Spring 2019, Math 8140: Topics in Algebraic Geometry

Syllabus:

Time/Location: MWF 1:50--2:45pm, Enarson Classroom building - Room: 358
Instructor: Hsian-Hua Tseng
Office: 642 Math Tower
Office Phone: 614-292-5581
E-mail: hhtseng-at-math-dot-ohio-state-dot-edu
Office Hours: By appointment.

Description: The goal of this course is to discuss aspects of symmetric functions and explore their connections with geometric objects, mainly Hilbert schemes of points.

Disability Statement.

Academic Misconduct Statement.

Course Plan:

Part I: Symmetric Functions

(1) Basic constructions of symmetric functions and their ring
(2) Schur functions
(3) Skew Schur functions
(4) Relations with characters of symmetric groups
(5) Littlewood-Richardson rule
(6) Hall-Littlewood functions
(7) Macdonald functions

Part II: Hilbert schemes of points in the plane

(1) Constructions of Hilbert schemes
(2) A digression to equivariant cohomology and localization
(3) Nakajima's result on (co)homology of Hilbert schemes of points
(4) Realization of Jack functions
(5) Realization of Schur functions
(6) Realization(s) of Macdonald functions

References and Comments:

Part I:

(a) We mainly follow the classic book on this subject:

I. G. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.

(b) The proof of Littlewood-Richardson rule is taken from the following paper:

V. Gasharov, A short proof of the Littlewood-Richardson rule, European J. Combin. 19 (1998), no. 4, 451--453.

(c) For more general constructions of Macdonald functions associated to other root systems, see the following book and paper:

I. G. Macdonald, Symmetric functions and orthogonal polynomials, University Lecture Series, 12. American Mathematical Society, Providence, RI, 1998.

I. G. Macdonald, Orthogonal polynomials associated with root systems, Sem. Lothar. Combin. 45 (2000/01), Art. B45a, 40 pp.

Part II:

(a) For the construction and basic properties of Hilbert schemes of points in the plane, we follow the classic book on this subject:

H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, 18. American Mathematical Society, Providence, RI, 1999.

A discussion on the existence of Hilbert (and Quot) shcemes in general can be found in the following paper:

N. Nitsure, Construction of Hilbert and Quot schemes, Fundamental algebraic geometry, 105--137, Math. Surveys Monogr., 123, Amer. Math. Soc., Providence, RI, 2005, arXiv:math/0504590.

(b) The standard reference for equivariant cohomology and localization is the following paper:

M. F. Atiyah, R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1--28.

(c) The realization of Jack functions in the equivariant cohomology of Hilbert schemes of points is discussed in Section 11.3 of the following book:

Z. Qin, Hilbert schemes of points and infinite dimensional Lie algebras, Mathematical Surveys and Monographs, 228. American Mathematical Society, Providence, RI, 2018.

Another reference for this is Section 3 of the following paper:

H. Nakajima, More lectures on Hilbert schemes of points on surfaces, Development of moduli theory--Kyoto 2013, 173--205, Adv. Stud. Pure Math., 69, Math. Soc. Japan, [Tokyo], 2016, arXiv:1401.6782.

(d) The realization of Schur functions in the equivariant cohomology of Hilbert schemes of points is discussed in Section 17.2 of the following paper:

D. Maulik, A. Okounkov, Quantum Groups and Quantum Cohomology, to appear in Asterisque, arXiv:1211.1287.

(e) One realization of Macdonald functions in the equivariant *K-theory* of Hilbert schemes of points is discussed in Equation (95) of the following paper:

M. Haiman, Combinatorics, symmetric functions, and Hilbert schemes, Current developments in mathematics, 2002, 39--111, Int. Press, Somerville, MA, 2003.

Another realization of Macdonald functions in the equivariant *cohomology* of Hilbert schemes of points is discussed in Theorem 4 of the following paper:

A. Okounkov, R. Pandharipande, The quantum differential equation of the Hilbert scheme of points in the plane, Transform. Groups 15 (2010), no. 4, 965--982.

The (not-so-surprising) relation between these two realizations is discussed in Section 4 of the following paper:

R. Pandharipande, H.-H. Tseng, The Hilb/Sym correspondence for C2: descendents and Fourier-Mukai, arXiv:1807.06969.