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