**Syllabus:**

**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.

**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.

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

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.

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.

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

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

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

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

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.

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