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)

The topics that will be covered this term include:

Multi-Dimensional Moment Problems,

Hilbert's 17th Problem and Sums of Squares,

Optimization and Control, Linear Matrix Inequality Representations.

Generalized Power Series models,

o-minimal Expansions of Valued Fields,

Exponential-Logarithmic power series,

Applications to solutions of Differential equations.

Thursday November 27th

Speaker: Ryan Siders-Bissell

University of Helsinki, Finland

Title: Permutations, Cycles, and Limits in the Infinitary Theory of Linear Order.

Abstract: we prove an effective normal form for any elementary class of linear orders, expressing membership in that class as the iterated formation of: 1. sequences, described by cycles in a graph, of 2. limit-types, indexed by hereditarily finite sets, of 3. iterated permutations of 1 (cycles, again) where the iteration loops through 1,2, and 3 2^k-many times, to describe an elementary class defined by a formula of quantifier rank k. The reader will notice that the theory of linear order interprets path connectedness, not the existence of a Hamiltonia cycle. Since the normal form caps what can be expressed in linear order, we bound the complexity of model-theoretic generation and model-checking on linear orders.

Thursday November 13th

Speaker: Murray Marshall

Title: Closures of preorderings in localisation in the unique finest convex topology

Abstract: ( view as pdf)

Thursday November 6th

Speaker: Dr. Kiran S. Kedlaya, MIT

Title: Stratification of valuation spaces by corank

Abstract: Let X be an irreducible variety over a field k. A fundamental inequality of Abhyankar bounds certain numerical invariants of a Krull valuation over k in terms of the dimension of X. The discrepancy in this inequality is what we call the corank; the larger the corank, the more difficult it is to describe the valuation using local coordinates. One may use the corank to stratify the Zariski-Riemann space of X, or the Berkovich analytic space defined by X by equipping k with the trivial absolute value. We describe some features of the resulting stratification: for instance, in the Berkovich space, the set of all points of corank at most m is open and path-connected. If time permits, we will reference some applications of this picture to analysis of D-modules in both the classical and p-adic settings.

Thursday October 9th

Speaker: Dr. Igor Klep, UCSD

Title: `Values of Noncommutative Polynomials, Lie Skew-Ideals and Tracial Nullstellens\"atze'

Abstract: A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for involutions of the first kind and four for involutions of the second kind) and this classification result is used to characterize noncommutative polynomials via their values in these algebras. As an application, we deduce that a polynomial is a sum of commutators and a polynomial identity of d by d matrices if and only if all of its values in the algebra of d by d matrices have zero trace.

Thursday September 11th

Speaker: Dr. M. Matusinski, (Universite de Bourgogne, France / U of S)

Title: Construction or Hardy type derivation in generalised series fields.

Abstract: Our purpose is to provide an arbitrary generalised series field with a Hardy type derivation, i.e. a derivation that behaves very like the one in a Hardy field. The first point consists in making sense of "defining a derivation" in the context of series fields even with infinite rank (that is with infinitely many comparability classes). We consider derivations that are strongly linear and for which a strong version of the Leibniz rule holds, what we call a series derivation. Then we will state several sufficient conditions so as such a series derivation is well-defined. The second point is devoted to state a necessary and sufficient condition for a series derivation on a generalised series field to be a Hardy type one.

Thursday September 18th and 25th

Speaker: Dr. Kasia Osiak (University of Katowice, Poland)

Title: The space of real places of a rational function field Part I and II

Abstract: ( view as pdf)

Tuesday April 22 and Thursday April 24.

Speaker: Dr Salih Azgin, McMaster University

Title: Model Theory Valued Difference Fields

Abstract: We will present some recent developments in the model theory of valued fields equipped with a distinguished automorphism as well as some open problems in the area. This study can be carried out in various interesting contexts which depend on the interaction between the valuation and the distinguished automorphism. We obtain various AxKochen type results for valued difference fields as well as relative quantifier elimination. Valued Difference Fields and Valued Fields in Positive Characteristic Considering valued fields in positive characteristic as valued diffence fields equipped with the Frobenius is an old but not-so-fruitful (yet) idea. We shall present yet another formalization of this idea and further connections to valued difference fields.

Tuesday April 15

Speaker: Andreas Fischer

Title: Algebraic models for o-minimal manifolds

Abstract: A differentiable manifold admits an algebraic model if it is diffeomorphic to some non-singular real algebraic set. We prove that every manifold whose underlying set is definable in some o-minimal structure admits an algebraic model, and the diffeomorphism can be chosen to be definable in this structure. For a large class of o-minimal expansions of the real exponential field, even definable smooth manifolds admit definably and smooth algebraic models.

Tuesday April 8 and Thursday April 10:

Speaker: Dr. Mickael Matusinski

Title: Differential equations with coefficients in generalized power series fields (III) and (IV)

Thursday April 3:

Speaker: Dr. Victor Vinnikov

Title: Noncommutative difference-differential calculus.

Abstract: In my previous talk, I have introduced backward shift operators for noncommutative rational functions. In this talk (which will be independent) I will discuss the general framework for noncommutative differential calculus and calculus of finite differences.

Tuesday April 1:

Speaker: Dr. Mickael Matusinski

Title: Differential equations with coefficients in generalized power series fields (II)

Thursday March 27:

Speaker: Dr. Victor Vinnikov

Title: Singularities of noncommutative rational functions, minimal realizations, and backward shifts.

