[English] I have decided to run a separate blog on mathematics for Brazilian students, the Matemática Replay!. This blog [Theorema Egregium] continues with the usual material. Thanks.
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[Português] Eu decidi manter um blog separado sobre matemática para estudantes Brasileiros, o Matemática Replay!. Este blog [Theorema Egregium] continua com o material usual. Obrigada.
Este “post” dá início (tentativamente) a uma série que irá contemplar problemas de matemática para alunos do 2o. grau (ensino médio brasileiro). O público-alvo é de alunos que estão passando por reais dificuldades em resolver problemas. As soluções serão apresentadas detalhadamente e, em seguida, serão propostos problemas similares.
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Nos problemas do post de hoje, você precisará ter os conceitos de progressão artimética e de números complexos.
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Você pode baixar gratuitamente os exercícios no link abaixo:
Problemas de Matemática do 2o. Grau – Como Resolvê-los Passo à Passo – Parte 1
[documento PDF, 3 páginas, 100Kb]
Atenção para Copyright na última página!
AVISO: Agora tenho um blog só de matemática, acesse: Matemática Replay!
Escrevi um tutorial de matemática para alunos do ensino fundamental brasileiro, para o 5o. ano (ou 4a. série, pelo regimento anterior), que pode ser baixado gratuitamente aqui.
Dicas de Matemática: Como resolver problemas?
Autora: Christine Córdula Dantas
Tamanho: 1.3 Mb
Número de páginas: 33
AVISO: Agora tenho um blog só de matemática, acesse: Matemática Replay!
Dear Readers,
Forgive my long delay in writing here. I spent the last few months wondering whether I would shutdown this blog or not. Well, I have decided that I will not, at least, not yet. So here are a few interesting links that I came across recently:
1) From the [PhilPhys] e-mail list, a talk by Stephen Summers (University of Florida).
“Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State”
Abstract: An overview is given of what mathematical physics can currently say about the vacuum state for relativistic quantum field theories on Minkowski space. Along with a review of classical results such as the Reeh-Schlieder Theorem and its immediate and controversial consequences, more recent results are discussed. These include the nature of vacuum correlations and the degree of entanglement of the vacuum, as well as the striking fact that the modular objects determined by the vacuum state and algebras of observables localized in certain regions of Minkowski space encode a remarkable range of physical information, from the dynamics and scattering behavior of the theory to the external symmetries and even the space-time itself. In addition, an intrinsic characterization of the vacuum state provided by the modular objects is discussed. [Foundations of Physics in Greater Paris]
2) A new issue of Philosophia Mathematica is available online [June 2009; Vol. 17, No. 2], in special:
- Mark Balaguer, Fictionalism, Theft, and the Story of Mathematics
- Francesco Berto, The Gödel Paradox and Wittgenstein’s Reasons
3) New edition of Edge:
What’s Next? Dispatches on the Future of Science (Edited by Max Brockman): “A preview of the ideas you’re going to be reading about in ten years.”
Just received.
…………………..
THE SCIENCE AND PHILOSOPHY OF UNCONVENTIONAL COMPUTING (SPUC09)
Cambridge (UK), March 23-25, 2009
SECOND CALL FOR PAPERS
We welcome submissions on topics normally classified under ‘natural computing’ or ‘unconventional computing’ or ‘hypercomputing’ including (but not restricted to) quantum computing, relativistic computing, biology-based computing, analogue computing, and also submissions on the philosophical implications of these new fields for topics including (but again not restricted to) philosophy of mind, philosophy of mathematics, the Church-Turing thesis.
Each presentation should last no more than 30 minutes; a further 10 minutes will be allowed for discussion.
Those wishing to make a presentation should submit by email a 250-word abstract of their paper to Mark Hogarth (mhogarth@cantab.net); enquiries to the same.
Registration fee (yet to be fixed) will be around £100.
Student bursaries are available.
Conference website: http://web.mac.com/mhogarth/Site/SPUC_Conference.html
ORGANISER
Mark Hogarth (Cambridge, UK)
CONFIRMED INVITED SPEAKERS
Selmer Brinsjord (New York, USA))
Jeff Barrett (Irvine, USA)
Philip Welch (Bristol, UK)
Tim Button (Harvard, USA)
Cristian Calude (Auckland, New Zealand))
István Németi (Budapest, Hungry)
Benjamin Wells (San Francisco, USA)
Hajnal Andréka (Budapest, Hungry)
Apostolos Syropoulos (Xanthi, Greece)
Susan Stepney (York, UK)
Bruce MacLennan (Tennessee, USA)
Peter Kugel (Boston, USA)
Mark Sprevak (Cambridge, UK)
Selim Akl (Kingston, Canada)
José Félix Costa (Swansea, UK)
ADVISORY PANEL
Mike Stannett (Sheffield, UK)
John Tucker (Swansea, UK)
Barry Cooper (Leeds, UK)
Sponsored by EPSRC through HyperNet (the Hypercomputation Research Network, EP/E064183/1)
Just received.
…………………………………………………….
Dear Christine,
I am writing to ask for your assistance in drawing the attention of exceptional, highly motivated students to the Perimeter Scholars International (PSI) program.
