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)

As I said, you don’t need to be interested in the Hodge Conjecture (although it’s one of the so-called millennium problems, and you would get a million $ for a proof or disproof). You’ll get a glimpse on some central ideas of geometry from this talk, including:

• Projective plane (where every two lines intersect in a unique point, but maybe a point at infinity)
• Cycles (certain “shapes” in spaces, like points, lines, circles)
• Complex numbers (you don’t need to remember i²=-1)
• Polynomial equations (like x²+y²=1)

At the end, it’s a bit fuzzy about integrals, differential forms and the actual Hodge Conjecture, but if you weren’t looking forward to learn anything about the Hodge Conjecture, this spares you about 30 minutes of the video.
If you’re really interested in the Hodge Conjecture, take any book about Algebraic Geometry and learn some of what’s in there. If you did this already – then what are you doing here?

When people ask me about the stuff I learn in mathematics, I try to explain something about homotopy/topology, circles defined by polynomial equations or projective space. This usually conveys the idea that mathematics is not all about number crunching or calculus. So this video is in the same spirit of explaining what mathematics is.

If you don’t want to know anything about geometry (maybe because you hate mathematics since school), then I recommend this video (10 minutes, but you can skip the last 5 minutes):
“The most important video you’ll ever see” [sic]

Let’s have a look at some of the best Web 1.0 math websites:

• The On-Line Encyclopedia of Integer Sequences – look up some short sequence of numbers to see in which patterns they fit.
• John Baez: This Week’s Finds in Mathematical Physics – John Baez has been writing his wonderful thoughts about mathematics, physics and the in-between for more years than I know what mathematics is. You can learn a lot from these notes. He has been posting it in sci.physics.research, sci.math.research, sci.physics and sci.math but now he also has an RSS feed, of course.
• The Mathematical Atlas – a hand-crafted tour through the various regions of mathematics, clustered along the AMS classification, spiced with many useful links. (I hope this will be relaunched as a community-based website one day. It deserves to survive).
• The Mathematics Genealogy Project – find out how half of all professors are descendants of Mersenne: 139335 mathematicians in the database, 61089 descendants of Mersenne. They have nice posters, too.

And now, before I sketch my vision of Math 2.0, for some of the best Web 2.0 math projects:

• The n-category café – a group blog about higher algebraic structures (especially n-categories) and physics. There are almost always interesting discussions going on.
• The Catsters – two mathematicians explain category theory (thus the name Catsters) in short, understandable snippets made exclusively for YouTube. Have you ever felt the need to learn what Monads are? String diagrams? Maybe you would be happy if someone would explain you limits and colimits. The Catsters do it, and they do it for free.
• The Open Problem Garden – a collectively maintained list of open problems in mathematics, ranked in difficulty. It’s still in an early phase of it’s life-time and somehow concentrated on problems with a combinatorial flavour, especially graph theory. Maybe you could enter your favourite open problem there?
• Wolfram Alpha – a mathematical data search and browse engine. You can look up statistics, perform comparisons and calculations and visualise this data. Very nice!
• Complexity Zoo – a website that collects computational complexity classes, with lots of helpful explanations and fact around them. At the moment of writing, they count over 480 complexity classes!
• Rigorous Trivialities – a group blog about algebraic geometry, with a huge series about “Algebraic Geometry from the Beginning” – which I recommend for it’s little intuitive text-snippets, where you can pick just what you need.
• The Secret Blogging Seminar – a group blog about algebraic geometry.
• The Unapologetic Mathematician – John Armstrong’s high-level educational math blog. You can pick some topic you want to learn and track back the links to the point where you’re on safe ground. This way, learning is much more efficient than using a linear book. Covers, for example, some category theory.
• What’s new – Terence Tao’s blog. He describes it with the words “Updates on my research and expository papers, discussion of open problems, and other maths-related topics”. Well said, worth a look!
• Timothy Gowers’s blog – currently obsessed with the PolyMath project (see below).

UPDATE: For a list less biased by my personal interests, see the thread “most helpful math resources on the web” on MathOverflow

Okay, now what is Math 2.0?

Math 2.0 is mathematics done collaboratively in genuine new ways over the internet.

This means, a website qualifies as Math 2.0 if it changed the way mathematicians collaborated.

