When using MathSciNet, clicking on the “**Author**” tab switches you from doing a search for published items to doing a search for authors. Author searches are very important, but getting the right name can be tricky. (See this earlier post on the subject of authors and names.) When you start typing a name in the Author search, after a couple of letters have been entered, MathSciNet begins offering suggestions. The suggestions are coming from instant queries of the Mathematical Reviews author database. For example, this is what what I now see when typing “Serre” into an Author search:

When you have a list of suggestions, you can either hit “Enter” to ignore the suggestion and search using what you have typed so far or click on one of them to execute an Author search using those terms. It is important to note that the result of choosing a selection is launching a search. Many times, you will have a unique match, but other times there will be multiple matches. For instance, selecting “Serre F” from the example above matches two authors: “Serre, F.” and “Serre, François“.

Help with Journal searches is particularly useful since many journal titles are similar. We have roughly 1,800 active journals in MathSciNet, plus many historical journals (which might just be older versions of current journals). Lots of journals have the word “Journal” in the title. “Mathematical” is also popular. Skimming this list can give you an idea of similarities of titles.

The auto-suggest feature for Journal searches works similarly to that for Author searches. Click on the “**Journal**” tab to switch to doing a search for journals. When typing in a search word or phrase, after you type a couple of letters, MathSciNet starts offering suggestions. The suggestions are coming from instant queries of the Mathematical Reviews journal database. For example, here is what I see after typing “functional” into a Journal search:

Once again, when you have a list of suggestions, you can either hit “Enter” to ignore the suggestion and use what you have typed so far or click on one of the suggestions to execute a Journal search using those terms. Again, the result of choosing a selection is launching a search. Many times, you will have a unique match, but other times, there will be multiple matches. For instance, clicking on “Journal of Functional Analysis” in the example produces six results. Some are older incarnations of the current “Journal of Functional Analysis“, but you also get Journal of Nonlinear Functional Analysis and Differential Equations.

We hope these two new features help you to use MathSciNet more effectively. We have been beta testing them in-house at Mathematical Reviews for a while now and have found both to be tremendously helpful.

And, if you are going to the Joint Mathematical Meetings in San Diego, I hope you will come see Mathematical Reviews and MathSciNet at the AMS booth in the exhibit hall. My previous post has information about our activities there.

]]>**Complimentary MathSciNet at the meeting**. By special arrangement, MathSciNet will be available when using the wifi at the conference site. This year, access to MathSciNet should also be available in the hotels!

**Demos of MathSciNet**. Wednesday, Thursday, and Friday of the meeting, at 2:15pm at the Mathematical Reviews area of the AMS booth, there will be demonstrations of how to use MathSciNet. The demos will be by experts, who know lots of great ways to use MathSciNet. Fill out the questionnaire and have a chance to win a $100 gift card!

**Update your Author Profile Page. **Representatives will be available at the booth to help you update your author profile page on MathSciNet, including the opportunity to add a photograph or your name in its native script. If you are an AMS member and have signed up for the professional photograph service at the JMM, we can use that picture. We will also be set up to take a picture, in case you don’t have a favorite photo available.

**Working at Mathematical Reviews. **If you are interested in possibly working at Mathematical Reviews as an Associate Editor, the JMM would provide a good opportunity to find out more about what we do. We are looking for someone with expertise in algebra and an interest in a range of algebraic topics. My earlier post explains the open position. Stop by the booth to talk with us!

**Mathematical Reviews Reception**: Friday, 6:00 pm– 7:00 pm. All friends of the Mathematical Reviews (MathSciNet) are invited to join reviewers and MR editors and staff (past and present) for a special reception in honor of all of the efforts that go into the creation and publication of the Mathematical Reviews database. Refreshments will be served. The location is the Presidio Room in the Marriott.

The move to the cloud means a new address for MathSciNet: https://mathscinet.ams.org/mathscinet. The old address redirects you to this URL, so there is no immediate need to change links or bookmarks — but updating links is not a bad idea. This change did affect some proxy settings for off-site users, however. We have contacted librarians and have provided instructions on how to reset the proxy settings. If you are having a problem, write to msn-support@ams.org.

The move to the cloud also means that we will be retiring the five mirror servers at Rice University (Houston), IMPA (Rio de Janeiro), IRMA (Strasbourg), MPI (Bonn), and the University of Bielefeld (Bielefeld). For twenty years, these institutions provided better global access to the Mathematical Reviews database by hosting servers around the world. Over time, though, the technology of the web has evolved to a point where distributed access is now best provided by hosting sites on cloud servers. We are truly grateful to the five hosts. Over the next six months, as the mirror sites retire, their URLs will begin redirecting to the cloud servers.

This is the latest of the enhancements rolling out for MathSciNet. Sometimes they are quite visible, like the faceted searches released in January. Sometimes they are infrastructure, like this move to cloud servers. The internal changes are often best appreciated by not being noticed at all, except that your searches are a little faster and MathSciNet is always available.

]]>The 2017 Nobel Prize in Physics was awarded to Rainer Weiss, Barry C. Barish, and Kip S. Thorne for their work on the detection of gravitational waves. (See Note 1.) The physics and engineering that go into this accomplishment are truly impressive. However, before anyone could imagine setting up the experiment, some mathematical questions needed to be answered. There are two articles in the August 2017 issue of the *AMS Notices* that give an overview the mathematics of gravitational waves. In this post, I crib from those two articles and provide a literature tour of some of the significant papers by relying on MathSciNet. A longer article by Bieri just published in the AMS Bulletin goes into more detail on a selection of the topics.

General Relativity (GR) depends heavily on mathematics, in particular, differential geometry and PDEs, since Einstein’s equations are nonlinear PDEs involving the curvature and the metric itself. The equations hold an inherent beauty that manifests itself not just in the mathematical description of space-time, but also in a rich realm of geometry: Einstein manifolds. This is digression from the theory of gravitational waves, but it is, in the language of Michelin guides, “Worth a trip.” As a tour guide for this trip, I heartily recommend the book *Einstein manifolds* written by a group of geometers under the pseudonym Arthur L. Besse.

There are many places for a mathematician to learn general relativity (GR). For a first read, Einstein’s own account in *The Meaning of Relativity* is extremely good. However it is deceptive: it seems like easy reading, but he is describing very deep ideas that are overturning our view of the world. You need to pay attention to understand what is going on. I didn’t pay attention the first time through and had to read it a second time. Here are three books that are suited to mathematicians for learning about general relativity.

- Hawking and Ellis,
*The large scale structure of space-time*. Cambridge Monographs on Mathematical Physics, No. 1. Cambridge University Press, London-New York, 1973. xi+391 pp. MR0424186 - Barrett O’Neill,
*Semi-Riemannian Geometry*, With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. xiii+468 pp. MR0719023 - Misner, Thorne, and Wheeler,
*Gravitation*. W. H. Freeman and Co., San Francisco, Calif., 1973. ii+xxvi+1279+iipp. MR0418833.

The book by Hawking and Ellis assembles the tools you need to understand general relativity, then applies them to discuss black holes, in particular the Hawking and Penrose singularity theorems. O’Neill’s book is written by a mathematician for mathematicians. The bulk of it is about differential geometry with a semi-Riemannian metric. You could give a complete course on differential geometry from O’Neill’s book and not cover relativity. But why would you leave it out? He sets everything up for you, then puts it to use. This is probably not the introduction that a physicist would want, but since it begins on familiar territory, it suits many mathematicians. Misner, Thorne, and Wheeler is often referred to as “the Bible” for studying general relativity. It is cited so often, that people often refer to it as MTW. It is a monster-sized book, but it is really two books in one: a one-semester introduction and a deeper look at the subject. MTW is the only one of these three books that really treats gravitational waves.

These three books provide a general introduction to GR. The two *Notices* papers are describing results that depend on the modern approach of geometric analysis in general relativity. For that, a good place to start would be the book:

MR2391586

Christodoulou, Demetrios(CH-ETHZ)

Mathematical problems of general relativity. I.

Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008. x+147 pp. ISBN: 978-3-03719-005-0.

I have not read it – I am relying on the review in MathSciNet! A short history of the mathematical study of Einstein’s equations can be found in the quite readable and very enjoyable article by Choquet-Bruhat:

MR3467361

Choquet-Bruhat, Yvonne(F-IHES)

Beginnings of the Cauchy problem for Einstein’s field equations.

Surveys in differential geometry 2015. One hundred years of general relativity, 1–16,

Surv. Differ. Geom., 20, Int. Press, Boston, MA, 2015.

The articles in the AMS Notices are:

MR3676432

Hill, C. Denson(1-SUNYS); Nurowski, Paweł(PL-PAN-CTP)

How the green light was given for gravitational wave search.

*Notices Amer. Math. Soc.* 64 (2017), no. 7, 686–692.

and

MR3676433

Bieri, Lydia(1-MI); Garfinkle, David(1-OAKL-P); Yunes, Nicolás(1-MTS-P)

Gravitational waves and their mathematics.

*Notices Amer. Math. Soc.* 64 (2017), no. 7, 693–707.

As the *Notices * articles point out, there are two big questions that had to be answered before we could have the spectacular results from LIGO. The paper by Hill and Nurowski addresses the first question: **1. Do gravitational waves exist?** The paper by Bieri, Garfinkle, and Yunes answers the second question: **2. How might one detect gravitational waves?**

As mentioned above, Einstein’s equations are nonlinear PDEs:

(1) $R_{i,j} – \frac{1}{2}R g_{i,j} = \kappa T_{i,j}$

where $g$ is the metric, $R_{i,j}$ is the Ricci curvature (which is a bunch of second derivatives of the metric), $R$ is the scalar curvature, and $T_{i,j}$ is the energy-momentum tensor that describes the matter or energy present in the space-time. There is a linearized form, which was known already to Einstein. First, think of the metric as a perturbation of the metric $\eta_{i,j}$ on Minkowski space:

$g_{i,j} = \eta_{i,j} + \epsilon h_{i,j}$.

To linearize, develop the left-hand side of (1) in powers of $\epsilon$, neglecting terms of order $2$ and above. Away from sources, the energy-momentum tensor is zero. So in these regions, the Einstein equations linearize to a system of decoupled relativistic wave equations in the unknowns $h_{i,j}$. Thus, at least in linearized GR, there are plane waves traveling at the speed of light. Because these are linear equations, superposition applies and you can add these plane waves to make any wave you want . These are the *gravitational waves* first described by Einstein. Far from sources, the linearized theory should coincide closely with the nonlinear theory. So one hopes to be able to detect gravitational waves by looking for phenomena that behave like the waves in the linearized model. But first one has to answer Question 1.

The question is whether or not waves exist *in the full theory*, not just the linearized version. Hill and Nurowski reduce this question to seven sub-questions. (Well that’s simpler, isn’t it?)

- What is the definition of a
*plane*gravitational wave in the full theory? - Does the so-defined plane wave exist as a solution to the full Einstein system?
- Do such waves carry energy?
- What is a definition of a gravitational wave with
*nonplanar front*in the full theory? - What is the energy of such waves?
- Do there exist solutions to the full Einstein system satisfying this definition?
- Does the full theory admit solutions corresponding to the gravitational waves emitted by bounded sources?

Sub-questions 1 and 4 look surprising at first – the answers ought to be “Of course!” But the mathematics of general relativity can be subtle, and simple things like bad choices of coordinates can mask important phenomena, as Einstein and Rosen discovered:

MR3363463

Einstein, A.; Rosen, N.

On gravitational waves.

J. Franklin Inst. 223 (1937), no. 1, 43–54.

(Well, actually, they made the bad choice and Howard Robertson discovered the badness of it when he refereed the paper.) Ignoring Robertson’s observation led Einstein and Rosen to conclude that plane waves don’t exist in *physical* space-times, since their coordinate choice indicated that the space-time had singularities.

The positive answer to sub-questions 1, 2, and 3 comes in a paper by Bondi in *Nature* and the paper:

MR0106747

Bondi, H.; Pirani, F. A. E.; Robinson, I.

