ALGEBRA SEMINAR
Centre for Algebra, Logic and Computation
Department of Mathematics and Statistics
University of Saskatchewan, Saskatoon, Canada
Coordinators:
M.R. Bremner, F.V. Kuhlmann, S. Kuhlmann, M.A. Marshall (Emeritus)
UPCOMING TALKS
None scheduled at present.
EARLIER TALKS
Thursday 20 December 2007
2:30 pm in McLean Hall 242.1
Speaker:
Irvin Roy Hentzel, Iowa State University, USA
Title:
Plenary train algebras
Abstract:
The Hardy-Weinberg result explains why a dominant gene does not eventually
overwhelm the entire population. Algebraically, the Hardy-Weinberg result is
(x^2)^2 = w(x)^2 x^2. This means that the ratio of alleles becomes steady
after the second generation. Algebras satisfying this identity are called
Bernstein algebras. This concept was generalized to higher degrees and
called plenary train algebras. We describe the generic plenary train algebra
and give a criterion which uses the roots of the plenary dependence equation
that tells when the only example for that plenary equation will be the generic
one.
Tuesday 7 August 2007
3:30 pm in McLean Hall 242.1
Speaker:
Victor Vinnikov, Ben Gurion University, Israel
Title:
Linear matrix inequality representation of convex sets
Abstract:
Which closed convex sets in ${\mathbb R}^m$ can be represented in the form
$$
\left\{
( x_1, \ldots, x_m ) \in {\mathbb R}^m
\colon
A_0 + x_1 A_1 + \cdots + x_m A_m \geq 0
\right\},
$$
where $A_0, A_1, \ldots, A_m$ are real symmetric matrices and $\geq 0$ means
that the symmetric matrix is positive semidefinite? Such a representation is
called a linear matrix inequality representation, and they came to prominence
in the last decade because of applications in control theory. As I will
explain, the problem really boils down to certain positive determinantal
representations of real polynomials. I will describe a complete solution in
case $m = 2$ (leading to a proof of a 1958 conjecture of P. Lax on real
hyperbolic polynomials) and a conjecture for higher dimensions. If time
allows, I will also discuss related problems: a ``lifted'' version -
representing a closed convex set as a projection of a set that has a linear
matrix inequality representation, and a noncommutative version - where
$m$-tuples of scalars are replaced by $m$-tuples of matrices of all
dimensions. While the commutative problem involves tools of classical
algebraic geometry, the noncommutative problem seems to lead to a newly
emerging area of (free) noncommutative function theory. Surprisingly, the
noncommutative problem is appearently much better behaved! The talk is based
on joint work with Bill Helton and Scott McCullough.
Thursday 26 July 2007
3:30 pm in McLean Hall 242.1
Speaker:
Mihai Putinar, University of California, Santa Barbara
Title:
Positive polynomials on fibre products of real algebraic varieties
Abstract:
A geometric interpretation of some recent advances in polynomial optimization
(due to Kojima and Lasserre) makes it possible to obtain a novel, very general
Striktpositivstellensatz. The novelty is two-fold: first, less than an
algebra structure is needed to have a weighted sum-of-squares decomposition
for positive polynomials with a "sparsity pattern", and second, the proofs
rely on a disintegration of measures technique, typically used in probability
theory. A large number of examples will illustrate the advantages and the
limits of this approach. All based on collaborative work with Salma Kuhlmann.
Tuesday 24 July 2007
2:30 pm in McLean Hall 242.1
Speaker:
Murray Marshall
Title:
Saturated preorderings in the power series ring in two variables
Abstract:
Thursday 19 July 2007
3:30 pm in McLean Hall 242.1
Speaker:
Jonathon Funk
Title: The universal covering geometric morphism of an inverse semigroup
Abstract:
Tuesday 17 July 2007
2:30pm in McLean Hall 242.1
Speaker:
Tim Netzer
Title:
Representing nonnegative polynomials using dimension reduction by group
actions
Abstract:
For a given semialgebraic set S, one would like the set of all nonnegative
polynomials to be a finitely generated preordering. However, in dimension
three or higher, this is never the case. Now one can try to find a group
action under which S invariant. If the orbit space has dimension two or
smaller and the problem is solvable there, then one has at least characterized
the nonnegative invariant polynomials on S by a finitely generated preordering.