However, it seems like the school education community, more focused on children, uses the term Math 2.0 as a buzzword for “learning math over the internet”.
From the Delicious tag Math2.0 you can see that the term is also used for math blogging.

Keep in mind that all this 1.0, 2.0 buzzword terminology is just tagging some websites. It’s not important, and as Tim Berners-Lee says, the web was always about communication from person to person, it’s nothing new.

My favourite Math 2.0 projects:

• The nLab – the wiki associated to the n-category café, an attempt to structure the discussions and facilitate re-use. This way, the nLab users build an expert encyclopaedia about their subject. Since it’s a subject with intense research going on, it’s more like their secret lab book than like the consensus-based Wikipedia. The rather inclusive viewpoint instead of the encyclopaedic exclusive viewpoint of Wikipedia has already created a very helpful collection of references. The nLab personal lab wikis have already shaped how people do their mathematical research, thus it truly qualifies for Math 2.0.
• MathOverflow – the mathematical question&answer web site, intended to be used by mathematicians (so, no homework questions on this site). Without MathOverflow you would have to know the right people. With MathOverflow you can just ask them.
• PolyMath – the first massively collaborative mathematics problem solving project. It was successful, so they’ve just recently started the next PolyMath project. Gowers and Nielsen have an article in Nature about PolyMath.
• The Tricki – a wiki of problem solving tricks. It’s somehow in the spirit of Polya’s book about mathematical problem solving, but much more practical, solution-centered in concrete situations. It’s something you couldn’t get with a book and it’s perpendicular to classical literature.

Usage of the terms “Web 2.0 math” and “Math 2.0″:

I have some ideas in my mind for a future Math 2.0 project, involving creative use of LaTeX, wikis and collaborative/social websites… but it will take another few months until the idea is ready to go public, and I still need to convince some collaborators to help me with the workload.

Where they talk about Web 1.0 and Web 2.0, the Web 3.0 is not far. Clearly, somebody must fill the buzzword Math 3.0 with some nonsense! Since this post is already long enough, I will speak about the semantic web, Web 3.0 and the great potential for mathematicians another time.

(And I’m really sorry that I didn’t list all good math blogs or other math projects, not even all my favourite ones, like this wonderful blog about motivic stuff. However, if I missed a popular one, I would be happy to hear about it in the comments.)

(*) in Germany, only parties that get at least 5% are considered for a seat in the Bundestag (with very few exceptions). There are numerous parties that didn’t reach this limit and they are subsumed under “other” and take 6% of all votes together.

As you can see, no party has more than 50%, not even closely. That’s why parties have to join together in a coalition, that reaches the 50% limit together. Technically, it’s possible that three parties join together in a coalition (then it would be effectively impossible to change politics via voting). It has happened just before the 2009 elections that the two biggest parties (CDU/CSU and SPD) formed a coalition. But that has not happened again, and as it’s usually the case, a small party joined a big party in coalition (FDP and CDU/CSU).

Maybe you notice that FDP and CDU/CSU don’t get 50% but only 14.6+33.8=48.4% of all votes. Well, this is correct (although you have to keep in mind that we are only talking about votes, not about people who are allowed to vote or inhabitants. I’m providing more numbers for this issue below). As I noted in the table above with a (*), the votes for parties that don’t reach 5% are excluded from the election counting. Yes, that’s true: If you vote for a party that’s unlikely to reach 5%, your vote is lost and your opinion who should be in the parliament doesn’t count any longer. But there are even more issues like that, so let’s wait for a moment and look at the table of votes given, excluding the “other” parties:

Now FDP and CDU/CSU have 15.5+36.0=51.5% of all votes. What about other coalition options? Just looking at the numbers, not politics, CDU/CSU+SPD would have been possible, too, with 60.5% of all votes. A coalition of three parties, FDP+SPD+GRÜNE would have collected 51.4% of all votes, so this would have been a possibility, too.

(comic licensed from Randall Munroe under a Creative Commons Attribution-NonCommercial 2.5 License)

Voting systems are designed (or at least, some people pretend they are) to give those power who are in favour of the masses. But what do the people really want? One (silly) way to look at the numbers would be: the more % a coalition has, the more people will be happy about it. Take another look at the numbers: in a coalition of CDU/CSU and FDP, there are 36.0 votes for CDU/CSU, thus 64.0 votes against CDU/CSU, and there are 15.5 votes for FDP, thus 84.5 votes against FDP. This means, strictly thinking about the numbers, that 64% of all votes are against this coalition.

