Overview

Overall, this is an edutainment book I would recommend to anyone who is remotely interested in either relativity theory, black holes, quantum mechanics, theories of everything or the nature of life.
Depending on your previous knowledge about physics, it will be a book you’ll read very fast or at your usual literature speed. It doesn’t contain any mathematics beyond talking about binary digits (0 and 1). In contrast to many other books, the passages about concepts I knew very well weren’t boring but written in a good expository way that will enable me to explain the concepts better to others in future. Each chapter contains some historical remarks and anecdotes (and not only the most commonly known stories). The passages which explained concepts that were new to me explained them very good and I didn’t have the feeling of missing anything (a problem I had with some parts of Penrose’s Road to Reality before I learnt the math elsewhere).
Perhaps I should stress that the ideas promoted in the book are fairly standard by now and there is not so much debate in the scientific community about it. It does not discuss string theory vs. loop quantum gravity or any other crackpot magnet debate. You can expect to get solid education from the text (at least most of it, see my discussion of chapter 9 below).

Now let’s see what the individual chapters are about:

1. Redundancy

Introduces codes and code-breaking (cryptography) in the context of the second world war, then uses this to illustrate redundancy in human languages. Contains some anecdotes about Turing.

2. Demons

First discusses Lavoisier’s caloric theory in chemistry, then its problems, the industrial revolution and Carnot’s theory of perfect (reversible) heat engines. The second law of thermodynamics (entropy increases) is presented, although entropy isn’t mentioned explicitly. Some stories about Boltzmann and Maxwell are told. The Bell curve (of a Gaussian distribution) is explained (with pictures and a throwing-marbles-in-a-box analogy). Then entropy is defined and its connection to time (reversibility) is stressed. Maxwell’s daemon is introduced.

Carnot engines are explained particularly well, you can’t get it wrong from this exposition.

3. Information

The history of information theory around Shannon. Definition of a bit (binary digit). Illustration of the importance of information with a story about Paul Revere and the American civil war. Then the relation between entropy and information is explained in-depth. Brillouin’s connection of thermodynamic entropy to Shannon’s information-theoretic entropy and Landauer’s work on limits of computation (cost of erasure) are presented, which ultimately leads to an explanation why Maxwell’s daemon is impossible.

The historical remarks about the importance of communication during the civil war and the discussion of the impossibility of Maxwell’s daemon were enlightening.

4. Life

This chapter starts with a discussion of Schrödinger’s lecture “What is life?” and continues with a discussion of the genetic code, DNA computers and the relation between evolution and information. The story of the Lemba Jews in Zimbabwe is told.

This chapter really takes its time to convince you that DNA is a storage device for information and can be used in a Turing machine device, so every cell is a computer of some kind.

5. Faster than light

Einstein and some history, special relativity and the double-slit experiment (wave-particle duality). The Michelson-Morley experiment. The spear-in-a-barn paradox. Quantum tunnelling.

This chapter might be a little bit boring if you know some special theory of relativity.

6. Paradox

More on wave-particle duality, self-interference and superposition. Schrödinger vs. Heisenberg and Schrödinger’s cat. Entanglement and the Einstein-Podolsky-Rosen (EPR) experiment. Gisin’s experiments in Geneva.

While the historical remarks and the explanation of Schrödinger’s cat is pretty similar to other books on the topic, I very much enjoyed the discussion of Gisin’s experiments and “spooky distance action”.

7. Quantum information

Qubits, more on Schrödinger’s cat, quantum computing, Shor’s and Grover’s algorithm. The quantum Zeno effect. Measurements undertaken by Nature and the Casimir effect (vacuum fluctuations). Decoherence. A new axiom: “Information can be neither created nor destroyed”.

The notion of qubits is explained with the example of Schrödinger’s cat and then the cat is debunked via decoherence occurring due to vacuum fluctuations. Nevertheless, decoherence doesn’t get enough room for a solid understanding.

8. Conflict

More on spooky distance action and Gisin’s experiments. Quantum teleportation and causality. Black holes, the no-hair theorem and Hawking radiation. Black holes as computers.

The author clearly favours the axiom that information can not be destroyed and lays out what you need to understand to get the next chapter’s ideas on information preservation in black holes. The passages on black holes as computers are interesting but slightly misleading, because it’s more a metaphor than a concept.

9. Cosmos

More on black holes and their entropy. The holographic principle. Discussion about the infinite universe and some version of the infinite monkey theorem, leading to a many–worlds theory. The Copenhagen interpretation and a many-worlds interpretation of quantum physics. The end of all life due to the second law of thermodynamics.

The holographic principle deserves more room than it is given in this chapter. The argumentation for many worlds because of an infinite universe is not very clear to me (I think it’s just wrong). The many-worlds interpretation of quantum physics is explained very nicely, although it contains no single word on science theory (which, in my opinion, dictates to abandon theories about the unfalsifiable…).

Appendices and bibliography

The first appendix explains logarithms and why the base is not that important. The second explains entropy in Shannon’s sense in more detail. The bibliography contains a lot of helpful references (including some ArXiv papers) but sadly, they aren’t references explicitly throughout the text.

Conclusion and recommendation

It was a nice to read survey of applications and instances of information theory and I guess for most people who are slightly interested in any of the topics mentioned, it would at least help to choose the next book to read. It’s not even expensive!

