Mathematics
of Materials and Macromolecules: Multiple Scales, Disorder,
and Singularities, September 2004 - June 2005
Abstracts:
IMA 2004 Summer Program:
June
7-18, 2004
Photo
Gallery Material
from Talks Impromptu
Talks
John
Baez (Department of Mathematics,
University of California at Riverside) baez@math.ucr.edu
http://www.math.ucr.edu/home/baez/
Why n-Categories?
Slides: pdf
ps
Research on n-categories is revolutionizing
our concept of mathematics by teaching us how to think of every
interesting equation as the summary of an interesting process.
Here we sketch how this approach leads to a new understanding
of even the simplest mathematical structures: in particular,
of the natural numbers. A topological study of the origin of
the natural number concept suggests that in the "true natural
numbers", addition satisfies an infinite hierarchy of coherence
laws for associativity, an infinite hierarchy of coherence laws
for commutativity, together with an infinite hierarchy of further
hierarchies of coherence laws. All these are built into the
concept of the "free k-tuply monoidal n-category on one generator",
and this should admit a description as the n-category of "n-braids
in codimension k." The objects here are elements of the space
of finite subsets of k-dimensional Euclidean space. The morphisms
are paths in this space, the 2-morphisms are paths of paths
in this space, and so on, with the n-morphisms being homotopy
classes of paths of paths of paths... in this space. Similarly,
the integers should be related to the n-category of "n-tangles
in codimension k."
John
Baez (Department of Mathematics, University
of California at Riverside) baez@math.ucr.edu
http://www.math.ucr.edu/home/baez/
What n-Categories Should Be Like
Slides: pdf
ps
We describe features that any useful theory
of n-categories should have. In particular, there should be
three specially nice sorts of n-categories: k-tuply monoidal
n-categories, n-groupoids and strict n-categories. We describe
the effects of imposing these extra conditions separately and
in combination, and conjecture the existence of a weakly commutative
cube of free and forgetful (n+1)-functors relating the resulting
eight classes of n-categories. Among other things, this cube
would give an explanation of the relation between n-categories,
homotopy theory, stable homotopy theory, and homology theory.
John
Baez (Department of Mathematics, University
of California at Riverside) baez@math.ucr.edu
http://www.math.ucr.edu/home/baez/
Space and State, Spacetime and Process
Slides: pdf
ps
General relativity may seem very different from
quantum theory, but work on quantum gravity has revealed a deep
analogy between the two, which is best understood using category
theory. General relativity describes space and spacetime in
terms of objects and morphisms in nCob, the category of n-dimensional
cobordisms. Quantum theory describes states and processes using
objects and morphisms in Hilb, the category of Hilbert spaces.
The analogy between general relativity and quantum theory is
made precise by the fact that both nCob and Hilb are "symmetric
monoidal categories with duals". Work on string theory and loop
quantum gravity suggests that the analogy goes deeper, in a
way that is best understood using n-categories. The "TQFT hypothesis"
is a preliminary attempt to make this precise.
Eugenia
Cheng (Department of Pure Mathematics
& Mathematical Statistics, Cambridge University) elgc2@cam.ac.uk
Multicategories and Related Definitions
and Opetopic Definitions
General: In
these talks I intend to give a brief overview and then let the
pace be set by the audience. I won't be producing detailed notes
because the details will depend heavily on what the audience
wants at the time. However, general notes can be found in Higher-Dimensional
Categories: an illustrated guidebook, Chapters 2, 3 and
4. This is available from http://www.dpmms.cam.ac.uk/~elgc2/guidebook
Multicategories and related definitions:
Tom Leinster will begin with an introduction
to generalised multicategories and operads. I will then explain
related definitions that use the notion of algebraic structure
on underlying data consisting of "globular" cells. These definitions
are expressed in the form: ``An $\omega$-category is an algebra
for a certain monad/operad". I will explain the various components
of the definition as necessary - I envisage this including magmas,
contractions and the construction of the initial operads in question,
possibly globular sets and probably not monads and algebras. (A
crash course on monads and algebras can be found in the Guide
Book, Section 2.4. I will be happy to talk about it informally
beforehand if anyone needs it.)