Gravitational waves in general relativity. III. Exact plane waves.

Proc. Roy. Soc. London Ser. A 251 1959 519–533.

See also the paper:

MR0096537

Pirani, F. A. E.

Invariant formulation of gravitational radiation theory.

Phys. Rev. (2) 105 1957 1089–1099.

Hill and Nurowski point out that there was a parallel discovery much earlier by a mathematician that included the Bondi-Pirani-Robinson waves as a special case, but the gravitational theorists overlooked it:

MR1512246

Brinkmann, H. W.;

Einstein spaces which are mapped conformally on each other.

Math. Ann. 94 (1925), no. 1, 119–145.

Oops.

Now you have a definition of a plane wave, but you still have a problem because full GR is nonlinear, so superposition doesn’t work — you cannot build arbitrary waves out of plane waves. However, Pirani gives a geometric definition of nonplanar waves by considering the principal null directions, which are the eigendirections of the Weyl tensor (the traceless part of the Riemann tensor). Specifically, waves occur for space-times that are (asymptotically) *algebraically special:*

MR0096537

Pirani, F. A. E.

Invariant formulation of gravitational radiation theory.

Phys. Rev. (2) 105 1957 1089–1099.

Positive answers to sub-questions 4 and 5 come from Andrzej Trautman in his two papers:

MR0097265

Trautman, A.

Boundary conditions at infinity for physical theories.

Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 1958 403–406.

and

MR0097266

Trautman, A.

Radiation and boundary conditions in the theory of gravitation.

Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 1958 407–412.

The basic idea is to say that a radiative spacetime should satisfy certain boundary conditions at infinity, in analogy with radiative fields in Maxwell’s theory of electromagnetism. (At some point, I probably should have introduced null infinity, but this post is already long enough.)

That leaves us with questions 6 and 7. These were answered by Robinson and Trautman by actually producing some solutions with the desired property.

MR0135928

Robinson, I.; Trautman, A.

Some spherical gravitational waves in general relativity.

Proc. Roy. Soc. Ser. A 265 1961/1962 463–473.

**Review**: The authors obtain a class of metrics of which some represent, in their own words, “a very simple kind of spherical radiation”. They begin by establishing

$ds^2=-\rho^2p^{-2}\{(d\xi-ad\sigma)^2+(d\eta-bd\sigma)^2\}+2d\rho\,d\sigma+c\,d\sigma^2$

($a$, $b$, $c$, $p$ functions of the coordinates $\xi$, $\eta$, $\rho$, $\sigma$, but with $\partial p/\partial\rho=0$) as a canonical form for a metric which admits a shear-free, diverging and hypersurface-normal null vector field $\sigma_i$ that satisfies $R_{ik}\sigma^i\sigma^k=0$. They then solve the remaining field-equations for empty space and show that the solutions define two families of $V_2$ and admit a number of local and integral invariants. Algebraic properties of the curvature tensor and its rate of change along propagation-rays are discussed, and several explicit solutions of the Einstein and Maxwell-Einstein equations are obtained. These include the Schwarzschild metric as well as some static degenerate solutions of Levi-Civita.

Reviewed by H. S. Ruse

So, after some hiccups and a lot of hard work, we know what gravitational waves are and even have some information about their existence.

Gravity appears to us as a fairly strong force. It sticks us to the Earth. The sun has a massive gravitational field that bends light (one thing that makes eclipses interesting to physicists). In the event detected by LIGO, about 3 times the mass of the sun was converted into gravitational waves in a fraction of a second. So if a gravitational wave is going to come our way, especially one caused by the collision of two black holes, you might expect it to arrive like the big waves on the North Shore of Oahu. But, alas, we are observing the waves far from the source, in an area that we are assuming is essentially flat. So, in order to observe gravitational waves, LIGO needed to detect a displacement that was on the scale of 1/1000 the charge diameter of a proton over the course of a 4 km baseline, or a change of about one part in $10^{21}$. In order to do this, the experimenters need to have a very precise picture of what to look for.

A good way to find solutions to the Einstein equations is to solve a Cauchy problem. The Einstein equations split into a set of evolution equations and a set of constraint equations. You want to solve this system by specifying initial data. Typically, you either prescribe data on a space-like hypersurface (the classical case) or on a null hypersurface (the characteristic case). In the classical case, the hypersurface is known as a Cauchy surface, which, at least locally, you can picture as a slice corresponding to a fixed time that will then evolve under the equation. For gravitational waves, though, the characteristic case is more suitable, as explained in the *Bulletin *paper by Bieri. In any case, a priori, there is no guarantee that if your initial data satisfy the constraints that the solution of the evolution equation also satisfies the constraints. But in the case of the Einstein equations, Choquet-Bruhat showed exactly that. This is great! Her local solution is in the paper

MR0053338

Fourès-Bruhat, Y.

Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. (French)

Acta Math. 88, (1952). 141–225.

The global solution is in her paper with Geroch:

MR0250640

Choquet-Bruhat, Yvonne; Geroch, Robert

Global aspects of the Cauchy problem in general relativity.

Comm. Math. Phys. 14 1969 329–335.

Once we know that solutions exist, we need to know more about them because we need to know what we are looking for. You can read about the long-term existence of solutions in the book by Christodoulou and Klainerman

MR1316662

Christodoulou, Demetrios(1-PRIN); Klainerman, Sergiu(1-PRIN)

The global nonlinear stability of the Minkowski space.

Princeton Mathematical Series, 41. Princeton University Press, Princeton, NJ, 1993. x+514 pp. ISBN: 0-691-08777-6

where they also investigate the asymptotic structure of these spacetimes. As stated in the review of the book, “It is shown that the laws of gravitational radiation discovered by Bondi and others more than thirty years ago using formal power series expansions are rigorously true in this class of spacetimes.”

