Tentative, yes, but the schedule will most likely stay the same, as long as I pass the written comps and the professors remain unchanged. I will be taking three courses: Kleiner‘s differental geometry, Tabak‘s partial differential equations, and Chatterjee‘s probability. I will also be doing a reading course with Germain: we will most likely cover Chapters 1, 2, 6, 7, 8, 9, and 10 in Eli Stein’s Harmonic Analysis. Sitting in on McKean‘s harmonic analysis class seems like a good idea—I could use a good review of the basics.

As for the seminars, I’d think it would be silly not to attend the analysis seminar and the graduate student / postdoc seminar. I will also try my damnedest to attend the colloquium regularly, but my enthusiasm might wane after a few weeks of not understanding anything. There are other seminars I might want to attend sometime in the future, but I think this is quite enough for the first semester.

• Advanced Calculus
  • Walter Rudin, Principles of Mathematical Analysis
  • Tom M. Apostol, Calculus, vol. 2
  • Tom M. Apostol, Mathematical Analysis
  • Michael Spivak, Calculus on Manifolds
  • James Munkres, Analysis on Manifolds
  • Wendell Fleming, Functions of Several Variables
• Complex Variables
  • Stein / Shakarchi, Complex Analysis
  • Ahlfors, Complex Analysis
  • Needham, Visual Complex Analysis
• Linear Algebra
  • Lax, Linear Algebra and Its Applications
  • Roman, Advanced Linear Algebra
  • Strang, Linear Algebra and Its Applications

As it turns out, I do have a habit of splurging on records every once in a while. Both the May 2010 purchase and the October 2010 purchase were followed by a few months of no purchases, presumably to digest the large number of records that I hoarded. This trend continued to this date, since the 8 new albums in January and the 11 new albums in February were a part of “one big purchase” that I had planned.

Now, this is troublesome. After a big purchase, I find myself playing a few albums repeatedly and not giving the rest the attention they deserve. Once I buy more albums, a good number of these “lesser records” will rarely enter my playlist again. Perfectly good dollars are wasted. Worse, perfectly fine albums are left on the shelf to collect dust: some records I genuinely dislike, but many others I just need enough listens to get used to them. Of course, I simply cannot afford “enough listens” for every album, if I buy 17 records at once.

Here is the new plan: I will buy three records each month. That way, I will be able to listen to a few more albums each year, while learning each album more thoroughly. In fact, I went ahead and ordered three just now:

• Dirty Projectors – Rise Above
• O.S.I. – Fire Make Thunder
• Jim Hall – Live

I might consider writing reviews later—but no promises!

I have been traveling in the past two weeks, visiting graduate schools and trying to decide where I want to spend my life for the next five years. I think it would be impolite to relay my impressions of the schools I visited on a blog, so I will resist the urge. Instead, I would like to share some information I have gathered at the visits that are of interest to fellow apprentice analysts.

On Friday, I learned that Professor Brett Wick at Georgia Tech is running the Internet Analysis Seminar, which, I believe, is on its third year. The idea is to collaborate on studying the basic material via the internet, and then to gather at a week-long conference to discuss the more advanced material. In a sense, it is fairly similar to Professor Christoph Thiele‘s Summer Schools in Analysis, except that Thiele’s summer school does not have the centralized coverage of the basic material that Wick’s seminar has. Indeed, Wick writes up lecture notes for the seminar, which can be found here and there.

Also, the University of Wisconsin at Madison has been awarded the NSF Research Training Groups grant in analysis. UW’s summary of the grant is as follows:

UW Math Professors Alexander Kiselev, Andreas Seeger and Leslie Smith have received an RTG grant from the NSF. The title of their grant is “Analysis and Applications”, and its approved budget is $1.8 million for 5 years. The funds are mostly for support of graduate students, postdocs, and undergraduate research. The proposal ranked first among RTG in Analysis submitted this year to the NSF. The grant involves faculty members in both analysis and applied mathematics groups.

I expect that UW will host many seminars, conferences, and workshops on harmonic analysis, partial differential equations, fluid dynamics, and mathematical biology.

* * * * *

Part 1. Triangular Fourier Series

What is a triangular Fourier series? Let us suppose that is a -periodic function on . If for some , we know that the classical Fourier series

converges to , both in the -sense (M. Riesz) and in the pointwise almost-everywhere sense (L. Carleson and R. Hunt). These results can be phrased in terms of boundedness of certain operators. For one, the -convergence happens if and only if the operator

is bounded on for all . Similarly, the almost-everywhere convergence happens if and only if the operator

is bounded on .

So then, we have a clear notion of the convergence

.

Why not take the square, and obtain:

.

We expand the left-hand side as follows:

.

The last term is the convolution .

We can now ask ourselves the following questions: (1) Does the following convergence happen?

(2) Similarly, does the following convergence happen?

where ?

As it turns out, we indeed have both convergence results, in the -sense (M. Lacey and C. Thiele), and in the almost-everywhere pointwise sense (C. Muscalu, T. Tao, C. Thiele). Equation (1) is, in fact, the simplest example of triangular Fourier series, whose name is perhaps best explained by the following graph of the range of summation:

, when .

To establish the -theorem, we need to prove that the operator

is bounded uniformly in .  Similarly, establishing the pointwise almost-everywhere convergence amounts to proving that the operator

is bounded uniformly in . More generally, we can consider the multilinear operators

.

Part 2. The Physical Reality

Why do we care about triangular Fourier series? Let us imagine that we have particles (bodies) on the plane, which move in circular orbits. For the sake of simplicity, we assume that the trajectories of the particles are concentric at 0. We can then write to denote the position of the jth particle on the complex plane; we note immediately that . Finally, let us also assume that each is distinct from one another.

Suppose that the particles influence each other in the following way, where for each :

.

In this case, we can write , where

We can now ask the following:

Question 1. Do we have for all ?

To answer this question, we shall embed the system into a more general system

,

where . We can then ask the similar question:

Question 2. Do we have for all and for almost every ?

These systems are called AKNS systems, which are connected to mathematical physics, KdV, NLS, and so on.

Let us now solved the problem posed above. An ansatz is that the solutions are of the form

for some . If this is the case, then we have

where is an n-by-n matrix such that

.

We consider first, as it is as difficult as the whole problem. In fact, if we can understand this case in full generality, then we will have obtained an understanding of the whole problem.

Since we hope to simplify the system further, we would like as many zeroes in our matrix as possible. In other words, we solve the following system:

.

This yields a 2-by-2 upper triangular system

,

whence we obtain

.

If we work in the simple subcase of , then we can solve this system by hand using linear algebra:

This yields, , which implies that

,

whence

.

The above identity yields the bound

.

We would like to have , so that the particles will not fly away. One way to obtain this bound is to have be an -function, whence our problem reduces to investigating the boundedness of

We can consider another continuous analogue of the Carleson operator introduced in the first part of the talk, known as Carleson’s operator:

By Carleson-Hunt, we know that is bounded for . Let us study this operator further.

If for some , and if is the conjugate exponent of , then Carleson-Hunt yields

,

whence we have

by the Hausdorff-Young inequality. Therefore, we have the following inequality:

.

We note, however, that the boundedness of

is given by Menshow-Zygmund (1930s) and so does not require Carleson-Hunt. (If , then we’d need Carleson, which requires much more effort.) So then, what does Carleson by itself give? First and foremost, we have the following inequality for :

.

If and for some , then once again Carleson gives the above estimate. Indeed, the Fourier transform of an function decays very, very slowly.

The conclusion is that the question posed above has an affirmative answer if we have a potential in for some . Furthermore, the question also has an affirmative answer if we have a potential which is the Fourier transform of an -function for some .

We might ask ourselves whether this is true in general. In other words, if, for some , the functions are either themselves or are Fourier transforms of -functions, could we say that no particles fly away? The first question has an affirmative answer, by a result of M. Christ and A. Kiselev from about ten years ago. The second question is a bit more complicated. For simplicity’s sake, we shall consider the problem in . The general case of our problem considers an n-by-n upper triangular matrix, which yields the operator

.

In general, the above operator is bounded. There is also the Carleson variation of this operator—this just means we put a hat on each .

Let us recall that a topological space is disconnected if there exists a disjoint pair of open subsets and of whose union is , and that is connected otherwise. A discrete-valued map is a continuous map from a topological space to a discrete space , which is a topological space in which every subset is open. An alternate characterization of connectedness is as follows:

Proposition 1 (Alternate characterization of connectedness). A topological space is connected if and only if every discrete-valued map on is constant.

The proof is quite simple. If is connected, then the preimage of an element in the image of is nonempty, open, and closed: therefore, must be the whole space. Conversely, if is not connected, then we can find a disjoint pair of open subsets and of whose union is , whence the map which is 0 on and 1 on is a nonconstant discrete-valued map on .

Why would anyone think of such an alternate definition? Glen Wilson offered the following perspective, which I very much like. First off, following the “categorical” way of thinking, we submit that describing a collection of object in terms of maps between them—as opposed to the objects themselves—is a good thing. But why constant maps? We can consider “labeling” each connected component in a topological space by quotienting out by its connected components. The resulting space is a discrete space, and the quotient map is a surjective continuous map that is constant if and only if is a singleton. Of course, this happens precisely when has one connected component, i.e., if is connected.

This alternate characterization leads us to swift proofs of the key properties of connected space. Let us first consider the following

Proposition 2. The continuous image of a connected space is connected.

Here is a one-liner for the proof: If is a connected space, a continuous map, and any discrete-valued map, then the composition must be constant, and so is connected. The following triumph of intuition also admits a devastatingly simple proof:

Proposition 3. If a collection of connected sets share a point, then the union is connected.

Here, any discrete-valued map must be constant on each connected set, and the value of must be the same because they all share a point: it follows that is constant on the union. Another illustrative example is as follows:

Proposition 4. If is a connected subset of a topological space , and if is a subset of such that , then is connected.

Again, the proof is very short. Any discrete-valued map on is a continuous, hence sequentially continuous, map that is constant on , whence we conclude that the value of on any limit point of is the same as the value takes on . It follows that is constant on , hence on .

Added 8/25/2011: I have purged the blog once again (I left one post—Connectedness and Discrete-Valued Maps—intact). Let’s see how long this lasts.