Opetopic definitions:
My aim in this talk will be to convey the idea of the opetopic
definitions while avoiding as much of the technical detail as
possible. Experience has shown that this is highly expedient
for those wishing to get a feel for the theory without necessarily
wanting to calculate with it. I will explain the ideas behind
the theory and the various components of the definition: opetopes,
opetopic sets, niches, and universal cells. I will show how
to recover a classical bicategory from an opetopic 2-category
as this tends to help shed a great deal of light on the various
components of the definition. If there is time I will discuss
various notions of strictness and points of comparison with
other definitions.
Note that for the benefit of overall understanding
I will aim to avoid talking in detail about: the use of multicategories
in the iterative construction of opetopes, the different technical
approaches to this construction, and the construction of morphisms
of opetopes. I will be happy to talk about this in an informal
session another day if anyone is keen to hear these details.
David Neil Corfield
(Oxford University)
N-Category
Theory as a Catalyst for Change in Philosophy
Paper:
pdf
Uli
Fahrenberg (Department of Mathematical
Sciences, Aalborg University, Denmark) uli@math.aau.dk
http://www.math.auc.dk/~uli/getco04
A Dihomotopy Double Category of a Po-Space
Slides: pdf
ps
We construct a double category on a po-space
capturing dihomotopy information in dimension 2. A po-space
is a topological space with a partial order defined on the points,
and dihomotopy is the po-space analogy of the usual homotopy
notion.
The 1-cells in our double category are dipaths
modulo reparametrisation, and the squares are maps from the
unit square into the po-space, modulo a notion of dihomotopy
taking the dipath reparametrisations into account. We show that
our dihomotopy double category obeys a van Kampen type property.
Thomas
M. Fiore (Department of Mathematics, University
of Michigan) fioret@umich.edu
Pseudo Algebraic Structures in Conformal
Field Theory (impromptu talk)
Introduction: dvi
pdf ps
Slides: dvi
pdf
ps
This talk deals with the pseudo algebraic structure
of gluing and disjoint union on the category of rigged surfaces
and its role in the definition of conformal field theory. Pseudo
algebras over Lawvere theories and 2-theories are treated in
order to capture the pseudo algebraic structure. This work is
an application of weak 2-categorical concepts to physics.
Philippe
Gaucher (Université Paris 7 Denis-Diderot)
http://www.pps.jussieu.fr/~gaucher
Towards a Homotopy Theory of Higher
Dimensional Automata
Slides:
pdf
We give an overlook of our work about globular
complexes and flows. Globular complexes, flows, S-homotopy and
T-homotopy are explained by examples. Several model categories
are presented. And some open questions are discussed.
Thomas
Leinster
(Department of Mathematics, University of Glasgow) tl@maths.gla.ac.uk
Survey
and Taxonomy
I
will describe in as non-technical a way as possible some of
the definitions of weak n-category that have been proposed,
drawing attention to similarities rather than differences. The
various definitions are placed in two groups, "algebraic" and
"non-algebraic". Broadly, I will follow Chapter 10 of my book
"Higher Operads, Higher Categories" (http://arxiv.org/abs/math.CT/0305049
or http://books.cambridge.org/0521532159.htm)
Thomas
Leinster
(Department of Mathematics, University of Glasgow) tl@maths.gla.ac.uk
Simplicial
Definition
Slides: streettalk1.pdf
streettalk2.pdf
Joint
with Nick Gurski.
Ross
Street has proposed two related definitions of n-category, one
in 1987 and one in 2002. Thomas Leinster will present the older
definition; Nick Gurski will present the newer one and some
of the motivating ideas.
Ross
Street (Mathematics Department,
Macquarie University) street@ics.mq.edu.au
http://www.math.mq.edu.au/~street/
An Australian Conspectus of Higher
Categories
Notes: pdf
Much Australian work on categories is part
of, or relevant to, the development of higher categories and
their theory. In this talk, I hope to describe some of the origins
and achievements of our efforts that they might perchance serve
as a guide to the development of aspects of higher-dimensional
work.
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