COLLOQUIUM
of the
Mathematical Sciences Group
at the Department of Computer Science

Colloquium chair: Salma Kuhlmann

## Past Events 2002/2003

Friday, September 6, 2002, Room 207 ARTS
3:30 p.m.

Professor Salma Kuhlmann
Mathematical Sciences Group, Department of Computer Science

gave a talk on

## Lexicographic exponentiation of chains *

* This talk is dedicated to Felix Hausdorff, in memory of the 60th anniversary of his tragic death. Lest we forget.

Abstract:
In [H] Hausdorff developed several arithmetic operations on totally ordered sets, generalizing many aspects of Cantor's ordinal arithmetic. He investigated in [H] the basic properties of this arithmetic.

Many open questions arise naturally: In [K], we studied lexicographic powers with base the set R of real numbers. We investigated whether the exponent of the power is an isomorphism invariant:

Theorem: Assume a is an ordinal, and J a chain in which the chain R does not embed. Assume that Ra embeds in RJ. Then a embeds in J. In particular, if a and b are distinct ordinals, then the chains Ra and Rb are nonisomorphic.

This theorem is used in [W] to classify the convex congruences of such powers, and to characterize when these chains have a 2-transitive automorphism group. This extends results of [H].

On the other hand, after establishing further arithmetic rules, we provide in [HKM] examples of nonisomorphic chains G and G' such that the lexicographic powers RG and RG' are isomorphic. Moreover, for a countable infinite ordinal a, we show that Ra* + a and Ra are isomorphic (a* denotes the reverse of the ordinal a).

We show that RR and RQ are nonisomorphic.

We show that DR has a 2-transitive automorphism group, (where D is a countable ordinal).

Further related open questions arise while studying the question of defining an exponential function on a power series field: in [KKS2] we study convex embeddings of a chain G in a lexicographic power DG, and apply the results in [KKS1] to prove that (full) power series fields never admit an exponential function. However, in [KS] we consider proper subfields consisting of series of length bounded by an uncountable regular k. We show that these subfields can be naturally endowed with 2k pairwise distinct exponentials making them into models of real exponentiation. This is done by constructing lexicographic chains with many increasing automorphisms of distinct sigma-ranks.

References:

[Gr] Trevor Green: Properties of Chains products and Ehrenfeucht-Fraissé Games on Chains, Msc. Thesis, University of Saskatchewan 2002.

[H] Felix Hausdorff: Grundzuge einer Theorie der geordneten Mengen, Math Annalen 65 (1908)

[HKM] Charles Holland - Salma Kuhlmann - Stephen McCleary: Lexicographic Exponentiation of chains, preprint 2002

[K] Salma Kuhlmann: Isomorphisms of Lexicographic Powers of the Reals, Proc. Amer. Math. Soc. 123, Number 9, September 1995

[KS] Salma Kuhlmann - Saharon Shelah: k-restricted power series fields with 2k exponentials, (work in progress)

[KKS1] Franz-Viktor Kuhlmann - Salma Kuhlmann - Saharon Shelah: Exponentiation in power series fields, Proc. Amer. Math. Soc. 125, Number 11, November 1997

[KKS2] Franz-Viktor Kuhlmann - Salma Kuhlmann - Saharon Shelah: Functorial equations for lexicographic products, to appear in Proc. Amer. Math. Soc.

[W] Pamela Warton: Lexicographic Powers over the Reals, Ph. D. Dissertation, Bowling Green State University 1998

Friday, September 13, 2002, Room 207 ARTS
3:30 p.m.

Professor Olivier Piltant
Université de Versailles, St Quentin, France

will talk on

## Zariski's theory of complete ideals

Abstract:
This story begins with the fundamental theorem of algebra: given complex numbers z1, ... , zn and positive integers a1, ... , an, any polynomial P with complex coefficients vanishing at z1, ... , zn at order at least a1, ... , an factors as
P(z) = Q(z).(z - z1)a1...(z - zn)an
for some polynomial Q.

