Rocky Mountain Algebraic Combinatorics Seminar

From Affine Weyl Groups to Discrete Painlev{\'e} Equations

Anton Dzhamay
University of Northern Colorado

The goal of this talk is to partially outline the role played by the affine Weyl groups in the theory of discrete Painlevé equations, using a discrete Painlevé equation known as q-PVI or q-P(A3(1)) as an example. This is a particular two-dimensional nonlinear recurrence relation that defines a birational map from C2 to itself. This map lifts to an isomorphism of a certain rational algebraic surface X, known as the Okamoto space of initial conditions, that is constructed from C2 using a sequence of blowups resolving the indeterminacies of the map. This surface is characterized by a certain sub lattice of affine type A3 in the Picard lattice. The complementary symmetry sub-lattice is given by the affine Dynkin diagram D5. Reflections in the roots of this sub-lattice generate elementary birational automorphisms of X, which, combined with the automorphisms of the Dynkin diagram, generate the extended affine Weyl symmetry group of the surface. It turns out that the original discrete dynamics corresponds to a translation in this symmetry sub-lattice, and the representation of the translation vector as a composition of elementary reflections and automorphisms allows us to recover back the coordinate description of our equation. The main objective of the talk would be to explain this inverse process.


Graphs with Integral Spectrum

Anton Betten
Colorado State University

The spectrum of a graph is the set of eigenvalues of the adjacency matrix of the graph, together with their multiplicities. In 1974, Harary and Schwenk initiate the study of graphs with integral spectra, that is, graphs whose eigenvalues are all integral.

In this talk, we will look at a lot of examples of graphs, many of which are indeed integral. We will discuss some of the methods for classifying integral graphs in certain cases. Finally, we will look at integral Cayley graphs.


Weber 223
4–6 pm
Friday, February 20, 2015
(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:

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