Meeting place and time: Thursdays 9:30am-11:30am in 509 Lake Hall
Organizers: Elie Casbi, Hunter Dinkins, Iva Halacheva, Valerio Toledano Laredo, Josh Wen, Yan Zhou
The main resource for the seminar will be the book: D. Maulik and A. Okounkov, Quantum Groups and Quantum Cohomology. (see AMS bookstore and the arXiv)
Schedule
Jan 19: LECTURE 1. Overview of the seminar theme. Speaker: Hunter Dinkins
Jan 26: LECTURE 2. Geometric invariant theory. Exercises Speaker: Josh Wen
Main references:
[BL, Section 2 (p.3-6)] (if time, also [D, Section 6.1, some of 6.2 (p.91-97)])
[G, Section 2.2 (p.6-8) ]
[Ki, Sections 9.1-9.4 (p.159-171)]
Further references:
[T, Sections 1-3 (p.1-13)]
[S, Sections 1.2-1.4]
[PV, Section 4.6] (See also: https://gauss.math.yale.edu/~il282/Inv.html)
[D, Ch 6-9, p. 121, examples]
[H1, H2]
Feb 2: LECTURE 3. Hamiltonian reduction. Exercises Notes Speaker: Hunter Dinkins
Main references:
[Ki, Sections 9.5-9.10 (p.171-191)]
[BL, Section 4 (p.10-17)]
[G, Section 4 (p.14-21)]
Further references:
[T, Section 4 (p.13-23)]
Feb 9: LECTURE 4. Quiver varieties I (definition, stability conditions). Notes Speaker: Sean Carroll Mentor: Iva Halacheva
Main references:
[Ki, Sections 10.1-10.4]
[MO, Sections 2.1-2.2 (p. 33-42)]
Further references:
[G, Sections 4-5 ]
Feb 16: LECTURE 5. Quiver varieties II (more on stability conditions). (See Lecture 4 Notes) Speakers: Sean Carroll and Ryan Kannanaikal
Main references:
[Ki, Sections 10.1-10.4]
[MO, Sections 2.1-2.2 (p. 33-42)]
Feb 23: Quiver varieties III (examples, tautological bundles). Notes Speaker: Ryan Kannanaikal Mentor: Josh Wen
Main references:
[Ki, Sections 10.5-10.7, 10.9, 11.1-11.2]
[MO, Section 2]
Mar 2: LECTURE 7. Equivariant cohomology I (definition, torus case examples, localization). Notes Speaker: Hongqin Zou Mentor: Valerio Toledano Laredo
Main references:
[B, Section 1]
[Ty, Section 1-2]
Further references:
[AB, Sections 1-3]
Mar 9: (Spring Break)
Mar 16: LECTURE 8. Equivariant cohomology II (further examples, generalizations, Chern classes of tautological bundles). Notes Speaker: Rahul Hirwani Mentor: Josh Wen
Main references:
[B, Section 2]
[Ty, Sections 3-6]
Further references:
[AB, Sections 1-3]
[Bo]
Mar 23: LECTURE 9. Geometry of stable envelopes I (definition, uniqueness). Speaker: Shengnan Huang Mentor: Yan Zhou
Main reference:
[MO, Sections 3.1-3.4, 4.1]
Further references:
[Mi, Feb 9 lecture]
[O, Section 1]
Mar 30: LECTURE 10. Geometry of stable envelopes II (examples, existence). Speaker: Hunter Dinkins
Main reference:
[MO, Sections 3.5-3.7, 4.1-4.2]
Further references:
[BMO]
Apr 6: LECTURE 11. Hopf algebras and quantum groups. Speaker: Aria Masoomi Mentor: Elie Casbi
Main reference:
[ES]
Further references
[M, Section 2]
[CP, Section 12]
[Mi, Mar 9 lecture]
Apr 13: LECTURE 12. Yangians and the (algebraic) FRT construction. Speaker: Anadil Saeed Rao Mentor: Elie Casbi
Main references:
[W, Sections 1-3]
[CP, Section 12.1]
[M, Sections 1, 2.1-2.5]
[Mi, Mar 9 lecture]
Further references:
[MO, Section 5.2]
[Mc, Sections 2, 3.3-3.4, 4.5]
Apr 20: LECTURE 13. Geometric R-matrices and the FRT procedure, properties of stable envelopes. Speaker: Ivan Karpov Mentor: Hunter Dinkins
Main references:
[Mc, Section 5]
[MO, Section 5]
Apr 27: LECTURE 14. ADE setting: algebraic vs geometric Yangians. Speaker: Vasily Krylov Mentor: Hunter Dinkins
Main reference:
[Mc, Section 6]
(Possible further topics: Bow varieties and 3D mirror symmetry, Quantum cohomology. )
-------------------------------------
References
[AB] M. Atiyah and R. Bott, The moment map and equivariant cohomology
[BL] B. Bolognese and I. Losev, A general introduction to the Hilbert scheme of points on the plane
[Bo] R. Bott, An introduction to equivariant cohomology
[B] M. Brion, Equivariant cohomology and equivariant intersection theory
[BMO] A. Braverman, D. Maulik, A. Okounkov, Quantum cohomology of the Springer resolution
[CP] V. Chari and A. Pressley, A guide to quantum groups
[D] I. Dolgachev, Lectures on Invariant theory
[ES] P. Etingof and M. Semenyakin, A brief introduction to quantum groups
[G] V. Ginzburg, Lectures on Nakajima’s quiver varieties
[H1] V. Hoskins, Moduli problems and geometric invariant theory
[H2] V. Hoskins, Geometric invariant theory and symplectic quotients
[Ka] J. Kamnitzer, Symplectic resolutions, symplectic duality and Coulomb branches
[Ki] A. Kirillov, Quiver representations and quiver varieties
[Mi] A. Minets, Notes from stable envelopes reading group
[MO] D. Maulik and A. Okounkov, Quantum Groups and Quantum Cohomology
[Mc] M. McBreen, Quantum cohomology of hypertoric varieties and geometric representations of Yangians
[M] A. Molev, Yangians and their applications
[O] A. Okounkov, Inductive construction of stable envelopes
[PV] V. Popov and E. Vinberg, Invariant theory in Algebraic geometry IV, Encyclopaedia of Mathematical Sciences, vol. 55, Springer Verlag.
[S] A. Schmitt, Geometric Invariant Theory and Decorated Principal Bundles. Zurich lectures in advanced mathematics, EMS, 2008.
[W] C. Wendlandt, The R-matrix presentation for the Yangian of a simple Lie algebra
[T] R. Thomas, Notes on GIT and symplectic reduction for bundles and varieties
[Ty] J. Tymoczko, An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson