Showing posts with label topology. Show all posts
Showing posts with label topology. Show all posts

Saturday, 11 May 2013

Topology, the periodic table and parasites

hree very interesting websites to amuse you. The first a homage to topology or as the author puts it:
"This is a toy for building complex 3D polyhedral shapes from simple ones by "recipes"
polyHédronisme allows you to construct various polyhedral shapes (only 3 dimensions though!) by composing rules over various bases, for example starting with a plain octahedron
and applying various rules we can get beauties such as this below using the rule "n18n18n9n9n9soxO" in the notation used on the site:



The second is the every popular Element of the Week via the The Guardian newspaper. A fun introduction of the basic building blocks of chemistry and each week a new element. This week, the famous and well known element: astatine .

I recommend you check out caesium, rubidium, potassium and sodium for the sheer fun of watching those being dropped into water and reacting violently. Such experiments are now banned from schools for safety reasons, along with many other aspects of science unfortunately...

Failing that there's the Royal Society of Chemistry's Visual Elements Period Table which is great fun.

And finally, for the squeamish of you stop reading now, for the less squeamish its probably best to avoid mealtimes, but this is great:

Parasite of the Day!

Introducing a new beast more or less regularly (if not every day). To understand this site you can read their introduction which helpfully has no pictures to put you off your food.

Sunday, 30 September 2012

Grothendieck Biography

Mathematics is full of "characters", Grigori Perelman, Pierre de Fermat, Évariste Galois, Paul Erdős, Andrew Wiles** to name just a few and each having their own, unique, wondrous story about their dedication to their mathematical work and life.

Perhaps none more so than Alexander Grothendieck exemplifies the mathematician and since 1991 lived as a recluse in Andorra. However since the body of work and his contribution to mathematics, particularly, category theory and topology has been almost legendary he retains a great mystery about him.

In order to understand Groethendieck and possibly the mind of the mathematician a series of biographies of Groethendieck are being written by Leila Schneps. The current draft and extracts can be found on her pages about this work.

I'll quote a paragraph from Chapter 1 of Volume II, that gives a flavour of Groethendieck's work and approach to mathematics:

Taken altogether, Grothendieck’s body of work is perceived as an immense tour de force, an accomplishment of gigantic scope, and also extremely difficult both as research and for the reader, due to the effort necessary to come to a familiar understanding of the highly abstract objects or points of view that he systematically adopts as generalizations of the classical ones. All agree that the thousands of pages of his writings and those of his school, and the dozens and hundreds of new results and new proofs of old results stand as a testimony to the formidable nature of the task.

This is truly a work at a scale a magnitude more detailed than, say Simon Singh's fascinating documentation about Wiles' work and proof of Fermat's Last Theorem. Suffice to say, I look forward to reading it. Maybe Simon Singh should make a documentary about Groethendieck?

** I credit Andrew Wiles with inspiring me to study for my PhD back in 1995

Wednesday, 30 May 2012

Semantic Isolation (Pt.1)

Working with Ora Lassila we have been discussing and working on the definition of the term "data silo" in order to clarify our ideas of semantic isolation when applied to databases, data assets and the interoperability and integration of the information contained within.

The term "silo" when applied to databases, data and information is interesting in that it occurs in a number of statements, such as, "my app data is siloed" or "we need to break the data silos" and so on.

The meaning of the term however is mixed in its usage and thus its usage is inconsistent and misleading in many cases; quite simply it is used to cover a large number of overlapping scenarios.

Understanding the scope and meaning of this term in its various contexts is central to understanding the interoperability problem in a practical sense. In a workshop today I have heard the term used in a large number of ways and also applied to the notion of interoperability: The term "silo" has been used to mean (at least!)
  • The data is siloed because it exists in its own database infrastructure
  • The data is siloed because it is on accessible via some access control
  • The data is siloed because it is in its own representation format
  • The data is siloed because it is not understandable/translatable (semantics)
We can present these are some kind of "lock-in" or "siloing continuum", where those usages on the left are more related to physical aspects and those on the right to more semantic in the information sense:




We obviously can create a more granular continuum (indeed that's what a continuum should allow) but the point here is to at least to present some kind of ordering over the differing uses of the term. The ordering runs from physical deployment and implementation through to abstract semantics.