You might say now: “maybe the votes for FDP have had CDU/CSU as their second choice?” and this is the right direction to look at. We’re not counting second or third choices now and this poses several problems. If we’d ask the 6% of voters who were ignored because they’ve voted a small party, what their second choice would be – it may change the outcome of the elections. Imagine that half of the other-party-voters would have chosen the LINKE as second choice (this is an estimate which might be close to reality) and the other half is distributed like the other votes. Look at the changed vote table then:

You see that FDP+CDU/CSU have now 15.1+34.8=49.9% of all votes. This is close, but not equal to 50%, so they could not form a coalition. This is just a little example to show what’s odd in voting systems.

Maybe you’re sitting in front of you computer, muttering “of course different voting systems lead to different outcomes, what’s wrong about it?”. In my personal opinion, a voting system should be designed in a way that the common wishes of the group are expressed in the result. Interestingly, mathematics tells us that this is impossible in many cases, especially if there are many options for a vote with a close outcome. The current voting system in Germany encourages strategic voting, which is voting a party (more generally an option) you don’t prefer most, knowing that the mathematics add up so your real preference will win more likely because of your vote. It’s even possible to weaken a candidate by voting for him. In the interesting Bush/Gore presidential election in the US, it’s pretty sure that Gore would have won if Nader (the third candidate) wouldn’t have been an option to vote for.

From the book “Chaotic Elections” you can learn that amongst all positional voting procedures, the one the least manipulatable by strategic voting is the Borda Count. For this procedure, you have to ask the voter about his second choice, and even his third, fourth (and so on). This sounds complicated and I don’t think it’s feasible yet for national votes. Maybe it suffices if we require to name a second choice. It does not suffice to open the possibility of naming a second choice – people don’t use this opportunity very often (this has been seen in experiments). That’s because almost nobody knows about voting paradoxes and how to create and avoid them. So we all are very manipulatable if we don’t learn how elections work, mathematically. However, manipulating elections by choosing a specific voting procedure or by strategic voting is not the only method of manipulation in democratic settings: imagine wrong statistics (or even no statistics) provided by bureaucracy or manipulative nomination, which includes strategies like “I propose something very close to what everybody likes, but I bundle it with something everybody hates – then nobody will vote for it”, which work if similar proposals are gathered to facilitate voting (in particular, the german e-petitioning system is vulnerable to this attack).

Some more numbers of general interest:

• total population, in millions: 81.8
• people allowed to vote, in millions: 62.2
• people who actually voted, in millions: 44.9
• people who did not vote but were allowed to, in millions: 17.3
• people who did not vote, including those not allowed to: 36.9
• percentage of actual voters in total population: 54.9
• percentage of people who voted for either CDU/CSU or FDP
  in total population: 26.6
• percentage of people who voted either CDU/CSU or FDP
  in population allowed to vote: 34.9
• percentage of people who didn’t vote either CDU/CSU or FDP
  in population allowed to vote: 65.0

the most interesting number, for me, is the last one. It indicates that 65% of all people that are allowed to vote could have changed the outcome of the vote if they would have had been asked for their second choice. Maybe asking them (and counting with Borda Count) would have shown that CDU/CSU+FDP was a common choice – maybe we would have had a completely different parliament in Germany now. It’s always questionable why 17.3 millions of Germans didn’t vote although they’ve had the right to do so. Maybe they knew that they couldn’t vote for their favourite party and change politics at the same time.

(comic licensed from Randall Munroe under a Creative Commons Attribution-NonCommercial 2.5 License)

I looked up the numbers used here from different statistical services and newspapers. All numbers are from 2009. Every number is either checked twice or calculated by myself. But beware: those are statistics, you’re not supposed to trust them!

There is a review about the book here, if you’re interested now.