So, now we are getting closer to the question of how you might detect gravitational waves. There is a good mathematical description of the waves. What is needed next is a description of what might actually be observable. The behavior of neighboring geodesics in any geometric setting is governed by the Jacobi equation. In typical differential geometry courses, the Jacobi equation is used to identify conjugate points. Here, though, the equation allows for a computation of the (small) displacement of test masses when a gravitational wave passes. Since gravitational waves move at the speed of light, it is helpful to know that there is a *memory effect* that lingers and can be computed. See, for instance,

MR1144215

Thorne, Kip S.(1-CAIT-TA)

Gravitational-wave bursts with memory: the Christodoulou effect.

Phys. Rev. D (3) 45 (1992), no. 2, 520–524

as well as more recent work as found in

MR3467364

Bieri, Lydia(1-MI); Garfinkle, David(1-OAKL-P); Yau, Shing-Tung(1-HRV)

Gravitational waves and their memory in general relativity. (English summary) Surveys in differential geometry 2015. One hundred years of general relativity, 75–97,

Surv. Differ. Geom., 20, Int. Press, Boston, MA, 2015.

What we need, though, is to be able to observe it and to measure it in the real world, thus confirming that it was caused by a gravitational wave. This uses the technique of *matched filtering*, meaning you have a collection of templates (derived from a solution) and you match the collected data to these templates.

Once again, the nonlinearity of the Einstein equations means that we don’t have many closed-form solutions to use for templates. Rather, we need simulations. The Choquet-Bruhat result tells us that for actual solutions, initial data that satisfy the constraints will evolve and still satisfy the constraints. But in a simulation — an approximation, even a tiny deviation can cause a violation of the constraint and invalidate the simulation. Fortunately, mathematical physicists are good at numerical solutions and simulations and came up with a suite of three techniques that overcome this: hyperbolicity, constraint damping, and excision. Here are some papers that are representative of each:

**Hyperbolicity**

MR2254287

Pretorius, Frans(3-AB-P)

Simulation of binary black hole spacetimes with a harmonic evolution scheme.

Classical Quantum Gravity 23 (2006), no. 16, S529–S552.

**Constraint damping**

MR3079062

Gundlach, Carsten(4-SHMP-SM); Martín-García, José M.(F-PARIS6-IAP); Garfinkle, David(1-OAKL-P)

Summation by parts methods for spherical harmonic decompositions of the wave equation in any dimensions.

Classical Quantum Gravity 30 (2013), no. 14, 145003, 31 pp.

**Excision**

MR2079939

Thornburg, Jonathan(D-MPIGP)

Black-hole excision with multiple grid patches.

Classical Quantum Gravity 21 (2004), no. 15, 3665–3691.

Finally, all the mathematical ingredients are in place: a detailed picture of gravitational waves in the full, nonlinear theory and a means of computing samples of gravitational waves to the level of accuracy needed for matched filtering. Performing the experiment requires turning the mathematics into machinery, which is in itself an exquisite feat of science and engineering.

Feynman used to boast that QED (quantum electrodynamics) was the physical theory that was able to make the most accurate predictions and that had been tested to the greatest level of precision, phrasing it as like measuring the distance from New York to LA to within the width of a human hair (one part in $10^{12}$). With the LIGO experiments, the general relativists have verified a theory using an instrument that is sensitive on the order of one part in $10^{21}$. (See Note 4).

**Acknowledgment**: I am grateful to Lydia Bieri for some extra comments on the work involved, in particular for helping me to understand Choquet-Bruhat’s work better.

**Notes**:

(1) The paper announcing the LIGO results is

MR3707758

Abbott, B. P.(1-CAIT-LIG); et al.;

Observation of gravitational waves from a binary black hole merger.

Authors include B. C. Barish, K. S. Thorne and R. Weiss.

Phys. Rev. Lett. 116 (2016), no. 6, 061102, 16 pp.

(2) Before Choquet-Bruhat, various people made some headway on the existence problem for Cauchy data for the Einstein equations. One such person was Cornelius Lanczos, who had a remarkable life. His mixture of bad luck and good luck led to him making contributions to several different areas of mathematics, including general relativity, but also numerical analysis and linear algebra. The AMS published a nice biography of Lanczos by Barbara Gellai where you can read the details.

(3) The experiments to detect gravitational waves are impressive and *expensive*. CERN’s search for the Higgs boson was even more expensive. These are two major accomplishments in experimental physics that don’t get off the ground without some good mathematics to indicate that there was something interesting out there and giving a clue as to how to look for it. But there are some mathematically inspired physics experiments that are much less expensive. For instance, you can build a magnetic monopole detector for about the cost of a cup of coffee at Starbucks. It is described in

MR0856880

Clifford Henry Taubes

Physical and Mathematical Applications of Gauge Theories,

*Notices Amer. Math. Soc.*, 33 (1986), no. 5, 707–715.

All you need is a battery, a lightbulb, and some wire. Oh, and the patience to sit unblinking for a very long time. The paper provides a very accessible introduction to gauge theory suitable for a general mathematical audience. Taubes also has a little fun with the writing. The article has passages like:

*These are the plans, good luck in the chase,
*

Earlier, discussing decay of quarks, Taubes writes:

*Please, don’t blink, as you might miss the prize*

*of seeing the lash of this year’s lone quark’s predicted demise.*

*After five years of patience, statistics are small;*

*no announcement’s been made of matter’s downfall.*

Other passages are in blank verse. That era of the AMS *Notices* is not (yet) available online. If you want to read the article, I am afraid you will have to go to the library. It is at least worth a detour as you walk from Starbucks to your office.

(4) I’ve skipped over something important in comparing the accuracy of QED to the sensitivity of the LIGO experiments. I refer you to a discussion on physics.stackexhange.com to address that.

]]>Emmanuel Candès has won a prestigious MacArthur Fellowship. The official announcement is here. The LA Times has a nice write-up. Both the Los Angeles Times and the MacArthur announcement highlight Candès’s work on compressed sensing. Terry Tao has a spot-on reaction to this work, quoted in the LA Times, typical of most mathematicians when you first hear about the method: *you can’t be getting be getting something for nothing. This can’t work.* But it does! Tao finally came around to believe it, as has the rest of the world.

