If is a positive semidefinite matrix (a symmetric matrix all whose eigenvalues are nonnegative), then the power method is simply to pick a random vector , and compute . If is of the order of , then one has a constant probability that

where is the largest eigenvalue of . If we are interested in the Laplacian matrix of a -regular graph, where is the adjacency matrix of the graph, this gives a way to compute an approximation of the largest eigenvalue, and a vector of approximately maximum Rayleigh quotient, which is useful to approximate Max Cut, but not to apply spectral partitioning algorithms. For those, we need a vector that approximates the eigenvector of the second smallest eigenvalue.

Equivalently, we want to approximate the second largest eigenvalue of the adjacency matrix . The power method is easy to adjust to compute the second largest eigenvalue instead of the largest (if we know an eigenvector of the largest eigenvalue): after you pick the random vector, subtract the component of the vector that is parallel to the eigenvector of the largest eigenvalue. In the case of the adjacency matrix of a regular graph, subtract from every coordinate of the random vector the average of the coordinates.

The adjacency matrix is not positive semidefinite, but we can adjust it to be by adding a multiple of the identity matrix. For example we can work with . Then the power method reduces to the following procedure: pick randomly , then subtract from every entry of , then repeat the following process times: for every entry , assign , that is, replace the value that the vector assigns to vertex with a convex combination of the current value and the current value of the neighbors. (Note that one iteration can be executed in time .

The problem is that if we started from a graph whose Laplacian matrix has a second smallest eigenvalue , the matrix has second largest eigenvalue , and if the power method finds a vector of Rayleigh quotient at least for , then that vector has Rayleigh quotient about for , and unless we choose of the same order as we get nothing. This means that the number of iterations has to be about , which can be quite large.

The video below (taken from this week’s lecture) shows how slowly the power method progresses on a small cycle with 31 vertices. It goes faster on the hypercube, which has a much larger .

A better way to apply the power method to find small eigenvalues of the Laplacian is to apply the power method to the pseudoinverse of the Laplacian. If the Laplacian of a connected graph has eigenvalues , then the pseudoinverse has eigenvalues with the same eigenvectors, so approximately finding the largest eigenvalue of is the same problem as approximately finding the second smallest eigenvalue of .

Although we do not have fast algorithms to compute , what we need to run the power method is, for a given , to find the such that , that is, to solve the linear system in given and .

For this problem, Spielman and Teng gave an algorithm nearly linear in the number of nonzero of , and new algorithms have been developed more recently (and with some promise of being practical) by Koutis, Miller and Peng and by Kelner, Orecchia, Sidford and Zhu.

Coincidentally, just this week, Nisheeth Vishnoi has completed his monograph Lx=b on algorithms to solve such linear systems and their applications. It’s going to be great summer reading for those long days at the beach.

The second-to-last time I had dinner in a place that gives fortune cookies, however, I had got this:

Oh well…

To keep a long story long, at the start of WW2, Italy, which was an ally of Germany, was initially neutral, in part because its armed forces were completely unprepared for war. At some point in the May of 1940, with German troops advancing into France, and British troops evacuating the continent, Italy decided to join what looked like a soon-to-end war, in order to claim some French territories and colonies.

But then, in 1941, Germany attacked Russia and Japan attacked the US, underestimating what they were getting into, and by the beginning of 1943 the tide was clearly turning against the “axis.” Italy’s king, who was definitely not the “fight until the last man” type, had Mussolini arrested, installed a general as prime minister, and started negotiating Italy’s surrender with the allies (even as Italian troops were fighting with the Germans in Russia and in Africa). Eventually, on September 8, 1943, the king announced a cease-fire. Because of the secrecy of the negotiations, nobody knew what was going in advance, and most of the Italian troops that were fighting with the Germans were taken prisoners, while the rest of the armed forces basically disbanded. German troops came into Italy from the North to occupy it, even as allied troops landed in Sicily and took control of most of Southern Italy. The king fled to the South, and the Germans freed Mussolini and installed him as head of a puppet government in the North.

With the Italian army disbanded, and with the allies neglecting the “Southern front” in Italy as they were plotting the landing in Normandy, guerilla groups were formed in Northern Italy to fight the Germans. Eventually, in April 1945 the German troops were retreating from the Eastern and Western fronts against the advancing American and Russian forces, and the allied made another push in Italy; concurrently, the resistance organizations planned an insurrection that, on April 25, liberated Torino and Milan. All the German forces in Italy surrendered on April 29.

