Seminar

Rocky Mountain Algebraic Combinatorics Seminar

Large Erd\H{o}s-Ko-Rado Sets in Polar Spaces

Ferdinand Ihringer
University of Regina

An Erdös-Ko-Rado set (EKR set) Y of { 1, …, n} is a family of k-sets, which pairwise intersect non-trivially. A non-trivial problem is to provide tight upper bounds on Y and classify all examples, which obtain that bound. Erdös, Ko and Rado proved |Y| ≤ \binomn−1k−1 for n ≥ 2k. Equality holds for n ≥ 2k+1 if and only if Y is the family of all k-sets, which contain one fixed element.

If we equip the vector space \mathbbFqn with a reflexive, non-degenerate sesquilinear form, then the subspaces that vanish on that form are a polar space, so-called isotropic subspaces. The largest isotropic subspaces of a polar space are called generators. We say that two generators intersect trivially if the dimension of their intersection is 0. An EKR set of a polar space is a set of pairwise non-trivially intersecting generators. We present various results on EKR sets for polar spaces, in particular we will discuss some recent so-called weak Hilton-Milner type results.

Searching for Balanced Sets

Gavin King
University of Wyoming

Let X be a finite set of unit vectors in some Euclidean space. Define Rα, β(x,y) for α, β ∈ \mathbb R and x, y ∈ X as Rα, β(x, y) = {z ∈ X: 〈z,x〉 = α,〈z,y〉 = β} satisfying:

• For each x,y, |Rα, β(x,y)|=|Rα, β(y,x)|.
• For any α, there is a constant pα such that for all x, ∑Rα, α(x,x)=pα x.
• For each α, β, γ there exists a constant mβ,γα such that for any pair of vectors vi, vj with 〈vi, vj 〉 = α, we have ∑Rβ, γ(vi,vj)−∑Rγ, β(vi, vj) = mβ, γα (vi−vj), regardless of our choice of vi and vj.
Balanced sets are a notion intricately tied to the concept of association schemes, and especially to the association schemes with the Q-polynomial property. I will be discussing the existing work on balanced sets as well as my own, such as a classification of balanced sets with small numbers of inner products, and ways to search for balanced sets connected to permutation groups.

Weber 223
4–6 pm
Friday, May 5, 2017
(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 21, 2017
April 7, 2017
Jason Williford, Anton Betten
March 24, 2017
Isabella Novik, Peter Brooksbank
March 3, 2017
Jintai Ding, Curtis Bennett
February 17, 2017
Fatma Karaoglu, Eric Moorhouse
February 3, 2017
Tim Penttila, James B. Wilson
December 2, 2016
Jim Fowler, Andrew Kelley
November 11, 2016
Joseph Gersch, Joshua Maglione
October 28, 2016
October 14, 2016
JM Landsberg, James B. Wilson
September 30, 2016
Alexander Hulpke, Oscar Levin
September 16, 2016
Delaram Kahrobaei, Amit Patel
June 23, 2016
April 29, 2016
Nick Loehr, Jason Williford
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