Michael Johnson,

*Oriented combinatorial topology and the specification of concurrent systems in the style of higher dimensional category theory*

Tuesday October 7, 2003,

Hongbin Cui, *Multiple-Categories, $\omega$-Complexes and Globes
*

**Abstract**: In this talk, I'll explain the concept of $\omega$-complexes, a
type of pasting diagrams representing compositions in multiple
categories. Then I'll describe some of my research on the product
structure of $\omega$-complexes.

Tuesday November 25, 2003,

Tony Thompson, *Duality in convexity: help still needed*

**Abstract**: In a paper in 1988, Erwin Lutwak gave a dictionary that included the
following correspondences:
Convex set/star-shaped set; support function/radial function; Minkowski
addition/radial addition; projection/cross-section; projection
body/intersection body; mixed volume/dual mixed volume.

This suggests a duality but, if so, it is not at all understood. I would like to look at more elementary duality in convexity theory and then explain the dictionary in the hope of throwing some light on the nature of the correspondence.

Tuesday January 20, 2004,

Yuri Bahturin, *Group Gradings on Algebras with Involution and Applications
*

Tuesday February 17, 2004,

Mitja Mastnak, *On Bimeasuring*

Tuesday March 2, 2004,

Bob Paré, *All About Span I: Basics*

**Abstract**: In this series of lectures I will discuss my joint work
with Robert Dawson and Dorette Pronk on the relationship between
the Span construction and the free adjoint construction.

The first talk will be basic definitions and motivation. No prerequisits (well almost none).

Tuesday March 10, 2004,

Bob Paré, *All About Span II: The Universal Property*

**Abstract**: Continuing last week's talk, I will discuss the universal
property of the Span construction, which will be generalized a few
times in upcomming talks.

Tuesday March 10, 2004,

Bob Paré, *All About Span III: The Improved Universal Property*

**Abstract**: Continuing last week's talk, I will introduce the free adjoint
construction \Pi_2 and compare it to Span thereby getting a better
universal property of Span.

Tuesday March 23, 2004,

Chris Kao, *Making Connections Between Fractal Trees and General
Iterative Function Systems*

**Abstract**: Fractal geometry is a relatively new area of mathematics.
One of the most striking characteristics of a fractal is its
self-similarity.Self-similarity enables us to investigate the general IFS
at ease for the complexity of these mathematical objects are the same at
every scale. One common way of defining fractals is as attractors of
iterated function systems, consisting of a finite set of contracting
mappings. The Collage Theorem describes how the self-similarity of the
fractal can be translated into the functions of such a system. In this
talk, we will discuss a certain type of "unorthodox fractals" called
fractal trees. A fractal tree is not strictly self-similar when one
examines the entire tree. However, if one were to just consider the tip
set, $J$, of the tree, one could see that $J$ is self-similar. A natural
question arised as to which IFS could be obtained from the tip sets of
fractal trees. We will demonstrate that by using the parameters of a given
tree to construct an IFS such that $J$ is the attractor and thus answer
the question. More specifically, we could answer this question for IFS
containing two contractions by looking at binary fractal trees. Lastly, we
will state some conjectures for IFS with more contractions.

Tuesday March 30, 2004,

Bob Paré, *All About Span IV: Paranormal Morphisms*

**Abstract**: I will introduce the notion of paranormal oplax morphism of
bicategories and give a better universal property of Span.

Tuesday March 30, 2004,

Bob Paré, *All About Span V: Paranormal Morphisms*

**Abstract**: This time I really will introduce the notion of paranormal oplax
morphism of bicategories and give a better universal property of Span.

Tuesday April 13, 2004,

Thomas Guedenon, *Localisation and catenarity in rings with f
initely generated nilpotent group action*

**Abstract**: Let G be a finitely generated nilpotent group acting on an
associative noetherian ring R and A=R*G the crossed product of R by
G. We will show that if R is G-hypercentral, then every G-prime ideal
P of B=R*G_i is classically localisable (here G_i is a term of a
composition series of G). If G is torsionfree, under some additional
hypotheses on the G-prime ideals of R, we will show that the localisation
B_P of B with respect to P is a regular semi-local ring. These results
will enable us to study the catenarity of A=R*G.

Tuesday June 1, 2004,

Richard Wood, *Generalizing the CCD lattice characterization theorem*

**Abstract**: In `CCD 4' Bob Rosebrugh and I proved that the full subcategory of
`sup-lattices' determined by the CCD lattices is equivalent to the
idempotent splitting completion of the bicategory of sets and relations.
(Much earlier Raney had proved some of our results, but without mention
of categories, for CD lattices.) The result was extremely useful in that
it had many corollaries.

Almost from the outset, we felt that our theorem was but a small specialization of a very general result. However, until recently even a precise statement, let alone a proof eluded us. We have now found both in a context that we had completely overlooked.

Let D = (D,d,m) be a monad on a category C in which idempotents split. Write kar(C_D) for the idempotent splitting completion of the Kleisli category. Write spl(C^D) for the category whose objects are pairs ((L,s),t), where (L,s) is an object of C^D, the Eilenberg-Moore category, and t:(L,s)--->(DL,mL) is a h o m o m o r p h i s m that splits s:(DL,mL)--->(L,s), with spl(C^D)(((L,s),t),((L',s'),t'))=C^D((L,s)(L',s')).

Theorem: kar(C_D) =~ spl(C^D)

In the talk I will prove this theorem, explain how it implies the CCD lattice characterization theorem, and time permitting, ramble on about related matters.