Math 5810, Spring 2026

Prof. Jim Haglund, jhaglund@math.upenn.edu
Course webpage: http://www.math.upenn.edu/~jhaglund/5810/

Office hours: F 10am-11am in DRL 4e2b

Office Phone: (215) 573-9093.

Lecture: TR 12 noon - 1:30pm in DRL 4E9

Course: Topics in combinatorial theory of current research interest. Specific topics to be covered in the course in Spring 2026 include:


Important Dates:
Wednesday, January 14: Classes begin
Monday, January 19: Martin Luther King day
Last Day to Add a Course: January 27
Last Day to Drop a Course: February 23
Spring Break: March 7-15
Grade mode change deadline March 23
Last Day of Classes: April 29

Reading Assignments:

Week of 1/19: Read pages 1-10 and 41-43, the 1st paragraph of p. 44, and the section "The Special values t=1 and t=1/q" on pp. 47-48 from the book "The q,t-Catalan Numbers and the Space of Diagonal Harmonics" (downloadable from my webpage)
Week of 1/27: For notes on q,t-Narayana Numbers and recursion for F_{n,k}(q,t,w,z), see slides at this link q,t-Narayana (ignore all identities from these slides involving symmetric function operators for the moment, just use as a source for the recursions and other combinatorial identities)
Week of 2/3: Look at paper "The combinatorics of knot invariants arising from the study of Macdonald polynomials" at link Rational Catalan and Knots
Week of 2/10: Read about parking functions and the Hilbert series on pages 77-85 of "The q,t-Catalan Numbers and the Space of Diagonal Harmonics" book.
Weeks of 2/17and 2/24: Read about the representtion theory of the symmetric group and the basic theory of symmetric functions on pages 10-25 and 33-34 of "The q,t-Catalan Numbers and the Space of Diagonal Harmonics" book. It would also be good to consult a standard book on the representation theory of the symmetric group like Sagan's "The Symmetric Group" or "The Representation Theory of Finitie Groups" by James and Liebeck. Also, a good source for an exposition of many of the topics we are covering this semester is Athina's Masters Thesis

Homework Assignments: HW1: Do exercises 1.10, 3.18, and 3.19 from my q,t-Catalan book. Due by Feb. 3.
HW2: Do exercises 5.7, 1.18 and 2.12 from my q,t-Catalan book. Due by Feb. 24.