There are many, many other voting paradoxes which can happen and affect our life, in every situation involving a democratic process with more than 2 options (**), which happens quite often. I recommend you to read something about this subject, maybe something shorter or better digestible than Saari’s book – but if you’re interested in politics, social phenomena or mathematics (or anything in-between), I guess it’s best to take a look at Saari’s book. It has parts which require you to understand some basic calculus, but in Germany you learn all this in the last years of highschool. Even if you ignore all calculations, the basic problems are well explained. I myself took some calculations for granted, since I began to trust the author after some examples turned out to be correct

The most provocative quote from Saari, from page 100 of the book:

For a price, I will serve as a consultant for your group for your next election. Tell me who you want to win. After talking to the members of your organization, I will design a ‘democratic procedure’ which ensures the election of your candidate.

and after reading the book I have to admit that it’s not a big challenge. Once you know the voter’s preferences with a certain probability, you’re able to design some voting system that “chooses” your favourite option. That’s why most democracies tend to stick to one voting procedure instead of changing it for each election (and gerrymandering is a popular example of such manipulations).

(**): Little exercise: why aren’t there any voting problems or paradoxes when there are only two options?

How to manage papers?

I have lots of PDFs on my hard-disk, and most of them is half-read or unread. Since I’m studying mathematics, these PDFs are lecture notes, research papers, my own notes and several more-or-less relevant books. How do I organise them? It’s a problem.

I separate my own notes from everything else, and keep my own PDFs with their LaTeX source in the same folder, these folders are sorted by subject. The other PDFs fall in 3 categories: the relevant, the irrelevant and the books. I keep all books in one big folder and use search to find something specific. The irrelevant PDFs are just somewhere floating in my computer – on my desktop and in various folders called “papers”, “some papers”, “more papers” and “maybe read”. The relevant PDFs are re-named to make them easier to find with a desktop search engine (I usually put the authors name, some keywords and the year in the file-name) and they’re sorted by subject in folders. I consider this a very bad solution.

So I have a second system: JabRef, which is an open source reference manager (and BibTeX database editor) written in Java, so you can use it on Windows, Linux and Mac OS X. In JabRef I put every book or paper I want to cite somewhere or that I think of being that interesting that I just have to keep one more safe reference. JabRef allows to link BibTex entries to files on your hard-disk. This way you can forget about where you stored your files.
Interesting fact: Google Scholar has a setting to enable BibTex export – which you can use together with JabRefs BibTex import for very fast database creation/updating. Sadly, the Google Scholar metadata is wrong sometimes…
Another interesting fact: there is a Bibsonomy plug-in for JabRef. Sadly, it doesn’t work for me (I don’t get it how to install it under Linux). Bibsonomy seems to be used almost exclusively by computer scientists, especially those working on semantic web stuff.

It’s strange, but I have a third system: a huge spreadsheet in Google Documents, where I keep all the papers and books that are related to something I want to learn in near future. I keep track of dependencies, that means I use this spreadsheet to find out which paper/book I have to read first in order to understand the second. I also attach some relevancy score to each item. Finally, some insane formula calculates a ranking across all items which tells me the top 3 papers I have to read next. This works (at least after I adjusted the formula long enough so that it displayed what I already knew was important). I’m currently thinking about replacing this system with CiteULike, because you can enter a reading priority there, too.

For Wikipedia users, the reference management system Zeteo might be of interest. You can’t have user accounts there, but easily manage references for later use in Wikipedia. Funny: it turns out it’s written by someone working in my math department!

(image “Dropbox Upgrade” licensed from Scott Beale under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Generic license)

It’s no fun to “maintain” three systems, one messier than the other. I’m currently thinking to just move everything over to JabRef, but this doesn’t resolve the “storage” problem. Maybe one big folder that contains it all, combined with desktop search is the best solution. Maybe it would be cool to store the papers on-line, in Ubuntu One, for example. But I wasn’t bold enough to try, yet. Maybe it’s good to keep track of references in some Web 2.0 tool, like Bibsonomy? I haven’t tried either. I think it is a good idea to remove any PDFs from your hard-disk that are available without restrictions (like arXiv papers) that you don’t need in the near future. The less there is, the easier it is to manage it. If you have any suggestions, I would be very happy to read your comments.

For quite some time, eknigu.com didn’t work for me any longer. I was annoyed by the 15 seconds of waiting before you could actually get something, so it was time I wrote a little userscript to get rid of the problems:

download the Eknigu Downloadr userscript

If you don’t know what a Greasemonkey userscript is, look here and here.