Candès is a collaborative researcher. The work on compressed sensing came out of conversations he was having initially with researchers in imaging science (MRIs). He tested out his ideas in conversations with Tao (while picking up their children at daycare, according to the legend). In MathSciNet, you can see that Candès has 48 coauthors, representing a wide variety of people: statisticians, analysts, younger collaborators, older collaborators.

One of his key publications on compressed sensing is his paper with Justin Romberg and Terry Tao in *Communications on Pure and Applied Mathematics,* Stable signal recovery from incomplete and inaccurate measurements. For your reading enjoyment, our review of it is reproduced below. The complete article is here.

Congratulations, Emmanuel Candès!

**MR2230846**

Candès, Emmanuel J.(1-CAIT-ACM); Romberg, Justin K.(1-CAIT-ACM); Tao, Terence(1-UCLA)

Stable signal recovery from incomplete and inaccurate measurements.

*Comm. Pure Appl. Math.* 59 (2006), no. 8, 1207–1223.

94A12

The authors consider the problem of recovering an unknown sparse signal $x_0(t)\in\Bbb{R}^m$ from $n\ll m$ linear measurements which are corrupted by noise, building on related results in [E. J. Candès and J. K. Romberg, Found. Comput. Math. 6 (2006), no. 2, 227–254; MR2228740; E. J. Candès, J. K. Romberg and T. C. Tao, IEEE Trans. Inform. Theory 52 (2006), no. 2, 489–509; MR2236170; E. J. Candès and T. C. Tao, IEEE Trans. Inform. Theory 51 (2005), no. 12, 4203–4215; MR2243152; “Near optimal signal recovery from random projections: universal encoding strategies?”, preprint, arxiv.org/abs/math.CA/0410542, IEEE Trans. Inform. Theory, submitted; D. L. Donoho, Comm. Pure Appl. Math. 59 (2006), no. 6, 797–829; MR2217606]. Here $x_0(t)$is said to be sparse if its support $T_{0}=\lbrace t\colon x_0(t)\neq0\rbrace$ has small cardinality. The measurements are assumed to be of the form $y=Ax_0+ e$, where $A$ is the $n\times m$ measurement matrix and the error term $e$ satisfies $\Vert e\Vert_{l_{2}}\le\epsilon$. Given this setup, consider the convex program $$ \min\Vert x\Vert_{l_{1}}\ {\rm subject\ to}\;\Vert Ax-y \Vert_{l_{2}}\le\epsilon.\;\;\;(\textrm {P}_{2}) $$

The authors define $A_{T}$, $T\subset\lbrace 1,\ldots,m\rbrace$, to be the $n\times\vert T\vert$ submatrix obtained by extracting the columns of $A$ corresponding to the indices in $T$. Their results are stated in terms of the $S$-restricted isometry constant $\delta_S$ of $A$ [E. J. Candès and T. C. Tao, op. cit., 2005], which is the smallest quantity such that $$ (1-\delta_{S})\Vert c\Vert_{l_{2}}^{2}\le\Vert A_{T}c \Vert_{l_{2}}^{2}\le(1+\delta_{S})\Vert c\Vert_{l_{2}}^2 $$ for all subsets $T$ with $\vert T\vert\le S$ and coefficient sequences $(c_{j})_{j\in T}$.

Their main results are below. The first describes stable recovery of sparse signals, while the second yields stable recovery for approximately sparse signals by focusing on the $S$ largest components of the signal $x_0$.

Theorem 1. Let $S$ be such that $\delta_{3S}+3\delta_{4S}<2$. Then for any signal $x_{0}$ supported on $T_0$ with $\vert T_0\vert\le S$ and any perturbation $e$ with $\Vert e\Vert_{l_{2}}\le\epsilon$, the solution $x^{\sharp}$ to $({\rm P}_2)$ obeys $$ \vert x^{\sharp}-x_0\vert_{l_{2}}\le C_{S}\cdot\epsilon, $$ where the constant $C_S$ depends only on $\delta_{4S}$. For reasonable values of $\delta_{4S}$, $C_S$ is well behaved; for example, $C_S\approx8.82$ for $\delta_{4S}=\frac{1}{5}$ and $C_{S}\approx 10.47$ for $\delta_{4S}=\frac{1}{4}$.

Theorem 2. Suppose that $x_0$ is an arbitrary vector in $\Bbb{R}^m$, and let $x_{0,S}$ be the truncated vector corresponding to the $S$ largest values of $x_0$ (in absolute value). Under the hypotheses of Theorem 1 in the paper, the solution $x^{\sharp}$ to $({\rm P}_{2})$ obeys $$ \Vert x^{\sharp}-x_0\Vert_{l_{2}}\le C_{1,S}\cdot\epsilon+ C_{2,S}\cdot\frac{\Vert x_0-x_{0,S}\Vert_{l_{1}}}{\sqrt{S}}. $$

For reasonable values of $\delta_{4S}$, the constants in the equation labelled 1.4 in the paper are well behaved; for example, $C_{1,S}\approx 12.04$ and $C_{2,S}\approx8.77$ for $\delta_{4S}=\frac{1}{5}$.

Reviewed by Brody Dylan Johnson

]]>*Mathematical Reviews* is a great place to work. You get to do something important and useful. You would also be working with great people. A list of the current editors is here. And here is a picture of some of us observing the eclipse in August 2017.

If you have any questions, drop me a line.

]]>I just came back from the Mathematical Congress of the Americas in Montreal. It was an intense week of mathematics. Besides having excellent invited and plenary lectures, there were 70 special sessions! There were five plenary lectures: Manuel del Pino (Universidad de Chile); Shafrira Goldwasser (MIT); Peter Ozsvath (Princeton University); Yuval Peres (Microsoft Research); and Kannan Soundararajan (Stanford University). Erik Demaine (Massachusetts Institute of Technology) and Étienne Ghys (École Normale Supérieure de Lyon) gave well-received public lectures. The list of invited speakers is here.