The ICTP

• The concept of the ICTP, founded 1964 by Nobel laureate Abdus Salam, seems to be, very broadly said, to give researchers in physics and mathematics from third-world-countries opportunities (including money) for research and learning with scientists from developed countries. To accomplish this goal, they have short-time visiting scientists (for about 6 months) and host a lot of summer schools, conferences and workshops in many different areas. Participants from not-that-much-developed countries get funding for housing and food, in the ICTP Guest Houses and restaurants. In my opinion, this works well, since I met several mathematicians from developing countries (like India or the USA ).
• The buildings I’ve heard of are: Leonardo Da Vinci Building, Enrico Fermi Building, Adriatico Guest House, Galileo Galilei Guest House. Just some small hints for anyone who intends to go to the ICTP:
  • In the Leonardo Building main lecture hall, there is only one point in the whole room where you can find outlets. So, if you intend to use a laptop the whole day, find this spot (above, in the centre) and stick to it
  • The coffee machine in the Leonardo Building (near the toilets, ground floor) isn’t that bad, if you choose the sugar level to be less than the default value of 6/8 points (more than 1/8 is already too sweet for me, and I like sweet coffee).
  • The library in the Leonardo Building is a very nice place to hang out and wander off.
  • In the Adriatico Guest House, there is a vending machine for beer. It costs only 60 cent and isn’t that bad. The problem is: if you come late, it’s empty.
• The food in the Leonardo restaurant at the ICTP was not as good as you might expect from Italy. Especially in the evening, you get exactly the same as for lunch – but now it’s probably cold and you have less choice. Therefore, a lot of people went to the next city to have dinner (which is, of course, more expensive).

Sightseeing around Miramare, Trieste and Venice

• Miramare: there is the famous Miramare castle, surrounded by an artificial (yet beautiful) park, which is a must-see (i.e. exotic birds, fishes, turtles, palm trees, small pathways etc.).
• Trieste: I have no idea what I saw, I just walked across the city randomly. It is a nice, small city with a lot of restaurants. The sea side is not that beautiful but the buildings often look like imported from Vienna. I totally recommend eating ice cream all the time.
• Walking from Miramare to Trieste: Probably a bad idea. I did it three times, but I have to admit that the last third of the way, which is about 30 minutes long, is not beautiful at all. If you don’t know where to go, you’ll end up walking down a big street with high walls to the left and the right side for the whole 30 minutes. It is impossible to walk this last part along the sea. The first part, speaking of the hour it takes to walk from Miramare to Barcola, is nice, although the way along the sea is also the way along a big street. Anyway, there is a bus.
• Grotta Gigante, a huge cave, probably (at least officially) the largest cave that you can visit as ordinary tourist, in the world. And it is really gigantic! I’ve never been to any cave before and it was impressive.
• Venezia – only 2 hours and 10 EUR away, very nice (I guess there is no need to say more).

Miramare park main gate, near the ICTP

View Larger Map

Summer School

The summer school took two and a half week, where the first week was devoted to (recall?) very basic material: the first lectures defined manifolds and varieties and rushed over to de Rham cohomology, Kähler forms, sheaves, schemes and by the end of the first week the lectures about variations of Hodge structures and mixed Hodge structures reached the limit of what I knew already before (since I used my train ride to skim through Voisin’s book).
Now seems to be a good time to thank Andreas Höring for having teached classical geometry of Kähler manifolds in Paris so well.
Since the lectures of Migliorini and de Cataldo on the Hodge theory of maps assumed knowledge of Abelian categories and spectral sequences, I still don’t understand why the summer school started with these basic courses. I can not imagine that anyone who understood any of the lectures in the second week, could have possibly needed to learn what a manifold or a variety is. Maybe there are some hidden motives behind this, I haven’t asked the organisers.
The organisation was very good, all course material was printed out in sufficient quantity, the lectures didn’t take more time than expected, there was coffee & cookies in sufficient quantity twice a day and the overall working and learning atmosphere was fine. I missed problem sessions a little bit, but the intended problem sessions might have just turned out to be example sessions instead because of people requesting this. I was impressed that the organisers found a quick replacement for Claire Voisin, who couldn’t come to Trieste. The replacement talks given by C. Schnell, F. Charles and M. Kerr were very good and understandable.
For the participants I recommend having a look at Charles Siegel’s blog and website for lecture notes he has taken.

(photo licensed from Mike Scoltock under a Creative Commons Attribution-NonCommercial 2.0 Generic License)

Conference

• D. Arapura: Beilinson-Hodge cycles on semiabelian varieties; his joint paper with Kumar on the Beilinson-Hodge-Conjecture is related. This was my favourite talk! (By the way, see also his (unrelated) exposition of D-Modules and related Hodge Theory which I didn’t know about before (thanks to Charles Siegel for pointing me to it)).
• L. Illusie: Semistable reduction and vanishing theorems, after Lan and Suh.
• P. Griffiths: Hodge domains and automorphic cohomology; there are some talk notes available. For the necessary background on Mumford-Tate groups, see Griffiths’ lecture notes from the summer school.
• M. Green: Vanishing of Chern Polynomials for Hodge Domains. He introduced his talk with a joke along the lines of “It’s my seventh talk since I came to the ICTP, and in many cultures, the number seven is special. After six days you should rest”. The talk was about his SIGMA joint paper with Carlson and Griffiths.
• C. Schnell: Néron models and Poincaré bundles. It wasn’t so much about Poincaré bundles than about mixed Hodge modules, Néron models and admissible normal functions.
• G. Pearlstein: The locus of the Hodge classes in admissible variations of mixed Hodge structure (joint work with Brosnan and Schnell).
• H. Movasati: Automorphic functions for moduli of polarized Hodge structures. He gave some intuition on modular forms as generating functions, then looked at the Hodge theory of elliptic curves, explained the notion of quasi/differential modular forms (the terminology seems to be unstable) and discussed some examples from physics.