Some technical info:

You could get the same result by downloading the page and manipulating the HTML code (you’ll know how to manipulate if you read the JavaScript sources). You have to set the form.action to the right URL and the form’s input field “id” to the value “go”. At the same time, you have to stop the internal JavaScript from resetting the input field “id” to the value “15″. This is essentially what the userscript does, although not by changing anything but by adding an additional layer which contains a second form not influenced by eknigu’s JavaScript.

I tried to give many more references I found useful.

Standard model structure on simplicial sets

Let be a morphism of simplicial sets. is said to be a topological weak equivalence if the geometric realization is a weak equivalence (that is, induces isomorphisms on all homotopy groups).

is said to be a Kan fibration if it has the right lifting property with respect to all horn inclusions. A horn inclusion is a map , where the k-th horn of the n-simplex is just the simplicial set generated by faces of the n-simplex except the k-th face (so the horn is a subcomplex of the boundary of the n-simplex).

The standard model structure on simplicial sets takes as weak equivalences the topological weak equivalences, as fibrations the Kan fibrations and as cofibrations the monomorphisms (which are just degreewise injective maps).

In the standard model structure, all simplicial sets are fibrant. A Kan complex is a simplicial set that satisfies the extension condition, which is, if you take (n+1) n-simplices that satisfy for all , that , then there exists a (n+1)-simplex whose faces are . The cofibrant objects in the standard model structure are exactly the Kan complexes. This standard model structure is sometimes called Kan model structure on simplicial sets. It is worth noting that the singular simplicial set of a topological space is always a Kan complex.

The cofibrant-fibrant replacement for a simplicial set is therefore a functor, that turn every simplicial set into a weakly equivalent Kan complex. This is achieved by either taking the singular simplicial set of the geometric realization of a simplicial set or via Kan’s functor.

More details can be found in May’s book “simplicial objects in algebraic topology”.

Monoidal categories

Monoidal categories generalize various notions of tensor-like operations in categories. They will be useful to define enriched categories, which are then used to define what a simplicial model structure is.

A (lax) monoidal category is a category equiped with a bifunctor, often denoted , an object called identity, and natural transformations that make this the identity of and the operation associative, up to isomorphism of functors. There is a coherence condition to be satisfied, so that all diagrams made out of the natural transformations corresponding to associativity, left unit and right unit, commute. It can be shown that every such lax monoidal category is equivalent to a strict one, where the natural transformations are identities. This equivalence can always be done via monoidal functors, which are those functors that respect the bifunctor , the identity and the natural transformations.

Good examples are the category of sets with cartesian product and the one-point-set as identity and the category of abelian groups with tensor product over the integers and the integers as identity. The category of small categories is a monoidal category, too, with the cartesian product of categories and the one-object-with-identity-category as identity.

The category of sets has the nice property that the functor has a right adjoint . If a monoidal category has this property of having a right adjoint to , it is called (left-)closed and the objects in the image of this right adjoint are called mapping objects, sometimes written as . The bifunctor that sends to the mapping object of the right adjoint of evaluated at is called internal Hom. It is important to differentiate between left-closed and right-closed categories but in many cases the monoidal structure is braided, which means there is a transformation (satisfying some commutative diagram), and for the category of sets this braiding is symmetric, which means it is an isomorphism, so left-closed and right-closed are equivalent notions. The category of sets and the category of small categories are examples of cartesian monoidal categories, because their monoidal product coincides with the categorical product and the identity is the final object. In cartesian closed categories, the mapping objects are written as exponentials .

Contravariant functors from a category to a monoidal category form a monoidal category with pointwise monoidal operation. This is the general way which makes the category of simplicial sets a monoidal category. Since the category of sets is cartesian closed, the inherited structure on simplicial sets is cartesian closed, too. It is an interesting fact, that geometric realization of simplicial sets is actually a monoidal functor, when we take the standard cartesian structure on the category of compactly generated weak Hausdorff spaces. In formula, this means in particular for any two simplicial sets and the geometric realization functor .

Now let’s turn to monoidal model categories. For these, we need the notion of a Quillen bifunctor.
Let be model categories. A left Quillen bifunctor is a functor that preserves small colimits in each variable (seperately) and satisfies this condition (sometimes called pushout-product axiom):
For all cofibrations in and in , the induced morphism is a cofibration in . If either or is, in addition, a weak equivalence, then is required to be a weak equivalence, too.

