Seminars and talks
Seminars and talks
Over the years I must have given hundreds of seminars, talks, and
short research courses. Most of these are lost forever. In January 03
I started to keep a record of these. Here is a
table of recent seminars
where you can find links to the slides and relevant notes where available.
Here is a brief description of the content these talks.
List of seminars given in reverse chronological order
27-may-07 (Manchester)
Galois connections done properly -- and --
Classical rings of fractions (two separate topics)
Slides available for first part.
15-May-07 and 22-may-07 (Manchester)
From rings of fractions to localizations
A wander around these topics as part of a longish teaching
seminar.
Slides available
02-May-07 (Leeds)
Is the Ackermann function optimal?
A standard construction, originally due to Ackermann, produces a
recursive function that is not primitive recursive. This can be
relativized to produce a jump on the poset of degrees up to primitive
recursive equivalence. What is there between a degree and its jump?
Quite a lot.
Slides and a write-up available.
18-April-07 (Swansea - BMC)
Is the Ackermann function optimal?
See next talk, above.
20-February-07 and 27-February-07 (Manchester)
Ranking techniques for posets, with an eye on modules
A short course of of two 90 minute sessions. Slides are available.
02-December-06 (Cambridge)
The topological content of Monotone Bar Induction and the Fan
Principle
Joint work with Nicola Gambino. A short account of the classical
meaning of the two choice principles, ending with an observation that
distinguished between them.
28-November to 01-December-06 (Cambridge)
An introduction to ordinal notations
A short course of about four hours. Slides are available and I intend
to write an account of the topic.
16-November-06 (MFG)
Ordinal notations
A run through of part of the material for the short course above.
23-May-06 (Lisbon)
Syntactic measures of complexity for numeric functions
There is a standard class of first order functions over the natural
numbers -- the Grzegorczyk hierarchy of moderate length -- that can
be stratified by the ordinals up to epsilon_0. These functions can be
seen in different ways.
- The functions dominated by some member of a fast growing hierarchy
- The provably total functions
- The functions named in a certain lambda calculus, a version of
G\"odels T
By modifying the lambda-calculus -- tiering the types -- we severely
weaken its strength. The ordinal measure epsilon_0 drops to 3, that is
we stay within the Kalmar elementary functions. I will explain the
background to this technique and try to indicate why it works.
19-May-06 (Coimbra)
Tiering without tears.
This talk is essentially the same as the one in Lisbon above.
15-May-06 (Coimbra)
One talk with three titles.
Boolean reflections of frames.
Higher level assemblies of frames.
Ranking techniques for frames.
The category Frm of frames is an algebraic (point-free) modifications
of the category of topological spaces. In particular, each topology is
a frame. A study of Frm gives us many algebraqic techniques not
available in the point-sensitive setting.
The category Frm includes the category CBA of
complete boolean algebras and complete morphisms. At first sight is
seems that CBA is a reflective subcategory of Frm, but
there is a mysterious set theoretic obstruction. Some frames can not
be reflected into CBA, and some not.
The category Frm is much richer than the category of
spaces. In particular, each frame A has an associated larger frame NA,
is assembly, the frame of all nuclei on A. When A is the topology of a
space S, a nucleus on A is essentially a Grothendieck topology for S.
The assembly construction N can be iterated through the ordinals
A --> NA --> N^2A --> N^3A --> ...
and this tower stabilizes precisely when A has a boolean
reflection.
Some properties of this tower can be measured by an extension of
the Cantor-Bendixson process on topological spaces. This extension
seems to be new even for spaces.
I will explain what I know about this tower, finishing with an
example where N^3A is boolean but N^2A is not. Not a lot seems to be
known beyond this level.
30-March-06 (MFG)
How long is a piece of string?
I describe a method of generating arbitrarily long strings of poset
adjunctions using pairwise distinct monotone maps. I show that a
particular instance of this method produces the simplicial category.
09-March-06 (MFG)
Styles of derivation for Propositional Calculus, and beyond.
After a brief mention of the Hilbert style I describe the Gentzen
and Natural styles, what they are good for, how they interact, and
how they relate to other parts of mathematics.
02-March-06 (MFG)
How big is the continuum?
Some `undergraduate' material leading up to K\"onig's little theorem
that, for instance, the continuum can not have size aleph_omega.
07-December-05 (MFG)
Trees, fans, football, and freedom of choice.
The first part of a joint seminar with Nicola Gambino on the use of
choice principles to prove spatiality.
27-July-05 (MFG)
What is a Grothendieck topology?
I may write up a set of notes covering this and the previous talk (below).
20-July-05 (MFG)
What is a sheaf?
I may write up a set of notes covering this and the next talk (above).
14-July-05 (MFG)
A rational approach to enumeration.
03-May-05 (MFG)
From G\"odel's Theorem to Topological Spaces with some Modal Logic for
softies.
Spring-Summer-04
During this period the MFG had a working
seminar on aspects of lattice constructions and event
structures. This was to help various research students with their
work, hence the rather odd mixture of topics. I gave quite a few
seminars/lectures in these sessions, as listed below. There were no
slides or written material for these sessions.
- 01-July-04 Ordinals done informally
- 24-June-04 The quest for an adjunction
- 17-June-04 Simple event structures an morphisms
- 09-June-04 Introduction to the coverage technique for
posets
- 03-June-04 More on event structures
- 27-May-04 More on lattice completions
- 20-May-04 Event structures based on consistency
properties
- 13-May-04 Universal algebra for Sup-semilattices
- 06-May-04 Event structures based on conflict relations
- 28-April-04 Various completions of posets
11-February-04 (MFG)
Ehrenfeucht-Fra\"{\i}ss\'e games for quantification
18-December-03 (MFG christmas oration)
A `constructive' proof of the completeness of propositional calculus
24-October-03 (MFG)
Scatological spaces
A short account of equilogical spaces as generalizations of
topological spaces based as neighbourthood spaces.