Demaine advocated for collaboration, pointing out that he had written papers with 429 different people. Not all of those papers are mathematical, but here is the list of his 299 co-authors in MathSciNet. In case you are wondering, MathSciNet’s collaboration distance between the two public lecturers is 4:

Erik D. Demaine | coauthored with | Noga Alon | MR3040956 |

Noga Alon | coauthored with | Alexander Lubotzky | MR1948752 |

Alexander Lubotzky | coauthored with | Rostislav I. Grigorchuk | MR2367034 |

Rostislav I. Grigorchuk | coauthored with | Étienne Ghys | MR3267516 |

One slide from Jeremy Kahn‘s invited lecture neatly sums up the Virtual Haken Conjecture:

I witnessed a new use of MathSciNet at the MCA. While sitting in one of the invited lectures, a person in front of me was researching the speaker on MathSciNet:

**The MCA prize winners were:**

**The Americas Prize**: José Antonio de la Peña – Universidad Nacional Autónoma de México.**The Solomon Lefschetz Medal**: Monica Clapp – Universidad Nacional Autónoma de México and Gunther Uhlmann – University of Washington.**The MCA Prize**: Héctor H. Pastén Vásquez – Harvard University; Vlad Vicol – Princeton University; Pablo Shmerkin – Torcuato Di Tella University and CONICET; Umberto Hryniewicz – Universidade Federal do Rio de Janeiro; Robert Morris – Instituto de Matemática Pura e Aplicada (IMPA).

The winner of the MathSciNet drawing was Matias Moya Giusti, from Villa Maria, Cordoba, Argentina, who was also the recipient of a travel grant from the AMS.

The AMS exhibit saw a lot of traffic, especially in the mornings. Many people stopped to look at the AMS books, find out about membership, or finish up some details with their travel grants. I talked with lots of people about MathSciNet. It was nice to see how many attendees not only used MathSciNet, but were also reviewers. One day at lunch, I happened to sit next to Hsian-Hua Tseng, who has written over 300 reviews!

The next MCA will be in Buenos Aires in 2021. I hope to see you there!

]]>Maryam Mirzakhani is known for her work on moduli spaces of Riemann surfaces. Some of her most cited work looks at the moduli space of a genus $g$ Riemann surface with $n$ geodesic boundary components. In two of her papers, she computes the volume of these moduli spaces, with respect to the Weil-Petersson metric (see below). In another, she provides a means for counting the number of simple closed geodesics of length at most $L$. Mirzakhani is also known for her work on billiards (see the review of her paper with Eskin and Mohammadi below), a subject closely related to moduli space questions. Teichmüller theory and the geometry of moduli spaces are famously deep subjects. Making progress requires mastering large areas of analysis, dynamical systems, differential geometry, algebraic geometry, and topology. I can only appreciate Mirzakhani’s work superficially, as I have not mastered those subjects. Instead, some reviews of her work are reproduced below.

**Notes:
1.** One of her biggest projects, joint work with Eskin studying the action of SL(2,R) on moduli space, is not published yet. So there is no item in MathSciNet for it. You can read the latest version on the arXiv.

**2.** Mirzakhani published three papers as an undergraduate: MR1366852, MR1386951, MR1615548. The second of these is regularly cited by combinatorists. The third paper was in the *Monthly. *

**3. **I started writing this post back in March, when I was highlighting the work of some remarkable mathematicians. It was delayed because describing her work is not simple: it is substantial and uses deep and difficult tools from several areas. Her papers are quite well written, with accessible introductions. However, the genius is in the details, which require real commitment to understand. The video produced for the ICM where she won her Fields Medal allows her to present something of her work. Amie Wilkinson describes Mirzakhani’s working style in this article in the NY Times. In a recent blog post, Terry Tao comments on how Mirzakhani was able to see disparate mathematical results through the lens of the mathematics she was developing herself.

**4**. Thank you to Tom Ward who spotted an inequality that was reversed in the original version of this post.

**MR2264808**

Mirzakhani, Maryam

Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces.

*Invent. Math.* 167 (2007), no. 1, 179–222.

32G15 (14H15)

For $L:= (L_1,\dots , L_n)\in \Bbb{R}_+^n$ one can define the moduli space $M_{g,n}(L)=M_{g,n}(L_1,\dots ,L_n)$ of hyperbolic Riemann surfaces of genus $g$ with $n$ geodesic boundary components of lengths $L_1,\dots ,L_n$. $M_{g,n}(L)$ carries a symplectic form called the Weil-Petersson symplectic form. The main object of study of the paper under review is the volume $V_{g,n}(L)=V_{g,n}(L_1,\dots ,L_n)$ of $M_{g,n}(L)$ calculated with respect to the volume form associated to the Weil-Petersson symplectic form. The main result of this paper is an explicit recursive formula (Section 5) for $V_{g,n}(L_1,\dots ,L_n)$ that allows one to effectively calculate $V_{g,n}(L_1,\dots ,L_n)$ from bottom-up. As an application, the author proves the following polynomial behavior of $V_{g,n}(L)$ (Theorem 1.1): $$V_{g,n}(L_1,\dots ,L_n)=\sum_{\alpha=(\alpha_1,\dots ,\alpha_n)\in \Bbb{Z}_+^n,\, \sum_{i=1}^n\alpha_i\leq 3g-3+n}C_\alpha \prod_{i=1}^nL_i^{2\alpha_i},$$ where $C_\alpha$ are some rational multiples of $\pi^{6g-6+2n-\sum_{i=1}^n2\alpha_i}$. In another work [J. Amer. Math. Soc. 20 (2007), no. 1, 1–23 (electronic); MR2257394] the author gave another proof of this result using symplectic reduction techniques. She also found a relation between $C_\alpha$ and intersection numbers on moduli spaces of Riemann surfaces.

