ABSTRACT: I will introduce recent work by Coecke, Pavlovic and Vicary on dagger-Frobenius algebras. In particular, I will show that there is a correspondence between dagger-Frobenius involutive monoids on a finite dimensional Hilbert space X and orthonormal bases on X. We shall see how this result allows us to describe quantum measurements in an arbitrary dagger-compact closed category and the implications of this for quantum information protocols.
ABSTRACT: A metric on the set of Borel probability measures on a metric space, introduced by Kantorovich in the early fourties, is shown to be the metric to model the computational effect of probabilistic nondeterminism. This metric gives rise to robust models, since small changes in the probabilities result in small changes in the distances.
ABSTRACT: My research project involves the study of digital representations for vectors. The most common digital representations are radix representations, such as the standard base 10 representation for real numbers. A umber, such as 567, has ones (7), tens (6), and hundreds (5) digits, and so on. This means that 567 is equal to 7 times 1, plus 6 times 10, plus 5 times 100. Each digit corresponds to a power of 10, thus 10 is called the base, or radix. The digits are chosen from a given digit set, such as 0, 1, ..., 9. There are many digital representations for real numbers, including continued fractions, beta transformations, Luroth series, and generalizations of these. In higher dimensions we have digital representations for vectors, including multidimensional continued fraction algorithms. My supervisor, Dr. Eva Curry, has also developed radix representations for vectors with integer entries. Here, the base is a square matrix with integer entries, and the digits are vectors. The inter-relations between different representations give important number theoretic and topological results about the behavior of certain transformations on vectors. Studying the digital representation of vectors will yield information about the topological properties of the set of "fractions". My goal is to determine the relationship between the dilation matrix and digit set used and topological properties of the resulting self-affine tile, such as connectedness or whether the tile is disc-like (topologically equivalent to a disc). In my presentation, I will summarize previous important results leading up to and which are relevant to our problem as well as outline recent results.
Rory Lucyshyn-Wright, York University, will give a talk entitled "Domains as Lax Algebras" at 2:30p.m. in N638 Ross.
ABSTRACT: The description of topological spaces as lax algebras of the filter monad yields a particularly effective interplay of order-theoretic and topological concepts. Thus, given also that the algebras of the filter monad over Set are the continuous lattices, we are drawn to seek application of the theory of lax algebras to domain theory, via the op-canonical extension of the filter monad. Strikingly, in this context the spaces described by lax algebras with strict multiplicative law turn out to be those which satisfy a certain domain-theoretic approximation property. Consequently, the sober spaces among these are in fact precisely the continuous dcpos under the Scott topology.
ABSTRACT: I will report on recent work with Chris Isham on the application of topos theory to physics. The Kochen-Specker theorem shows that a naive realist description of quantum systems is impossible. This can be understood as the inapplicability of Boolean logic to quantum systems. In order to arrive at a more realist description, one can use the internal logic of a certain topos of presheaves. The choice of this topos is directly motivated from the Kochen-Specker theorem. I will show which structures within this topos are of physical significance and how propositions about physical quantities are assigned truth-values. The whole topos scheme is not bound to quantum theory, but allows for major generalisations, potentially in the direction of a future theory of quantum gravity.
ABSTRACT: We show how the P-theorem in semigroup theory, due to McAlister, is related to the universal convering of a topos.
ABSTRACT: We propose a convenient theory, which can be used for reformulation of classical results about algebraic theories and varieties into the homotopy context. It follows the ideas of J. Rosicky used in the article On homotopy varieties, arXiv:math/0509655.
ABSTRACT: Model-Driven Engineering (MDE) is an approach to building complex software systems, where models rather than code are primary artifacts of software development. In MDE, code production is just the final (and presumably automatic and simple) step of a long and complex process of model operation: their building, integration, validation, transformation, refinement…. Calls for a basic theory of modeling and model management are a common place in the literature, and mathematics, as one of the oldest and deserved modeling disciplines, has much to offer to the community. One goal of the talk is to outline the mathematical basics of model management and show its essentially categorical nature. Somewhat surprisingly, the most abstract to date formulations of algebraic and logical concepts turn out really adequate to practical engineering needs. Another goal is to outline some interesting mathematical questions posed by MDE applications.
Bio: Zinovy Diskin received his PhD in math from the University of Latvia in 1994 and worked in industry and academia in Latvia and US. He has been dealing with models and modeling in different contexts: logic management in algebraic logic, view and schema integration in database design, metadata management in enterprise data integration. Recently he has been working in an IBM-CITO funded project on formal semantics for UML (one of industrial standard for MDE), and currently is with Model Merging Group at Dept. of Computer Science, UofT.
ABSTRACT: Algebraic topology can be done in the category of simplicial sets. Graph theory can be done in the related categories of presheaves. How to fit algebraic graph theory, especially the characteristic polynomial of adjacency matrices, into the categorical framework? I will sketch a possible approach, which uses a subcategory of coverings of graphs. Examples come from work of undergraduate strudents at Canisius College last summer.
ABSTRACT: We will describe the "basic distributive law" r:UD --> DU of Marmolejo, Rosebrugh, and Wood. Here D and U are the monads on the category ORD of ordered sets given by down- and up-closed subsets, respectively. This distributive law and its restrictions to ideals/filters or finitely generated downset/upsets governs many aspects of contemporary constructive order (enriched category) theory. As an example, we will show its relation to the problem of extending a functor on ORD to the category of order ideals.
ABSTRACT: It is well known that in a topological space, a filter converges to a point if and only if it is finer than the neighborhood filter of that point. Under certain hypotheses, this simple remark may be extended to lax algebras, and leads by way of the Kleisli category of the associated monad to a certain "neighborhood presentation" of the theory of lax algebras. A distinct advantage of this approach is that it does not explicitly require the use of a lax extension of the monad functor, but at the same time it suggests that an "associated lax extension" must always exist.
