Category Archives: offhand

Missing LeBron

When I was a postdoc in Princeton I subscribed to the Trenton Times, because I felt it was important to be in touch with what was going on in my local community and not just follow national news.  The only story I remember was one that said “hey, a basketball team from Akron is coming to play against a top prep-school team in Trenton, and they’ve got this kid LeBron James they say is incredible, you should come check it out.”  And I really did think about it, but I was a postdoc, I was trying to write papers, I was busy, too busy to drive into Trento for a high-school basketball game.

So I guess what I’m trying to say is, yes, subscribe to your local paper because local journalism badly needs financial support, and maybe actually take seriously the local events it alerts you to.

 

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The first pancake is always strangely shaped

Alena Pirutka gave a great algebraic geometry seminar here last week, about (among many other things!) families of smooth projective varieties containing both rational and non-rational members.   We were talking about how you have to give a talk several times before it really starts to be well-put together, and she told me there’s a Russian proverb on the subject:  “The first pancake is always strangely shaped.”  I am totally going to go around saying this from now on.

 

Ocular regression

A phrase I learned from Aaron Clauset’s great colloquium on the non-ubiquity of scale-free networks.  “Ocular regression” is the practice of squinting at the data until it looks linear.

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What is the Lovasz number of the plane?

There are lots of interesting invariants of a graph which bound its chromatic number!  Most famous is the Lovász number, which asks, roughly:  I attach vectors v_x to each vertex x such that v_x and v_y are orthogonal whenever x and y are adjacent, I try to stuff all those vectors into a small cone, the half-angle of the cone tells you the Lovász number, which is bigger and bigger as the smallest cone gets closer and closer to a hemisphere.

Here’s an equivalent formulation:  If G is a graph and V(G) its vertex set, I try to find a function f: V(G) -> R^d, for some d, such that

|f(x) – f(y)| = 1 whenever x and y are adjacent.

This is called a unit distance embedding, for obvious reasons.

The hypersphere number t(G) of the graph is the radius of the smallest sphere containing a unit distance embedding of G.  Computing t(G) is equivalent to computing the Lovász number, but let’s not worry about that now.  I want to generalize it a bit.  We say a finite sequence (t_1, t_2, t_3, … ,t_d) is big enough for G if there’s a unit-distance embedding of G contained in an ellipsoid with major radii t_1^{1/2}, t_2^{1/2}, .. t_d^{1/2}.  (We could also just consider infinite sequences with all but finitely many terms nonzero, that would be a little cleaner.)

Physically I think of it like this:  the graph is trying to fold itself into Euclidean space and fit into a small region, with the constraint that the edges are rigid and have to stay length 1.

Sometimes it can fold a lot!  Like if it’s bipartite.  Then the graph can totally fold itself down to a line segment of length 1, with all the black vertices going to one end and the white vertices going to the other.  And the big enough sequences are just those with some entry bigger than 1.

On the other hand, if G is a complete graph on k vertices, a unit-distance embedding has to be a simplex, so certainly anything with k of the t_i of size at least 1-1/k is big enough.   (Is that an if and only if?  To know this I’d have to know whether an ellipse containing an equilateral triangle can have a radius shorter than that of the circumcircle.)

Let’s face it, it’s confusing to think about ellipsoids circumscribing embedded graphs, so what about instead we define t(p,G) to be the minimum value of the L^p norm of (t_1, t_2, …) over ellipsoids enclosing a unit-distance embedding of G.

Then a graph has a unit-distance embedding in the plane iff t(0,G) <= 2.  And t(oo,G) is just the hypersphere number again, right?  If G has a k-clique then t(p,G) >= t(p,K_k) for any p, while if G has a k-coloring (i.e. a map to K_k) then t(p,G) <= t(p,K_k) for any n.  In particular, a regular k-simplex with unit edges fits into a sphere of squared radius 1-1/k, so t(oo,G) < 1-1/k.

So… what’s the relation between these invariants?  Is there a graph with t(0,G) = 2 and t(oo,G) > 4/5?  If so, there would be a non-5-colorable unit distance graph in the plane.  But I guess the relationship between these various “norms” feels interesting to me irrespective of any relation to plane-coloring.  What is the max of t(oo,G) with t(0,G)=2?

The intermediate t(p,G) all give functions which upper-bound clique number and lower-bound chromatic number; are any of them interesting?  Are any of them easily calculable, like the Lovász number?

Remarks:

  1.  I called this post “What is the Lovász number of the plane?” but the question of “how big can t(oo,G) be if t(0,G)=2”? is more a question about finite subgraphs of the plane and their Lovász numbers.  Another way to ask “What is the Lovász number of the plane” would be to adopt the point of view that the Lovász number of a graph has to do with extremizers on the set of positive semidefinite matrices whose (i,j) entry is nonzero only when i and j are adjacent vertices or i=j.  So there must be some question one could ask about the space of positive semidefinite symmetric kernels K(x,y) on R^2  x R^2 which are supported on the locus ||x-y||=1 and the diagonal, which question would rightly be called “What is the Lovász number of the plane?” But I’m not sure what it is.
  2. Having written this, I wonder whether it might be better, rather than thinking about enclosing ellipsoids of a set of points in R^d, just to think of the n points as an nxd matrix X and compute the singular values of X^T X, which would be kind of an “approximating ellipsoid” to the points.  Maybe later I’ll think about what that would measure.  Or you can!

 

 

 

 

 

 

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The chromatic number of the plane is at least 5

That is:  any coloring of the plane with four colors has two points at distance 1 from each other.  So says a paper just posted by Aubrey de Grey.

The idea:  given a set S of points in the plane, its unit distance graph G_S is the graph whose vertices are S and where two points are adjacent if they’re at distance 1 in the plane.  If you can find S such that G_S has chromatic number k, then the chromatic number of the plane is at least k.  And de Grey finds a set of 1,567 points whose unit distance graph can’t be 4-colored.

It’s known that the chromatic number of the plane is at most 7.  Idle question:  is there any chance of a “polynomial method”-style proof that there is no subset S of the plane whose unit distance graph has chromatic number 7?  Such a graph would have a lot of unit distances, and ruling out lots of repetitions of the same distance is something the polynomial method can in principle do.

Though be warned:  as far as I know the polynomial method has generated no improvement so far on older bounds on the unit distance problem (“how many unit distances can there be among pairs drawn from S?”) while it has essentially solved the distinct distance problem (“how few distinct distances can there be among pairs drawn from S?”)

 

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David English Revisited

I never realized that David English of Somerville MA, besides being a prolific writer of letters to the editor, was a weirdo artist of the 1950s!

 

Not even the most poorly paid shipping clerk

One more from Why Men Fail:

Not even the most poorly paid shipping clerk would dream of trying to make his own shirts, and confidential investigation would probably reveal that mighty few darn their own socks.  Yet the cities are full of women on march larger salaries who not only make their own clothes, but cook their own meals and do their own laundry.

So in 1927, it was more unusual to cook for yourself than it was to make your own clothes?  When did that flip?

 

I’m tryin’, I’m tryin’, I’m tryin’, I’m tryin’

“I’m tryin, I’m tryin’, I’m tryin’, I’m tryin'”

reappears, 25 years after Slanted and Enchanted, in Selena Gomez’s “Bad Liar””:

Both songs are lopey and talky.  Stephen Malkmus is talking over the Fall’s “A New Face in Hell.”   Gomez is talking over “Psycho Killer.” Gomez, unlike Malkmus, tells you what she’s trying to do, or trying to not do.  I don’t think this blunts the basic ambiguity of the line — I’m trying to do something, but also, yeah, I’m a little trying, aren’t I?

Bonus track:  Julian Cope, “Try Try Try.”  Your famous victory will be no victory!

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Why Men Fail

That’s the book I picked up off the shelf while working in Memorial Library today.  It’s an book of essays by psychiatrists about failure and suboptimal function, published in 1936.  In the introduction I find:

We see what a heavy toll disorders of the mind exact from human happiness when we realize that of all the beds in all the hospitals throughout the United States one in every two is for mental disease; in other words, there are as many beds for mental ailments as for all other ailments put together.

That’s startling to me!  Can it really have been so?  What’s the proportion now?

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