05-September-03 (St Andrews)
The intrinsic complexity of natural number functions
Abstract:
In 1936 Turing (who later spent some time in Manchester)
characterized the `computable natural number functions'. He did this
using a `model of computation' by describing what we now call a turing
machine. Since then many other models of computation have been
described, some easier to use than others. Arising out of this we now
have a highly structured notion of the `complexity' of natural number
functions, at least in the first order setting. At the higher levels of
complexity, that is beyond the turing computable functions, this is
concerned with the turing degrees. At the lower levels, that is
within the turing computable functions, this complexity is usually
attached to some model of computation and measures how much resources
are needed to evaluate the function.
Around the same time that Turing was working in England, a group
(Church, Kleene, Rosser, and others) were working on the same problem
in Princeton. Originally they developed a $\lambda$-calculus to
analyse the notion, but abandoned that approach when Turing's result
appeared. Since then until quite recently that tool has been in
abeyance (but used for other purposes).
This $\lambda$-calculus approach is quite attractive since it
seems to lead to a notion of `intrinsic complexity' that is a measure
that doesn't depend on a model of computation. It also handles higher
order functions quite easily. Essentially the very syntax used to
specify the functions is a (rather too fine) measure of the
complexity.
In this talk I will outline the milestones in this approach,
describe some of the recent developments, and suggest some possible
paths for future research.
28-August-03 (MFG)
Why some people can't count (Tiering without tears)
A run through of the above talk.
07-August-03 (MFG)
Quantales and linear logic
A talk based on part of the paper LINK
05-June-03 (MFG)
Categories of contexts
30-May-03 (MFG)
Uniform Type Systems
08-may-03 (Bath)
The tiering method of controlling complexity
Abstract:
If left to their own devices recursions (on the natural numbers)
can very quickly produce extremely fast growing and complex
functions. However, when set up in a formal way, the very syntax of
the specifying recursions gives an accurate calibration of the
complexity of the specified function. It would be nice if this kind of
calibration method could be used for function that never get much
beyond the feasible ones. In recent year several methods of taming
recursion have been investigated, one of which is tiering (sometimes
known as stratification).
I will first outline the pre-history of this topic (the untyped
and the simply typed lambda calculus, and G\"odel's T), and then I
will describe how tiering works.
If you don't know anything about the lambda calculi, the first
part of the talk can be seen as an introduction to these.
07-May-03 (Swansea)
Notations for iterations
Abstract:
Some of you may have seen some of the topic of `ordinal
notations' and been impressed, or perhaps not. The standard approach
to this topics obscures a lot of what is going on, and often misses
the point. In fact, it is about the nesting of iterations and how
complicated thees can become.
In this talk I will explain some of the background from the stand
point of iterations. I will then go on to describe an applied
lambda-calculus with explication notations for these gadgets. Finally,
I will indicate how these notations are related to the standard
Bachmann notations.
The first part of the talk should be understandable without a
knowledge of ordinals.
06-May-03 (Swansea)
Forms of recursion
Abstract:
Recursion theory as currently understood is concerned with the
nalysis of Turing degrees. In this topic any two Turing computable
functions are deemed to be equivalent. However, there is an earlier,
less expensive, set of material which is concerned with recursion and
is in danger of being forgotten.
Some of the better know forms of recursion go by the names
`Primitive recursion', `Course of value recursion', `Tail recursion',
and the like. These are all concerned with converting given natural
number functions into natural number functions. However, there are
many other kinds of recursion, not all concerned with natural number
functions, and some concerned with higher order functions.
In this talk I will try to put these method into a general
setting, and suggest ways that this material could be turned into at
least one course that could be taught at a postgraduate level (and
perhaps earlier).
16-April-03 (MFG)
What is the second order lambda calculus?
20-February-03 (MFG)
!What is Linear Logic?
31-January-03 (Birmingham)
What is and what is not Point-free and Point-sensitive topology
Abstract:
In this context point-sensitive topology is what is usually called
point-set topology (as developed in Kelley's book). This is part of a
larger subject `topology' which includes algebraic topology,
homotopy theory, homological algebra, perhaps some algebraic
geometry, and other things. Of course, it impinges on many other
topics, and it is not clear where `topology' ends and other subjects
begin.
Along side this there is another area of mathematics which is even
less well defined. It impinges on such topics as ring theory,
algebraic number theory, module theory, algebraic geometry, and some
parts of `topology'. For instance, the study and use of quantales is
a part of this area. This area of mathematics uses entirely
algebraic methods. Point-free topology is a part of this area, and
consist of the study of certain lattices called frames.
There is a direct relationship between point-free and
point-sensitive, but the information flow goes entirely one way. I
will show, by an example, how the entirely algebraic view of the
point-free subject leads to a much deeper understanding to that part
of the point-sensitive subject which instigated, and made necessary,
its development. The secret is to remember a part of the larger
algebraic area which seems to have nothing to do with topology.
Before this I did not keep systematic records. The following
is what I can remember doing or is recorded elsewhere (and neither
of these implies the other).
16-May-02 (MFG)
Some observations on the Vietoris construction
Since then I have since written a set of notes on this, GIVE LINK.
Early 02 (MFG) As part of a short series on monads
and triples I gave two talks
- 08-March-02
The $\beta$-triple, including a lesson in the history of topology
- 15-Februrary-02
A quick survey of adjunctions
23-November-01 (MFG)
Number structures and recursion
SORT OUT
30-August,10,14,18-September-01 (MFG-short course)
$\Omega$-valued sets
Notes available on web page -- SORT OUT
Harold Simmons
Last modified: Wednesday 19 September 2007