# Seminar

## Rocky Mountain Algebraic Combinatorics Seminar

### Algebraic and Combinatorial Macdonald Polynomials

Nick Loehr
Virginia Polytechnic Institute and State University

Macdonald polynomials, which are symmetric polynomials involving two extra parameters, have played a central role in algebraic combinatorics ever since their introduction by Ian Macdonald in 1988. By specializing the parameters in various ways, one obtains many classical and modern bases for the vector space of symmetric functions. Unfortunately, the original algebraic definition of Macdonald polynomials are complicated, non-explicit, and difficult to work with. This talk contrasts the algebraic definition with Jim Haglund's revolutionary combinatorial formula for Macdonald polynomials involving explicit objects and permutation statistics parametrized by integer partitions. I will outline the proof (due to Haglund, Haiman, and myself) that the combinatorial formula agrees with the algebraic definition, as well as describing some ramifications of Haglund's formula. The talk assumes no specific prior knowledge of Macdonald polynomials or symmetric functions.

### Extremal Problems on Cycles and Theta Graphs

Jason Williford
University of Wyoming

Let n be a positive integer and G a graph. We define ex(n,G) to be the largest number of edges possible in a graph on n vertices that does not contain G as a subgraph.

The problem of determining ex(n,G) for general n and G officially began with Turán's theorem that solves ex(n, Km) for all n and m, a result that is striking in its precision. The celebrated Erdös-Stone-Simonovits theorem generalizes this asymptotically to all G with chromatic number at least 3. However, the magnitude of ex(n,G) is unknown for most bipartite G.
In this talk we will discuss some of the recent progress on ex(n,G) when G is a cycle or a theta graph (a collection of edge disjoint paths between two vertices). This will involve constructions of graphs defined on vector spaces over finite fields.

Weber 223
4–6 pm
Friday, April 29, 2016
(Refreshments in Weber 117, 3:30–4 pm)

This is a joint Denver U / UC Boulder / UC Denver / U of Wyoming / CSU seminar that meets biweekly. Anyone interested is welcome to join us at a local restaurant for dinner after the talks.

### Previous Seminars:

April 15, 2016
Alexander Hulpke, Klaus Lux
April 1, 2016
Eamonn O'Brien, Izabella Stuhl
February 19, 2015
James Wilson, Anton Betten
December 4, 2015
Maria Monks Gillespie, Dane Flannery
November 13, 2015
Richard Green, Tim Penttila
October 23, 2015
Christina Boucher, Sylvia Hobart
October 9, 2015
Josh Maglione, Ghodratollah Aalipour
September 25, 2015
September 11, 2015
James B. Wilson, Tim Penttila
May 8, 2015
Amanda Schaeffer Fry, Peter Brooksbank
April 24, 2015
March 6, 2015
Felice Manganiello, Eric Moorhouse
February 20, 2015
Anton Dzhamay, Anton Betten
February 6, 2015
Alexander Hulpke, Morgan Rodgers
December 5, 2014
Stefaan De Winter, Gretchen Matthews
November 14, 2014
Greg Coxson, Tom Dorsey
October 31, 2014
Octavio Paez Osuna, Sylvia Hobart
October 10, 2014
Takunari Miyazaki, Eric Moorhouse
September 26, 2014
Elissa Ross, Anton Betten
September 12, 2014
Petr Vojtěchovský, Alexander Hulpke
May 9, 2014
Philip DeOrsey, Tim Penttila
April 25, 2014
William J Martin, Jason Williford
April 11, 2014
Victor Pambuccian, George Shakan
March 7, 2014
Nathan Lindzey, Jens Harlander
February 21, 2014
Ross McConnell, Anton Betten
November 22, 2013
Justin Hughes, Josh Maglione

Department of Mathematics