Rocky Mountain Algebraic Combinatorics Seminar

Resolvable designs and maximal arcs in projective planes

Vladimir D. Tonchev
Michigan Technological University

Let D={ X, B } be a Steiner 2-(v,k,1) design with point set X, collection of blocks B, and let v be a multiple of k, v=nk. A parallel class is a set of v/k=n pairwise disjoint blocks, and a resolution is a partition R of B into disjoint parallel classes. A design is resolvable if it admits a resolution.

Two resolutions are compatible if they share one parallel class, and all other classes pairwise have at most one block in common. In this talk, we discuss an upper bound on the maximum number of mutually compatible resolutions of a resolvable 2-(nk,k,1) design D. The bound is attainable if and only if D is embeddable as a maximal (kqq +k,k)-arc in a projective plane of order q=(vk)/(k−1).
We will report on some recent work on maximal sets of mutually compatible resolutions associated with maximal (52,4)-arcs in the known projective planes of order 16.


Commutative Semifields

Morgan Rodgers
Lake Superior State University

Finite semifields are well studied objects in combinatorics and finite geometry and have many connections to other interesting geometric structures. They play a central role in the theory of projective planes, generalised quadrangles, and polar spaces, and have applications to cryptography, and maximum rank distance codes. Of particular interest are commutative semifields with odd order, for which very few constructions are known. The property of being commutative implies that these semifields have applications to perfect nonlinear functions.

We will first consider commutative semifields which have rank two over their middle nucleus. These objects are equivalent to semifield flocks of a quadratic cone in projective 3-space, and are therefore also equivalent to translation ovoids of Q(4,q). We will also consider commutative semifields with rank 3 over their middle nucleus. In both of these situations, a computer search for new examples can be conducted by searching for certain linear sets. This is joint work with Michel Lavrauw.


Weber 223
4–6 pm
Friday, April 21, 2017
(Refreshments in Weber 117, 3:30–4 pm)
Colorado State University

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.

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Previous Seminars:

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