Mathematics Seminar

Rocky Mountain Algebraic Combinatorics Seminar

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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 \mathbbF_qⁿ 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 R_{α, β}(x,y) for α, β ∈ \mathbb R and x, y ∈ X as R_{α, β}(x, y) = {z ∈ X: 〈z,x〉 = α,〈z,y〉 = β} satisfying:

• For each x,y, |R_{α, β}(x,y)|=|R_{α, β}(y,x)|.
• For any α, there is a constant p_α such that for all x, ∑R_{α, α}(x,x)=p_α x.
• For each α, β, γ there exists a constant m_β,γ^α such that for any pair of vectors v_i, v_j with 〈v_i, v_j 〉 = α, we have ∑R_{β, γ}(v_i,v_j)−∑R_{γ, β}(v_i, v_j) = m_{β, γ}^α (v_i−v_j), regardless of our choice of v_i and v_j.

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)
Colorado State University

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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.

PDF version Previous Seminars: April 21, 2017 Vladimir D. Tonchev, Morgan Rodgers 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 John MacLaren Walsh, Henry Adams 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 Jason Cantarella, Michal Adamaszek 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 Ross McConnell, Henry Adams September 11, 2015 James B. Wilson, Tim Penttila May 8, 2015 Amanda Schaeffer Fry, Peter Brooksbank April 24, 2015 Heide Gluesing-Luerssen, Phil DeOrsey 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 Fort Collins, Colorado 80523