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David Roberts commented on Elementary topology of champs (1 day ago)
This looks like an algebro-geometric version of what I did in my thesis (, and a little bit more.

Title: Elementary topology of champs
Author: Michael McQuillan

Abstract: Broadly speaking the present is a homotopy complement to the book of Giraud, albeit in a couple of different ways. In the first place there is a representability theorem for maps to a topological champ (a.k.a. stack) and whence an extremely convenient global atlas, i.e. the path space, which permits an immediate importation of the familiar definitions of homotopy groups and covering spaces as encountered in elementary text books. In the second place, it provides the adjoint to Giraud's co-homology, i.e. the homotopy 2-group Π2, by way of the 2-Galois theory of covering champs. In the sufficiently path connected case this is achieved by much the same construction employed in constructing 1-covers, i.e. quotients of the path space by a groupoid. In the general case,so inter alia the pro-finite theory appropriate for algebraic geometry, the development parallels the axiomatic Galois theory of SGA1. The resulting explicit description of the homotopy 2-type can be applied to prove theorems in algebraic geometry: optimal generalisations to Π2 (by a very different method, which even gives improvements to the original case) of the Lefschetz theorems (over a locally Noetherian base) of SGA2, and a counterexample to the extension from co-homology to homotopy of the smooth base change theorem. These limited goals are achieved, albeit arguably at the price of obscuring the higher categorical structure, without leaving the 2-category of groupoids.

#arXiv arXiv:1507.00797 #spnetwork  
Richard Green commented on The Majority Illusion in Social Networks (3 days ago)
Why your friends, on average, have more friends than you do

The friendship paradox is the observation that your friends, on average, have more friends than you do. This phenomenon, which was first observed by the sociologist Scott L. Feld in 1991, is mathematically provable, even though it contradicts most people's intuition that they have more friends than their friends do.

Wikipedia gives a nice intuitive explanation for this phenomenon: People with more friends are more likely to be your friend in the first place; that is, they have a higher propensity to make friends in the first place. However, it is also possible to explain the phenomenon using graph theory and mathematical statistics. I give an outline of the mathematical proof at the end of this post for those who are interested, but the upshot is that if we look at everybody's numbers of friends, and these numbers have a mean of μ and a variance of σ^2, then the average number of friends that an average friend has is μ + (σ^2/μ), which will be greater than μ assuming that someone has at least one friend and that not everyone has the same number of friends.

The recent paper The Majority Illusion in Social Networks ( by Kristina Lerman, Xiaoran Yan, and Xin-Zeng Wu explores some phenomena that are related to the friendship paradox. The authors explain how, under certain conditions, the structure of a social network can make it appear to an individual that certain types of behaviour are far more common than they actually are.

The diagram here, which comes from the paper, illustrates this point. It shows two social networks, (a) and (b), each containing 14 individuals. In each case, three vertices are marked in red; let's suppose that these correspond to the heavy drinkers in the group. In social network (a), the heavy drinkers are three of the most popular people, and the configuration of the network means that each of the other eleven individuals observes that at least half of their friends are heavy drinkers. This will lead these eleven people to think that heavy drinking is common in their society, when in fact it is not: only 3/14 of the group are heavy drinkers. In social network (b), there are the same number of heavy drinkers, but they are not particularly popular, and nobody in the group will have heavy drinkers as most of their friends. 

Network (a) is experiencing what the authors call the majority illusion, whereas network (b) is not. The illusion will tend to occur when the behaviour in question is correlated with having many friends. The paper shows that the illusion is likely to be more prevalent in disassortative networks, which means networks in which people have less tendency to be friends with people like themselves. Observe that in network (b), many of the non-heavy drinkers are friends with each other, whereas in network (a), they are not. This suggests that network (a) is more disassortative, and thus more susceptible to the majority illusion.

The authors also study the phenomenon using three real-world data sets: (a) the coauthorship network of high energy physicists (HepTh), (b) the social network Digg, studying only the mutual-following links, and (c) the network representing the links between political blogs. It turns out that networks (a) and (b) are assortative, and (c) is disassortative. They also look at the the case of Erdős–Rényi-type networks, which can be thought of as random. 

As the paper points out, the friendship paradox has real life applications. For example, if one is monitoring a contagious outbreak, it is more efficient to monitor random friends of random people than it is to monitor random people. This is because the friends are more likely to be better connected, are more likely to get sick earlier, and are more likely to infect more people once sick. The reason for this has to do with the fact that these attributes are positively correlated with having many friends. 

If you're wondering why your coauthors are on average cited more often than you are, or why your sexual partners on average have had more sexual partners than you have, now you know. I found out about this paper via my Facebook friend +Paul Mitchener, who has more Facebook friends than I do.

Relevant links

Wikipedia's page on the Friendship Paradox contains much of the information in this post, including a sketch of the proof given below:

Wikipedia on assortativity:

Here's another post by me about the mathematics of social networks, in which I explain why it is impossible for everyone on Google+ to have more than 5000 followers. It provoked a surprisingly hostile reaction:

A post by me about Erdős and Rényi's construction of the random graph:

Appendix: Mathematical proof of the friendship paradox

Assume for simplicity that friendship is a symmetric relation: in other words, that whenever A is a friend of B, then B is also a friend of A. We can then model a friendship network with an undirected graph G, with a set of vertices V and a set of edges E. Each vertex v in V represents an individual, and each edge e in E connects a pair of individuals who are friends. For each vertex v in V, the number d(v) (the degree of v) is the number of edges connected to v; in other words, the number of friends v has. 

The average number of friends of everyone in the network is then given by summing d(v) over all vertices v of V, and then dividing by |V|, the total number of people. Using basic graph theory, this number, μ, can be shown to be equal to 2 times |E| divided by |V|, where |E| is the number of edges.

In order to find the number of friends that a typical friend has, one first chooses a random edge of E (which represents a pair of friends) and one of the two endpoints of E (representing one of the pair of friends); the degree of this latter vertex is the number of friends that a friend has.  Summing these degrees over all possible choices amounts to summing d(v)^2 over all possible vertices, and since the number of choices is 2 times |E|, it follows that the average number of friends a friend has is the sum of d(v)^2 divided by 2 times |E|. Using the formula for μ above, it follows that μ times the average number of friends that a friend has is equal to the average value of d(v)^2. However, the average value of d(v)^2 is also equal, by basic mathematical statistics, to the sum of the square of the mean of the d(v) plus the variance of the d(v). The result follows from this.

#mathematics #scienceeveryday #spnetwork arXiv:1506.03022
Thomas Scoville replied RE: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture (3 days ago)
Ya know, even with the Internet, we're still resolving difference with mayhem. Especially in the Middle East -- which is tragic, as so much math&science emanated from there in the first place. Enantiodromia strikes again.
John Baez replied RE: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture (4 days ago)
"Brahe was far from a dry scholar. In 1566 at the age of 20, he lost part of his nose in a duel with another Danish nobleman named Manderup Parsbjerg. The duel is said to have started over a disagreement about a mathematical formula. Because 16th century Denmark didn't have resources like the internet to figure out who was right, the only solution was to try to kill each other. For the rest of his life, Brahe wore a prosthetic nose. His fake nose was likely made of copper, although he probably also had gold and silver noses around for special occasions."
John Baez replied RE: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture (4 days ago)
+Scott Vorthmann - cool!

+Lars Fosdal - I'll check out some Tycho Brahe stuff; I'm a big fan of him and his golden nose.
Lars Fosdal replied RE: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture (5 days ago)
Nyhavn is a nice place for a stroll, but Copenhagen has lots of stuff to see. Tycho Brahe is another keyword.
Scott Vorthmann replied RE: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture (5 days ago)
The harbor looks similar to Bergen, Norway, where the buildings date from the time of the Hanseatic League and the salted cod trade.
#globalAttractorConjecture new topic created (5 days ago)
John Baez commented on Toric Differential Inclusions and a Proof of the Global Attractor Conjecture (5 days ago)
Chemical reactions in Copenhagen

This is a famous harbor called Nyhavn.  I haven't been there yet!  I'm in Copenhagen at a workshop on Trends in Reaction Network Theory, and I've been sweating away in hot classrooms listening to talks. 

But don't feel sorry for me!  (You probably weren't.)  I've been loving these talks, loving the conversations with experts and the new ideas — and after the workshop is over, I'm going to spend a few days walking around this town.

A reaction network is something like this:

2 H₂ + O₂ → 2 H₂O
C + O₂ → CO₂

just a list of chemical reactions, which can be much more complicated than this example.   If we know the rate constants saying how fast these reactions happen, we can write equations saying how the amounts of all the chemicals changes with time! 

Reaction network theory lets you understand some things about these equations just by looking at the reaction network.  It's really cool.

The biggest open question about reaction network theory is the Global Attractor Conjecture, which says roughly that for a certain large class of reaction networks, the amount of chemicals always approaches an equilibrium. 

It's a hard conjecture: people have been trying to prove it since 1974.  In fact, two founders of reaction network theory believed they'd proved it in 1972.  But then they realized they had made a basic mistake... and the search for a proof started. 

The most exciting talk so far in this workshop — at least for me — was the one by Georghe Craciun.  He claims to have proved the Global Attractor Conjecture!  He's a real expert on reaction networks, so I'm optimistic that he's really done it.  But I haven't read his proof, and I don't know anyone who says they follow all the details. 

So, there's work left for us to do.  His paper is here:

• Georghe Craciun, Toric differential inclusions and a proof of the global attractor conjecture,

There's a branch of math called 'toric geometry', which his title alludes to... but I asked him how much fancy toric geometry his proof uses, and he laughed and said "none!"   Which is a pity, in a way, because it's a cool subject.  But it's good, in a way, because it means mathematical chemists don't need to learn this subject to follow Craciun's proof.