We introduce the method and give examples.
Thursday 5 July 2007
2:30 pm in McLean Hall 242.1
Speaker:
Tim Netzer
Title:
The invariant moment problem
Abstract: For a given semialgebraic set K in R^n, one can try to characterize
the linear forms on the real polynomial ring which are integration with
respect to a measure on K. This is the so-called moment problem. If K is
invariant under some group action, one can ask for the invariant linear forms
which are integration with respect to some invariant measure on K. This is the
invariant moment problem. We will introduce the topic and give results and
examples. We show that neither one of the two properties implies the other.
Thursday 7 June 2007
2:30 pm in McLean Hall 242.1
Speaker:
Manuela Haias
Title:
Contraction groups and asymptotic couples - the construction of a differential
field with given value group
Abstract (PDF)
Tuesday 15 May 2007
4:00 pm in McLean Hall 242.1 - NOTE NEW TIME
Speaker:
Mikhail Kochetov (joint work with Y. Bahturin and S. Montgomery)
Title:
Group gradings and Hopf algebra actions
Abstract: Gradings on associative, Lie, and Jordan algebras arise in various
contexts of mathematics and theoretical physics. Recently there has been a
considerable progress in the classification of gradings on simple algebras of
these types. A group grading on an algebra $A$ is a vector space
decomposition $A = \bigoplus_{g \in G} A_g$ where $G$ is a group and
$A_g \cdot A_h \subseteq A_{gh}$ for all $g, h \in G$. If $G$ is a finite
abelian group whose order is not divisible by the characteristic of the ground
field, then a $G$-grading on $A$ is equivalent to an action of the dual group
$\widehat{G}$ on $A$ by automorphisms. This duality of gradings and actions
has been used in recent work of Y. Bahturin, M. Zaicev, and I. Shestakov on
the classification of gradings on simple Lie and Jordan algebras over a field
of characteristic zero. We will discuss the difficulties that arise in the
case of positive characteristic and how to overcome them using Hopf algebras.
Thursday 10 May 2007
2:30 pm in McLean Hall 242.1
Speaker:
Andrew Douglas
Title:
Finite dimensional representations of the Euclidean algebra
Abstract: We will discuss the finite dimensional representations of the
Euclidean algebra e(2) and consider the classification of certain families of
these representations. We will then investigate the finite dimensional
representations of e(2) that are obtained by embedding e(2) into sl(3). We
will show that the finite dimensional, irreducible representations of sl(3)
restricted to e(2) are indecomposable, and, when possible, we will give a
graphical description of these e(2) representations.
Tuesday 8 May 2007
2:30 pm in McLean Hall 242.1
Speaker:
Andreas Fischer
Title:
Definable smooth manifolds are affine
Abstract:
M. Shiota showed that smooth semialgebraic manifolds are not necessarily
affine, even if they are compact. This is due to the quasi-analyticity of
definable smooth functions in polynomially bounded o-minimal structures.
That is, a definable smooth function with open connected domain is completely
determined by its Taylor series at a single point. This quasi-analyticity no
longer holds if the exponential function is definable. We show that if
additionally a smooth cell decomposition holds, then every definable smooth
manifold of dimension n is definably diffeomorphic to a definable smooth
submanifold of 2n+1 dimensional Euclidean space.
Thursday 4 January 2007
10:15 am in McLean Hall 242.1
Speaker: Irvin Roy Hentzel, Iowa State University
Title:
The Albert Program, its structure and various applications
Abstract: ALBERT is a computer program designed to study identities in
non-associative algebras. It was created by Dave Jacobs at Clemson
University. The user gives ALBERT a list of identities, followed by a
non-associative polynomial. ALBERT will determine if that non-associative
polynomial is implied by the list of identities. We discuss how the program
makes the determination, and we show how this procedure can be used to detect
ideals, elements of the nucleus, elements of the center, and zero divisors.