Zariski's theory of complete ideals was begun in 1938 in order to provide an algebraic framework for studying birational correspondences in algebraic geometry. His famous unique factorization theorem' for complete ideals in dimension two can be viewed as a local extension in two variables of the above theorem. It states that any ideal defined by assigned tangency conditions at the origin of C2 factors in a unique way as a product of ideals defined by simple assigned tangency conditions. A basic example goes as follows: an algebraic curve in C2 which is smooth and tangent to the x-axis at the origin has an equation of the form

P(x,y) := Q1(x,y)y + Q2(x,y)x2 = 0,
where Q1 and Q2 are polynomials with Q1(0,0) not equal to 0. Similarly for an algebraic curve which has an ordinary cusp tangent to the y-axis at the origin whose equation has the form
P'(x,y) := Q'1(x,y)x2 + Q'2(x,y)xy2 + Q'3(x,y)y3 = 0,
where Q'1, Q'2, Q'3 are polynomials with Q'1(0,0), Q'3(0,0) not equal to 0. A very special case of Zariski's theorem is that the equation of any algebraic curve having one smooth branch tangent to the x-axis and an ordinary cusp tangent to the y-axis at the origin belongs to the product ideal I:=(y,x2)(x2,xy2,y3).

In this talk, I will present Zariski's theorem and Lipman's theory of finitely supported complete ideals which extends Zariski's theory to higher dimensions. My contribution consists in stating and proving a three dimensional version of Zariski's unique factorization theorem. This result draws an interesting connection with the Abhyankar-Moh theorems around embeddings of the complex affine line into the complex affine plane.

Professor Piltant is visiting the Research Unit "Algebra and Logic" for the month of September 2002.

Friday and Saturday, September 20 and 21, 2002

### Saskatchewan Algebra and Number Theory Mini-meeting University of Regina Main Campus

Friday, September 27, 2002, 3:30 p.m.

Professor Murray Bremner
Department of Math & Stats

gave a talk on

## Quantization of Lie and Jordan triple systems

Abstract:
This talk will show how the decomposition of the group ring of the symmetric group into a direct sum of full matrix subrings can be used to give a complete classification of n-ary operations. Roughly speaking, row equivalence of matrices corresponds to quasi-equivalence of operations. In particular, the Lie and Jordan products represent the two non-trivial quasi-equivalence classes of binary operations. For ternary operations, there are infinitely many quasi-equivalence classes, which divide into eight classes, and four infinite families of classes each with a single parameter. The Lie triple product is contained in one of the infinite classes, and the other operations in the class can be regarded as quantizations of that product. Similar remarks apply to the Jordan triple product. For special values of the parameter, the operation satisfies an identity of degree 5. This identifies some new ternary operations which define varieties of triple systems, similar to Lie and Jordan triple systems, which seem to be an interesting direction for further research.

This is the talk I gave at the International Workshop on Polynomial Identities in Algebras at Memorial University (August 29 - September 3, 2002) and at the Conference on Topics in Linear Algebra at Iowa State University (September 12-13, 2002). The slides are available in pdf format on my website at

Friday, October 4, 2002, 3:30 p.m.

Professor Keith F. Taylor
Department of Math & Stats

gave a talk on

## Type structure of von Neumann algebras generated by discrete groups

Abstract:
The theory of continuous unitary representations provides some of the most useful approaches to analysis of, and on, locally compact topological groups. The theory is strikingly successful in classes such as semisimple Lie groups, reductive algebraic groups, connected nilpotent groups and compact groups. This success rests on the fact that, for any group in each of these classes, the set of equivalence classes of irreducible unitary representations can be parametrized with a reasonable parameter space. By a theorem of Glimm, such a parametrization exists iff the group is Type I (that is, every representation of the group generates a Type I von Neumann algebra).

If the topology on the group is actually discrete (effectively there is no continuity restriction on the representations), then Thoma proved that the group is Type I iff it has an abelian subgroup of finite index. Thus, for discrete groups, the unitary representation theory is either essentially trivial or it is intractable. In the talk, these concepts will be introduced and a somewhat elementary proof of a stronger version of Thoma's theorem, based on the use of polynomial identities, will be sketched.

Friday, October 18, 2002, 3:30 p.m.

Dr. Mikhail Kotchetov
Research Unit Algebra and Logic, U of S

gave a talk on

## Formal power series and identities of hyperalgebras

Abstract:
We consider the problem of determining when a cocommutative Hopf algebra is PI, i.e., satisfies a nontrivial polynomial identity (as an algebra). By the classical decomposition theorem of Kostant-Cartier-Gabriel, any pointed cocommutative Hopf algebra can be represented as the smash product of a group algebra and a connected Hopf algebra. A well-known theorem of Passman gives necessary and sufficient conditions for group algebras to be PI. We will look at the other factor of the decomposition, namely, at connected cocommutative Hopf algebras, which we call "hyperalgebras" for brevity.