The author’s approach to the recursive formula for $V_{g,n}(L)$ begins with a generalization of McShane’s identity [G. McShane, Invent. Math. 132 (1998), no. 3, 607–632; MR1625712]. We summarize this result (Theorem 1.3) as follows. Let $X$ be a hyperbolic Riemann surface with $n$ geodesic boundary components $\beta_1,\dots ,\beta_n$ of lengths $L_1,\dots ,L_n$. Then the following holds: $$ \sum_{\{\gamma_1,\gamma_2\}}\scr{D}(L_1,l_{\gamma_1}(X), l_{\gamma_2}(X))+\sum_{i=2}^n\sum_\gamma \scr{R}(L_1,L_i, l_\gamma(X))=L_1.\tag1 $$ The ingredients of this formula are explained below:

- for a geodesic $\gamma\subset X$, its length is denoted by $l_\gamma(X)$;
- the first sum is over all unordered pairs of simple closed geodesics $\{\gamma_1, \gamma_2\}$ bounding a pair of pants with $\beta_1$;
- the second sum is over all simple closed geodesics $\gamma$ bounding a pair of pants with $\beta_1,\beta_i$.
- the functions $\scr{D}, \scr{R}\colon \Bbb{R}_+^3\to \Bbb{R}_+$ are defined in terms of the geometry of a pair of pants (see Section 3). These functions can be explicitly calculated (Lemma 3.1): $$\scr{D}(x,y,z)=2\log\left(\frac{e^{\frac{x}{2}}+e^{\frac{y+z}{2}}} {e^{\frac{-x}{2}}+e^{\frac{y+z}{2}}}\right),$$ $$\scr{R}(x,y,z)=x-\log \left(\frac{\cosh(\frac{y}{2})+\cosh (\frac{x+z}{2})}{\cosh(\frac{y}{2})+\cosh(\frac{x-z}{2})} \right).$$

The author’s proof of (1) is based on a detailed analysis of geodesics and pairs of pants on $X$, carried out in Sections 3 and 4.

To prove the recursive formula for $V_{g,n}(L)$ the author develops a method to integrate functions given in terms of hyperbolic lengths. Note that the functions involved in (1) are of this kind. The author finds a way to express integrals of such functions over moduli spaces of Riemann surfaces using Weil-Petersson volumes. This is Theorem 7.1. The recursive formula for $V_{g,n}(L)$ is then obtained by integrating (1) against the Weil-Petersson volume form and applying Theorem 7.1.

Reviewed by Hsian-Hua Tseng

**MR2257394**

Mirzakhani, Maryam

Weil-Petersson volumes and intersection theory on the moduli space of curves.

*J. Amer. Math. Soc.* 20 (2007), no. 1, 1–23.

14H15 (14N35 32G15)

For $b_1,\dots,b_n\in\Bbb{R}_+$, put $b=(b_1,\dots,b_n)$ and let $M_{g,n}(b)=M_{g,n}(b_1,\dots,b_n)$ be the moduli space of hyperbolic Riemann surfaces with geodesic boundary components of lengths $b_1,\dots,b_n$. On $M_{g,n}(b)$ there is a symplectic form called the Weil-Petersson symplectic form. Let $V_{g,n}(b)=V_{g,n}(b_1,\dots,b_n)$ denote the volume of $M_{g,n}(b)$ calculated using the volume form associated to the Weil-Petersson symplectic form. The paper under review presents an explicit relationship between the Weil-Petersson volume $V_{g,n} (b)$ of $M_{g,n} (b)$ and the intersection numbers of tautological classes on the moduli space of stable curves. To achieve this, the author expresses the compactified moduli space $\overline{M}_{g,n}(b)$ as a symplectic quotient, as follows: consider the following moduli space of bordered Riemann surfaces with marked points: $$ \widehat{M_{g,n}}:=\{(X,p_1,\dots,p_n)\mid X\in\overline{M}_{g,n}(b_1,\dots,b_n),\ b_i\geq 0,\ p_i\in\tilde{\beta}_i\}. $$ Here $\tilde{\beta}_i$ is a parallel curve to the $i$-th boundary component $\beta_i\subset X$. The moduli space $\widehat{M_{g,n}}$ carries a symplectic form, naturally induced from the Weil-Petersson form. For a bordered Riemann surface $X$, denote by $l_{\beta_i}(X)$ the length of its $i$-th boundary component $\beta_i$. This gives a map $l^2/2\colon \widehat{M_{g,n}}\to \Bbb{R}_+^n$ defined by $$(l^2/2)(X, p_1,\dots,p_n) =(l_{\beta_1}(X)^2/2,\dots ,l_{\beta_n}(X)^2/2).$$ The author proves that $l^2/2$ is the moment map associated to the $T=(S^1)^n$ action on $\widehat{M_{g,n}}$ defined by rotating the points $p_1,\dots,p_n$, and that the symplectic quotient at value $(b_1,\dots,b_n)$ is symplectomorphic to $\overline{M}_{g,n}(b_1,\dots,b_n)$. Then, using the relationship between symplectic forms of reduced spaces at different values, she proves that the volume $V_{g,n}(b_1,\dots,b_n)$ is a polynomial in the $b_i$’s for $b_i$ sufficiently small, and the coefficients of this polynomial are explicitly given (see Theorem 4.4). In particular, the leading coefficients are, up to some prefactors, of the form $\int_{\overline{M}_{g,n}}\psi_1^{\alpha_1}\cdots\psi_n^{\alpha_n}$ with $\alpha_1+\dots +\alpha_n=3g-3+n$.

In a previous work [Invent. Math. 167 (2007), no. 1, 179–222; MR2264808], the author found a recursive formula for the volumes $V_{g,n}(b)$. A review of this formula is given in Section 5 of the paper under review. In the present paper the author uses this recursive formula and the result about coefficients of the volume polynomial to derive a proof of Witten’s conjecture (Kontsevich’s theorem) in the form of Virasoro constraints of a point. This is done by substituting the volume polynomials into the recursion, then extracting a recursion for the leading coefficients. By relating the coefficients with descendant integrals over $\overline{M}_{g,n}$, Virasoro constraints appear immediately.

Reviewed by Hsian-Hua Tseng

**MR2415399**

Mirzakhani, Maryam

Growth of the number of simple closed geodesics on hyperbolic surfaces.

*Ann. of Math. (2)* 168 (2008), no. 1, 97–125.