The resistance was the training ground of some of the first generation of politicians of the new Italian Republic (a referendum to abolish the monarchy passed in 1946, and a new Republican constitution was approved in 1948), and it brought people who were willing to die for their ideals into politics. That spirit didn’t last very long, but it remains one of the few bright spots in recent Italian history.

In the first week of class: what are the topics of the course, and how to prove that the eigenvalues of the adjacency matrix of a regular graph tell you how many connected components there are in the graph.

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.

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 the Cheeger constant of to be

where 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 as

where 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 6 Let 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 10 Let 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 .

]]> http://lucatrevisan.wordpress.com/2013/03/21/proof-of-the-cheeger-inequality-in-manifolds/feed/ 1 luca http://lucatrevisan.wordpress.com/2013/03/20/the-cheeger-inequality-in-manifolds/ http://lucatrevisan.wordpress.com/2013/03/20/the-cheeger-inequality-in-manifolds/#comments Wed, 20 Mar 2013 18:16:28 +0000 luca http://lucatrevisan.wordpress.com/?p=2642 ]]> Readers of in theory have heard about Cheeger’s inequality a lot. It is a relation between the edge expansion (or, in graphs that are not regular, the conductance) of a graph and the second smallest eigenvalue of its Laplacian (a normalized version of the adjacency matrix). The inequality gives a worst-case analysis of the “sweep” algorithm for finding sparse cuts, it shows a necessary and sufficient for a graph to be an expander, and it relates the mixing time of a graph to its conductance.

Readers who have heard this story before will recall that a version of this result for vertex expansion was first proved by Alon and Milman, and the result for edge expansion appeared in a paper of Dodzuik, all from the mid-1980s. The result, however, is not called Cheeger’s inequality just because of Stigler’s rule: Cheeger proved in the 1970s a very related result on manifolds, of which the result on graphs is the discrete analog.

So, what is the actual Cheeger’s inequality?

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 the Cheeger constant of to be

where the is the boundary of , is the -dimensional measure, and is -th dimensional measure defined using . Then

The purpose of this post is to describe to the reader who knows nothing about differential geometry and who does not remember much multivariate calculus (that is, the reader who is in the position I was in a few weeks ago) what the above statement means, to describe the proof, and to see that it is in fact the same proof as the proof of the statement about graphs.

In this post we will define the terms appearing in the above theorem, and see their relation with analogous notions in graphs. In the next post we will see the proof.

First we recall the definitions in the case of graphs, which are good “models” to keep in mind as we will give the definitions of a manifold and of the Laplacian of a manifold.

Let be a -regular graph and be the adjacency matrix of . If is a subset of vertices, its volume is defined as the sum of the degrees of the elements of , . The boundary of is the set of edges that have one endpoint in and one endpoint in ; the conductance of is

and the conductance of is defined as

The Laplacian matrix of is defined as , where is the adjacency matrix of . Since is a symmetric real matrix, all its eigenvalues are real, and, if we call them , the Cheeger inequality in graphs is

1. Manifolds, gradient, and divergence

1.1. What is a manifold?

First let us discuss (non-rigorously) what is a manifold: we can think of an -dimensional manifold as a subset , for some , such that for every point , if we look at a small ball around , the intersection of the ball with looks like a ball in . For example, consider a two-dimensional sphere in : for every point on the sphere, a small neighborhood around it looks like a flat disc.

The formalization of this notion, which will turn out not to require us to think of as a subset of is rather technical, and it has quite a lot of pieces. Instead of introducing the rigorous definition piece by piece, and trying to understand what a manifold is, I think it’s better to start by thinking about what a manifold does, that is, what kind of properties we want from the definition, and what kind of calculations we want the definition to allow, and then the details of the definition is just whatever works to capture these intentions.

Basically, we would like to be able to do with an -dimensional manifold most of the things we would do with . We would like to have a distance function between elements that satisfies the triangle inequality and that is the “length of the shortest path” from to in , we would like to have a measure , that gives us the “volume” of a subset . If we define functions , we would like to integrate them, and compute , or , where . We would also like to have a definition of continuous functions , or , or , and, finally, we would like to define derivatives of functions .

On the other hand, we will not try to think of the elements of as vectors, and, in particular, we will not have a definition of adding elements of , or multiplying them by constants, much less of taking an inner product of them. (This means that the distance function will not come from a norm.) To see why not, consider a 2-dimensional sphere in . We want to think of it as completely symmetric under rotations, so there can be no meaningful notion of a point on the sphere being “bigger” than another, so if is a point on the sphere, there isn’t a meaningful notion of . (You could try to say , but then this would imply .)

Because a manifold is not a vector space, it is tricky to define derivatives. We want to think of an -dimensional manifold as a generalization of , and a function does not have a derivative. Instead, for every direction , it has a partial derivative at in the direction

and the above expression requires us to add elements of and to multiply them by constants.

However, if we look at a point , the immediate neighborhood of does look like a small piece of , which is a vector space. To make this intuition formal, instead of trying to formalize the notion of “small piece of a vector space,” one defines directional derivatives where the direction is not in but in the tangent space to at , which is a honest-to-God vector space.

In the rigorous definition, we have a topology on , which allows to talk about “small intervals around” a point , and to talk about “continuous functions” where can be , , , and so on. Then, for every small interval around a point we have an -dimensional coordinate system, which we can think of as an approximation of the interval around as a piece of ; these coordinate systems are called “charts” and their collection is called an “atlas.” Charts of nearby points are supposed to be “consistent in their intersection,” which is assured by consistent mappings between them. For every point we also have an -dimensional vector space, which is the tangent space at . There are also mapping that relate the tangent spaces of nearby points. Finally, we have a “metric,” that, to make things more confusing, is not a distance function that satisfies the triangle inequality, but is the definition of an inner product between vectors of the tangent spaces. From the metric, it is possible to define a distance function on that satisfies the triangle inequality, and a measure , and from the measure we can define integrals. Taking partial derivatives is still a bit tricky, and we will return to it when we talk about the gradient.

If is a subset of the manifold, then the boundary of is the set of points such that there are arbitrarily small intervals around that contain both elements of and elements of . For nice enough , will be an -dimensional object, so we will have , but we can use the metric to define an -dimensional measure .

So we have a general sense of the definition of the Cheeger constant in manifold. For the analogy to graphs, we should think of the points of as , of as , of infinitesimally close points as an edge, and of the boundary of a set as the set of edges leaving .

1.2. What is the Laplacian?

Now we have to define the Laplacian. The Laplacian is a linear operator that maps a function to a function , and it is defined as

where is the gradient operator and is the divergence operator. (All the functions that we talk about are infinitely differentiable.)

To see what these operators are, and what they have to do with the Laplacian of graphs, recall that, in a graph,

the Laplacian of at is the difference between the value of at and the average value of at neighbors of . Similarly, in a manifold, measures how much bigger tends to be on average than at a random point close to .

We will give some intuition about how the definition of the Laplacian in matches the intuition from the case of graphs. The generalization from to an -dimensional manifold is quite technical, but the general intuition is similar.

1.3. The Laplacian in one dimension

To see how to formalize this intuition, suppose and so . Then, our intuition for the Laplacian of at is that, for a small , we should look at

Qualitatively, this quantity is positive if is concave and negative if is convex, so it seems related to , where is the second derivative of . Indeed, the Taylor expansion of at is

and so

And, indeed, for functions both the gradient operator and the divergence operator are just the derivative operator, and so and we have

which matches our intuition from the case of graphs.

For functions , the operator is linear, and we may ask if it has eigenvalues and eigenfunctions, that is, if there are numbers and functions for which the equation

holds. Let’s see: what functions are equal to their second derivative, up to a multiplicative constant? For example, and are, where is an integer. (So is .)

Now suppose that our manifold is a unit circle in the plane, that is, points of are of the form . This is a (one-dimensional) manifold because a small arc around a point is quite close to being a straight segment. If is periodic with period , we can use it to define a function on , where ; similarly, if is defined on the circle we can think of it as a periodic function with period , where . In fact, there isn’t really any difference between thinking of functions defined over and periodic functions of period , and the Laplacian of will be the operator that takes a periodic function to minus its second derivative. So and will be eigenvectors of the Laplacian of the circle, with eigenvalues . Note the extreme similarity to the spectrum of the circle as a graph, where the eigenvectors are vectors whose -th coordinate is or and ranges in . The main difference is one of normalization: in the circle graph, all eigenvalues are at most 2, at the smallest positive one is about ; in the circle manifold the eigenvalues are arbitrarily large, and the smallest positive one is 1. However, if we call the -th smallest eigenvalue, in both cases we have that is about .