In the case of zero characteristic, hyperalgebras are just universal envelopes of Lie algebras, which are PI only when commutative (by a result of Bahturin and Latyshev). In the case of prime characteristic, however, there are hyperalgebras other than universal or restricted envelopes. An extensive class of such hyperalgebras are the so called (co)reduced hyperalgebras, which are characterized by the property that their dual is isomorphic to a power series algebra. They are also precisely the hyperalgebras that arise from formal group laws by (a version of) Lie theory for prime characteristic. So it is not surprising that for coreduced hyperalgberas, we have the same result as for universal envelopes in characteristic zero: they are PI only when commutative. The proof is based on using the corresponding formal group law to construct a certain group and applying Passman's theorem to its group algebra.

Mikhail Kotchetov defended his Ph.D. at the Memorial University of New Foundland in September 2002. He is now a post-doctoral fellow of the Research Unit "Algebra and Logic" (2002--2003). Misha is co-supervised by S. Kuhlmann, M. Marshall and K. Taylor.

Friday, November 1, 2002, Room 207 ARTS
5:00 p.m.

Dr. Mikhail Kotchetov
Research Unit Algebra and Logic, U of S

gave a talk on

## Formal power series and identities of hyperalgebras II

Friday, November 15, 2002, 5:00 p.m.

Dr. Roland Auer
Research Unit Algebra and Logic

gave a talk on

## Ray class fields of global funcion fields, and curves with many points

Abstract:
Curves over finite fields with many rational points have their applications in Coding Theory and to (Quasi-)Monte Carlo Methods.

While there are theoretical upper bounds for the number of rational points on a (smooth, projective, absolutely irreducible, algebraic) curve of a given genus over a fixed finite field, establishing meaningful lower bounds would actually involve constructing curves with many points.

In the present talk, we will consider abelian covers of explicitly given low genus curves in which many of the rational points are forced to split. The existence of these covers, their degree and the genus of the covering curve is derived from (the function field case) of global Class Field Theory, a classical topic in Algebraic Number Theory.

By construction, the covering curves are expected to have many rational points, and some of them actually beat previously known records.

Friday, November 22, 2002, 3:30 p.m.

Professor Murray Bremner
Department of Mathematics and Statistics

gave a talk on

## Using linear algebra to discover the defining identities for Lie and Jordan algebra

Abstract:
In any associative ring R with product xy, we can define two new operations: the Lie bracket [x,y] := xy-yx and the Jordan product x o y := (xy+yx)/2. It is easy to check directly that the Lie bracket satisfies anticommutativity and the degree-3 Jacobi identity

[x,x] = 0, [[x,y],z] + [[y,z],x] + [[z,x],y] = 0,

and that the Jordan product satisfies commutativity and the degree-4 Jordan identity

y o x = x o y, ((x o x) o y) o x = (x o x) o (y o x).

Most textbook accounts of nonassociative algebras merely state these identities and verify them directly, without showing how they could have been discovered in the first place. This talk will show how elementary combinatorics and linear algebra can be used to find these identities given only the definitions of the Lie bracket and Jordan product. This method generalizes naturally to much larger and harder problems in the study of identities for nonassociative algebras, in which computing the row canonical form of a large matrix (at least one million entries) and the representation theory of the symmetric group can be used to discover new identities. This talk was originally given at the Canadian Undergraduate Mathematics Conference in Calgary last July, and hence should be accessible to a broad audience.

Friday, November 29, 2002, we had two talks:

at 3:30 p.m.,

Professor Mik Bickis
Mathematical Sciences Group, Department of Computer Science

gave a talk on

## Why do they all come at once? Reflections on the distribution of spacings

Abstract:
An empirical situation for observing spacings would occur if one were to note the lengths of time between consecutive arrivals at the finish line of a marathon. The distribution of such spacings tends to be "heavy-tailed", meaning that one would expect many short times with occasional very long waits. It has long been known that for uniformly distributed random variables, the distribution of spacings is asymptotically exponential. In 1965, Pyke published a key paper examining the probabilistic aspects of spacings in a general framework.