We will also discuss the limitations and other problems associated with this
current version of ALBERT.
Thursday 14 December 2006
10:15 am in McLean Hall 242.1
Speaker: Lisa Hayden
Title:
Galois groups of low order torsion points on elliptic curves and Drinfeld modules
Abstract:
This talk will summarize my master's thesis where I calculate the Galois
groups of extensions generated by torsion points of low order on elliptic
curves and Drinfeld modules through their corresponding division polynomials.
I investigate division polynomials of degree up to and including four, which
correspond to 2-torsion and 3-torsion points on elliptic curves and
(T+a)-torsion and (T2+aT+b)-torsion points on Drinfeld modules
of rank 1 and 2.
These calculations depend on the invariants that classify elliptic curves and
Drinfeld modules up to an isomorphism.
I will also present a result that gives the analogy of the x-coordinate
for Drinfeld modules.
Thursday 7 December 2006
10:15 am in McLean Hall 242.1
Speaker: Hamid Usefi
Title:
The isomorphism problem for restricted enveloping algebras
Abstract:
The isomorphism problem for integral group rings of nilpotent groups was given
a positive solution independently by Roggenkamp & Scott and Weiss in late
1980.
However the modular isomorphism problem for p-groups remains an
outstanding open problem.
It is well-known that there are natural connections between the modular
isomorphism problem and the isomorphism problem for restricted enveloping
algebras.
Let L be a restricted Lie algebra with restricted enveloping algebra
u(L).
The isomorphism problem asks what invariants of L are determined by
u(L).
In this expository talk I shall first talk about the roots and motivations for
this problem.
In particular, I shall mention some of the connections between the isomorphism
problem and the modular isomorphism problem for p-groups.
Next I shall consider Abelian restricted Lie algebras and provide some
positive answers in this case.
Thursday 30 November 2006
10:15 am in the Lounge (McLean Hall 201)
Speaker: Andreas Fischer
Title:
On the constant for Lipschitz stratifications
Abstract:
A Lipschitz continuous function f: R^n --> R^m is a function which satisfies
\norm{f(a)-f(b)} \leq L\norm{a-b}. L is called the Lipschitz constant of f.
Lipschitz cells are defined by induction: A Lipschitz cell in R is either
a singleton or an open interval. A Lipschitz cell in R^n is either a singleton,
or a set of the form { (x,y): x \in X, y = h(x) } for some Lipschitz cell X of
R^m and a semialgebraic Lipschitz continuous function h: X --> R^{n-m}; or a
set of the form { (x,y): x \in X, f(x) < y < g(x) } for some Lipschitz cell X
of R^{n-1} and semialgebraic Lipschitz continuous functions g,f: X-->R which
satisfy f(x) < g(x) for all x \in X. It is a well known fact that any
semialgebraic set can be partitioned into finitely many semialgebraic sets
which are, after some linear orthogonal change of variables, Lipschitz cells.
We will sketch the idea how to prove that, in the above case, all the
functions f,g,h can be assumed to have Lipschitz constant (at most)
L = n^{3/2}.
Thursday 23 November 2006
10:15 am in the Lounge (McLean Hall 201)
Speaker: Wei Fan
Title:
Nonnegative polynomials on compact semi-algebraic sets in the one-variable
case
Abstract:
We investigate the relationship between the quadratic module $M_S$ and the
preordering $T_S$ generated by a finite subset $S$ of $\mathbb{R}[x]$, the
polynomial ring in one variable.
It is easy to see that $M_S \subseteq T_S$, but $M_S \ne T_S$ in general, as
$M_S$ may not be closed under multiplication.
Now assuming the associated basic semi-algebraic set $K_S$ is compact in
$\mathbb{R}$, we are asking does this imply that $M_S = T_S$?
The answer turned out to be yes.
Scheiderer first settled this problem.
I will give another proof.