The Claremont Colleges
Algebra/Combinatorics Seminar
Spring 2007

Wednesdays 12:05 - 1:05 PM
Millikan 211
Pomona College, Department of Mathematics
610 N. College Ave. (Corner of 6th and College Ave.)
Claremont, CA 91711

• Wednesday, January 31, 2007
  Russ Merris (California State University East Bay & Pomona College)
  Graph Immanants
  
• Wednesday, February 7, 2007
  No Meeting
  
• Wednesday, February 14, 2007
  Russ Merris (California State University East Bay & Pomona College)
  Graphic Sequences
  
• Wednesday, February 21, 2007
  William Watkins (California State University, Northridge)
  The maximum sum of squares of the degrees of a graph
  
• Wednesday, February 28, 2007
  Gizem Karaali (Pomona College)
  A Beginner’s guide to Hopf algebras
  
• Wednesday, March 7, 2007
  No Meeting
  
• Wednesday, March 14, 2007
  No Meeting
  
• Wednesday, March 21, 2007
  Yi Chen (Pomona College)
  Graph algorithms for preconditioning sparse linear systems
  
• Wednesday, March 28, 2007
  Stephan Garcia (Pomona College)
  Complex symmetric matrices
  
• Wednesday, April 4, 2007
  No Meeting
  
• Wednesday, April 11, 2007
  Stephan Garcia (Pomona College)
  Complex symmetric matrices II
  
• Wednesday, April 18, 2007
  Geoff Buhl (California State University, Channel Islands)
  Moonshine: a meandering history
  
• Wednesday, April 25, 2007
  Mihai Putinar (University of California, Santa Barbara)
  Positivity in a free *-algebra
  
• Wednesday, May 2, 2007
  Yesem Kurt (Pomona College)
  Cryptography over Non-commutative Groups
  

• Graph Immanants
  Russ Merris (California State University East Bay & Pomona College)
  Much of algebraic graph theory involves the use of matrices and functions of matrices to study graphs. The heart of this first talk involves matrix functions, called immanants--a generalization of determinant and permanent--applied to Laplacian matrices of graphs--a discrete analogue of Laplacian differential operators of surfaces.
  
• Graphic Sequences
  Russ Merris (California State University East Bay & Pomona College)
  Partially ordered by majorization, Par(k), the set of partitions of k, is a (nonmodular) lattice. The primary topic of conversation will be the subposet of this lattice consisting of those partitions that are realizable as the degree sequence of some (simple) graph. A nice characterization of the maximal elements of this subposet leads naturally to the secondary topic of Laplacian graph spectra and some open problems.
  
• A Beginner’s guide to Hopf algebras
  Gizem Karaali (Pomona College)
  Hopf algebras are the appropriate mathematical structures to develop the theory of quantum groups, so it is natural to study them in order to understand the whole field of quantum algebra. In this talk we will provide one such exposition. In particular we will describe what a Hopf algebra is and how it behaves, and see several examples. If time remains, I will also compare and contrast various generalizations of this very useful notion. My own motivation for finding the “correct” generalization comes from my interest in objects known as dynamical quantum groups, but other researchers have other reasons for being interested in generalizations. These will hopefully be mentioned along the way.
  
• Complex symmetric matrices I & II
  Stephan Garcia (Pomona College)
  In this elementary talk we will discuss the class of complex symmetric matrices (i.e. matrices with complex entries which are self-transpose). We will highlight some classical results which we believe deserve to be more widely known. For instance, we prove that every square matrix is similar to a complex symmetric matrix -- in particular, there are no restrictions on the Jordan canonical form.
  
• Moonshine: a meandering history
  Geoff Buhl (California State University, Channel Islands)
  The story of the Moonshine conjectures involves both colorful characters and unique structures. In this talk I will present the history of the Moonshine conjectures: the amazing connection between the Monster, the largest sporadic simple group, and the modular function j. The proof of the Moonshine conjectures by Frenkel-Lepowsky-Meurman and Borcherds introduced a new algebraic objects, vertex operator algebras, which continue to provide unexpected connections with modular functions, sporadic simple groups, and string theory.
  
• Positivity in a free *-algebra
  Mihai Putinar (University of California, Santa Barbara)
  This is an overview of recent Positivstellensatze in a free *-algebra, compared to the classical commutative ones.
  
• Cryptography over Non-commutative Groups
  Yesem Kurt (Pomona College)
  We will talk about key agreement schemes, encryption methods, and signature algorithms that work over non-commutative groups, in particualr over braid groups. We will look at the properties of braid groups that make them desirable for practical systems. Most of the cryptosystems based over braid groups have weaknesses, and even though not completely broken, they are not recommended for future use. We will discuss the weaknesses and talk about a new system that seems to resist the attacks against earlier systems.