Although the explicit distribution of spacings is complicated, limit theorems can be obtained when the number of observations goes to infinity. This so-called asymptotic spacing distribution can be described by the Laplace transform of a distribution derived from the parent distribution. Tauberian theorems then allow one to relate the tail of the spacings distribution to the order of contact to the x-axis of the parent density function. Polynomial and exponential parent densities lead to a negative-power tail for the spacings distribution whereas a Gaussian density gives a spacings distribution with a heavy tail that is not a negative power.

These ideas will be illustrated with computer-simulated examples.

at 4:30 p.m.,

Professor Murray Marshall
Mathematical Sciences Group, Department of Computer Science

gave a talk on

## The Elementary Type Conjecture

Abstract:
The systematic study of quadratic forms over an arbitrary field of characteristic not equal to 2 was initiated by Witt in 1937. Two milestone papers in the area are the paper of Pfister relating quadratic forms and orderings and the paper of Milnor pointing out a possible relationship between the Witt ring of quadratic forms and the Galois cohomology of the field - a relationship which was later verified by Voevodsky.

At the same time, it was noted early on that different fields can be quadratically equivalent' in the sense that their quadratic form theory is `the same'. In particular, for fixed n >= 0, there are only a finite number of quadratically inequivalent fields K with |K*/K*2| = 2n. The talk is a survey of the attempts to classify these.

Friday, December 6, 2002, 3:30 p.m.

Tzvetalin Vassilev
Department of Computer Science

gave a talk on

## Algorithms for Efficient Enumeration of Triangulations of a Planar Point Set

Abstract:
Triangulations of the planar point sets was one of the defining areas of the field of Computational Geometry in early 70's. At the same time it is one of the most active research areas within Computational Geometry nowadays with hundreds of papers published every year. Enumeration of objects/structures is a classical combinatorial problem that has its roots in the works of ancient mathematicians.

In the last decade there was a substantial interest in the enumeration of the triangulation of a given planar point set. Considering the fact that the number of different triangulations of a set of n points is exponential in n (currently best lower and upper bounds are 23n and 28n), this is a challenging problem from the algorithmic point of view, and the emphasis is on discovering the efficient algorithm for enumeration of triangulations. The talk will overview three of the algorithms that have been developed:

- Reverse search (Avis and Fukuda, 1996)
- Path of the triangulation, based on Divide-and-Conquer technique (Aichholzer, 1999), and
- The triangulation tree (by Bespamyatnikh, 2001).

Timing and bounds issues will be discussed, and the speaker will provide a collection of open problems in this area.

Friday, January 10, 2003, 3:30 p.m.

Jingxiang Luo
Department of Computer Science

gave a talk on

## The Null Value Problem and Effective Simulation of Level/Phase Process

Abstract:
The null value problem is as follows: given a series of matrices ${Q_j}, -g \le j \le h$, where $g,h$ are natural numbers, each $Q_j$ is of size $m \times m$, what is the solution of the following equation?

\vec{0} = \vec{f} \cdot \sum_{-g \le j \le h} x^{-j} Q_j

For any solution, $x$ is called a null value (also the generalized eigenvalue), and $\vec{f}$ the associated null vector. A level/phase process is a Markov process that satisfies the condition of level invariance. It is stable if descending to a lower level is always more likely than ascending to a higher level. In a stable level/phase process, reaching a very high level will be a rare event. The effective simulation concerns performance metric that depends heavily on some rare occurring event. The usual Monte Carlo simulation will be very inefficient due to the lack of observation of rare event of interest. Hence, effective simulation is desired.

There are classical results that reveal the link of the solution of the null value problem with some asymptotics of the equilibrium distribution of a stable level/phase process. The new result is that the solution of the null value problem can also be employed to derive a fast simulation strategy "rate tilting", based on the importance sampling principle in variance reduction techniques. In this talk, we will show the effectiveness of rate tilting under some conditions, and compare this strategy with others. Finally, we address how to find the solution of null values and null vectors with algorithms of low computation complexity, or pilot simulations.

Friday, January 17, 2003, Room 207 ARTS
3:30 p.m.

Professor Bruce Watson
University of Witwatersrand, South Africa (presently visiting U of S)

gave a talk on

## Isospectral transformations between Sturm-Liouville problems

Abstract:
The Crum-Darboux transformations is considered. It will be illustrated that it isospectral maps certain classes of non-standard Sturm-Liouville problems to standard Sturm-Liouville problems. It will also be displayed that it not only preserves the spectrum but also the so called norming constants.

