The Third Road

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  • #151
Kea
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event in July

Oxford have something cool coming up: an event under the name CATS, KETS and CLOISTERS. See

http://se10.comlab.ox.ac.uk:8080/FOCS/CKCinOXFORD_en.html [Broken]
 
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  • #152
selfAdjoint
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Before there was Cambridge and All That, there was Oxford and Merton College, and its Mean Speed Rule (c. 1350:-> [tex]s = \frac{1}{2}at^2[/tex]). Long may it wave.
 
  • #154
Kea
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selfAdjoint said:
Warms the cockles of my old topologist's heart.

With good reason, selfAdjoint. :smile: The statement is highly non-trivial. Deeper than the deep blue sea. What is cohomology about, really? One certainly needs some sort of [itex]d[/itex] operator with a property like
[itex]d^2 = 0[/itex]. Well, let's use the symbol [itex]1[/itex] instead of [itex]0[/itex] because, after all, we don't just want Abelian cohomology. Some things in mathematics are non-commutative. Then clearly we are talking about monads. Yes, the category theoretic kind.

An example of a nice monad is the double power set functor, which takes a set [itex]S[/itex] to subsets of subsets of [itex]S[/itex] (that's not a typo). Being a monad means that, somehow, each identity arrow on a set, meaning the set [itex]S[/itex] itself, factors into a kind of square root of [itex]1[/itex]. Simple? Hardly. But certainly cohomology.

There is a progression of ideas here:
1. everything is an object (in a set)
2. everything is an arrow (in a category, or points becoming Strings)
3. everything is a functor (by lifting an arrow into Cat)
4. everything is a ....

:smile:
 
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  • #155
selfAdjoint
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Kea said:
With good reason, selfAdjoint. The statement is highly non-trivial. Deeper than the deep blue sea. What is cohomology about, really? One certainly needs some sort of operator with a property like
. Well, let's use the symbol instead of because, after all, we don't just want Abelian cohomology. Some things in mathematics are non-commutative. Then clearly we are talking about monads. Yes, the category theoretic kind.

That's great Kea! Does the monad concept also embody the cobordism aspect of cohomology?
 
  • #156
Kea
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A little more about monads....

selfAdjoint said:
Does the monad concept also embody the cobordism aspect of cohomology?

Oh, yes, although as a measly physicist I don't claim to understand much. But the following might be of some interest:

Imagine we could show that the category of vector spaces was a quantum topos. The truth arrow would be a map from [itex]\mathbb{C}[/itex] into [itex]\mathbb{C} \oplus \mathbb{C}[/itex], the qubit. It's OK to projectivise and get the 2-sphere going to [itex]\mathbb{C}\mathbb{P}^{3}[/itex], which might be familiar to people who know a little bit about classical causality.

Anyway, if one has a pair of monads, [itex]a[/itex] and [itex]b[/itex], then the commutativity of them is a distributive law in an abstract setting. Now take a pen and paper. Draw the Venn diagram for three intersecting circles. Classical distributivity is just a little region in this diagram. But in quantum toposes we are allowed to (i) knot things and (ii) add in a String direction. In other words we get 6 special points on a kind of trefoil knot and the circles become tubes.

Exercise: recover the Sundance preons from these sort of diagrams. Of course, mass hasn't been taken into account yet so there will only be one generation.

Note that the D-brane people have also vigorously studied the recovery of the SM from [itex]\mathbb{Z}_{6}[/itex] orbifolds and products of three 2-tori. See Honecker-Ott and Bailin.

:smile:
 
  • #157
Kea
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P.s.

You know, octopi are very strange creatures. o:)
 
  • #158
Kea
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Cool!! I don't know how anyone is supposed to keep up with this guy Lauda - I just found this:

Frobenius algebras and ambidextrous adjunctions
A. D. Lauda
http://www.tac.mta.ca/tac/volumes/16/4/16-04.pdf

Anyway, he looks at these ambijunctions in 2-categories first, which are secretly 2D TFTs, and then he categorifies them. This involves defining Frobenius pseudomonoids and understanding them in the context of Gray type structures, which we are rather fond of here.

