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The Simons Institute for the Theory of Computing at Berkeley has started operations a couple of months ago, it has been off to a great start. This semester there is a program on applications of real analysis to computer science, in which I am involved, and one on “big data,” whose workshops have been having such high attendance that they had to be organized offsite.

The institute itself is housed in a beautiful circular three-story building, half of whose second floor is an open space with sofas, whiteboards, tall ceilings, big windows, and exposed pipes on the ceiling, for an added loft-like look. If it had a ball pit, a foosball table, and free sushi it would look like the offices of a startup.

Next year, there will be programs on spectral graph theory, on applications of algebraic geometry, and on information theory.

Junior people (senior graduate students, postdocs, and junior assistant professors who have received their PhD no longer than six years ago) can participate in the program of the institute as “fellows.” Information on the fellowships is at simons.berkeley.edu/fellows2014; the deadline to apply is December 15.

This year is the centennial of Paul Erdős’s birth. Erdős lived most of his adult life as a traveling mathematician, “couchsurfing,” as we would say now, from place to place and from mathematical conference to mathematical conference. He wrote more than 1,500 papers with more than 500 different coauthors, introduced the probabilistic method and was the defining figure of the “Hungarian approach” to combinatorics. He died at age 83 while attending a mathematical conference.

Last year, we celebrated the centennial of Alan Turing’s birth. Turing and Erdős have become such iconic figures both for the impact of their work and for the fascinating facts of their lives. I would like to argue that the cultural archetype through which we interpret their lives is that of the *saint*. It is clearly that of the *martyr saint* in the case of Turing, while Erdős gave up material possessions and devoted his life to others, traveling everywhere and “preaching” to everybody, much in the mold of Saint Francis.

(A comparison of the Turing centennial celebration and Erdős’s, and a look at the frescoes of Medieval Catholic churches will show which kind of saint people are more interested in.)

The first step to become a saint of the Catholic church is to establish that the person exhibited “heroic virtues,” which is a great expression. This is an archetype that is not restricted to religion: you see it occurring in communist propaganda (Stakhanov, Lei Feng) and in every civil rights movement.

Saints were the “celebrities” of the Middle Ages, those whose life people liked to talk about. But contemporary celebrities come from a totally different archetype, that of the Greek God. Greek (and Roman) gods were petty and jealous, they cheated on their spouses, they were terrible parents, but there were good stories to be told about them. We don’t want (at least, I don’t) to live the life of a saint, but thinking about them is certainly inspirational and it makes us think that if someone can be so much better than us, maybe we can be a little better ourself in the practice of “virtues”, whatever this may mean to us. And we don’t admire gods, but, well, it’s probably fun to be one.

As usual, I have lost track of what I was trying to say, but I think that it speaks well of the academic community that we are more interested in saints than in gods, I will close by saying that my favorite saint of complexity theory is Avi Wigderson, I will keep to myself who my favorite god of complexity theory is, and I will leave it to the readers to contribute their picks.

*[I have been asked by the office of public affairs of the Institute for Advanced Study to publicize the following press release. L.T.]*

April 1, 2013. For immediate release.

Cofounders Jean Bourgain and Peter Sarnak announce today the launch of *eXpandr*, a new venture that aims to become the world’s leading provider of expander graphs.

“We are excited about our mission to change the way the world uses expanders.” said CEO Guli Mars, who joined eXpandr after a distinguished career in several leading technology companies. “Expanders are vital to revenue-generating logarithms, and our technology will revolutionize a multi-billion dollar market.”

“Big data, disruption”, said Juan Raman, senior vice president for marketing. “Innovation, cloud computing”, Mr. Raman continued.

“Let p be a prime congruent to 1 modulo 4″ said Jean Bourgain, cofounder and senior vice-president for analytic number theory, “and consider the irreducible representations of PSL(2,p).”

*About the Institute for Advanced Study*. The Institute for Advanced Study is one of the world’s leading centers for theoretical research and intellectual inquiry. The Institute exists to encourage and support fundamental research in the sciences and humanities—the original, often speculative thinking that produces advances in knowledge that change the way we understand the world. Work at the Institute takes place in four Schools: Historical Studies, Mathematics, Natural Sciences and Social Science. It provides for the mentoring of scholars by a permanent Faculty of no more than 28, and it offers all who work there the freedom to undertake research that will make significant contributions in any of the broad range of fields in the sciences and humanities studied at the Institute.

The Institute, founded in 1930, is a private, independent academic institution located in Princeton, New Jersey. Its more than 6,000 former Members hold positions of intellectual and scientific leadership throughout the academic world. Some 33 Nobel Laureates and 38 out of 52 Fields Medalists, as well as many winners of the Wolf or MacArthur prizes, have been affiliated with the Institute.