Friday, January 24, 2003, Room 207 ARTS
3:30 p.m.

Professor John Martin
Department of Mathematics & Statistics

gave a talk on

## Fixed Point Sets of Spaces

Abstract:
A topological space X is said to have the complete invariance property (CIP) if every nonempty closed subset of X is the fixed point set of a self-mapping of X . If we replace "self-mapping" by "autohomeomorphism" in the definition, then we say that X has the complete invariance property with respect to homeomorphisms (CIPH). Some conditions which ensure that a class of spaces have CIP or CIPH will be discussed.

## January 31, 3:30-5:30 and February 1, 9:45-4:30The Second Saskatchewan Mathematics Mini-meeting

Friday, February 28, 2003, 3:30 p.m.

Warren Code

gave a talk on

## Examples of Sturm-Liouville problems with eigenparameter-dependent boundary conditions

Abstract:
There have been several Colloquium talks examining the modified Sturm-Liouville (S-L) problem:

-(py')' + qy = kry
y'(0)/y(0) = cot(t)
y'(1)/y(1) = R(k),

where p > 0, r > 0 and q are real-valued functions on the interval [a,b], t is in the interval [0,p), and R(k) is a function of the eigenparameter (my thesis focusses on the case R(k) = Ak2 + Bk + C in particular).

This talk will discuss a few motivating examples for the study of such problems. These will include selected examples from heat transfer, vibration and acoustic phenomena. It will be demonstrated how physical conditions at the boundary may be modified such that an eigenparameter-dependent condition replaces the constant one produced in elementary treatments.

Friday, March 7, 2003, Studio B, Division of Media and Technology, Education Building
3:30 p.m.

Professor Franz-Viktor Kuhlmann
Research Unit Algebra and Logic

gave a talk on

## The mathematical background of asymmetric cryptosystems

Abstract:
Cryptography is used to safeguard information that is stored or sent through channels like the internet. Using asymmetric cryptosystems, one can make the encryption procedure public while keeping the corresponding decryption algorithm secret. Alternatively, one can use an asymmetric cryptosystem to safely exchange keys for symmetric cryptosystems (which in general are more effective), and then send the information using the latter.

I will give a survey on the algebraic and number theoretical methods used for asymmetric cryptosystems. I will explain the basic idea of asymmetric cryptosystems. Then I will describe the Merkle-Hellman Knapsack cryptosystem, RSA, the discrete logarithm problem, Diffie-Hellman key exchange and the El Gamal cryptosystem. Finally, I will quickly describe the use of elliptic curves in connection with the discrete logarithm problem. If time permits, I will also discuss possible attacks on these cryptosystems.

The talk was broadcasted to the University of Regina. I would like to thank the Division of Media and Technology at the U of S for their generous support, and Cheryl Piché and Frank Harrington for their invaluable help.

### Transparencies for this talk (pdf file)

Tuesday, March 18, 2003, 4:00 - 5:20 p.m.

Professor Sarah Witherspoon
Department of Mathematics and Computer Science, Amherst College

gave a talk on

## Algebraic Deformation Theory

Abstract:
A deformation of a ring is a one-parameter family of rings, the original ring corresponding to one value of the parameter. Such deformations give rise to quantum groups from groups and Lie algebras, and to noncommutative spaces from ordinary spaces. However, deformations are notoriously difficult to find in general. In this introductory talk, we will start with the definition of a deformation and some examples, and then we will describe Hochschild cohomology and its relationship to deformations. Finally we will explain some recent theoretical results and apply them to new examples arising from finite group actions on spaces, that is orbifolds. This talk will be accessible to an audience that would include interested undergraduate and graduate students.

## March 21 and 22The Fourth Annual Colloquiumfest

### Talk in the Computer Science Seminar Series

Monday, March 24, 2003, 3:30 p.m.

Professor Klaus Keimel

gave a talk on

## Ordered cones, finitary approximation, semantics of probabilistic phenomena in programming

Abstract:
For the purposes of denotational semantics, Dana Scott has introduced the notion of a semantic domain (see [1]). These are ordered sets which are directed complete and in which every object can be approximated from below by elements of finite type. The order is supposed to model the increase of information content, the elements of finite type should represent objects computable in finite time, and the directed completeness assures the presence of ideal objects but still finitarily approximable.