I wonder what a categorified CFT would be physically?
:biggrin:
 
  • #160
OK, after your done with all the quoting other people and thinking inside the "box".... this is something that has been making me wonder, just hear me out; I have read a few of the Doc's books. I am thinking of making "device" we will call it. I really don't mean to go out to far on a limb hear; if you have a magnet that is round and is that interesting, and if you spin them in 3 fields "axis". All am wondering right now is 10 to the 25th power sounds big; so you have 25 magnets and ten rings, all spinning, now what..... if you spin them in the right rotation and or speed what happens?.... this is what I have been thinking about for the past few nights
 
  • #161
OK, after your done with all the quoting other people and thinking inside the "box".... this is something that has been making me wonder, just hear me out; I have read a few of the Doc's books. I am thinking of making "device" we will call it. I really don't mean to go out to far on a limb hear; if you have a magnet that is round and is that interesting, and if you spin them in 3 fields "axis". All am wondering right now is 10 to the 25th power sounds big; so you have 25 magnets and ten rings, all spinning, now what..... if you spin them in the right rotation and or speed what happens?.... this is what I have been thinking about for the past few nights
 
  • #162
oops sorry about that
 
  • #163
arivero
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selfAdjoint said:
Before there was Cambridge and All That, there was Oxford and Merton College, and its Mean Speed Rule (c. 1350:-> [tex]s = \frac{1}{2}at^2[/tex]). Long may it wave.

It is OT, I know, but you have intriged me: are you sure about date and formula?
 
  • #164
selfAdjoint
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arivero said:
It is OT, I know, but you have intriged me: are you sure about date and formula?

As you say, OT. Maybe we should move these two posts to the History forum?

They didn't have the notation, and their expression was in terms of "forms" which were functions over the Euclidean (or Eudoxian) proportions. A modern view of the proportions sees them as positive real numbers, but that was way in the future for these medieval thinkers. Because the proportions are developed rigorously in Euclid's book, the Merton scholars were able to do careful thinking about their forms. They had names instead of notation: A "uniform form" was a constant function from the proportions to the proportions; a "diffform form" was a linear function, and a difformly difform" form was a constantly accelerated function, i.e. a quadratic, although I don't think they ever spotted the link between difformlly difform and the square function.

The domain variable of the form (corresponding to t in the formula) was called Latitude, and the range (corresponding to v) was called Longitude. And the Merton College mean speed rule was "The total Longitude made good by a difform form (i.e. linear function) over a range of Latitude is the same as made good by a uniform form (i.e. constant function) of value the mean of the two Longitudes at the end of the range". Work it out.

Nicole Oresme, a wonderful French mathematician contemporary with these Merton scholars, proved this theorem by graphing the Latitude versus the Longitude; the graph came out a right triangle (on top of a rectangle) and he applied the rule for the area of a triangle (area = one half base X height) to demostrate the mean speed rule.

{Added} Another Frenchman, the philosopher Jean Buridan, asserted that objects in motion have a quality he called impetus which was propotional to the speed of the object and to its weight. This is not to be confused with the stupid thing also called impetus taught by the scholastics of Galileo's time. And Buridan also claimed that the impetus of a falling body was a difform form of time. All any of these people would have had to do would be to apply the Merton rule to Buridan's impetus to derive the law of falling bodies. But right about this time two things happened. The Black Death raged across Europe, and it is conjectured that Buridan died of it. And the long standing schism in the Catholic Church, which saw two separate papacies, was healed and the reunified church took the opportunity to crack down on dangerous thought. Oresme was offered a bishop's hat, at a remote country diocese in Normandy - he was a Norman by birth. And Merton College, whose statutes were much more liberal than the typical Oxford college, was "normalized", ending its research program.
 
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  • #165
marcus
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Kea said:
Care of David Corfield, there is a new Category blog: http://bosker.wordpress.com/
Kea, if you have a free moment you might want to glance at
http://arxiv.org/abs/quant-ph/0606114
q-Deformed Spin Networks, Knot Polynomials and Anyonic Topological Quantum Computation
Louis H. Kauffman, Samuel J. Lomonaco Jr
87 pages, 58 figures
========
this isn't a recommendation but it has 58 figures
and the title contains several hot keywords.:smile:
besides being by Louis Kauffman how can you miss, it's got to be worth a look
 
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  • #166
Kea
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While we're busy looking at all the cool stuff out there, let's not forget:

Open-closed TQFTs extend Khovanov homology from links to tangles
Aaron D. Lauda, Hendryk Pfeiffer
http://www.arxiv.org/abs/math.QA/0606331

This has to do with loop+arc diagrams for state sums - really cool stuff - and the mathematicians should be impressed too!