• Doran: Normal forms for lattice polarized K3 surfaces and Siegel modular forms (I had to skip this talk in favour of sleep – but I was told that it was good).
• S. Usui: Neron Models in log mixed Hodge theory by weak fans; slightly related preprint.
• J. Carlson: Further speculation and progress on Hodge theory for cubic surfaces (joint work with D. Toledo); related preprints. He introduced the talk with a story which concluded by “if you’re confused, just keep going”.
• F. Charles: Remarks on the Lefschetz standard conjecture and hyperkähler varieties, see his preprint on the topic.
• L. Maxim: Characteristic classes of complex hypersurfaces, see related paper. He introduced the virtual tangent bundle of a (possibly singular) hypersurface in a smooth manifold (the difference between tangent and normal bundle in K-Theory) and functorial homology characteristic classes (like Todd, L, Chern, but on homology); the general case are Hirzebruch-type invariants. He proceeded to express the (complicated) Brasselet-Schürmann-Yokura “Milnor-Hirzebruch”-classes in terms of virtual Milnor-Hirzebruch classes and invariants of the singularities.
• M. Kerr: Mumford-Tate groups and the classification of Hodge structures (more accurately, classification of Mumford-Tate subdomains, joint work with Griffiths and Green).
• C. Siegel: The Schottky Problem. He explained the well-known genus 3 and 4 cases and his approach to genus 5.
• P. Dalakov: Deformations of the Hitchin section and DGLA’s.

• sadly, I missed the conference talks of E. Cattani, S. Cautis and L. Migliorini because I travelled back to Germany on Friday morning.

Again, Charles Siegel took notes for some of the conference talks, see here and here.

Conclusion

I would be happy to go to the ICTP again in this life. Also, Hodge theory seems to be nice (at least parts of it).

small remark: I heard some rumour about a conference last year where they decided about the pronounciation of “Hartshorne”. Clearly, the person is called Harts-horne, as several Australian mathematicians told me, but now the books name was decided to be Hart-shorne. Hilarious!

I wrote a JabRef export filter that takes a BibTex file with file links (so, BibTex fields of the form file={somefile.pdf}) and writes a linux shell script to rename the files systematically according to the scheme [bibtexkey] – [authors] – [title].[extension]. Then JabRef can find the file again via its automatic file association mechanism. I use lower-case bibtexkeys but the export filter is easily adaptable, read about it on the JabRef custom export filter documentation page.

Just create (or download) a file named “renamer.layout” and fill in this line:
\begin{file}mv "\format[FileLink]{\file}" "\format[ToLowerCase,FormatChars]{\bibtexkey} - \format[AuthorNatBib,ToLowerCase,FormatChars,RemoveBrackets]{\author} - \format[FormatChars,RemoveBrackets,ToLowerCase]{\title}.\format[Replace(.*:,),ToLowerCase]{\file}"\end{file}
then open JabRef and go to the menu entry Options->Manage custom exports->Add new where you enter (for example) “renamer” as Export name, the full path to your renamer.layout file in the Main layout file field and “sh” as File extension.

Then open your BibTex file (.bib) with JabRef and then select the menu entry File->Export and select in the drop-down-menu Files of Type your newly created export filter renamer (*.sh). This gives you a shell script which, if executed, renames all files linked from the BibTex document into a standardised format (and moves all into the directory from where you execute the script).

This is only useful if you have files linked from your BibTex file, so you might need to do this first. If you already have filenames that contain some metadata, like author names or document titles, you might be very happy with JabRef’s RegEx-capable automatic file finder, which can be configured in the menu entry Options->Preferences->External Programs->External file links.

Even if you don’t use JabRef, you can use this process as described by exploiting the export-as-BibTex-capabilities of your favourite reference management system.

You might ask “why”, and I respond: my files are all organised in a way from which I can easily extract metadata using only the tools some operating system provides, so in case I don’t have access to my BibTex file, I can still find the desired files using the GNU/Linux command locate. Of course, I also have included the BibTex information in XMP into the PDF files (which is another feature of JabRef that I like a lot), so nothing is lost if I ever switch the reference management system.

Another lesson learned from this blogpost: writing specific JabRef export filters is very easy. Another one I wrote is able to download automagically entries from the arXiv when the URL is supplied in the url BibTex field. I won’t post it here because you need to disguise wget as “Mozilla 5.0″, otherwise the arXiv won’t let you download stuff (robot protection). I hope those who are able to figure out the details are also responsible enough to not download huge amounts of papers from the arXiv.

Putting it together, this provided a convenient approach to get arXiv papers with full metadata included in filename, PDF and BibTex on my computer. The still-not-perfect part is the first, getting the metadata from arXiv in BibTex format – I use CiteULike as proxy (and would be happy to hear about better solutions with JabRef).

You might also ask why I keep copies of my references on my computer (or why they have to be linked from my reference management system). I just find it very convenient to use my laptop as eReader, even when no internet is available, and given that I have 100+ references in the system, it is good to have metadata such as keywords, abstract, reviews, annotations and so on.

I learned about JabRef export filters somehow by accident because of another project related to reference management, which you might hear about soon (not yet).