PSI is an innovative, Masters level course designed to prepare students for cutting-edge research in theoretical physics. It provides a broad overview, allowing students to choose their preferred specialisation, and extensive tuition in formulating and solving interesting problems.
The due date for applications is February 1st: applications received after this date may still be considered but only as long as places remain available.
A number of outstanding lecturers have already signed up to teach, including for example Yakir Aharonov, Phil Anderson, Matt Choptuik, Nima Arkani-Hamed, John Cardy, Ruth Gregory, Michael Peskin, Sid Redner, Xiao-Gang Wen, and a number of Perimeter Institute research faculty. They will be supported by full-time tutors dedicated to the course.
All accepted students will be fully supported.
For further details, see www.perimeterscholars.org.
Thank you in advance for helping us to make this exciting opportunity known as widely as possible.
With my best wishes,
Neil Turok
Director
Perimeter Institute for Theoretical Physics
Waterloo, Ontario, Canada
I have submitted an essay to the FQXi competition. If you are interested in reading it, click here.
Title: On the Nature of Time – Or Why Does Nature Abhor Deadlocks?
Essay Abstract
This essay aims at introducing a novel point of view on the nature of time, inspired by a synthesis of three seemingly unrelated concepts: Bergson’s notion of duration, Dijkstra’s notion of concurrency, and Mach’s notion of inertia.
Edit (June 9th 2009): Apparently, the essays on the nature of time are no longer available at the FQXi site. I have made a very few small corrections and modifications in my essay and a new version is available here (pdf file).
There is a review at Nature’s News and Views section by P.-M. Binder about a recent article by David H. Wolpert from NASA Ames Research Center, entitled “Physical limits of inference“. Binder writes:
A provocative contribution to the logic of science extends the theorems of Kurt Gödel and Alan Turing, and bears on thinking about prediction, the standard model of particles, and quantum gravity.
From the abstract of the paper, one reads
We show that physical devices that perform observation, prediction, or recollection share an underlying mathematical structure. We call devices with that structure “inference devices”. We present a set of existence and impossibility results concerning inference devices. These results hold independent of the precise physical laws governing our universe. In a limited sense, the impossibility results establish that Laplace was wrong to claim that even in a classical, non-chaotic universe the future can be unerringly predicted, given sufficient knowledge of the present. Alternatively, these impossibility results can be viewed as a non-quantum-mechanical “uncertainty principle”.
[Yeah, Laplace was wrong even classically, according to my SF novel... 
]
and
(…) We informally discuss the philosophical implications of these results, e.g., for whether the universe “is” a computer.
I find it very surprising that this was published in Physica D: Nonlinear Phenomena, and not in a philosophical journal. I have no criticisms against this work in particular (I did not read the paper in full), it is just that it does not seem, from a first impression, a physics paper per se, as much as interesting as it may seem.
Another (somewhat funny, I must admit, but it may be a reflection of my present pessimistic/sarcastic mood) excerpt from Binder’s review is this:
The other limitation is our inability to bring quantum mechanics and gravity into a single theory, although several viable alternative theories are being studied [9]. Quantum electrodynamics, a refinement of quantum mechanics, is defined by just two parameters (the charge and mass of the electron), whereas quantum gravity would require infinitely many parameters, and hence infinite experiments to determine those parameters, making it so far a meaningless theory.
BTW, Ref. [9] above is Wilczek’s book, The Lightness of Being.
The prize goes to three Japanese:
Yoichiro Nambu (1/2) “for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics” and to Makoto Kobayashi (1/4) and Toshihide Maskawa (1/4) “for the discovery of the origin of the broken symmetry which predicts the existence of at least three families of quarks in nature”.
It is somewhat foolish, but I find exciting to see Nobel prizes announced. Congratulations!
Edit: Does anyone have a clue why Nicola Cabibbo did not receive the prize along with Kobayashi and Maskawa?
Edit: See also:
If you would like to suggest and/or vote for some out of print math book to come back to print, visit here:
Back in 2005, when I started to read more and more papers on quantum gravity, I have found one paper that really interested (and still interests) me a lot, and was the subject of one of my first posts back at my older blog.
- The Continuum Limit of Discrete Geometries, by Manfred Requardt [http://arxiv.org/abs/math-ph/0507017]
It is a well written paper, with lots of interesting mathematics (Gromov’s geometric group theory, random graphs, etc). As far as I know, it has not been much discussed in other blogs.
Recently, Requardt posted another interesting one, which I am presently reading:
- About the Minimal Resolution of Space-Time Grains in Experimental Quantum Gravity, by Manfred Requardt [http://arxiv.org/abs/0807.3619]
Abstract:
We critically analyse and compare various recent thought experiments, performed by Amelino-Camelia, Ng et al., Baez et al., Adler et al., and ourselves, concerning the (thought)experimental accessibility of the Planck scale by space-time measurements. We show that a closer inspection of the working of the measuring devices, by taking their microscopic quantum many-body nature in due account, leads to deeper insights concerning the extreme limits of the precision of space-time measurements. Among other things, we show how certain constraints like e.g. the Schwarzschild constraint can be circumvented and that quantum fluctuations being present in the measuring devices can be reduced by designing more intelligent measuring instruments. Consequences for various phenomenological quantum gravity models are discussed.
