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Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao

# Category Archive

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## (Ingrid Daubechies) Planning for the World Digital Mathematical Library

8 May, 2013 in guest blog, media | Tags: Ingrid Daubechies, World Digital Mathematical Library | by Terence Tao | 39 comments

[This guest post is authored by Ingrid Daubechies, who is the current president of the International Mathematical Union, and (as she describes below) is heavily involved in planning for a next-generation digital mathematical library that can go beyond the current network of preprint servers (such as the arXiv), journal web pages, article databases (such as MathSciNet), individual author web pages, and general web search engines to create a more integrated and useful mathematical resource. I have lightly edited the post for this blog, mostly by adding additional hyperlinks. - T.]

This guest blog entry concerns the many roles a World Digital Mathematical Library (WDML) could play for the mathematical community worldwide. We seek input to help sketch how a WDML could be so much more than just a huge collection of digitally available mathematical documents. If this is of interest to you, please read on!

The “we” seeking input are the Committee on Electronic Information and Communication (CEIC) of the International Mathematical Union (IMU), and a special committee of the US National Research Council (NRC), charged by the Sloan Foundation to look into this matter. In the US, mathematicians may know the Sloan Foundation best for the prestigious early-career fellowships it awards annually, but the foundation plays a prominent role in other disciplines as well. For instance, the Sloan Digital Sky Survey (SDSS) has had a profound impact on astronomy, serving researchers in many more ways than even its ambitious original setup foresaw. The report being commissioned by the Sloan Foundation from the NRC study group could possibly be the basis for an equally ambitious program funded by the Sloan Foundation for a WDML with the potential to change the practice of mathematical research as profoundly as the SDSS did in astronomy. But to get there, we must formulate a vision that, like the original SDSS proposal, imagines at least some of those impacts. The members of the NRC committee are extremely knowledgeable, and have been picked judiciously so as to span collectively a wide range of expertise and connections. As president of the IMU, I was asked to co-chair this committee, together with Clifford Lynch, of the Coalition for Networked Information; Peter Olver, chair of the IMU’s CEIC, is also a member of the committee. But each of us is at least a quarter century older than the originators of MathOverflow or the ArXiv when they started. We need you, internet-savvy, imaginative, social-networking, young mathematicians to help us formulate the vision that may inspire the creation of a truly revolutionary WDML!

Some history first.  Several years ago, an international initiative was started to create a World Digital Mathematical Library. The website for this library, hosted by the IMU, is now mostly a “ghost” website — nothing has been posted there for the last seven years. [It does provide useful links, however, to many sites that continue to be updated, such as the European Mathematical Information Service, which in turn links to many interesting journals, books and other websites featuring electronically available mathematical publications. So it is still worth exploring ...] Many of the efforts towards building (parts of) the WDML as originally envisaged have had to grapple with business interests, copyright agreements, search obstructions, metadata secrecy, … and many an enterprising, idealistic effort has been slowly ground down by this. We are still dealing with these frustrations — as witnessed by, e.g., the CostofKnowledge initiative. They are real, important issues, and will need to be addressed.

The charge of the NRC committee, however, is to NOT focus on issues of copyright or open-access or who bears the cost of publishing, but instead on what could/can be done with documents that are (or once they are) freely electronically accessible, apart from simply finding and downloading them. Earlier this year, I posted a question about one possible use on MathOverflow and then on MathForge, about the possibility to “enrich” a paper by annotations from readers, which other readers could wish to consult (or not). These posts elicited some very useful comments. But this was but one way in which a WDML could be more than just an opportunity to find and download papers. Surely there are many more, that you, bloggers and blog-readers, can imagine, suggest, sketch. This is an opportunity: can we — no, YOU! — formulate an ambitious setup that would capture the imagination of sufficiently many of us, that would be workable and that would really make a difference?

## An introduction to special relativity for a high school math circle

22 December, 2012 in expository, math.MP, non-technical, teaching | Tags: mass-energy equivalence, special relativity | by Terence Tao | 25 comments

Things are pretty quiet here during the holiday season, but one small thing I have been working on recently is a set of notes on special relativity that I will be working through in a few weeks with some bright high school students here at our local math circle.  I have only two hours to spend with this group, and it is unlikely that we will reach the end of the notes (in which I derive the famous mass-energy equivalence relation E=mc^2, largely following Einstein’s original derivation as discussed in this previous blog post); instead we will probably spend a fair chunk of time on related topics which do not actually require special relativity per se, such as spacetime diagrams, the Doppler shift effect, and an analysis of my airport puzzle.  This will be my first time doing something of this sort (in which I will be spending as much time interacting directly with the students as I would lecturing);  I’m not sure exactly how it will play out, being a little outside of my usual comfort zone of undergraduate and graduate teaching, but am looking forward to finding out how it goes.   (In particular, it may end up that the discussion deviates somewhat from my prepared notes.)

The material covered in my notes is certainly not new, but I ultimately decided that it was worth putting up here in case some readers here had any corrections or other feedback to contribute (which, as always, would be greatly appreciated).

[Dec 24 and then Jan 21: notes updated, in response to comments.]

## Lars Hormander

30 November, 2012 in math.AP, obituary | Tags: correspondence principle, fourier integral operators, lars hormander, pseudodifferential operators | by Terence Tao | 4 comments

Lars Hörmander, who made fundamental contributions to all areas of partial differential equations, but particularly in developing the analysis of variable-coefficient linear PDE, died last Sunday, aged 81.

I unfortunately never met Hörmander personally, but of course I encountered his work all the time while working in PDE. One of his major contributions to the subject was to systematically develop the calculus of Fourier integral operators (FIOs), which are a substantial generalisation of pseudodifferential operators and which can be used to (approximately) solve linear partial differential equations, or to transform such equations into a more convenient form. Roughly speaking, Fourier integral operators are to linear PDE as canonical transformations are to Hamiltonian mechanics (and one can in fact view FIOs as a quantisation of a canonical transformation). They are a large class of transformations, for instance the Fourier transform, pseudodifferential operators, and smooth changes of the spatial variable are all examples of FIOs, and (as long as certain singular situations are avoided) the composition of two FIOs is again an FIO.

The full theory of FIOs is quite extensive, occupying the entire final volume of Hormander’s famous four-volume series “The Analysis of Linear Partial Differential Operators”. I am certainly not going to try to attempt to summarise it here, but I thought I would try to motivate how these operators arise when trying to transform functions. For simplicity we will work with functions ${f \in L^2({\bf R}^n)}$ on a Euclidean domain ${{\bf R}^n}$ (although FIOs can certainly be defined on more general smooth manifolds, and there is an extension of the theory that also works on manifolds with boundary). As this will be a heuristic discussion, we will ignore all the (technical, but important) issues of smoothness or convergence with regards to the functions, integrals and limits that appear below, and be rather vague with terms such as “decaying” or “concentrated”.

A function ${f \in L^2({\bf R}^n)}$ can be viewed from many different perspectives (reflecting the variety of bases, or approximate bases, that the Hilbert space ${L^2({\bf R}^n)}$ offers). Most directly, we have the physical space perspective, viewing ${f}$ as a function ${x \mapsto f(x)}$ of the physical variable ${x \in {\bf R}^n}$. In many cases, this function will be concentrated in some subregion ${\Omega}$ of physical space. For instance, a gaussian wave packet

$\displaystyle f(x) = A e^{-(x-x_0)^2/\hbar} e^{i \xi_0 \cdot x/\hbar}, \ \ \ \ \ (1)$

where ${\hbar > 0}$, ${A \in {\bf C}}$ and ${x_0, \xi_0 \in {\bf R}^n}$ are parameters, would be physically concentrated in the ball ${B(x_0,\sqrt{\hbar})}$. Then we have the frequency space (or momentum space) perspective, viewing ${f}$ now as a function ${\xi \mapsto \hat f(\xi)}$ of the frequency variable ${\xi \in {\bf R}^n}$. For this discussion, it will be convenient to normalise the Fourier transform using a small constant ${\hbar > 0}$ (which has the physical interpretation of Planck’s constant if one is doing quantum mechanics), thus

$\displaystyle \hat f(\xi) := \frac{1}{(2\pi \hbar)^{n/2}} \int_{\bf R} e^{-i\xi \cdot x/\hbar} f(x)\ dx.$

For instance, for the gaussian wave packet (1), one has

$\displaystyle \hat f(\xi) = A e^{i\xi_0 \cdot x_0/\hbar} e^{-(\xi-\xi_0)^2/\hbar} e^{-i \xi \cdot x_0/\hbar},$

and so we see that ${f}$ is concentrated in frequency space in the ball ${B(\xi_0,\sqrt{\hbar})}$.

However, there is a third (but less rigorous) way to view a function ${f}$ in ${L^2({\bf R}^n)}$, which is the phase space perspective in which one tries to view ${f}$ as distributed simultaneously in physical space and in frequency space, thus being something like a measure on the phase space ${T^* {\bf R}^n := \{ (x,\xi): x, \xi \in {\bf R}^n\}}$. Thus, for instance, the function (1) should heuristically be concentrated on the region ${B(x_0,\sqrt{\hbar}) \times B(\xi_0,\sqrt{\hbar})}$ in phase space. Unfortunately, due to the uncertainty principle, there is no completely satisfactory way to canonically and rigorously define what the “phase space portrait” of a function ${f}$ should be. (For instance, the Wigner transform of ${f}$ can be viewed as an attempt to describe the distribution of the ${L^2}$ energy of ${f}$ in phase space, except that this transform can take negative or even complex values; see Folland’s book for further discussion.) Still, it is a very useful heuristic to think of functions has having a phase space portrait, which is something like a non-negative measure on phase space that captures the distribution of functions in both space and frequency, albeit with some “quantum fuzziness” that shows up whenever one tries to inspect this measure at scales of physical space and frequency space that together violate the uncertainty principle. (The score of a piece of music is a good everyday example of a phase space portrait of a function, in this case a sound wave; here, the physical space is the time axis (the horizontal dimension of the score) and the frequency space is the vertical dimension. Here, the time and frequency scales involved are well above the uncertainty principle limit (a typical note lasts many hundreds of cycles, whereas the uncertainty principle kicks in at ${O(1)}$ cycles) and so there is no obstruction here to musical notation being unambiguous.) Furthermore, if one takes certain asymptotic limits, one can recover a precise notion of a phase space portrait; for instance if one takes the semiclassical limit ${\hbar \rightarrow 0}$ then, under certain circumstances, the phase space portrait converges to a well-defined classical probability measure on phase space; closely related to this is the high frequency limit of a fixed function, which among other things defines the wave front set of that function, which can be viewed as another asymptotic realisation of the phase space portrait concept.

If functions in ${L^2({\bf R}^n)}$ can be viewed as a sort of distribution in phase space, then linear operators ${T: L^2({\bf R}^n) \rightarrow L^2({\bf R}^n)}$ should be viewed as various transformations on such distributions on phase space. For instance, a pseudodifferential operator ${a(X,D)}$ should correspond (as a zeroth approximation) to multiplying a phase space distribution by the symbol ${a(x,\xi)}$ of that operator, as discussed in this previous blog post. Note that such operators only change the amplitude of the phase space distribution, but not the support of that distribution.

Now we turn to operators that alter the support of a phase space distribution, rather than the amplitude; we will focus on unitary operators to emphasise the amplitude preservation aspect. These will eventually be key examples of Fourier integral operators. A physical translation ${Tf(x) := f(x-x_0)}$ should correspond to pushing forward the distribution by the transformation ${(x,\xi) \mapsto (x+x_0,\xi)}$, as can be seen by comparing the physical and frequency space supports of ${Tf}$ with that of ${f}$. Similarly, a frequency modulation ${Tf(x) := e^{i \xi_0 \cdot x/\hbar} f(x)}$ should correspond to the transformation ${(x,\xi) \mapsto (x,\xi+\xi_0)}$; a linear change of variables ${Tf(x) := |\hbox{det} L|^{-1/2} f(L^{-1} x)}$, where ${L: {\bf R}^n \rightarrow {\bf R}^n}$ is an invertible linear transformation, should correspond to ${(x,\xi) \mapsto (Lx, (L^*)^{-1} \xi)}$; and finally, the Fourier transform ${Tf(x) := \hat f(x)}$ should correspond to the transformation ${(x,\xi) \mapsto (\xi,-x)}$.

Based on these examples, one may hope that given any diffeomorphism ${\Phi: T^* {\bf R}^n \rightarrow T^* {\bf R}^n}$ of phase space, one could associate some sort of unitary (or approximately unitary) operator ${T_\Phi: L^2({\bf R}^n) \rightarrow L^2({\bf R}^n)}$, which (heuristically, at least) pushes the phase space portrait of a function forward by ${\Phi}$. However, there is an obstruction to doing so, which can be explained as follows. If ${T_\Phi}$ pushes phase space portraits by ${\Phi}$, and pseudodifferential operators ${a(X,D)}$ multiply phase space portraits by ${a}$, then this suggests the intertwining relationship

$\displaystyle a(X,D) T_\Phi \approx T_\Phi (a \circ \Phi)(X,D),$

and thus ${(a \circ \Phi)(X,D)}$ is approximately conjugate to ${a(X,D)}$:

$\displaystyle (a \circ \Phi)(X,D) \approx T_\Phi^{-1} a(X,D) T_\Phi. \ \ \ \ \ (2)$

The formalisation of this fact in the theory of Fourier integral operators is known as Egorov’s theorem, due to Yu Egorov (and not to be confused with the more widely known theorem of Dmitri Egorov in measure theory).

Applying commutators, we conclude the approximate conjugacy relationship

$\displaystyle \frac{1}{i\hbar} [(a \circ \Phi)(X,D), (b \circ \Phi)(X,D)] \approx T_\Phi^{-1} \frac{1}{i\hbar} [a(X,D), b(X,D)] T_\Phi.$

Now, the pseudodifferential calculus (as discussed in this previous post) tells us (heuristically, at least) that

$\displaystyle \frac{1}{i\hbar} [a(X,D), b(X,D)] \approx \{ a, b \}(X,D)$

and

$\displaystyle \frac{1}{i\hbar} [(a \circ \Phi)(X,D), (b \circ \Phi)(X,D)] \approx \{ a \circ \Phi, b \circ \Phi \}(X,D)$

where ${\{,\}}$ is the Poisson bracket. Comparing this with (2), we are then led to the compatibility condition

$\displaystyle \{ a \circ \Phi, b \circ \Phi \} \approx \{ a, b \} \circ \Phi,$

thus ${\Phi}$ needs to preserve (approximately, at least) the Poisson bracket, or equivalently ${\Phi}$ needs to be a symplectomorphism (again, approximately at least).

Now suppose that ${\Phi: T^* {\bf R}^n \rightarrow T^* {\bf R}^n}$ is a symplectomorphism. This is morally equivalent to the graph ${\Sigma := \{ (z, \Phi(z)): z \in T^* {\bf R}^n \}}$ being a Lagrangian submanifold of ${T^* {\bf R}^n \times T^* {\bf R}^n}$ (where we give the second copy of phase space the negative ${-\omega}$ of the usual symplectic form ${\omega}$, thus yielding ${\omega \oplus -\omega}$ as the full symplectic form on ${T^* {\bf R}^n \times T^* {\bf R}^n}$; this is another instantiation of the closed graph theorem, as mentioned in this previous post. This graph is known as the canonical relation for the (putative) FIO that is associated to ${\Phi}$. To understand what it means for this graph to be Lagrangian, we coordinatise ${T^* {\bf R}^n \times T^* {\bf R}^n}$ as ${(x,\xi,y,\eta)}$ suppose temporarily that this graph was (locally, at least) a smooth graph in the ${x}$ and ${y}$ variables, thus

$\displaystyle \Sigma = \{ (x, F(x,y), y, G(x,y)): x, y \in {\bf R}^n \}$

for some smooth functions ${F, G: {\bf R}^n \rightarrow {\bf R}^n}$. A brief computation shows that the Lagrangian property of ${\Sigma}$ is then equivalent to the compatibility conditions

$\displaystyle \frac{\partial F_i}{\partial x_j} = \frac{\partial F_j}{\partial x_i}$

$\displaystyle \frac{\partial G_i}{\partial y_j} = \frac{\partial G_j}{\partial y_i}$

$\displaystyle \frac{\partial F_i}{\partial y_j} = - \frac{\partial G_j}{\partial x_i}$

for ${i,j=1,\ldots,n}$, where ${F_1,\ldots,F_n, G_1,\ldots,G_n}$ denote the components of ${F,G}$. Some Fourier analysis (or Hodge theory) lets us solve these equations as

$\displaystyle F_i = -\frac{\partial \phi}{\partial x_i}; \quad G_j = \frac{\partial \phi}{\partial y_j}$

for some smooth potential function ${\phi: {\bf R}^n \times {\bf R}^n \rightarrow {\bf R}}$. Thus, we have parameterised our graph ${\Sigma}$ as

$\displaystyle \Sigma = \{ (x, -\nabla_x \phi(x,y), y, \nabla_y \phi(x,y)): x,y \in {\bf R}^n \} \ \ \ \ \ (3)$

so that ${\Phi}$ maps ${(x, -\nabla_x \phi(x,y))}$ to ${(y, \nabla_y \phi(x,y))}$.

A reasonable candidate for an operator associated to ${\Phi}$ and ${\Sigma}$ in this fashion is the oscillatory integral operator

$\displaystyle Tf(y) := \frac{1}{(2\pi \hbar)^{n/2}} \int_{{\bf R}^n} e^{i \phi(x,y)/\hbar} a(x,y) f(x)\ dx \ \ \ \ \ (4)$

for some smooth amplitude function ${a}$ (note that the Fourier transform is the special case when ${a=1}$ and ${\phi(x,y)=xy}$, which helps explain the genesis of the term “Fourier integral operator”). Indeed, if one computes an inner product ${\int_{{\bf R}^n} Tf(y) \overline{g(y)}\ dy}$ for gaussian wave packets ${f, g}$ of the form (1) and localised in phase space near ${(x_0,\xi_0), (y_0,\eta_0)}$ respectively, then a Taylor expansion of ${\phi}$ around ${(x_0,y_0)}$, followed by a stationary phase computation, shows (again heuristically, and assuming ${\phi}$ is suitably non-degenerate) that ${T}$ has (3) as its canonical relation. (Furthermore, a refinement of this stationary phase calculation suggests that if ${a}$ is normalised to be the half-density ${|\det \nabla_x \nabla_y \phi|^{1/2}}$, then ${T}$ should be approximately unitary.) As such, we view (4) as an example of a Fourier integral operator (assuming various smoothness and non-degeneracy hypotheses on the phase ${\phi}$ and amplitude ${a}$ which we do not detail here).

Of course, it may be the case that ${\Sigma}$ is not a graph in the ${x,y}$ coordinates (for instance, the key examples of translation, modulation, and dilation are not of this form), but then it is often a graph in some other pair of coordinates, such as ${\xi,y}$. In that case one can compose the oscillatory integral construction given above with a Fourier transform, giving another class of FIOs of the form

$\displaystyle Tf(y) := \frac{1}{(2\pi \hbar)^{n/2}} \int_{{\bf R}^n} e^{i \phi(\xi,y)/\hbar} a(\xi,y) \hat f(\xi)\ d\xi. \ \ \ \ \ (5)$

This class of FIOs covers many important cases; for instance, the translation, modulation, and dilation operators considered earlier can be written in this form after some Fourier analysis. Another typical example is the half-wave propagator ${T := e^{it \sqrt{-\Delta}}}$ for some time ${t \in {\bf R}}$, which can be written in the form

$\displaystyle Tf(y) = \frac{1}{(2\pi \hbar)^{n/2}} \int_{{\bf R}^n} e^{i (\xi \cdot y + t |\xi|)/\hbar} a(\xi,y) \hat f(\xi)\ d\xi.$

This corresponds to the phase space transformation ${(x,\xi) \mapsto (x+t|\xi|, \xi)}$, which can be viewed as the classical propagator associated to the “quantum” propagator ${e^{it\sqrt{-\Delta}}}$. More generally, propagators for linear Hamiltonian partial differential equations can often be expressed (at least approximately) by Fourier integral operators corresponding to the propagator of the associated classical Hamiltonian flow associated to the symbol of the Hamiltonian operator ${H}$; this leads to an important mathematical formalisation of the correspondence principle between quantum mechanics and classical mechanics, that is one of the foundations of microlocal analysis and which was extensively developed in Hörmander’s work. (More recently, numerically stable versions of this theory have been developed to allow for rapid and accurate numerical solutions to various linear PDE, for instance through Emmanuel Candés’ theory of curvelets, so the theory that Hörmander built now has some quite significant practical applications in areas such as geology.)

In some cases, the canonical relation ${\Sigma}$ may have some singularities (such as fold singularities) which prevent it from being written as graphs in the previous senses, but the theory for defining FIOs even in these cases, and in developing their calculus, is now well established, in large part due to the foundational work of Hörmander.

## Spending symmetry

18 November, 2012 in admin, book | by Terence Tao | 9 comments

I recently finished the first draft of the last of my books based on my 2011 blog posts (and also my Google buzzes and Google+ posts from that year), entitled “Spending symmetry“.    The PDF of this draft is available here.  This is again a rather  assorted (and lightly edited) collection of posts (and buzzes, and Google+ posts), though concentrating in the areas of analysis (both standard and nonstandard), logic, and geometry.   As always, comments and corrections are welcome.

## UCLA Math Undergraduate Merit Scholarship for 2013

15 November, 2012 in advertising | Tags: scholarship, UCLA, undergraduate study | by Terence Tao | 1 comment

[Once again, some advertising on behalf of my department, following on a similar announcement in the previous three years.]

Two years ago, the UCLA mathematics department launched a scholarship opportunity for entering freshman students with exceptional background and promise in mathematics. We have offered one scholarship every year, but this year due to an additional source of funding, we will also be able to offer an additional scholarship for California residents.The UCLA Math Undergraduate Merit Scholarship provides for full tuition, and a room and board allowance for 4 years. In addition, scholarship recipients follow an individualized accelerated program of study, as determined after consultation with UCLA faculty.   The program of study leads to a Masters degree in Mathematics in four years.
More information and an application form for the scholarship can be found on the web at:
and
To be considered for Fall 2013, candidates must apply for the scholarship and also for admission to UCLA on or before November 30, 2012.

## Garth Gaudry

18 October, 2012 in obituary | by Terence Tao | 3 comments

Garth Gaudry, who made many contributions to harmonic analysis and to Australian mathematics, and was also both my undergradaute and masters advisor as well as the head of school during one of my first academic jobs, died yesterday after a long battle with cancer, aged 71.

Garth worked on the interface between real-variable harmonic analysis and abstract harmonic analysis (which, despite their names, are actually two distinct fields, though certainly related to each other).  He was one of the first to realise the central importance of Littlewood-Paley theory as a general foundation for both abstract and real-variable harmonic analysis, writing an influential text with Robert Edwards on the topic.  He also made contributions to Clifford analysis, which was also the topic of my masters thesis.

But, amongst Australian mathematicians at least, Garth will be remembered for his tireless service to the field, most notably for his pivotal role in founding the Australian Mathematical Sciences Institute (AMSI) and then serving as AMSI’s first director, and then in directing the International Centre of Excellence for Education in Mathematics (ICE-EM), the educational arm of AMSI which, among other things, developed a full suite of maths textbooks and related educational materials covering Years 5-10 (which I reviewed here back in 2008).

I knew Garth ever since I was an undergraduate at Flinders University.   He was head of school then (a position roughly equivalent to department chair in the US), but still was able to spare an hour a week to meet with me to discuss real analysis, as I worked my way through Rudin’s “Real and complex analysis” and then Stein’s “Singular integrals”, and then eventually completed a masters thesis under his supervision on Clifford-valued singular integrals.  When Princeton accepted my application for graduate study, he convinced me to take the opportunity without hesitation.  Without Garth, I certainly wouldn’t be where I am at today, and I will always be very grateful for his advisorship.  He was a good person, and he will be missed very much by me and by many others.

## Bill Thurston

22 August, 2012 in math.GT, math.HO, obituary | Tags: geometrization conjecture, hyperbolisation theorem, Outside in, William Thurston | by Terence Tao | 31 comments

Bill Thurston, who made fundamental contributions to our understanding of low-dimensional manifolds and related structures, died on Tuesday, aged 65.

Perhaps Thurston’s best known achievement is the proof of the hyperbolisation theorem for Haken manifolds, which showed that 3-manifolds which obeyed a certain number of topological conditions, could always be given a hyperbolic geometry (i.e. a Riemannian metric that made the manifold isometric to a quotient of the hyperbolic 3-space $H^3$).  This difficult theorem connecting the topological and geometric structure of 3-manifolds led Thurston to give his influential geometrisation conjecture, which (in principle, at least) completely classifies the topology of an arbitrary compact 3-manifold as a combination of eight model geometries (now known as Thurston model geometries).  This conjecture has many consequences, including Thurston’s hyperbolisation theorem and (most famously) the Poincaré conjecture.  Indeed, by placing that conjecture in the context of a conceptually appealing general framework, of which many other cases could already be verified, Thurston provided one of the strongest pieces of evidence towards the truth of the Poincaré conjecture, until the work of Grisha Perelman in 2002-2003 proved both the Poincaré conjecture and the geometrisation conjecture by developing Hamilton’s Ricci flow methods.  (There are now several variants of Perelman’s proof of both conjectures; in the proof of geometrisation by Bessieres, Besson, Boileau, Maillot, and Porti, Thurston’s hyperbolisation theorem is a crucial ingredient, allowing one to bypass the need for the theory of Alexandrov spaces in a key step in Perelman’s argument.)

One of my favourite results of Thurston’s is his elegant method for everting the sphere (smoothly turning a sphere $S^2$ in ${\bf R}^3$ inside out without any folds or singularities).  The fact that sphere eversion can be achieved at all is highly unintuitive, and is often referred to as Smale’s paradox, as Stephen Smale was the first to give a proof that such an eversion exists.  However, prior to Thurston’s method, the known constructions for sphere eversion were quite complicated.  Thurston’s method, relying on corrugating and then twisting the sphere, is sufficiently conceptual and geometric that it can in fact be explained quite effectively in non-technical terms, as was done in the following excellent video entitled “Outside In“, and produced by the Geometry Center:

In addition to his direct mathematical research contributions, Thurston was also an amazing mathematical expositor, having the rare knack of being able to describe the process of mathematical thinking in addition to the results of that process and the intuition underlying it.  His wonderful essay “On proof and progress in mathematics“, which I highly recommend, is the quintessential instance of this; more recent examples include his many insightful questions and answers on MathOverflow.

I unfortunately never had the opportunity to meet Thurston in person (although we did correspond a few times online), but I know many mathematicians who have been profoundly influenced by him and his work.  His death is a great loss for mathematics.

## Call for nominations: National Academy of Sciences Award for Scientific Reviewing

13 July, 2012 in advertising | by Terence Tao | 15 comments

The National Academy of Sciences award for Scientific Reviewing is slated to be given in Mathematics (understood to include Applied Mathematics) in April 2013.  The award cycles among many fields, and the last (and only) time it was given in Mathematics was 1995.  This year, I am on the prize committee for this award and am therefore circulating a call for nominations.

This award is intended “to recognize authors whose reviews have synthesized extensive and difficult material, rendering a significant service to science and influencing the course of scientific thought”.   As such, it is slightly different in focus from most awards in mathematics, which tend to focus more on original research contributions than on synthesis and exposition, which in my opinion is an equally important component of mathematical research.

In 1995, this prize was awarded to Rob Kirby “For his list of problems in low-dimensional topology and his tireless maintenance of it; several generations have been greatly influenced by Kirby’s list.”.

Instructions for how to submit nominations can be found at this page.  Nominees and awardees do not need to be members of the Academy, and can be based outside of the United States.   The award comes with a medal and a \$10,000 prize.  The deadline for nominations is 1 October 2012.

## Mini-polymath4 discussion thread

12 July, 2012 in admin, polymath | Tags: mini-polymath4 | by Terence Tao | 7 comments

I’ve just opened the research thread for the mini-polymath4 project over at the polymath blog to collaboratively solve one of the six questions from this year’s IMO.  This year I have selected Q3, which is a somewhat intricate game-theoretic question.  (The full list of questions this year may be found here.)

This post will serve as the discussion thread of the project, intended to focus all the non-research aspects of the project such as organisational matters or commentary on the progress of the project.    The third component of the project is the wiki page, which is intended to summarise the progress made so far on the problem.

As with the previous mini-polymath projects, I myself will be serving primarily as a moderator, and hope other participants will take the lead in the research and in keeping the wiki up-to-date.

## Mini-polymath4 begins in three hours

Just a reminder that the mini-polymath4 project will begin in three hours at Thu July 12 2012 UTC 22:00.

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