplectic homology of affine varieties qsms 2023.05.25 1074
26 [QSMS Geometry seminar 2023-05-25] Symplectic field theory and codimension-2 stable Hamiltonian submanifolds qsms 2023.05.25 1071
Board Pagination Prev 1 2 3 4 5 6 7 8 Next
/ 8