*About eXpandr*. eXpandr aims to disrupt the way the world uses expander graphs, and to become the leading commercial provider of expanders. eXpandr received $2 million in angel investing and will launch its first product by Summer 2013.

Having (non-rigorously) defined the Laplacian operator in manifolds in the previous post, we turn to the proof of the Cheeger inequality in manifolds, which we restate below.

Theorem 1 (Cheeger’s inequality)Let be an -dimensional smooth, compact, Riemann manifold without boundary with metric , let be the Laplace-Beltrami operator on , let be the eigenvalues of , and define theCheeger constantof to bewhere the is the boundary of , is the -dimensional measure, and is -th dimensional measure defined using . Then

We begin by recalling the proof of the analogous result in graphs, and then we will repeat the same steps in the context of manifolds.

Theorem 2 (Cheeger’s inequality in graphs)Let be a -regular graph, be its adjacency matrix, be its normalized Laplacian matrix, be the eigenvalues of , and define for every subset of vertices . Define the conductance of aswhere is the number of edges with one endpoint in and one endpoint in . Then

**1. Proof the Cheeger inequality in graphs **

We will use the *variational characterization* of the eigenvalues of the Laplacian of a graph .

and if is a minimizer in the above expression then

Following the definition of we see that

and so the minimum in (3) is 0, and it is achieved for . This means that

The expression in the right-hand-side of (4) is an important one, and it is called the Rayleigh quotient of , which we will denote by :

It is also useful to consider the variant of the Rayleigh quotient where there are no squares; this does not have a standard name, so let us call it the Rayleigh quotient and denote it by :

The proof of the graph Cheeger inequality now continues with the proof of the following three facts.

Lemma 3 (Rounding of embeddings)For every non-negative vector there is a value such that

Lemma 4 (Embedding of into )For every non-negative vector , we have

Lemma 5 (From an eigenvector to a non-negative vector)For every such that there is a non-negative such that and such that

Now let us start from a function that optimizes (4), so that and , then apply Lemma 5 to find a function such that the volume of the vertices having positive coordinates in is at most and such that . Then consider the vector such that ; by Lemma 4, we have , and by Lemma 3 there is a threshold such that the set has conductance . Since is a subset of the vertices having positive coordinates in , we have , and so

which is the Cheeger inequality for graphs. It remains to prove the three lemmas.

*Proof:* of Lemma 3. For each threshold , define the set

The idea of the proof is that if we pick at random then the probability that an edge belongs to is proportional to and the probability that is proportional to , so that the expected number of edges in is proportional to the numerator of and the expected number of vertices in is proportional to the denominator of ; if is small, it is not possible for to always be large for every .

To avoid having to normalize the range of to be between and , instead of taking averages over a random choice of , we will consider the integral over all values of . We have

because we can write , where if and otherwise, and we see that only the values of between and make , so .

We also have

and if we denote by the threshold such that is smallest among all the , then

so that

*Proof:* of Lemma 3. Let us consider the numerator of ; it is:

(we used Cauchy-Swarz)

(we used the definition of and Cauchy-Swarz again)

And so

*Proof:* of Lemma 5. Let be the median of , and consider defined as . We have

because the numerators of and are the same (the additive term cancels). The denominators are such that

because and the vector are orthogonal, and so by Pythagoras’s theorem the length-squared of equals the length-squared of plus the length-squared of .

Let us define and so that . We use the following fact:

Fact 6Let be disjointly supported non-negative vectors (“disjointly supported” means that they are non-zero on disjoint subsets of coordinates), then

*Proof:* The numerator of is

and, using orthogonality and Pythagoras’s theorem, the denominator of is

The fact now follows from the inequality

The lemma now follows by observing that and are non-negative and disjointly supported, so

and that both and have at most non-zero coordinate.

**2. Proof of the Cheeger inequality in manifolds **

We will now translate the proof of the graph Cheeger inequality to the setting of manifolds.

As you may remember, we started off by saying that is symmetric and so all its eigenvalues are real and they are given by the variational characterization. Now we are already in trouble because the operator on manifolds cannot be thought of as a matrix, so what does it mean for it to be symmetric? The consequence of symmetry that is exploited in the analysis of the spectrum of symmetric matrices is the fact that if is symmetric, then for every we have

and the property makes no references to coordinates, and it is well defined even for linear operators over infinite-dimensional spaces, provided that there is a notion of inner product. If we the define the inner product

on functions , and more generally

for functions , where is a vector space with inner product , then we can say that an operator is *self-adjoint* if

for all (appropriately restricted) functions . If is compact, this property is true for the Laplacian, and, in particular, and are adjoints of each others, that is,

(The discrete analog would be that is the transpose of .)