I like his writting style and the various points that he covers in his papers with sensible criticism.
I include here an email from Prof. Bertram Kostant to Ben Wallace-Wells on his view about the E(8) group and some considerations on Lisi’s theory. According to Lisi, this part of the email was permitted to become publically available.
Dear Ben Wallace-Wells,
The following is my response to your queries. In order to answer your question about the Lie group E(8), I found it necessary in the first paragraph to add some historical context. I hope it is not too burdensome to read.
Lie (pronounced Lee) group theory was developed by mathematicians towards the end of the 1800′s. An important accomplishment at that time was also a classification of the simple Lie groups. It turned out there were 4 infinite families and 5 exceptional Lie groups, the largest (containing all the others) of which is E(8). There is an unfortunate double usage here of the word “simple”. There is of course, the everyday usage (eu) meaning easy to understand and a technical use (tu) meaning not built up from other groups. For example, the title of Lisi’s paper is “An Exceptionally Simple Theory of Everything”. His use of Exceptionally Simple is a pun. The exceptional refers to the exceptional Lie groups and simple is (tu). Lie groups started entering physics in a serious way at the beginning of the twentieth century. Perhaps more prominent was Einstein’s theory of special relativity, where the Lie group involved was the Lorentz group. This is a (eu) group and occupies only a very tiny sliver of something as sophisticated as E(8). Also Bohr’s theory of atomic spectra uses the rotation group SO(3) and again is an (eu) and a very tiny sliver of E(8). For the most part, Lie groups were more or less put on the “back burner” by both mathematicians and physicists until the middle of the twentieth century, At that time, it became a serious object of study by mathematicians. I should make it perfectly clear that I am a research mathematician and not a physicist. My speciality is Lie groups and any use of physics terminology here is only what is common knowledge. On occasion I have been motivated by physics – for example, the marvelous development of quantum mechanics by physicists in the 1920′s. I believe that there were some stirrings about Lie groups by physicists in the middle of the twentieth century. I have the following prescient story to tell. I was a visiting member of Princeton’s Institute for Advanced Study in 1955. It was a Good Friday in April and Einstein was looking for the Institute bus to take him back home to 112 Mercer Street. Being Good Friday, the driver was on holiday amd I offered to drive him home. We had a wonderful conversation and at one point he asked me what I was working on. I told him Lie groups. He then remarked, wagging his finger, that that will be very important. Actually, I was quite surprised that he knew who Lie was. About a week later Einstein was dead. In the middle of the twentieth century, physicists developed what is called quantum field theory (Feynman, Schwinger, etc.) Also at that time, the powerful accelerators were producing a zoo of new particles. To deal with this menagerie of particles and to carry forward Einstein’s program of finding a unified field theory (unifying all 4 forces of nature), physicists came up with what is called the Standard Model (Weinberg, etc.) This involved what is called a gauge group. In fact, in the Standard Model, the gauge group is a (eu) simple Lie group. A more refined development was the grand unified theory (GUT) of Glashow and Georgi. Here the gauge group (SU(5)) was more interesting. The GUT theory happily confers a desired fractional electric charge on such exotic particles as quarks. These theories also unified three of the four forces of Nature.
The latter part of the twentieth century also saw the development, by physicists, of string theory. String theory has had vast consequence for mathematics (excluding Lie groups). However, as far as I know, there have been no experimental verifications of the physics involved. (For his work in this area, the mathematical physicist Ed Witten was awarded the most prestigious prize in mathematics.)
A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent “object” in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure. It is easy to arrive at the feeling that a final understanding of the universe must somehow involve E(8), or otherwise put, (tongue in cheek) Nature would be foolish not to utilize E(8). There was a good deal of publicity about E(8) in the last few years when a team of about 25 mathematicians, using the power of present computers and a very complicated program, succeeded in determining all of the vast number of (to use a technical term) characters associated with it. Incidentally, one of the main leaders of the team was an ex-student of mine, David Vogan. It was Vogan who told me about Lisi’s paper. Another person involved here is John Baez. Baez, (a relative of the singer Joan Baez) is a professor of mathematics at the Riverside campus of the University of California. Baez performs a great service to the Math-Physics community by publishing a very engaging weekly report on doings of mathematicians and physicists – explaining latest results in physics to mathematicians and latest results of mathematics to physicists. His week 253 report deals with Lisi’s paper. In effect Lisi is saying that E(8) is the ultimate gauge group. Lisi’s theory makes some astounding claims. Among them is that E(8) “sees” all the elementary particles in the Universe. In addition, Lisi claims that his theory unifies all 4 forces in Nature (the last being gravity) and thereby achieves Einstein’s dream of a unified field theory. String theorists, by and large, heartily dismiss Lisi’s theory. But among some prominent nonstring theorists (e.g., Lee Smolin), the paper has been acclaimed. Incidentally, string theorists utilize E(8), but not as a gauge group. According to Baez’s week 253 report, one of Lisi’s motivations in going to E(8) was that the Glashow-Georgi GUT theory “sees” only one generation of fermions. Apparently there are 3 “generations” of such particles. The Lie group E(8) has a triality construction and I believe that Lisi thought that this may be used to give all 3 generations. Since I had something to do with this triality construction, I became interested in Lisi’s paper. I remind you, I am not a physicist and cannot comment one way or another on the physics involved. However, mathematically, I was able to show, using beautiful results of such finite group theorists as John Thompson, Robert Griess, Alex Ryba and an important input from Jean-Pierre Serre, together with some old results of mine, that E(8) not only “sees” GUT in a natural way, but in fact is itself (viewed through one of the facets) a composite of two copies of GUT. This is the subject matter of what I have been lecturing about. One such lecture was at UC Riverside, which was filmed and put on line by John Baez.