Now it seems that when people talk about "breaking the [data] silos" they are actually referring to enabling interoperability of the data between differing services; and often this is addressed at the physical database or access control level. Occasionally the discussion gets mixed and syntax and representation of data is addressed.

Interoperability of information starts at the semantic level and works in reverse (right to left) through the above continuum; physical, logical, access control and syntax should not prevent sharing and common understanding of data. For example, if one tackles interoperability of information by standarising on syntax or representation (eg: JSON vs XML) then the resultant will be two sets of data that can't be merged because they don't have the same meaning; similarly at the other end of the continuum centralising databases (physically or logically) doesn't result in interoperability - maybe easier system management but never interoperability of information.

Interestingly I had an extremely interesting discussion about financial systems and that interoperability between these is extremely high even at the application (local usage) level and this is simply because the underlying semantics of any financial system is unified. The notions of profit, loss, debit, credit and translations between the meanings of things such as dollars, yen, euros, pounds and the mathematics of financial values is formally defined and unambiguously understood; even if the mechanics if financial and economic systems isn't, but that's a different aspect altogether.

Also an important points here is that the link between financial concepts to real-world concepts and objects is well relatively easily definable. Indeed probably all real-world concepts and objects have their semantics defined in terms of financial transactions and concepts. Thus siloing of data probably can only occur in the financial world at the access control level.

The requirements for breaking the silos is easily understood as the ability to cross-reference two different data-sets and be sure (within certain bounds) that the meaning of the information contained there is is compatible. We want to perform things such as "1 + one equals 2" and be sure that the concept of "one" is the same as "1", the definition of "+" matches the concept of "+" applied to things such as "1","2" etc as well as things such as "one","two" etc. In this case the common semantics of "1" and "one" has been defined...fortunately.

It is vitally important to understand that if we can unify data sets through translations via common semantics then the siloing of data breaks and we get data liberation or what some call data democratisation. Unification of semantics however is faught with difficulties [1] but is the key prerequisite to integration and interoperability and ultimately a more expansive usage of that information.


References:

[1]Ian Oliver, Ora Lassila (2011). Integration "In The Large". Position paper accepted at the W3C Workshop on Data and Services Integration, October 20-21 2011, Bedford, MA, USA


Saturday, 19 May 2012

Odd topology...

Saw this on a wall in Brighton, got me wondering...



...what topological space are they using or did they divide by zero somewhere?

Tuesday, 16 August 2011

Topology

Just discovered this site on topology: Sketches of Topology, visualizations of low dimensional topology.

What makes this site different is the emphasis on visualisation of some very abstract but stunningly beautiful mathematical structures. I'd post some pictures but the author's (Kenneth Baker)  Flickr site while providing links to sharing the pictures, Blogger won't link to them...




Friday, 9 April 2010

Graph cohomology

Graph cohomology. Just a link for future reference (and a reminder to do a search), though this paper> looks interesting, especially the embedding graphs in Rn...

Friday, 19 March 2010

My Numbers, Higher Dimensional Categories, Sheaves and Modal Logic

Erdos: 5
n-Category: 2  (actually I think it's 1 and a half....it tends towards 0 depending on the time of day :-) )*
Bacon: infinite

*Dan Freed claims his is still 1

So what the hell is an n-Category anyway?
Higher-Dimensional Categories: an illustrated guide book
Eugenia Cheng and Aaron Lauda
University of Cambridge, 2004
and I happen the like this neat mix of sheaves and modal logic found here:

Topology and First-Order Modal Logic

 

Thursday, 7 January 2010

Physics, Topology, Logic and Computation: A Rosetta Stone

This has been around for a while in the draft form, but now there's a (re)post on Lambda the Ultimate for the forthcoming book(?):
Physics, Topology, Logic and Computation: A Rosetta Stone
John C. Baez and Mike Stay, 2009.
Mentioned in the discussions are two other papers linked here: