"Noncommutative algebra and comninatorial topology"
Time: 4:50 PM
Location: Hill 705
Monday, November 16th
Ravi Srinivasan, Rutgers University (Newark)
"Picard-Vessiot Theory"
Time: 4:50 PM
Location: Hill 705
Abstract: Let F be a characteristic zero differential field with an algebraically closed field of constants C. I will describe the construction of a Picard-Vessiot Extension (PVE) for a linear homogeneous differential equation over F. The group of differential automorphisms of a PVE fixing F is called the differential Galois group; there is a Galois correspondence between its algebraic subgroups and intermediate differential subfields. Examples of PVEs for F=C(x) with the usual derivation will be discussed, and we will also compute the differential Galois group for our examples.
Monday, November 9th
Paul Ellis, University of Connecticut
"The classfication problem for finite rank dimension groups "
Time: 4:50 PM
Location: Hill 705
Abstract: An unperforated partially ordered abelian group A is a dimension group if A satises the Riesz interpolation property (given a,a' ≤b,b' there is a c with a,a' ≤ c ≤b,b'). These are related to "Bratteli diagrams". Paul will discuss the difficulty of classifying them when the rank is at least 3, and show that the problem for a given rank cannot be reduced to the classification problem for a smaller rank.
Monday, October 19th
Ken Johnson, Penn-State-Abington
"Mathematics arising from a new look at the Dedekind-Frobenius group matrix and group determinant "
Time: 4:50 PM
Location: Hill 705
Abstract: Frobenius invented group character theory in order to solve the problem of the factorization of the group determinant. His papers are hard to understand and when the modern methods for group representation theory were introduced his initial work was largely forgotten. To each representation of a (finite) group there is associated a polynomial which is a factor of the group determinant, and Frobenius introduced "k-characters" to describe this polynomial. Professor Gelfand has commented that perhaps physicists might benefit from looking at these polynomials. Among other places these k-characters have occurred in work of Buchstaber and Rees and also are related to work of Wiles and Taylor on "pseudocharacters" of finite dimensional representations of infinite groups.
I will describe the early work from an elementary point of view and give an account of some of the new ideas coming from it, and also indicate some of the connections with probablity.
Monday, October 12th
Bob Guralnick, USC
"Derangements in Finite and Algebraic Groups "
Time: 4:50 PM
Location: Hill 705
Abstract: A permutation on a set is called a derangement if it has no fixed points. The study of the proportion of derangements in finite transitive groups has a long history and the problem has many applications. We will discuss this as well as the analogous problem for algebraic and show the connection between the two. In particular, we will discuss recent results (joint with Fulman) about conjugacy classes in finite Chevalley groups and the solution of a conjecture made independently by Aner Shalev and Nigel Boston.
Monday, October 5th
Lourdes Juan, Texas Tech
"Differential Central Simple Algebras and Picard-Vessiot representations"
Time: 4:50 PM
Location: Hill 705
Abstract: A differential field is a field K with a derivation, that is, an additive map D:K → K satisfying D(fg)=D(f)g+fD(g) for f,g in K.
The field of constants C of K are the zeros of D. A differential central simple algebra (DCSA) over K is a pair
(A,mathcal D) where A is a central simple algebra and $mathcal D$ is a derivation of A extending the derivation D of its
center. Any DCSA, and in particular a matrix differential algebra over K, can be trivialized by a Picard-Vessiot (differential
Galois) extension E of K. In the matrix algebra case, there is a correspondence between K-algebras trivialized by E and
representations of the differential Galois group of E over K in PGLn(C) that can be interpreted as cocycles equivalent up to co-boundaries. I will start with a brief introduction to differential Galois theory.
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