Having seem the film, Lisi sent me an enthusiastic E-mail, saying my results were brand new to him and speculating on what the meaning of the second GUT might be. I am too happy to forward Lisi’s E-mail letter to you, if you wish to see it. At any rate, if there is any physical validity to E(8) as a gauge group, the ball is in the court of physicists to interpret what this doubling up of GUT might mean.
I am happy to cooperate with you on your New Yorker article. However, I think it is best to do this by E-mail and not via phone conversations. I wish to avoid all the misquotations attendant to the New Yorker publication having to do with the solution of the Poincare conjecture.
Bertram Kostant Professor Emeritus of Mathematics at MIT
Debates on Lisi’s theory are back being discussed over at n-Category Café from this post on chronologically, so that you can check the exchange progress.
However, it is difficult to tell ahead the end of this story. It appears that some sort of agreement is slowly and painfully being reached, but it is clear that there is still a long way to go.
[Previous posts on Lisi's theory can be found here.]
[Edit 19-Jul-08: An interesting summary by Distler is found in a previous entry from the above mentioned exchange, dated May 22, 2008, and pointed out by Urs Schreiber. It is interesting not only for being a summary of what he interprets from Lisi's theory and the points that he indicate as being problematic, but also because he actually finds what he calls "something kinda cool about the elements of the construction" and that it "would be mildly interesting to see what sort of actions one could build with this construction".]
Mathematician Terence Tao will be giving the course:
Perelman’s proof of the Poincaré conjecture
He will also include blog posts on the course.
PS – As far as I know, this is *not* an April Fools’ Day joke.
Recently, John Baez posted over at his blog a talk by Bertram Kostant on the exceptional group E8. It is interesting to follow it and the discussions over there if you are studying Lisi’s recent work. Or not. E8 is interesting per se.
Previous posts on this (on inverse chronological order): here, here and here.
A result still coming out from my short time in software engineering… My contribution was mostly from the mathematical side, and the first steps for making a connection with the MSCs.
I’m writting this post to keep track of books on quantum gravity (and closely related/helpful books). I own only the first one, which I recommend if you are interested in the main conceptual problems of quantum gravity. Regarding the others, I am presently considering purchasing Thiemann and Henneaux & Teitelboim’s books.
Rovelli – Quantum Gravity (errata)
Thiemann – Modern Canonical Quantum General Relativity
Kiefer – Quantum Gravity
Amelino-Camelia – Planck Scale Effects in Astrophysics and Cosmology
Gomberoff – Lectures on Quantum Gravity
Ambjørn et al. – Quantum Geometry: A Statistical Field Theory Approach
Carlip – Quantum Gravity in 2+1 Dimensions
Gambini & Pullin – Loops, Knots, Gauge Theories and Quantum Gravity
Henneaux & Teitelboim – Quantization of Gauge Systems
Rickles et al. – The Structural Foundations of Quantum Gravity
Callender – Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Gravity
There is also Daniele Oriti’s book, which is an edition of several contributions about various approaches to quantum gravity. As it appears, the book has not been released yet. You can learn more about it over at my old blog, Background Independence (scroll down after my two book reviews to find Oriti’s invited post). At that blog you can also find my “Basic Curriculum for Quantum Gravity” (scroll down a little more, after Oriti’s post, or use this direct link to a backup copy of that post, with comments). There, I link to several other helpful books and downloadable tutorials/papers. That list have not been updated. I believe these two papers by Ashtekar are useful recent reviews:
- Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions
- An Introduction to Loop Quantum Gravity Through Cosmology
There is also Smolin’s 2006 Lectures on quantum gravity. See here for more details.
(Via Woit’s Blog). This is also an interesting paper by Alexander Maloney and Edward Witten: http://arxiv.org/abs/0712.0155
We consider pure three-dimensional quantum gravity with a negative cosmological constant. The sum of known contributions to the partition function from classical geometries can be computed exactly, including quantum corrections. However, the result is not physically sensible, and if the model does exist, there are some additional contributions. One possibility is that the theory may have long strings and a continuous spectrum. Another possibility is that complex geometries need to be included, possibly leading to a holomorphically factorized partition function. We analyze the subleading corrections to the Bekenstein-Hawking entropy and show that these can be correctly reproduced in such a holomorphically factorized theory. We also consider the Hawking-Page phase transition between a thermal gas and a black hole and show that it is a phase transition of Lee-Yang type, associated with a condensation of zeros in the complex temperature plane. Finally, we analyze pure three-dimensional supergravity, with similar results
Witten has written some months ago a paper in 3D quantum gravity and the monster group (see previous post here). Apparently, such ‘unphysical’ models may be (are supposed to be?) useful to learn about full quantum gravity.