The first posts (from Feb 2010 to June 2010):

• Grand opening
• Schreier theory
• Jordan type of a modular representation
• Bundles for adults
• Primes and probability
• On an isomorphism from G_T to G_1
• Adversarial Statistics
• Devinatz-Hopkins-Smith, I, II, III
• Complex orientations and the Steenrod algebra
• Eilenberg-Mac Lane spaces in the chromatic picture

Enjoy reading!

By the way: the ICTP Summer School on Hodge Theory is still teaching the basics needed to start doing Hodge theory – I’m looking forward to see Migliorini’s lecture on the Hodge theory of maps this afternoon! See also the short lecture notes from Charles Siegel at Rigorous Trivialities (and the notes from day 2 here), he will post something every day.

Bayes (English mathematician, 1702-1761) proved a theorem about conditional probabilities, nowadays called “Bayes’ theorem“. Suppose there are two statements A and B, which might overlap (e.g. A=”it’s raining today” and B=”it’s raining the whole week”¹, where the truth of B implies the truth of A). Now imagine these statements are more or less likely, so you attach some probability to these statements, p(A) and p(B), with values in 0-100% (or, for the mathematically oriented readers: let p be a probability measure on some discrete -algebra containing A and B). It’s not only the probability of A and B we might be interested in, but also the conditional probability “How likely is A when B is true?”, which we write p(A|B). Bayes’ theorem now reads:
, and this means in words, that the probability of A under the condition that B is true, multiplied by the probability of B, is the same as the probability of B under the condition that A is true, multiplied by the probability of A.

Let me put this in context. Let A be the statement “There will be a big volcano eruption in 2010″ and let B be the statement “Someone predicted that there will be a big volcano eruption in 2010″. Then we can talk about the probabilities of A and B (although we don’t know them exactly) and about the conditional probabilities, how likely the volcano eruption is, under the condition that someone predicted it, and the conditional probability how likely it it that someone predicted it, under the condition that it happens. If we believe that predicting volcano eruptions is possible, then we think that the conditional probability that it happens if someone predicted it, is higher than the probability that it happens with or without someone predicting it. Looking at Bayes’ formula, we see
, which tells us in words, that the probability of a volcano eruption under the condition that someone predicted it, is proportional to the probability of a volcano eruption and anti-proportional to the probability of someone predicting it. We see also, that the probability of a volcano eruption under the condition that someone predicted it is greater than the probability of a volcano eruption only if , that means, only if the probability of someone predicting the eruption is strictly smaller than the probability of someone predicting it under the condition that it happens.

Now you might know that there are some ways to predict volcano eruptions (I’m no expert). So the probability that someone predicts it under the condition that it happens is relatively high, but since there is someone claiming to forecast volcano eruptions every year (whether it happens or not), the absolute probability of someone predicting a volcano eruption for this year is 100%. So we can’t infer that volcano eruptions are likely just because someone predicted volcano eruptions.

Substitute volcano eruptions with your favourite Doomsday scenario and choose some arbitrary probability for this. The probability of someone predicting this scenario is close to 100% and therefore you can’t infer that it’s likely to happen just because someone told you so.

Substitute volcano eruptions with a war in Middle East and someone predicting it with an oil company doing business there after the war. If we realise that oil companies are pretty likely to do business in oil-rich countries, even more likely if there is no war going on, then we see (via Bayes’ theorem), that it’s not likely that the war was started just because of the oil business.

I don’t want to say that conspiracies don’t exist or that there are no wars about resources (like oil). I just want to point out that in each case, one has to find more evidence and stronger arguments than just coincidence of events. Test your argument against Bayes’ theorem!
If someone tells you his latest conspiracy theory, you might have been thinking “it might be true or false but I can’t prove him wrong and the probability that he’s right is not zero”. This is not a good response. Instead, you should always ask: “and why don’t you think it’s all just coincidence and happened by chance?”². This hypothesis will save you from the Bayesian fallacy.

You can use Bayes’ theorem to strengthen your arguments: If for two events A and B the conditional probability P(B|A) is really greater than the absolute probability P(B), then the probability P(A|B) is strictly greater than the probability P(A), which means that from measuring B you can infer that A is much more likely now. This is called “Bayesian inference” and it’s really important, for example, to find out which medicinal treatments cause more good than harm.

If you want to know more about argumentational fallacies of a similar kind, take a look at this paper (Khalil 2008) I found googling for “Bayesian fallacy”, although the author uses these words (completely) differently.

¹ – by the way, it has been raining the whole week here in Freiburg…

² – If people don’t like the thought that something happens “by chance”, they might not understand how order arises from chaos. This is another problem (which causes a lot of confusion), which I want to discuss separately (later).

Jet Nestruev: Smooth Manifolds and Observables, Springer, 2003

about a month ago (after I stumbled over a question on MO) and there was an exercise that resisted solution for more than a week.

Well…. now I found out that I have just misread the exercise. However, this way I basically did several exercises at once. Here comes the problem and its solution:

The problem

(inspired by page 28, chapter 3, exercise 3.17.5 in Nestruev)

Find a smooth (real) manifold such that it’s algebra of smooth functions is isomorphic to the algebra of all smooth functions that have some rational period (i.e. there exists such that for all x). Note that we don’t fix a period here. Let’s call the algebra in question (smooth functions on the real line with some rational period) .

You might want to stop reading here and think for a second (or minutes) about the solution or similar problems that have easier solutions. A more vague problem would be

Find a space such that the functions correspond to functions that are periodic with some rational period.

The manifold version of the problem has no solution (there doesn’t exist such a smooth manifold), as I will prove. I don’t know if there is some precision of the vague version (e.g. some additional structure on the space) which provides a solution but I don’t think so. If we fix a rational period (say 1), everything is nice and good in the world of smooth manifolds and smooth functions.