In this talk, we will present the correspondence between both approaches to the theory of lax algebras, suggest how they may benefit from each other, and discuss a number of related results.
ABSTRACT: Following the description by Manes [1] of the category of compact Hausdorff spaces as the Eilenberg-Moore category for the ultrafilter monad, Barr [2] showed that by weakening the axioms for a monad and the subsequent algebras, the Eilenberg-Moore category could be seen to be isomorphic to the category of topological spaces. In this talk, we will explain why the ultrafilter monad may be replaced by the filter monad in Barr's result without any loss of information, and how this may be related to the existing theory of lax algebras as presented in [3].
[1] E.G. Manes, "A triple theoretic construction of compact algebras",
Springer Lecture Notes in Math. 80 (1969) 91-118.
[2] M. Barr, "Relational algebras", Springer Lecture Notes in Math.
137
(1970) 39-55.
[3] M.M. Clementino, D. Hofman and W. Tholen, "One setting for all:
metric, topology, uniformity, approach structure", Appl. Cat. Struct. 12
(2004) 127-154.
This is joint work with B.B. Banaschewski and M.M. Ebrahimi.
ABSTRACT: Crossed modules were introduced by Whitehead in 1946 to study higher homotopy groups. They have been used successfully in various areas by many other authors.
We present the category of crossed modules and use them to generalize some well-known exact sequences of abelian groups to a class of nonabelian groups. A kernel-cokernel sequence is associated to a short exact sequence of crossed modules.
We study homology of crossed chain complexes and obtain a long exact homology sequence. Also homotopy can be defined for crossed chain maps in such a way that homotopic crossed chain maps induce the same homology maps.
With every crossed module we associate a categorical group. The kernel-cokernel sequence for crossed modules will be interpreted in this context.
Hans-E. Porst, Univeristy of Bremen, will continue to talk about "From varieties to convarieties" at 3:00p.m. in N638 Ross.
ABSTRACT: A concept of equation morphism is introduced for every endofuctor $F$ of a cocomplete category $\Ce$. Equationally defined classes of $F$--algebras for which free algebras exist are called varieties. Every variety is proved to be monadic over $\Ce$, and conversely, every monadic category is equivalent to a variety. And the Birkhoff Variety Theorem is proved for ``\Set--like'' categories.
By dualizing, we arrive at a concept of coequation such that covarieties, i.e., coequationally specified classes of coalgebras with cofree objects, precisely correspond to comonadic categories. Natural examples of covarieties are presented.
2:00p.m.: Tibor Beke, University of Michigan, will speak on ?b>Calculus of fractions and homotopy theory?
3:30p.m.: Jiri Rosicky, Masaryk University and York University, will speak on ?b>Left determined model categories and universal homotopy theories?
4:30p.m.: Walter Tholen, York University, will speak on ?b>Lax factorization algebras?
Everybody is welcome.
Professor Maria Manuel Clementino, Univeristy of Coimbra, Portugal, will give a talk entitled "On Triquotient Maps" at 2:30p.m. in N638 Ross.
ABSTRACT: Joyal and Tierney proved (AMS Memoirs, vol. 309) that a morphism of commutative monoids (in sup-lattices) is effective for descent iff it is pure. In this talk we want to characterize a certain class of effective morphisms, by showing that pure morphisms are effective for descent w.r.t. some cofibrations of accessible categories with pushouts.
George Janelidze, Georgian Academy of Sciences and York University, will give a series of lectures on "Semi-Abelian Categories" commencing today at 3:00p.m. in N638 Ross.
The lectures should be accessible to anybody with a background in algebra and/or algebraic topology.
ABSTRACT: There is a tendency in General Topology to study particular kinds of spaces much more intensively than special types of maps. In this talk we wish to demonstrate how beneficial it is to think of fundamental object notions as being induced by particular types of morphisms, and to study the interaction between both, the object and the morphism notions. Our "role model" is the class of closed maps of topological spaces which triggers a remarkably complete abstract theory of separation, compactness and perfectness. But applications are also to be found in Algebra which we shall discuss as well.
ABSTRACT: In this second, more technical, talk, I will discuss an action of the absolute Galois group of the rationals--a poorly understood group important in number theory--on a class of braided monoidal categories, and use it to construct a Galois action on knots and their invariants. This action was found recently by me and exploring its properties and consequences is one of my top priorities. I will assume the audience is familiar with the content of my previous talk.
Talk 1: The modern theory of knot invariants at 4:00p.m. in N638 Ross.
ABSTRACT: Work by Vaughan Jones in the 1980s related knot theory to more mainstream areas of math and initiated the current interest in knots and their invariants. His work was generalised and reinterpreted in the following years, and what has resulted is an elegant conceptual relationship between knot invariants and braided monoidal categories. This first talk reviews this relationship; it will be kept at an introductory level which everyone should be able to follow.
Back in 1963 Benabou introduced the notion of a monoidal category (a category with multiplication). Twenty years later Joyal and Street initiated the study of monoidal categories with a braiding. Surprisingly, their braided monoidal categories have played a starring role in the recent resurgence of interest n knot theory led by the work of Vaughan Jones. In my first of two talks, I will review the elegant relationship between braided monoidal categories and modern knot theory. In the second talk, I will discuss an action of the absolute Galois group of the rationals on a class of these categories, and use it to construct a Galois action on knots and their invariants. This action was found recently by me and exploring its properties and consequences is one of my top priorities (after teaching of course). Depending on the audience, I will aim to keep my talks at a more conceptual level which everyone should be able to follow.