(Via Physics Forums). This is the new paper by Lee Smolin: http://arxiv.org/abs/0712.0977.
We study a unification of gravity with Yang-Mills fields based on a simple extension of the Plebanski action to a Lie group G which contains the local lorentz group. The Coleman-Mandula theorem is avoided because the theory necessarily has a non-zero cosmological constant and the dynamics has no global spacetime symmetry. This may be applied to Lisi’s proposal of an E8 unified theory, giving a fully E8 invariant action. The extended form of the Plebanski action suggests a new class of spin foam models.
Notice the citation to Lisi’s recent paper.
I read Lisi’s paper and I’ll read this one. Sounds very interesting indeed, and a quite “fast link” to Lisi’s idea, that is, a connection of that idea with the LQG framework.
If you would like to discuss that paper in technical terms, I’ll be glad to host your comments here. Recall that this is a highly moderated, very high signal to noise ratio blog. Be educated.
Update: A personal view can be found here.
A finite set of mathematical relations is the foundation of reality.
The objetive of science is to find those relations. As a consequence, science also attempts to clarify the ontological status of those mathematical relations. That is, to bring the metaphysical frontier in which they are deeply buried closer into the physical frontier.
If mathematics were reality itself, then reality would be an infinite substance, or an infinite number of substances, which would reflect in the impossibility of building physical laws.
(Ok, I want to believe that).
In any case, somehow, reality emerges to us as a deeply constrained thing. At the same time, science seems to have intrinsic limits to probe reality. Whether the former is a reflection of the latter, or vice-versa, seems to be our inescapable metaphysical ‘karma’.
.
.
The Universe is all that physically exists, its the totality of all things: matter (energy), spacetime; all entities that can be realized, observed, sensed, deduced, measured. All within and outside horizons.
It is the whole, and hence only one by definition.
It is the ultimate reality, and hence only one by definition.
The mathematical — or, for what is worth, even mental — picture of many universes (the multiverse) can only be conceived into a larger “universe” in which the set of multiple universes belong, and hence the multiverse concept inexorably has the limit of an ultimate, “whole universe” or “meta universe”, or “final universe”.
There is only one universe.
The hypothesis of the multiverse has an intrinsic imperfection from its very conception. It is a fragile concept since from simple logic reasoning it always necessarily reduces to an ultimate, “one universe”.
It is an unnecessary hypothesis that I abhor and reject.
Tegmark’s new paper… I have no comments for the moment. I didn’t read it.
Update: Just finished reading it. Bad philosophy. Sorry.
Rubik’s Cube has approximately 43 quintillion possible configurations (that’s about 
I’ve just learned from John Baez that he will be participating in the Abel Symposium this year. The Abel Symposia are organized by the Norwegian Mathematical Society, and you can learn about previous editions here. The program for 2007 is here.
John Baez will be giving a survey lecture on Higher Gauge Theory and Elliptic Cohomology.
Since this is all too much advanced mathematics for me, I content myself with listing below some references that seem to be suitable for my stage of understanding:
- The Heart of Cohomology, by Goro Kato.
- From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes, by Madsen and Tornehave.
These will add to my huge wish list.
BTW, check out Baez’s course on Quantization and Cohomology here:
- Fall 2006, Winter 2007, Spring 2007.
I’m helping with preparing some figures for this course and trying at the same time to learn something from the process.
Update: John Baez has a new post on his talk over at n-Category Café.
Update: More on Abel Symposium on Baez’s TWF 255.
Via philphys list, I have just learned that
An extensive archive of Imre Lakatos’s papers and letters – mostly in English, but some in Hungarian – is held in the British Library for Political and Economic Science at the London School of Economics. Thanks to the generosity of an anonymous donor, Research Fellowships to a total amount of US$ 25,000 will be available in the calendar year 2007 for scholars wishing to pursue some research project on Lakatos and/or his contemporaries that requires consultation of the archive. The Fellowships will be held at the LSE Department of Philosophy, Logic and Scientific Method which will also provide facilities for the Fellows.
And also, to make thinks more interesting:
Notes for prospective candidates: The LSE Lakatos Archive contains Lakatos’s notes and working papers dating from 1945 and other personal documents, including his correspondence from 1956 with some 1000+ correspondents including Agassi, Carnap, Feyerabend, Kalmar, Koestler, Kuhn, Polanyi, Polya, Popper, Quine, Szabo, Tarski, and many other figures of the philosophical, intellectual and academic establishments of the time.
I think this is an interesting opportunity for a good candidate willing to go a little deeper on the ideas of Lakatos. Since I really knew very little about them, except for a few glimpses from memory about this philosopher, I checked out a Wikipedia article on him, which has a nice summary of his life and ideas.
An interesting passage is:
For Lakatos, what we think of as a ‘theory’ may actually be a succession of slightly different theories and experimental techniques developed over time, that share some common idea, or what Lakatos called their ‘hard core’. Lakatos called such changing collections ‘Research Programmes’. The scientists involved in a programme will attempt to shield the theoretical core from falsification attempts behind a protective belt of auxiliary hypotheses. Whereas Popper was generally regarded as disparaging such measures as ‘ad hoc’, Lakatos wanted to show that adjusting and developing a protective belt is not necessarily a bad thing for a research programme. Instead of asking whether a hypothesis is true or false, Lakatos wanted us to ask whether one research programme is better than another, so that there is a rational basis for preferring it.