Now a monoidal model category is a closed monoidal category equipped with a model structure such that the unit object is cofibrant and the tensor functor is a left Quillen bifunctor. This definition ensures that the homotopy category will be a closed monoidal category. In some rare cases, the unit object is not cofibrant and one uses a slightly weaker condition, but this isn’t necessary for our purposes here.

The category of simplicial sets, with the usual cartesian monoidal structure and the standard model structure, is a monoidal model category. The (in my personal perspective) hardest part of the proof is to see that the tensor functor preserves the trivial cofibrations (that are exactly the anodyne extensions). Hovey (see below) does a very good job at explaining this.

Further reading:

• nLab: monoidal category
• nLab: closed monoidal structure on presheaves
• nLab: monoidal model category
• nLab: simplicial set
• Mark Hovey: Model Categories (especially page 107 and following)

Enriched category theory

The definition of a category enriched over some monoidal category is a priori not directly related to the definition of a category, but a posteriori it’s just “ordinary category + extra structure”.

A category enriched over a monoidal category is a class of objects (as usual) and for each two objects an object . The analog of identities are morphisms in the category and the composition is defined as morphism . Of course, associativity of composition and identity axioms are required to hold.

The usual definition of a category is included in the enriched definition if we look at categories enriched over (well, depending on your definition of a category, you get only locally small categories this way).

Every enriched category has an underlying ordinary category where the Hom-sets are given by , so one can speak of giving an ordinary category an enriched structure.

A category which is enriched over is usually called (strict) 2-category. Of course, is itself a 2-category. This is very common: every closed symmetric monoidal category is enriched over itself, since it has internal Hom-functors.

One can define enriched functors and enriched transformations in the obvious manner, so it’s possible to speak of functor categories and therefore the enriched categories over a fixed monoidal category form a 2-category.

Since in a -enriched category we have (morphisms from object to object ) being an object of , we can for every object of consider the morphisms . If this has an adjoint, namely , then the functor is called the copower of by . It is actually a bifunctor. In the case where , this is always the monoidal product functor of , and thus it’s often called tensor. A category is copowered if it has copowers, that is there is a copower bifunctor satisfying the adjoint relation. The dual notion of power is sometimes called cotensor. I think speaking of tensors, in general, is not a good idea but it won’t hurt us for -homotopy theory.

To get a better feeling for copowers, look at the category of topological spaces (which carries a natural monoidal model structure, see Quillen’s Homotopical Algebra for details). The copower of a topological space by a simplicial set is just the topological space and the power of a topological space by a simplicial set is just the topological space .

An enriched model category , enriched over a monoidal model category is defined to be a category enriched over , powered and copowered, whose underlying ordinary category has a model structure such that the copower functor is a left Quillen bifunctor.

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

Further reading:

• nLab: enriched category
• nLab: copower
• nlab: enriched model category
• G.M. Kelly: “Basic Concepts of Enriched Category Theory”, London Mathematical Society Lecture Note Series No.64 (1982) – free PDF available!
• Daniel G. Quillen: Homotopical Algebra (especially chapter II, paragraph 3)

Topos theory

The topoi we’re talking about are Grothendieck topoi. Those are, by definition, categories equivalent to the category of sheaves on a small site. A site is a category equipped with a Grothendieck topology. A Grothendieck topology can be given by a pretopology although many different pretopologies may yield the same Grothendieck topology.

A pretopology consists of a set for each object, called the set of covering families. Each such covering family is supposed to be a set of morphisms into the object in question, such that these morphisms are stable under refinement and pullback and contain all isomorphisms into the object. Refinement is, if you have a covering family and for each a covering family then is supposed to be a covering family as well. Pullback is, if you have a morphism then the covering family obtained by pullback of each morphism of a covering family is a covering family is a covering family of .