A simple sub-problem

If we would fix a rational number , and look at all smooth functions with period , these f would factor through and provide functions such that . Finally we see that they all factor through so there is an isomorphism .
This even works for any non-rational .
Let’s call the algebra of smooth functions on the real line with period .

From simple to hard

For a rational number , the algebra of smooth functions with period is clearly a sub-algebra of the algebra of all rational-periodic functions . For two rational numbers , we even have and isomorphic (via the pullback along ).
On the geometric side, this should correspond to a homeomorphism and the induced morphism is indeed the identity. To see what I mean precisely by “induced morphism”, look at the section “The Nestruev approach” below.

Furthermore, for each rational and each natural number , is a sub-algebra of . Since every rational number can be written as reduced quotient , every -periodic function is also -periodic, hence -periodic, and is the smallest natural number for which is -periodic.

Looking at the geometric picture again, the inclusion induces a map (in the other direction). This map is the n-fold covering of by (proof left to the reader).

A maybe-not-solution

If we say a non-periodic function has the rational period 0, the manifold we’re looking at is just the real numbers. This notion of rational periodic function is surely not a useful one, since every function satisfies , thus every function is 0-periodic. Note also that constant functions should have smallest natural period 1, not 0.

A misleading idea

Take the space , the disjoint union of a circle with a point. Every element of defines a function on : it has a smallest natural period , which we take as value on the point , and we use the values on to define values on the (by factorizing through the quotient map that identifies and ).

Now there are functions on that assign to some non-rational value, and this shows that this approach fails somehow. This cannot even be saved by taking for a locally ringed space, putting on the ring or because periods are always non-negative and is not a ring. You might take a “locally monoided space”… but then you could as well just take as ringed space the pair . This can indeed be considered as one step in the “solution” of the problem by Nestruev below.

The Nestruev approach

Nestruev wants us to think of differential geometry in terms of observables and states. This approach looks at differential geometry in a kind-of-scheme-theoretic way, taking a manifold to be something glued from local function algebras.

An algebra defines a topological space with underlying set the dual space of all -algebra homomorphisms (functionals), and the weak topology for all maps that come from evaluations at elements of .

Define an algebra
Now there is an obvious homomorphism , by sending each to the evaluation at and this is clearly surjective. It is not always injective. The necessary and sufficient condition to is to be geometric, which is defined to be the property

Dual spaces

So let’s have a look at the dual spaces of and . This was the actual exercise I misread first

(page 28, chapter 3, exercise 3.17.4 and 3.17.5 in Nestruev)

Describe the dual spaces of and .

The dual space of is simply , as described above. A detailed proof is contained in Nestruev. The most difficult thing to prove is that doesn’t contain more points than those coming from evaluations. To prove this, compactness of is used.

The dual space of consists of all -algebra homomorphisms . At least we have all evaluations at points . In the case of these evaluations coincide for two if the difference was an integer. This can’t happen here because for each two there is a function with period greater than such that .

To see that there aren’t more functionals than those coming from evaluations, suppose having such a functional . Then for each , there is some with . Note that we could actually choose the to lie in and the whole proof works for , too. The complements of the pre-images are non-empty open sets which form a cover of . Choose a finite subcover and define a function by

Geometric algebras

Nestruev also poses a follow-up problem (strangely, earlier in the text)

(page 24, chapter 3, exercise 3.9.5)

Decide whether and/or are geometric -algebras.

is clearly geometric, since as a set and a 1-periodic function on that vanishes everywhere on is always the constant zero function, so contains only the zero function, thus is the zero ideal. So we know now that is isomorphic to .

Similarly, if a rational-periodic function on vanishes everywhere on , which is equivalent to being in the kernel of all elements of , then it must be the constant zero function. So is geometric, too, and is isomorphic to .

This is a good point to get confused. Take care of the definition of – these are exactly those maps that are \emph{given by evaluations}, nothing more. Of course, we would like to have some smooth-manifold-like object instead of and look at smooth functions on instead of this algebra .

For geometric algebras , an algebra homomorphism induces a dual morphism (we have already looked at this in the case of the inclusion , which has as dual map the n-fold covering of , and the isomorphism , which has as dual map the identity of ). It is easy to prove that an isomorphism has a homeomorphism as dual map. The inclusion has the dual map given by the quotient , which is an -fold covering map.

Complete algebras

The next step in the Nestruev definition of smooth manifolds is the notion of complete algebras, a technical notion required to be able to talk about restrictions which are required to define what a smooth algebra should be.
For any subset of the dual space of a geometric -algebra , the elements can be restricted to , which yields the restriction homomorphism . The space has to be defined carefully, as the space of functions on that can locally be written as restrictions of functions in :

Of course, Nestruev wants us to check if our algebras are complete:

(page 32, chapter 3, exercise 3.28.3)

Decide whether and/or are complete -algebras.

is clearly complete, since a function locally defined by the restriction of smooth 1-periodic functions can be seen as a smooth function on that satisfies , so it’s an element of again.

is incomplete. For example, define a function by choosing for each interval with a smooth periodic function with period such that for . The functions can now be chosen such that is not periodic, for example by letting the maxima of the grow without boundary, letting be unbounded, too.

The algebras of smooth functions on manifolds are always complete (exercise!), so we have seen that the manifold we were looking for doesn’t exist.