So would the conception of a theory be a limiting process? I do not refer here to the process of developing a theory per se, like the struggling years that took Einstein to construct his general theory of relativity. But to the behaviour of paradigm shift: would it be a limiting process, even though sometimes pumby, uneven?
More links on Lakatos:
- Science and Pseudoscience: you can hear the mp3 file or read the transcript.
- Lakatos award: given for an outstanding contribution to the philosophy of science, widely interpreted, in the form of a book published in English during the previous six years. (From this site I have learned about an interesting book, whose author was awarded part of the 2006 Lakatos prize — Harvey Brown, Professor of Philosophy of Physics, University of Oxford, for his book Physical Relativity: Space-time Structure from a Dynamical Perspective).
Are 3D Quantum Gravity and the Monster group fundamentally related?
I’ll take some time to figure this out , so for the moment I’ll be collecting here some links of interest.
Witten’s paper and talk at Strings 07 about 3D Quantum Gravity. [I've posted about this here, where I also indicate links to blogs that are discussing the talks over at Strings 07, so you can send comments to that post as well].
Wikipedia article on the Monster Group.
MathWorld article on the Monster Group.
Baez’s This Week’s Finds # 66, where he gives an introduction to the Monster Group.
Solomon’s paper On Finite Simple Groups and Their Classification.
The Atlas of Finite Group Representations. See, in special, the sporadic groups — the Monster group.
A paper by T. Gannon: Monstrous Moonshine: The first twenty-five years.
Some books:
- Sporadic Groups by Aschbacher.
- Atlas of Finite Groups by Conway.
- Geometry of Sporadic Groups by Ivanov.
- Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics by Gannon (added later).
- Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics by Ronan; non-technical account (added later).
Update: Here are some very basic questions that I have posted over at PF. Answers are very welcomed.
1. Why is k “an integer for topological reasons”? (k is a parameter that appears in a second term — a multiple of the Chern- Simons invariant of the spin connection — added to the action).
2. Further, what is “holomorphic” factorization? (A pointer to the basic literature on this will suffice). Is it the only possible constraint?
3. He argues that the (naive) partition function Z_0(q) differs from the “exact” Z(q) by terms of order O(q). Would this be correct for any k?
4. He finds that for k=1 the monster group is interpreted as the symmetry of 2+1-dimensional black holes. How sensitive is this result with respect to the value of k, and to respect to the other assuptions used in the derivation?
Update on question #1: Here is the answer by John Baez (as posted over at PF) –
The Chern-Simons action S is invariant under small gauge transformations (those connected to the identity by a continuous path), but changes by multiples of a certain constant c under large gauge transformations. What shows up in path integrals is the exponentiated action exp(ikS) where k is some coupling constant. The consequence is clear: exp(ikS) remains unchanged under large gauge transformations if and only if exp(ikc) = 1, meaning that k has to be an integer multiple of 2 pi / c.
If you set up all your normalization conventions nicely, c = 2 pi, so k has to be an integer.
This stuff is explained a bit more in my book Gauge Fields, Knots and Gravity, in section II.4, Chern-Simons Theory. Also see the end of section II.5.
In 3d quantum gravity, the consequence is that the cosmological constant can only take certain discrete values!!!
It’s quite clear now! It was the phrase “topological reasons” in his talk that seemed mysterious to me.
Update: Question # 2 can be elucidated in this Wikipedia article on the Weierstrass factorization theorem.
Update: There is of course, Lieven le Bruyn’s excellent blog, formely known as “Neverendingbooks”, but recently reformulated into MoonshineMath, focused on the Monstrous Moonshine.
Update: John Baez also indicates this paper by Gannon.
Update: John Baez just wrote about Witten’s new paper on his TWF254 (see the blog entry here).
I have just learned (via Peter Woit’s blog) about a recent article in Wikipedia, written by Christina Sormani, concerning the Poincaré Conjecture and its solution, given by Griori Perelman and Richard Hamilton (Columbia University).
The article is quite pedagogical and contains excellent graphics, with pointers to other articles, on-line lectures and websites.
Let us attempt a somewhat primitive reinterpretation of a quantum state as a set of processes.
You can see a process as a function that gives some output from a given input. It’s somewhat like an unitary operator. A given particle state evolves to another as a combined result of a set of many processing, “internally active elements”. Imagine a qubit. A set of quantum states encompassing any possible state between |0> and |1> is here what I call a set of “fundamental processes”.
Internally, a lot of activity is happening to the qubit. The processes are not simply independent agents, but active elements which share finite mutual “resources”. I’m not sure at the present stage what to make of these “shared resources” in such a reinterpreted quantum theory (in LQG, they could be seen as the edges of a spin network: nodes *share* common edges representing spin, so the evolution of spin network states could be seen as a result of how these processes act and share resources, and in the present case is to maintain gauge invariance). In other words, the processes do not act completely independently, but are concurrent in the sense that they need to access some shared resources in order to evolve.