A sheaf on a category equipped with a pretopology is a presheaf that satisfies for each object and each covering family that

Topoi have many useful categorical properties. To name same of them: they have all finite limits and all finite colimits and they are cartesian closed monoidal categories (so you can do some kind of Lambda calculus inside a topos). Consider “broadening” a category by using the category of presheaves on it (via Yoneda embedding). The choice of a topology and therefore what we call a sheaf, thus object of our topos, ensures categorical properties nice enough to think about the objects in our topos as the real “spaces” to define -homotopy theory. Look, for analogy, at topological spaces, which can be rather ill-behaved. Topologists work instead with the category of compactly generated spaces, which behave more like CW complexes. In this category, we know some nice (classical) homotopy theory, while this is not the case with the category of all topological spaces. For more heuristic arguments why this is the “right” way to proceed, look at Voevodsky’s paper in Documenta Mathematica.

The most common examples of topoi are the category of small sets (figure out how this is a topos as an exercise!) and the sheaves on the small/big Zariski sites of schemes. However, we’re interested in the sheaves on Nisnevich sites, which I will therefore describe here:

The big Nisnevich site of a scheme is the category of smooth schemes over the fixed base scheme equipped with the Nisnevich topology. The Nisnevich topology is in-between the Zariski and the étale topology, so I want to describe those three topologies at once, for comparison. Nisnevich called his topology the completely decomposed topology, or just cd-topology.

The canonical topology is the biggest topology that makes all representable presheaves actually sheaves. All topologies finer than that are called subcanonical. Now look at three examples of subcanonical pretopologies, ordered from coarsest to finest:

The Zariski topology is given by covering families that are surjective families of scheme-theoretic open immersions (by open immersion I mean a morphism that decomposes uniquely into an isomorphism and the inclusion of an open subscheme; open immersions are always étale morphisms, that means flat and unramified).

The Nisnevich topology is given by covering families that are surjective families of étale morphisms with the property that for every point , there exists an and a point such that the induced map of residue fields is an isomorphism.

The étale topology is given by covering families that are surjective families of étale morphisms.

Further reading:

• Saunders MacLane, Ieke Moerdijk : Sheaves in Geometry and Logic (1992)
• nLab: Grothendieck Topos
• John Baez: Topos theory in a nutshell (comes with lots of extra references)
• Jacob Lurie: Higher Topos Theory (For higher dimensional analogues. I recommend the appendices) – free PDF available!
• Yevsey A. Nisnevich: “The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory” (1989). in J. F. Jardine and V. P. Snaith. Algebraic K-theory: connections with geometry and topology. Proceedings of the NATO Advanced Study Institute held in Lake Louise, Alberta, December 7–11, 1987. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 279. Dordrecht: Kluwer Academic Publishers Group. pp. 241-342.
• Nisnevich’s website has this paper, a huge bibliography and more for you – for free!
• Vladimir Voevodsky: -homotopy theory (in: Documenta Mathematica, Extra Volume ICM I (1998), 579-604) – free PS available!

If someone would appreciate a posting about algebraic geometry related stuff (such as étale morphisms), leave a comment and I see what I can do.

Jörg Rothe: “Some Facets of Complexity Theory and Cryptography: A Five-Lectures Tutorial”

from the abstract:

In this tutorial, selected topics of cryptology and of computational complexity theory are presented. We give a brief overview of the history and the foundations of classical cryptography, and then move on to modern public-key cryptography. Particular attention is paid to cryptographic protocols and the problem of constructing the key components of such protocols such as one-way functions. A function is one-way if it is easy to compute, but hard to invert. We discuss the notion of one-way functions both in a cryptographic and in a complexity-theoretic setting. We also consider interactive proof systems and present some interesting zero-knowledge protocols. In a zero-knowledge protocol one party can convince the other party of knowing some secret information without disclosing any bit of this information. Motivated by these protocols, we survey some complexity-theoretic results on interactive proof systems and related complexity classes.

It’s accessible with a minimum amount of knowledge in computer science and/or mathematics. I guess it’s even funny without knowing anything (but then you should skip the “mathematical” parts). A man-in-the-middle-attack on an insecure channel is illustrated by a picture of Erich Mielke

If you ever wondered how PGP works, this is a good place to learn it!

some more bibliographic info:
57 pages, 17 figures, Lecture Notes for the 11th Jyvaskyla Summer School
Journal reference: ACM Computing Surveys, volume 34, issue 4, pp. 504--549, December 2002
Cite as: arXiv:cs/0111056v2 [cs.CC]