-closed algebras

Returning to the general setting, we would like to have a homeomorphism for any -algebra and subset . If is geometric, there is always a continuous map which is a homeomorphism onto a subset of , but fails to be surjective in general.

A geometric -algebra is -closed, if for all finite families of functions and all smooth maps , the composite is an element of (considered as function on ).

Nestruev proves that for each basis open set of a -closed geometric algebra , the map is surjective.

Smooth algebras

A complete, geometric -algebra is called smooth if there exists a countable open cover of such that the algebras are isomorphic to for some fixed natural number , called the dimension of . Of course, the space has now a structure of a smooth manifold of dimension and .

is a smooth algebra.

Repairing defects

Any -algebra yields a geometric one, by quotienting out the ideal (at the risk of getting just the trivial algebra). A geometric algebra is obviously not changed in this process.

Any geometric algebra yields a complete one, by defining the completion to be the algebra . A complete algebra remains untouched by this process. The completion of is just the algebra of smooth functions on the real line.

Any geometric algebra yields a -closed one, by adding all functions of the form for and . This closure is definable via abstract nonsense, too: it is the unique smooth envelope (which is defined by the universal property, that morphisms to -closed algebras should factor uniquely through the smooth envelope). This way, the smooth envelope acts as mediator between non-closed algebras with smooth algebras.

Some comments

If anything here remains unclear, leave a comment. If something is wrong, please leave a comment. I also recommend reading Nestruev. It’s a nice elementary textbook (translated from Russian) that could be interesting for anyone who does either differential geometry, algebraic geometry or theoretical physics and of course for those who like all of these topics and their intersection. The algebra is frequently used in other examples throughout the book, but never appears again.

Jet Nestruev is a collective pseudonym, like Nicolas Bourbaki but a little bit less influental (and they didn’t write as much). The members of this group are A.Astashov, A.Bocharov, S.Duzhin, A.Sosinsky, A.Vinogradov and M.Vinogradov. If you like this kind of mathematics, take a look at this page about the works of A. Vinogradov.

The main theme, guessing from a rather philosophical paper, seems to be the notion of diffiety, a geometric object that plays for quantum field theories the same role that smooth manifolds play for classical mechanics. However, this is not just some abstract nonsense but actually something about partial differential equations. I guess I will have to learn about jet bundles some day…

1. Easy video recording – let the user take videos with every webcam within seconds, then upload to YouTube or similar. A similar proposal (a simple video editor) is on the lifehacker.com five feature request list.
2. Stream capturing – saving streamed video data doesn’t work so easily with all those different streaming formats. For some, you need a RTSP stream catcher, then maybe a RTMP stream catcher and for some you seem to be able to use just mplayer. And then there are many cases where all fails. Technically, what can be played can also be saved. But then there is the big Flash Player obstacle – some Flash videos are well-protected. Gnash may help there.
3. PDF reader&editor – one tool that allows for reading PDFs, annotating them, publishing&sharing the comments, manipulating the PDF itself, adding additional layers, manipulating PDF metadata, etc. Just like the Adobe Acrobat Reader in its more expensive variant – but as open source tool with the ability to write plug-ins and integration into Gnome or KDE (or any) desktop. Okular is already on the right track! See also my article about editing PDF metadata.
4. Centralised instant messenger and (video)telephony – unite Pidgin, Skype and other Videochat and IM apps in one UI. Maybe put this together with microblogging tools, since people use their IM status messages like microblogging anyway. Skype has announced to open-source parts of their Linux client, so this is not totally out of reach. Open source alternatives to TweetDeck are also there, for example Gwibber. See also my article on microblogging and news.
5. Metadata in file browser – make the file explorer a metadata editor, paving the road for a semantic desktop. Even the Windows Explorer can do better than Nautilus for now! But then I haven’t tried KDE’s Dolphin for a while and this might be the right thing to do… See also my article on music metadata as well as my article on photo metadata.
6. Asset manager – even one step further, make the file browser a pluggable asset manager, that can take the shape of a photo collection manager, a scientific paper organiser or a website bookmark manager. So far I know only of commercial asset managers and haven’t yet investigated which one runs on Linux and might be useful for me. Do you have any recommendations?
7. Annotation everywhere – a note-taking application that can annotate every single file or item on the desktop. This way you can relate a specific email to a task, to a note, to a website, to an application and a specific file – thus documenting entire work-flow states for later continuation. Well, there is Tomboy for now. See also my article on note-taking.
8. Private browsing – create for Firefox or any other browser a CSS/Javascript security model that avoids CSS privacy hacks by not letting any information about how the HTML rendered leak into the Web. That would include creating an open source Flash plug-in that doesn’t publish all Font and SuperCookie information. See EFF’s PanoptiClick for more information about this. The SuperCookie issues can be softened with the BetterPrivacy Add-On for Firefox.
9. Easy emulation – integrate a Dalvik VM naturally into the desktop, so that it’s very easy to install&run Android apps from the applications menu. Maybe the project to implement a Dalvik VM in Java is the right way to do this. Of course, the same would be nice for Wine but I don’t consider this an option because Dalvik is open and Windows isn’t.
10. Synchronise data with external sources – I want to backup all configuration and some data files with a variety of places: external hard-disks and remote storage services in the Web (encryption is necessary here). Ubuntu One is already a big step forward but I really want to backup all configuration so I could crash my computer, buy a new one, hit the “reinstall the software that was there before” button and then everything is back to normal. This is (almost) technically possible. Another road is, that I want to backup the data stored elsewhere (Delicious bookmarks, Google Reader Shared news, Facebook comments, etc.) to my home computer so I’m not stuck with one provider forever (so I can quit Facebook some day). This seems to be impossible for now, but the problem lies in coding “adapters” that take data from one service and move it to the other one.