I’m not sure what to do with when you observe the system in such a framework.
Imagine now a n-dimensional space in which every orthogonal axis represents a process. Every point along an axis is a representation of a given quantum state (input) evolving to another (output). For instance, in one axis you could set up the following “scheduling”:
|0> -> |1/sqrt(2)> -> |1> -> |0> -> …
in another axis, this one:
|1/sqrt(2)> -> |1> -> |0> -> |1/sqrt(2)> -> |1> -> …
and so on, so you see there is a quite a large number of possible schedules for the qubit. But, say, two processes could not be at the same “time” sharing the same resources: this translates to some constraint that represents the forbidden usage of (or action upon) the same common resource by different processes which are competing at the same “time” (here, “time” is also to be interpreted in some partially ordered sense).
All possible scheduling (histories) of each of the processes form, combined, a directed topological manifold that encodes *all* the possible histories of a given particle. “Directed” in the sense that there is a local partial order structure imposed on the manifold as the quantum states evolve (a direction of “time”). A point in this manifold represents a superposition of processes (state functions). But since the processes that evolve quantum states must share common resources, there are forbidden regions on the manifold because the processes cannot access the same resource at the same “time”. So there are natural constraints that must be obeyed, and these constraints determine a topological, typical signature of the quantum system in question.
These constraints (that actually forbid the system to go into some kind of “deadlock”) could be seen as correlations/anti-correlations between quantum states, thus providing an interpretation of why energy levels of a harmonic oscillator are quantized, for instance.
There is an emergent field joining topology and concurrency theory — “di”topology — that study these various ditopological manifolds, which carry this extra structure (“di”rection).
It turns out that the idea seems to be easier to grasp when you include gravity. The reason comes from the fact that the scheduling of concurrent processes can be described, as I said, in topological terms by a manifold with a local partial order, a ditopological manifold. And pictorially, spin networks (or spin foams, for what is worth) seem adequate to fit this idea in a more immediate sense because of causality issues.
But that is another story.
I’ve been working on the idea that quantum systems are inherently concurrent systems (see this previous post under construction). They can be examined under directed algebraic topology (or ditopology) tools (e.g., see here, or google about it).
The most interesting problem to model is quantum entanglement: underlying this phenomenon one abstracts out a quantum state as fundamentally corresponding to a set of concurrent processes. It would be interesting to examine how to restore local realism under this general hypothesis.
One can go further into this and ask whether quantization arises from the idea that, given that quantum systems are seen as concurrent systems, there are naturally forbidden regions in the multi-dimensional state space, where each dimension represents a concurrent process, defined by the fact that these processes locally share “resources” and no two processes can “act” on the shared resources at the same “time”. Forbidden regions correspond to the discretization of nature in the quantum limit.
So nature would fundamentally be a huge deadlock avoidance system.
Update: discussions can be found in the comment section of this blog entry over at n-Cateogry café.
Update: Below follows, for the record, a cut and paste from the discussion over at n-Category café. For the detailed links, please refer to that blog entry (see link above), where the comments appeared.
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Christine, I assume the application of directed algebraic topology to concurrent systems is an outgrowth of the work of Herlihy and Shavit, which was done in the late 90s and won the 2004 Gödel Prize.
You might want to communicate with Prakash Panangaden at McGill about your work. He appears to have worked, as a theoretical computer scientist, on concurrent and distributed systems and also on quantum computing, and has a strong interest in the causal structure of general relativity and its connection to abstract notions arising in computer science.
Posted by:
Chris W. on January 17, 2007 1:58 AM |
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In terms of applications of general geometrical methods to concurrency, the idea goes back to the 70’s. See Goubault, E. Geometry and Concurrency: A User’s Guide. But you are right to cite the work of Herlihy and Shavit as an evidence of how this subject is getting a lot of attention recently, and giving rise to important developments in the field. It looks a beautiful paper, but I have never studied it in detail.Panangaden also published in gr-qc, and this is one of his most interesting papers, I guess… I don’t know how far this is getting attention, but sounds an original and promissing line of research.
Best regards,
Christine
Posted by:
Christine Dantas on January 17, 2007 11:51 AM
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Hi Christine, you probably already know that Carl Petri, one of the founders of Concurrent Systems as a field of study, used to propound the view that an adequate semantics for concurrency would need to be sufficiently broad to also encompass physics.
From what I’ve been reading recently on Anima ex Machina blog he’s updated this position somewhat to a more “out-there” Universe-is-a-Petri-net kind of stance. He might be right! (as might Smolin – I saw your Amazon review!) On the same page I linked you can also find thoughts of Seth Lloyd and some others on this topic from a conference in Berlin last year.
You’re probably also familiar with Vaughan Pratt who clearly thinks very deeply on this sort of topic (although his publications can be frighteningly dense).
When I was a CS student as Glasgow they used to ask us questions like the “big simulation” argument, just to freak us out. I was never able to see why this would confuse a hardy theoretical physicist though — surely they’d be interested (in theory) in the turtle at the bottom of the tower, ie the real simulation that presumably has to simulate itself?!