Abstract: In my previous series of talks on noncommutative convexity, I have introduced noncommutative rational functions and their realizations. I have also indicated that one of the main issues involved was whether the singularities of a noncommutative rational function coincide with the singularities of the ``long resolvent'' in a minimal realization. (I will review all the relevant concepts so that this talk will be independent of the previous series.) In this talk, I will discuss a result (joint with Dmitry Kalyuzhniy-Verbovetskiy) that the answer to the above question, in a fairly general setup, is YES. The method of proof is quite interesting: we introduce noncommutative backward shifts, and more generally a difference--differential calculus for noncommutative rational functions.

Tuesday March 25:

Speaker: Dr. Mickael Matusinski

Title: Differential equations with coefficients in generalized power series fields

Abstract: Our subject is to prove a generalization of the classical Newton-Puiseux theorem, that is to express the connection between the support of some equations and those of generalized series solutions. We consider the field Mr of series with well-ordered support included in the Hahn product Hr with finite rank r (i.e. the lexicographic product of r copies of the reals), and we equip it with a Hardy type derivation. The differential equations F(y,.,y(n))=0 we consider are formal series with coefficients in Mr. Thus we show that there exists some well-ordered subsets T1, ., Tr of Hr , such that any solution y0 in Mr with positive valuation has its exponents belonging to a positive well-ordered subset of Hr obtained from Supp F, T1, ., Tr by a finite number of elementary transformations.

Tuesday March 18:

Speaker: Dr. Andreas Fischer

Title: O-minimal analytic separation is dimension 2

Abstract: We study the Hardy field associated with an o-minimal expansion of the real numbers with the following result. The set of definable analytic germs at infinity is dense in the Hardy field if and only if closed definable disjoint subsets of 2-space can be separated by a definable analytic function.

Thursday January 31:

Speaker: Professor Murray Marshall

Title: Weak closures of quadratic modules (Part III)

Tuesday January 29:

Speaker: Dr. Victor Vinnikov, Ben Gurion University, Israel.

Title: Convex noncommutative rational functions (Part IV)

Thursday January 24:

Speaker: Professor Murray Marshall

Title: Weak closures of quadratic modules (Part II)

Tuesday January 22:

Speaker: Dr. Victor Vinnikov, Ben Gurion University, Israel.

Title: Convex noncommutative rational functions (Part III)

Thursday January 17:

Speaker: Professor Murray Marshall

Title: Weak closures of quadratic modules

Abstract ( view as pdf)

Let $A$ be a finitely generated $\Bbb{R}$-algebra (there is no harm in assuming here that $A$ is the polynomial ring $\Bbb{R}[\underline{x}] := \Bbb{R}[x_1,\dots, x_n]$) and let $M$ be a finitely generated quadratic module of $A$. Let $\overline{M}$ denote the closure of $M$ in the unique finest locally convex topology on $A$. Interest in the closure of a quadratic module arises in studying the multidimensiona moment problem and also in polynomial optimization via semidefinite programming. $\overline{M}$ is understood in various special cases but is not very well understood in general. We consider results of Schm\"udgen (and also of Scheiderer and others) and extensions of these results. Various examples are given. We define recursively something which we call the weak closure of $M$, denoted $\widetilde{M}$, which satisfies $M \subseteq \widetilde{M} \subseteq \overline{M}$, and show that $\overline{M} = \widetilde{M}$ holds in a large number of cases. (In fact, we know of no example where $\overline{M} \ne \widetilde{M}$, although such an example must certainly exist.) The reader should note that the weak closure$\widetilde{M}$ is not the same thing as the sequential closure $M^{\ddagger}$ of $M$, which also satisfies $M \subseteq M^{\ddagger} \subseteq \overline{M}$. For sequential closure there is an example known already, due to Netzer, where $\overline{M} \ne M^{\ddagger}$.

Tuesday January 15:

Speaker: Dr. Victor Vinnikov, Ben Gurion University, Israel.

Title: Convex noncommutative rational functions (Part II)

Thursday January 10:

Speaker: Professor Danielle Gondard, Universite Paris 6

Title: Real Holomorphy Rings and the complete Real Spectrum (joint work with Muray Marshall)

Abstract:

After recalling known facts on the real holomorphy ring of a field, we present possible corresponding definitions for a ring, and deal with what we called "Real Holomorphy Rings" and "Complete Real Holomorphy Rings". Then we introduce and study a new notion of "Complete Real Spectrum".

Tuesday January 8:

Speaker: Dr. Victor Vinnikov, Ben Gurion University, Israel.

Title: Convex noncommutative rational functions (Part I)

Abstract:

I will discuss convex noncommutative rational functions, and show that their sublevel sets always admit a linear matrix inequality representation. In other words, highly unmanageable convex rational inequalities with noncommuting (matrix) variables can be always replaced by linear matrix inequalities. The proof uses a blend of techniques: noncommutative realization theory, which goes back to the work of Schutzenberger on finite automata in the early 1960s, and a variety of recent results which aim at developing function theory and algebraic geometry in the setting of free noncommuting variables.

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

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

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.

Page maintained by Salma Kuhlmann. Last Update: November 21 2008.