Each construct in a programming language asks for a construction on semantic domains which models this construct. Non-determinism - like nondeterminstic choice - is modelled by sets of possible results. It is mathematically challenging that the constructions on semantic domains should remain within the same category.

Probabilistic features - like probabilistic choice - are modelled by the probabilistic power domain.

If both kinds of non-determinism occur (see [2]), there is a need for a mixed power domain. After introducing semantic domains, we will present an approach to this mixed power domain based on [3], and we will illustrate it by giving a semantics for the toy language considered in [2].

Literature:
1. G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott: Continuous Lattices and Domains, Encyclopaedia of Mathematics and its Applications, vol. 93, Cambridge University Press (2003), xxx+590pp.
2. A. McIver and C. Morgan: Partial correctness for probabilistic demonic programs, Theoretical Computer Science 266 (2001), 513-514.
2. R. Tix: Continuous D-cones: Convexity and Powerdomain Constructions, PhD Thesis, Darmstadt (1999).

Friday, March 28, 2003, 3:30 p.m.

Professor Robert Fitzgerald
Southern Illinois University

gave a talk on

Abstract:
For a non-degenerate quadratic form q over the reals, we have:

(1) q has a non-trivial zero iff |sgn q | < dim q.
(2) q has the maximal number of zeros iff sgn q =0.

Here dim q is the number of variables and sgn q is the signature. I will discuss recent attempts to extend some form of (1) and (2) to other fields. For example, in (1), the inequality is replaced by a new inequality involving various invariants of the underlying field.

Dr. Fitzgerald was visiting the Research Unit Algebra and Logic from March 17 to March 31.

Friday, April 4, 2003 - Double Colloquium:
3:30 p.m.

Leipzig, Germany

gave a talk on

## The Classical Moment Problem and Positive Polynomials

Abstract:
The talk begins with an overwiew on the history and on classical results of the moment problem and the theory of positive polynomials. Then the moment problem for closed semi-algebraic sets of the d-dimensional space is discussed. The talk closes with some new results on the moment problem for non-compact closed semi-algebraic sets which have "sufficiently many" bounded polynomials.

Dr. Schmuedgen was visiting the Research Unit Algebra and Logic April 1st-8th.

4:30 p.m.

Professor Frank Sottile
University of Massachusetts, Amherst, USA

gave a talk on

## Enumerative Real Algebraic Geometry

Abstract:
Consider the following hard problem: Give meaningful information about the number r of real solutions to a system of multivariate polynomial equations. Quite often much more is known than merely that r is between 0 and the number d of complex solutions. This stronger information is typically available when the system has some geometric structure. Enumerative real algebraic geometry treats this hard problem for systems coming from geometry.

In this talk, I will survey some of what is known. In particular, I will discuss recent better understanding of the bounds of 0 and d. A picture is emerging: for systems from geometry, the upper bound of d can always be obtained, and in many cases there are non-trivial lower bounds on the number of real solutions, with certain gaps that have been discovered experimentally.

Wednesday, April 9, 2003, 3:30 p.m.

Dr. Mohammed El Bachraoui
Vrije Universiteit, Amsterdam, The Netherlands

gave a talk on

## Representation theorems for relation algebras

Abstract:
I will discuss the development of relation algebras from A. de Morgan until A. Tarski: from concrete binary relations to the abstract theory of relation algebras. The standard model of relation algebras is the algebra of binary relation on some set. These algebras and their subdirect product are referred to as representable relation algebras. Unlike Boolean algebras and groups, R. Lyndon showed that not every relation algebra is representable. As a consequence of this fact, representation theorems are worth investigating.

In the talk I will give a new representation theorem for relation algebras. Combining this result with a result of R. Maddux and a result of S. Givant, we arrive at a more general representation theorem.

I will finally list some open problems which are suggested by this work.

Dr. El Bachraoui was visiting the Research Unit Algebra and Logic March 31 - April 29.

Friday, April 25, 2003, Room 208 ARTS
3:30 p.m.

Professor Salma Kuhlmann
Research Unit Algebra and Logic

gave a talk on

## Representation of Polynomials Positive on Subsets of the Real Line, with Applications to the Multidimensional Moment Problem

Abstract:
We analyse the preorderings and quadratic modules of R[X] associated to semi algebraic subsets of the real line. We give criteria, in terms of the chosen description of the subset, for the preorderings or modules to be closed or saturated.