Self-adjointness (and appropriate conditions on ) imply a version of the spectral theorem and of the variational characterization. In particular, all eigenvalues of are real, and if there is a minimum one then it is

and if is a minimizer of the above, then

(The minimization is quantified over all functions that are square-integrable, and the minimum is achieved because if is compact then the space of such functions is also compact and the cost function that we are minimizing is continuous. In this post, whenever we talk about “all functions,” it should be understood that we are restricting to whatever space of functions makes sense in the context.)

From the property that and are adjoint, we have

so

where the Rayleigh quotient

is always non-negative, and it is zero for constant , so we see that and

By analogy with the graph case, we define the “ Rayleigh quotient”

And we can prove the analogs of the lemmas that we proved for graphs.

Lemma 7 (Rounding of embeddings)For every non-negative function there is a value such that

where the Cheeger constant of a subset of the manifold is

Lemma 8 (Embedding of into )For every non-negative function , we have

Lemma 9 (From an eigenfunction to a non-negative function)For every function such that there is a non-negative such that and such that

Let us see the proof of these lemmas.

*Proof:* of Lemma 7. For each threshold , define the set

Let be a threshold for which is minimized

We will integrate the numerator and denominator of over all . The *coarea formula* for nonnegative functions is

and we also have

which combine to

so that

*Proof:* of Lemma 8. Let us consider the numerator of ; it is:

We can apply the chain rule, and see that

which implies

and, after applying Caucy-Swarz,

And so

*Proof:* of Lemma 9. Let be a median of , and consider defined as . We have

because the numerators of and are the same (the derivatives of functions that differ by a constant are identical) and the denominators are such that

where we used the fact the integral of is zero.

Let us define and so that . We use the following fact:

Fact 10Let be disjointly supported non-negative functions (“disjointly supported” means that they are non-zero on disjoint subsets of inputs), then

*Proof:* We begin with the following observation: if is a non-negative function, and , then , because has to be a local minimum.

Consider the expression occurring in the numerator of . We have

But

because for every at least one of or is zero, and so at least one of or is zero.

Using this fact, we have that the numerator of is equal to the sum of the numerators of and :

and the denominator of is also the sum of the denominators of and :

because for every . The fact now follows from the inequality

The lemma now follows by observing that and are non-negative and disjointly supported, so

and that both and have a support of volume at most .

If anybody is still reading, it is worth observing a couple of differences between the discrete proof and the continuous proof.

The Rayleigh quotient is defined slightly differently in the continuous case. It would correspond to defining it as

in the discrete case.

If are disjointly supported and nonnegative, the sum of the numerators of the Rayleigh quotients and can be strictly smaller than the numerator of , while we always have equality in the continuous case. In the discrete case, the sum of the numerators of and can be up to twice the numerator of (this fact is useful, but it did not come up in this proof), while again we have exact equality in the continuous case.

The chain rule calculation

corresponds to the step

In the continuous case, and are “infinitesimally close”, so we can approximate by .

I would like to thank all those that contributed to the Turing Centennial series: all those who wrote posts, for sure; but also all those who spread the word about this, on blogs, twitter, facebook, and in real life; those who came to read them; and those who wrote lots of thoughtful comments. In a community where discussions over conference organizational issues or over the importance of matrix multiplication algorithms can become very acrimonious, I am impressed that we could have such a pleasant and troll-free conversation. This goes to show that *in theory* has not only the smartest and most handsome readers of the whole internet, as was well known, but also the nicest ones!

We will definitely do this again in 2054, to mark the centennial of Turing’s death.

A few days ago, I was very saddened to hear of the death of Sally Ride. A Stanford Alumna, Sally Ride became to first American woman to travel in space, she served on both the investigative committees after the two Shuttle disasters, and dedicated the past decade to the goal of getting young kids, and girls in particular, interested in science and technology. She cofounded, and directed, a non-profit foundation to further these goals, and wrote several books. After her death, it was revealed that she had been in a 25-year relationship with another woman, who was also the coauthor of her books and a partner in her foundation.

I think it is significant that a person that certainly had a lot of courage, determination, willingness to defy stereotypes, and to be an inspiration for people like her, felt that she could not be publicly out during her life. (In interviews about their books, Ride and her partner Tam O’Shaughnessy referred to each other as “friends”.)

Let’s hope that in 2054 it’s not just computer science professors in the West that are confortable being out, but also astronauts, movie stars, professional athletes, and so on.

Two more posts are coming soon. Meanwhile, here is a wonderful interview with Robert MacPherson, which is part of an interview series by the Simons Foundation. Although the interview does not mention Turing, it does mention Kolmogorov.

(via Not Even Wrong)

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