Am I the only one who wants these features? Are they that hard to implement? (Yes) Hey, for most of these features, I would pay some money (depending on how well it’s implemented). Oh well, and I admit that these features are not really Linux-related. It’s just that I use Ubuntu and would want to have solutions available on open platforms. I guess web-apps and Java- or .NET-based apps would be OK for me, too – but then look again at the wish-list and you’ll see that most features require desktop applications.

If you have suggestions for applications that solve one of those problems at least somehow a little bit, please leave a comment.

What is your favourite not-yet-there Desktop/Ubuntu/GNU/Linux feature?

The penguin image (Tux) is licensed from linux.org under a CC-BY-SA license

I’m not going to do this, but in this post I’ll present some of my favourites from the last months at MathOverflow (omitting the more subject-specific ones):

• Success stories: Kevin Buzzard asked for errata on Cassel-Fröhlich’s Algebraic Number Theory book and the London Maths Society (LMS) is going to reprint that book, including an erratum. See also the discussion on meta.MO if errata requests are appropriate for MO and the Errata Wiki, an attempt to provide a common place for these things.
• Success stories: Timothy Gowers asked if MO has led to mathematical breakthroughs, and the answers list some cases where it at least helped a lot.
• Tools: Ilya Nikokoshev asked for tools that help in organising research notes.
• Tools: Elisha Peterson asked for tools that help in organising papers (toread, tocite, etc.) and the answers were helpful for my discussion on how to manage papers.
• Tools: Anton Petrunun asked for tools that help in collaborative paper writing.
• ToRead: Ilya Nikokoshev asked for “a single paper everyone should read” and there are some nice suggestions, if you don’t have enough to read yet
• ToRead: David Hansen asked for papers that maximise the ratio importance:length and there are some suggestions which won’t take much time to read!
• Teaching: Gil Kalai asked for fundamental examples in different branches of mathematics. The thread seems to me to be a very useful source to look for motivation.
• Teaching: Michael Hoffman asked what the undergraduate curriculum is missing and there are various answers and interesting controversies.
• Teaching: Ryan Budney asked for success stories with curriculum reforms (at average research universities).
• Fun: Mathematics in the real world – Theo Johnson-Freyd explains how to calculate using a very long frictionless one-dimensional billiard table.
• Fun: Andrew Stacey asked how to respond to “I was never much good at maths at school” and there are lots of serious answers, like this one by Andrew Tuggle which I like most.
• Understanding: Reid Barton explains how spectral sequences generalise long exact sequences.
• Understanding: Andrew Critch explains a geometric picture of flat modules where “flat” really means “flat” in an intuitive sense.
• Understanding: Scott Carnahan asked for a motivation for (higher) algebraic K-Theory.
• Understanding: Javier gives a rough idea about the field with one element (F_un en français)
• Understanding: Yuhao Huang asked “Why is the decomposition theorem awesome?”, but there are not many helpful answers despite links to the de Cataldo and Migliorini article. Have a look at the “related questions” section in the sidebar. I put this question here only because of the decomposition theorem winter school in Freiburg, Germany this month (Feb 2010).

The software MO uses is very well suited for this kind of excerpt, since there are permalinks not only for questions but for answers, too.

So maybe someone who is already thinking about being a math blogger will adopt this idea and watch out for nice general-interest questions&answers on MO, to blog about them occasionally (I won’t).

a simplicial model category is just an enriched model category which is enriched over the monoidal model category of simplicial sets.

but details are also to be found below.

The simplicial model structure on simplicial sheaves on a topos

In Definition 1.2, for every small site , a model structure on is defined:

1. The weak equivalences are the stalkwise (pointwise) weak equivalences
2. The cofibrations are the monomorphisms
3. The fibrations are defined via the right lifting property with respect to acyclic cofibrations

Remark 1.3 is a technical subtlety. If you happen to have a conservative set of points of a topos , then weak equivalence of a morphism of sheaves on can be tested pointwise: , where denotes the weak equivalences in the standard model structure of simplicial sets. A conservative set of points is just a set of points that is a conservative family of functors, which is by definition, that the product functor is a conservative functor.
A functor is conservative if it reflects isomorphisms. That means, isomorphism implies isomorphism for each morphism .
This technical lemma is used later in the text, but the homotopy sheaves are not, so I guess you can forget the proof details when reading the text for the first time.

See also: conservative functor in nLab

Theorem 1.4 (the structure defined by is a model category structure) cites the result of Corollary 2.7 in Jardine: Simplicial Presheaves, in no. 47 J.Pure Applied Math, 1987 which is originally due to Joyal. Since the article is behind a paywall, I’ll give you a rough idea:

• (MC1), (MC2) and (MC3) are deduced from the model structure on simplicial sets.
• (MC4) relies on the fact that the morphism from a presheaf to its associated sheaf is a weak equivalence and then applying the axiom for with the global fibration and topological weak equivalence model structure. (MC4) for is proved with a trick that uses (MC5).
• (MC5) is essentially a small object argument.

The corresponding homotopy category of on is written .

See also: small object argument in nLab

Proper model categories

Remark 1.5 states that the model structure is a proper one. The proof is available in Jardine, J.F.: Stable homotopy theory of simplicial presheaves, in no. 39 Can. Math. J, 1987 which is available for free here.