There have been a lot of other good talks here.  You can read about some on my blog:

including the comments, where I'm live-blogging. 

I gave a talk called 'Probabilities and amplitudes', about a mathematical analogy between reaction network theory and particle physics, and you can see my slides.  Alas, the talks haven't been videotaped, and most of the other speaker's slides aren't available.  I have, however, collected links to some papers.

I've gotten at least two ideas that seem really promising, both from a guy named Matteo Polettini, who is interested in lots of stuff I'm interested in.  I won't tell you about them until I work out more details and see if they hold up.  But I'm excited!  This is what conferences are supposed to do.   They don't always do it, but when they do, it's really worthwhile.

The picture here was taken by a duo called angel&marta.  You can see more of their fun photos of Europe here:

Finally, here is the abstract of Craciun's paper:

Abstract. The global attractor conjecture says that toric dynamical systems (i.e., a class of polynomial dynamical systems on the positive orthant) have a globally attracting point within each positive linear invariant subspace -- or, equivalently, complex balanced mass-action systems have a globally attracting point within each positive stoichiometric compatibility class. We introduce toric differential inclusions, and we show that each positive solution of a toric differential inclusion is contained in an invariant region that prevents it from approaching the origin. In particular, we show that similar invariant regions prevent positive solutions of weakly reversible k-variable polynomial dynamical systems from approaching the origin. We use this result to prove the global attractor conjecture.

#spnetwork arXiv:1501.02860 #chemistry #reactionNetworks #globalAttractorConjecture   #mustread  
#highergeometry new topic created (Jun. 26, 2015)
David Roberts commented on The WZW term of the M5-brane and differential cohomotopy (Jun. 26, 2015)
When I first (quickly) read this, I mistakenly thought the cohomotopy in question was valued in a differential refinement of the rational homotopy type of S^4. I couldn't see where the switch happened to the full homotopy type, which Urs assured me it did (it starts at the bottom of page 8, continues on page 9).

This is very cool, in that it is about nonabelian cohomology valued in a extremely complicated (parameterised) homotopy type, denoted S^4_conn. Passing from a more naive object (the locally constant sheaf of Kan complexes given by the singular set of S^4) to S^4_conn is analogous to passing from (p_1)/2 to the Chern-Simons 2-gerbe with its connective structure.

Title: The WZW term of the M5-brane and differential cohomotopy
Authors: +Domenico Fiorenza​ , +hisham sati​ , +Urs Schreiber

Abstract: We combine rational homotopy theory and higher Lie theory to describe the WZW term in the M5-brane sigma model. We observe that this term admits a natural interpretation as a twisted 7-cocycle on super-Minkowski spacetime with coefficients in the rational 4-sphere. This exhibits the WZW term as an element in twisted cohomology, with the twist given by the cocycle of the M2-brane. We consider integration of this rational situation to differential cohomology and differential cohomotopy.

#arXiv #spnetwork arXiv:1506.07557 #highergeometry  
#CombinatorialGameTheory new topic created (Jun. 24, 2015)
Dana Ernst commented on Impartial avoidance games for generating finite groups (Jun. 24, 2015)
"Impartial avoidance games for generating finite groups" by +Bret Benesh, +Dana Ernst, and +Nandor Sieben now on the arXiv.

#CombinatorialGameTheory   #spnetwork  arXiv:1506.07105 
Richard Green replied RE: 10 Questions about Boggle Logic Puzzles (Jun. 24, 2015)
/msg +Yonatan Zunger did Pavlov's dogs know IRC commands?
John Baez replied RE: 10 Questions about Boggle Logic Puzzles (Jun. 24, 2015)
Yay, a +1!  Ah, I feel better now.
Yonatan Zunger replied RE: 10 Questions about Boggle Logic Puzzles (Jun. 24, 2015)
/me +1's +John Baez's comment to provide him with positive feedback. Then eats a food pellet. :9
John Baez replied RE: 10 Questions about Boggle Logic Puzzles (Jun. 24, 2015)
I agree.  But there's a kind of Pavlovian reaction to getting +1's that hard for me to resist.  At this point, having learned a bit about how to get a high score, I wish those +1's weren't there at all. 
Richard Green replied RE: 10 Questions about Boggle Logic Puzzles (Jun. 24, 2015)
That's definitely true, +Russ Abbott. Having a post go hot and get a lot of meaningless engagement is a very empty experience. If every day felt like that on Google+, I would have stopped posting.
Russ Abbott replied RE: 10 Questions about Boggle Logic Puzzles (Jun. 23, 2015)
+Richard Green I have found that interacting with a few people who care about what they say is more important that producing high "engagement" statistics. A worthwhile discussion with a few people is more valuable (to me) than a post that gets lots of +1's but no thoughtful comments.