:smile:
 
  • #167
Kea
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While we're busy looking at all the cool stuff out there, let's not forget:

Open-closed TQFTs extend Khovanov homology from links to tangles
Aaron D. Lauda, Hendryk Pfeiffer
http://www.arxiv.org/abs/math.QA/0606331

This has to do with loop+arc diagrams for state sums - really cool stuff - and the mathematicians should be impressed too! Roughly speaking, it's about piecing together two Frobenius type structures - one commutative and one symmetric - and characterising their interaction. The author's call this latter bit knowledgable Frobenius. So it's quite like linear logic, really!

:smile:
 
  • #168
  • #169
selfAdjoint
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Abstract Differential Geometry and QG

Interesting new paper on the arxiv:

gr-qc/0607038
From: Ioannis Raptis
Date: Mon, 10 Jul 2006 14:38:35 GMT (28kb)
A Dodecalogue of Basic Didactics from Applications of Abstract Differential Geometry to Quantum Gravity
Authors: Ioannis Raptis
Comments: 16 pages, preliminary version
We summarize the twelve most important in our view novel concepts that have arisen, based on results that have been obtained, from various applications of Abstract Differential Geometry (ADG) to Quantum Gravity (QG). The present document may be used as a concise, yet informal, discursive and peripatetic conceptual guide-cum-terminological glossary to the voluminous technical research literature on the subject. In a bonus section at the end, we dwell on the significance of introducing new conceptual terminology in future QG research by means of `poetic language'


From "lessonet 2:

The sole dynamical variable in ADG-gravity is an algebraic A-connection field D5 acting (on the local sections of) a vector sheaf E defined on an in principle arbitrary topological space X. The physical kinematical configuration space in the theory is the moduli space AA(E)/AutE of the affine space of connections A modulo the (local) gauge transformations in the principal sheaf AutE. The ADG-formalism on gravity is called halforder formalism in order to distinguish it from the first-order one of Palatini-Ashtekar, and from the original second-order one of Einstein, both of which depend on a background smooth manifold for their differential geometric expression. From the ADG-theoretic vantage, gravity is regarded as a pure gauge theory since only the connection, and not the smooth metric (or equivalently, the smooth tetrad field), is a dynamical variable. It follows that the connection D represents the gravito-inertial field and, unlike the gμ of GR, not the chrono-geometrical structure. There is no ‘spacetime geometry’ in ADG-gravity, or rather more mildly, if there is any ‘space’ (:‘geometry’) at all, it is already encoded in the A chosen. ‘Geometry’ (or indeed, ‘spacetime’) is completely encoded in our (generalized) measurements in A. There is no geometry without measurement, without the production (:recording) of numbers of some
sort. At the same time, (the products of) measurements (:numbers) are our own actions (and numbers our own artifacts/inventions), hence no physical reality, and no interpretation as the gravitational field living ‘out there’, should be given to the spacetime metric, like in the original formulation of GR. This is consistent with our viewing gravity as a pure gauge theory—ie, that the gravitational field is simply the connection D.

(The bolding is mine).
 
  • #170
Kea
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New Coecke Paper

Marcus has found a new gem

Quantum measurements without sums
http://arxiv.org/abs/quant-ph/0608035
Bob Coecke, Dusko Pavlovic
36 pages, 46 pictures

which definitely belongs here. It extends the categorical diagram techniques of Kindergarten Quantum Mechanics to discuss measurement in a novel way in terms of special internal objects. This really is a good way to think of quantum mechanical information processes.

:smile:
 
  • #171
Kea
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new paper

There is a new paper by Fuchs et al (the categorical CFT people)

Duality and defects in RCFT
http://www.arxiv.org/PS_cache/hep-th/pdf/0607/0607247.pdf [Broken]

:smile:
 
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  • #173
marcus
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Kea said:
Hey, guys! I caved in ... and created a blog:

http://kea-monad.blogspot.com/

:smile:
I've been reading your posts there with considerable interest. You've recorded your thoughts there for some time without making a general announcement of it. Thanks for sharing with us!
 
  • #175
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whoa! nice blog Kea!
 

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