The results are used to investigate the solvability of the Moment Problem in higher dimensions. We provide criteria for the existence of a positive solution to the Moment Problem on subsets of cylinders with compact base. We apply these results to the representation of polynomials positive on non-compact polyhedra. We present open problems concerning the connection between various representations and the Moment Problem.

(Joint Work with M. Marshall and Niels Schwartz.)

Friday, May 2, 2003 - Double Colloquium:
3:30 p.m.

Professor Renate Scheidler
Dept. of Mathematics & Statistics, University of Calgary

gave a talk on

## How to exchange secrets - communication of secret cryptographic keys

Abstract:
Conventional (one-key) cryptographic systems, such as the Data Encryption Standard (DES) or the new Advanced Encryption Standard (AES), are the secure communication schemes of choice for many institutions. This is because they are both fast and sufficiently secure for most applications. The real difficulty in employing such cryptosystems is the problem of securely transmitting a secret cryptographic key between communicants. This talk describes a solution to this problem -- a means by which a secret key can be safely transmitted across an insecure channel. Our key exchange protocol is based on the algebra and arithmetic of reduced principal ideals in a real quadratic number field; the technique can be extended to real hyperelliptic function fields.

This research was conducted in collaboration with M. J. Jacobson, Jr., and H. C. Williams, both at the University of Calgary. Despite the algebraic and number theoretic nature of the topic, this presentation is designed to be accessible to a general mathematics-trained audience.

Professor Scheidler's visit was partially supported by the Role Model Speaker Fund of the College of Arts and Science.

4:30 p.m.

Dr. Nikolaos Galatos
Vanderbilt University, Nashville

gave a talk on

## Decision problems on commutative distributive residuated lattices

Abstract:
Residuated lattices constitute algebraic semantics for sub-structural logics. Brouwerian algebras and Abelian lattice-ordered groups are examples of residuated lattices that have a distributive lattice and a commutative monoid reduct.

We investigate the decidability of the equational and quasi-equational theory of the variety CRL of distributive commutative residuated lattices. In particular, we reduce the solvability of the word problem for CRL to that of semigroups, known to be unsolvable, thus obtaining the undecidability of the quasi-equational theory as a corollary. Moreover, in joint work with James Raftery, we establish the decidability of the equational theory of CRL, using results from relevance logic.

Friday, May 9, 2003, 3:30 p.m.

Professor Brett Stevens
School of Mathematics and Statistics, Carleton University

gave a talk on

## Mathematical software testing: an exploration of diverse viewpoints on an applied problem

Abstract:
Recently mathematical objects called covering arrays have been successfully used in reliability testing to great advantage. These objects, under many names, have been studied for over 100 years. We will examine their application strengths and the many approaches used to study and construct them. This includes extremal set theory: Sperner systems and the Erdos-Ko-Rado result; information theory approaches producing asymptotic results; constructions motivated from latin squares and design theory and, recently, more industrially relevant models using graph homomorphisms and meta-heuristic searches. The talk will attempt to give the breadth of flavors of approaches that are used to study one simply defined object.

Professor Stevens' visit was supported by the Colloquium Fund of the Mathematical Sciences Group.

Friday, May 16, 2003, 3:30 p.m.

Professor Melvin Henriksen
Harvey Mudd College, Claremont, California

gave a talk on

## What we know and do not know about the space of minimal prime ideals of a ring of continuous functions

Abstract:
The space of minimal prime ideals of a commutative semiprime ring A with the hull-kernel topology (alias the Zariski topology) has long been known to be a zero-dimensional Hausdorff space and there are characterizations of when it is compact. In case A = C(X) is the ring of all continuous real-valued on a Tychonoff space (i.e., a subspace of of a compact (Hausdorff) space), while there are many simply stated characterizations of spaces X with this property, they are not easy to use to answer natural questions about them. For example: What can be said about subspaces, products, or images under various kinds of continuous mappings of spaces with a compact space of minimal prime ideals? Recent results obtained jointly with R.G. Woods will be presented along with some unsolved problems.

### Departmental Research Seminars

Last update: May 20, 2003 --------- created and maintained by Franz-Viktor Kuhlmann