1.4. The gradient in

For the higher-dimensional case, let us start from the case . In this case, let us fix the orthonormal basis for where is the vector that has zeroes everywhere except a 1 in the -th position. For a function , its gradient is a function that maps a point to an -dimensional vector, which contains the partial derivatives of , that is

where, for a vector , the partial derivative is the function

If and are orthogonal then

this means that for every vector we have

It is important to note that there was nothing special about the use of the base . Take an arbitrary orthonormal base , then define the “base-” gradient as

then we again have that for every , after writing ,

so, in fact, is always the same vector regardless of the choice of .

Indeed, we could take as the definition of the gradient to say that is the unique vector that satisfies the equation

for every .

It is also worth nothing that, among all unit vectors , the one that maximizes is the one parallel to , so points in the direction in which grows the most near , which is the most intuitive interpretation.

If in general, then is a vector in the tangent space of at , and the directional derivative , where is a vector in the tangent space at , satisfies

where the inner product is in the tangent space. The definition of partial derivative, and the fact that there is a vector that satisfies the above equation for every is beyond the scope of these notes.

1.5. The divergence in

The definition of divergence is a bit more tricky. If , then takes in input a function , and returns a function . The definition of , (up to scaling) can be thought of as follows: consider a small sphere of radius around , pick a random element of the sphere by picking a random unit vector and considering the vector , and then look at the average value of

which is the component of that “points away” from in the direction of the line that joins to .

This will be positive if the vector tends to point away from , for random close to , and negative otherwise.

1.6. The Laplacian in

Now let us see what happens when we instantiate this definition to . We pick a random unit vector , and we compute

For small ,

so

which is the average rate of growth of in a random direction. So we have that is the rate of decrease of in a random direction. This is roughly the equivalent of the discrete case where is the difference between and the average value of for a random neighbor of .

It turns out that we can also make the correspondence between the discrete and the continuous case closer, by finding linear operators in the discrete case whose composition gives the Laplacian, and that are somewhat analogous to and .

Consider the incidence matrix of defined as follows:

where we have fixed an arbitrary orientation of the edges of . Then is an -dimensional vector such that equals for every (directed) edge . We can also think of as mapping a vertex to a -dimensional vector , whose -th coordinate is where is the -th outgoing edge from . In this view, acts somewhat like the , giving us all the “derivatives” of in all directions. (The minus sign is because in the derivative at we consider where is close to , but “at ” gives us .)

If is an -dimensional vector, then

is the flow through , if we think of as defining a flow on the (directed) edges of . This is the analog of the divergence , which also computes the net flow of away from a point .

The reader can verify that we indeed have .

So it was with much joy, this early morning in Japan, that I heard the news that Shafi Goldwasser and Silvio Micali have been named as recipients of this year’s Turing award.

Omer Reingold has more information about their work. With no offense to colleagues around my age and younger, Shafi and Silvio are also representative of a time when leading theoretical computer scientists were more interesting people. They both have incredible charisma.

My favorite memory of Shafi and Silvio is from the time I interviewed for a faculty job at MIT. Shafi was in the last weeks of her pregnancy and did not make an appointment to see me, but then the day of my interview she changed her mind and showed up in Silvio’s office halfway through my meeting with him.

Silvio had been looking at my schedule and was giving me advice on how to talk to various people. Shafi asked what we were talking about, and then proceeded to give the opposite advice that Silvio had been giving me. The two of them spent the rest of the meeting arguing with each other.

The course page is up for people to sign up. A friend of mine has compared my camera presence to Sheldon Cooper’s in “Fun with Flags,” which is sadly apt, but hopefully the material will speak for itself.

Meanwhile, I will be posting about some material that I have finally understood for the first time: the analysis of the Arora-Rao-Vazirani approximation algorithm, the Cheeger inequality in manifolds, and the use of the Selberg “3/16 theorem” to analyze expander constructions.

If you are not a fan of recorded performances, there will be a live show in Princeton at the end of June.

(Chart by Hilary Sargent Ramadei)