I’m looking forward to seeing where the quantum lambda-calculus course goes with this kind of thing. Quantum computation seems to rely on arbitrary-precision complex amplitudes — but having gone to all the trouble of defining True and False as morphisms in the course, surely we won’t now be allowed to pull high-precision complex numbers out of the hat? If you were designing a programming language they’d need to be defined and computed themselves somehow… who “computes” these amplitudes that they’re relying on?
Posted by:
Allan E on January 17, 2007 2:09 AM
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Hi Allan,
And according to Nielsen and Chuang,
“Quantum computation and quantum information has taught us to think physically about computation (…) we can also learn to think computationally about physics.”
This view is attractive and I believe physics is heading towards this broad idea. I do not have a clue how far this will lead us.
Concerning Petri nets, these and other classical concurrency models have been generalized to the concept of po-spaces (“po” from “partial order”). See Sokolowski, S., Directed topology and concurrency a short overview.
Best,
Christine
Posted by:
Christine Dantas on January 17, 2007 12:21 PM
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And concerning the question in your last paragraph, I don’t know how to answer to that… There is a huge gap between a broad idea and the technical details that one has to face in order to make the idea actually work or make sense! But thanks for pointing that out, I’ll have to think about it.
Posted by:
Christine Dantas on January 17, 2007 12:31 PM
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Let me see if I understand what you’re getting at. If there are correlations among a set of concurrent processes, they must be constrained to avoid deadlocks. Perhaps the necessary constraints induce correlations and anti-correlations among states and state transitions in the system that resemble correlations among quantum states. These correlations also imply that the system avoid parts of its state space.
You seem to be assuming discretization at the outset, insofar as the number of concurrent processes is finite, and the associated state space is finite. Should I assume that you have in mind a continuous analog of a set of concurrent processes?
Here is a simple and fairly concrete model you might want to examine. Consider the state space to include a set of n Boolean variables, and the concurrent processes to be n Boolean transition functions of two variables that yield a “subsequent” state for each of the variables. The processes (transition functions) share resources in the significant sense that each of the functions operates on two variables, at least one of which might be used as an input by another function. What would constitute a deadlock in such a system, and what is required to avoid it? Is there necessarily a stochastic component to the system’s behavior? That is, does the transition structure have to rearrange itself every so often, in a way that can’t be described deterministically? I realize this is a very sketchy problem formulation.
By the way, with respect to the relevance of algebraic topology, the set of transition functions in this simple model can be considered as defining an abstract simplicial complex; the Boolean variables are the vertices and the functions define the edges. (Remember however that these functions have some internal structure, since we have 16 distinct, albeit interrelated, Boolean functions to choose from.) My previous questions can then be posed as questions about the “moves” that can and must occur within this complex, in order to avoid certain “pathologies” in the evolution of the system. (By the way, in this light, the collection of transition functions can be loosely regarded as a “gas” of edges [1-D!] which is evolving in parallel with and necessarily coupled to a “gas” of Boolean state variables.)
One more point: The Boolean variables are of course undergoing transitions with successive “time steps”. One might ask if one can define a causal order on these transitions or “events”. That is, given one such event, can one say that it was preceded in a well-defined way by another set of events, and followed accordingly by yet another set of events? It is not immediately evident how such a description is to be derived from the transition functions, although one might plausibly expect that it should be possible.
(I have in mind here a complementary description in terms of causal sets. By the way, regarding the compatibility of discreteness based on causal sets with Lorentz invariance, see Discreteness without symmetry breaking: a theorem [1 May 2006].)
Posted by:
Chris W. on January 17, 2007 4:26 AM
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Dear Chris W.,
Thanks for this elaborate comment. Your first paragraph summarizes well the broad idea.
And yes, one could assume a continuous “scheduling” of the processes, and also the space could be assumed to be infinite dimensional. What I think is important here is to assume a local partial order – a “po-space”… (I’m interested in ergodic moves…) Examples of such spaces are found here:
* Sokolowski, S., Classifying holes of arbitrary dimensions in partially ordered cubes. Tech. Rep. 2000-1, Kansas State University, Computing and Information Sciences, Aug. 2000. linke here.
* Raussen, M., Geometric investigations of fundamental categories of dipaths. Unpublished, 2001. (sorry, can’t find the link now).
* Gaucher, P., About the globular homology of higher dimensional automata. Cahiers de Topologie et Geometrie Differentielle Categoriques XLIII-2 (2002), 107156. link here.
And thanks for proposing a problem. I guess we all have sketchy ideas for the moment! No problem about that, in fact it is great to know that this is quite an open field for research.
Your comment has given me months to think over! Don’t have much valuable to add for the moment, but thanks a lot.
Christine
NOVO!! (New!! Poetry book in Portuguese) Livro de Poemas escrito em Português por Christine Córdula (Dantas):
Clique na imagem para acessar.
Livro de Ficção-científica escrito em Português por Christine Córdula (Dantas):
This is my first Science Fiction novel, entitled "Open Time" ("Tempo Aberto"), released in Portuguese. I am studying the possibility of translating it into English.
Click on the cover image to learn more about it: summary and preface (by clicking "Preview this book" over at the lulu site). Warning: all contents are in Portuguese.
A summary in English can be found here.