A simplicial model category is proper if

• (P1) the pullback of a weak equivalence along a fibration is always a weak equivalence,
• (P2) the pushout of a weak equivalence along a cofibration is always a weak equivalence.

(P1) is proved for simplicial sets via fibrant replacement, such that one has a cartesian diagram up to weak equivalence, and then application of K. Brown’s coglueing lemma, which is Lemma 1 on page 428 of Brown, K.: Abstract Homotopy Theory and Generalized Sheaf Cohomology, in Vol. 186 Transactions of the American Mathematical Society, 1973 which you can download from the nLab for free.
(P2) is proved for simplicial sets in a dual fashion, using the fact that simplicial sets are always cofibrant and a dual of Brown’s coglueing lemma.

For simplicial presheaves on a topos, the proofs are similar. For (P1), fibrant replacement yields a cartesian diagram (up to weak equivalence) in which all objects are locally fibrant simplicial presheaves (which form a category of fibrant objects) and the coglueing argument can be applied. For simplicial sheaves, (P1) and (P2) follow since the associated sheaf morphism is a weak equivalence.

It should be mentioned that (P1) is also called right proper and similarly (P1) left proper.

See also: proper model category in nLab

Functorial fibrant replacements (1.6)

(MC5) demands in particular, that every morphism is functorially factorizable into a fibration after an acylic cofibration.
A resolution on a site (which carries a model structure) is defined to be a functor and a transformation such that for every simplicial sheaf , the object is fibrant and is an acyclic cofibration.
Indeed, if is a morphism, we can factorize it into an acyclic cofibration followed by a fibration. Rename the acyclic cofibration and the object , then is a fibration, thus fibrant. Voilà – since (MC5) demands this to be functorial, the functor/transformation conditions for a resolution are fulfilled.
It should be clear that this works the same way for cofibrant replacements, although we won’t need this here, since in the simplicial model structure we’re looking at on , all objects are cofibrant.

See also: Kan fibrant replacement in nLab

Simplicial model categories

For every two objects , we defined

If you already know about the “subtleties” in the definition of simplicial model categories (maybe from my article about simplicial model categories), skip the next two paragraphs.

A category is a simplicial model category if it is a model category that is enriched over simplicial sets, that satisfies the additional axioms (Quillen):

• (SM0): for all and all finite simplicial sets , and exist.
• (SM7): If is a cofibration and a fibration, then is a fibration of simplicial sets, which is trivial if either or is trivial. (The S denotes the simplicial mapping object of ).

(SM0) is also phrased “ is powered and copowered” and sometimes already included in the definition of an enriched model category (like I did in my article about simplicial model categories). (SM7) is also phrased “the copower functor is a left Quillen bifunctor” and sometimes already included in the definition of an enriched model category (like I did, again). So, if you take the “modern” definition of a model category enriched over a monoidal model category, those axioms are already included (I put them in here just because they will show up in the literature and also because you might not have read my article about the definition of simplicial model categories).

Lemma 1.10, different notions of equivalence are the same

For fibrant and a morphism, these three statements are equivalent:

1. is a simplicial homotopy equivalence,
2. is a weak equivalence,
3. is a weak equivalence.

The proof indication is mostly a list of references, so let’s have a more detailed look, which will then finish this posting.

• (2)=>(1)
  factorise the weak equivalence into a cofibration followed by an acyclic fibration . Then is a weak equivalence again (by 2-out-of-3). By an argument in Quillen’s Homotopical Algebra (Corollary 2.5), obtain a retraction of by the lift in the diagram
  
  and then get a simplicial homotopy from to by the lift in the diagram
  
  and now is a simplicial homotopy inverse of . To actually obtain a simplicial homotopy inverse of , we’re going to build a simplicial homotopy inverse of . For this, observe that all objects are cofibrant (since cofibrations are by definition just monomorphisms), and that the dual statement to what we just proved is that a trivial fibration between cofibrant objects is a simplicial homotopy equivalence.
  What is ? What is ? you might ask. The object is just the simplicial set , whose geometric realisation in looks like the interval , hence the name (and I used this notation here because it’s the same as in Quillen’s book). The object is the internal mapping object . If this remains unclear, you might want to read some introduction to enriched category theory.
• (1)=>(3)
  We will not try to construct a weak homotopy equivalence but a homotopy equivalence:
  Using the definition of for and , you’ll see the canonical isomorphism . Now take a simplicial homotopy inverse to the map and choose a simplicial homotopy between and . This yields a map which, composed with the canonical isomorphism above, is the homotopy between and we’re looking for. The other composition is handled the same way.
• (3)=>(2)
  From SGA4 6.8.2 we learn that every point of has an associated functor , where is the category of neighbourhoods (French: voisinages) of . A neighbourhood is a couple where and . The cofiltrant category of neighbourhoods of admits a small cofinite full subcategory, so by abstract nonsense the functor is a pro-object in . A pro-object is, by definition, just a functor from a small cofiltered category to (think of it as a diagram to form a projective limit, hence the name). Let’s write the pro-object , hiding the small cofinal full subcategory of in the indices.
  Now for a point , is a filtering colimit (=projective limit) of all , thus a filtering colimit of weak equivalences. We conclude that is itself a weak equivalence. Since this holds for every point, is a weak equivalence.

This is something not to miss if you’re interested in mathematics, and it might also be valuable if you’re taking a first course in complex analysis. Even after you’ve taken a course on complex analysis, you might enjoy the animation of the Hopf fibration (which I liked most).

Go straight to watching the videos in English.

via The Math Less Traveled (via Wadler’s Blog)