32G15

Let $X$ be a complete hyperbolic Riemann surface of genus $g$, with finite area and $n$ cusps. The paper under review studies the growth of the number $s_X(L)$ of simple closed geodesics of length at most $L$. In fact the author studies a more refined problem, as follows. Let $S_{g,n}$ be a closed surface of genus $g$ with $n$ boundary components. The mapping class group ${\rm Mod}_{g,n}$ acts on the set of isotopy classes of simple closed curves on $S_{g,n}$, and each isotopy class of a simple closed curve contains a unique simple closed geodesic on $X$. For a simple closed geodesic $\gamma$ the author considers the following counting function: $$ s_X(L,\gamma):=\#\{\alpha\in {\rm Mod}_{g,n}\cdot\gamma\,|\, l_\alpha(X)\leq L\}, $$ where $l_\alpha(X)$ is the length of $\alpha$. Note that $s_X(L)=\sum_\gamma s_X(L,\gamma)$. The definition of this counting function can be extended to multi-curves $\gamma=\sum_{i=1}^k a_i\gamma_i$. Here, by definition, multi-curves $\gamma_i$’s are disjoint, essential, nonperipheral simple closed curves which are pairwise non-homotopic, and $a_i>0$. The length of a multi-curve $\gamma$ is defined to be $l_\gamma(X):=\sum_{i=1}^k a_il_{\gamma_i}(X)$. The first main result of this paper asserts that for a rational multi-curve $\gamma=\sum_i a_i\gamma_i$ (i.e. $a_i\in \Bbb{Q}$), the limit $n_\gamma(X):=\lim_{L\to\infty}\frac{s_X(L,\gamma)}{L^{6g-6+2n}}$ defines a continuous proper function $n_\gamma\colon\scr{M}_{g,n}\to\Bbb{R}_+$.

The main tool used in this paper is the space $\scr{ML}_{g,n}$ of compactly supported measured laminations on $S_{g,n}$. There is a length function $\scr{ML}_{g,n}\to \Bbb{R}_+$ induced by the hyperbolic metric $X$ on $S_{g,n}$. The Thurston measure $B(X):=\mu_{Th}(B_X)$ of the unit ball $B_X$ with respect to this length function defines a function $B\colon \scr{M}_{g,n}\to\Bbb{R}_+$. The author proves that this function $B$ is integrable with respect to the Weil-Petersson volume form. Set $b_{g,n}:=\int_{\scr{M}_{g,n}}B(X) dX$. The next main result of this paper states that for each rational multi-curve $\gamma$ there is a number $c(\gamma)\in \Bbb{Q}_{>0}$ such that $n_\gamma(X)=\frac{c(\gamma)B(X)}{b_{g,n}}$. The proofs of these statements rely heavily on a study of measures on $\scr{ML}_{g,n}$. In fact the two main results are direct consequences of a statement about the asymptotic behavior of some discrete measures. The second main result has the following corollary: for rational multi-curves $\gamma_1,\gamma_2$ we have $\lim_{L\to\infty}\frac{s_X(L, \gamma_1)}{s_X(L,\gamma_2)}=\frac{c(\gamma_1)}{c(\gamma_2)}$. This may be rephrased as saying that the relative frequencies of different types of simple closed curves on $X$ are universal rational numbers.

Reviewed by Hsian-Hua Tseng

**MR3418528**

Eskin, Alex; Mirzakhani, Maryam; Mohammadi, Amir

Isolation, equidistribution, and orbit closures for the ${\rm SL}(2,\Bbb R)$ action on moduli space. (English summary)

*Ann. of Math. (2)* 182 (2015), no. 2, 673–721.

58D27 (22F10 32G15 37C85 37D40 60B15)

The results in this paper are analogous to the theory of unipotent flows and concern orbit closures and equidistribution for the $\textrm{SL}(2,\Bbb R)$-action on the moduli space of compact Riemann surfaces. Their number is such that a review can only give an impressionistic sampling. The proofs rely on the measure-classification theorem from [A. Eskin and M. Mirzakhani, “Invariant and stationary measures for the $\textrm{SL}(2,{\Bbb R})$ action on moduli space”, preprint, arXiv:1302.3320], which is a partial analogue of M. Ratner’s measure-classification theorem in the theory of unipotent flows [Ann. of Math. (2) 134 (1991), no. 3, 545–607; MR1135878], which was in turn motivated by the Raghunathan Conjecture [S. G. Dani, Invent. Math. 64 (1981), no. 2, 357–385; MR0629475; G. A. Margulis, in Number theory, trace formulas and discrete groups (Oslo, 1987), 377–398, Academic Press, Boston, MA, 1989; MR0993328]. The second major ingredient is the main technical result of this paper (Proposition 2.13), an isolation property of closed $\textrm{SL}(2,\Bbb R)$-invariant manifolds, whose proof takes up of Sections 4–10.

The proofs of the principal results are in Section 3; these are actually simpler than the proofs of the analogous results in the theory of unipotent flows, due in no small part to the fact (Proposition 2.16, a consequence of the isolation property) that there are at most countably many affine invariant submanifolds in each stratum, while unipotent flows may have continuous families of invariant manifolds (which involve the centralizer and normalizer of the acting group). The proof of the isolation property in turn is based on the recurrence properties of the $\textrm{SL}(2,\Bbb R)$-action from [J. S. Athreya, Geom. Dedicata 119 (2006), 121–140; MR2247652] and on the uniform hyperbolicity in compact sets of the Teichmüller geodesic flow [G. Forni, Ann. of Math. (2) 155 (2002), no. 1, 1–103 (Corollary 2.1); MR1888794].

Some terminology to give formal statements: $H(\alpha)$ denotes a stratum of Abelian differentials, i.e., the space of pairs $(M,\omega)$ where $M$ is a Riemann surface and $\omega$ is a holomorphic 1-form on $M$ whose zeros have multiplicities $\alpha_1\cdots\alpha_n$ with $\sum\alpha_i=\chi(M)\ge0$. The form $\omega$ defines a canonical flat metric on $M$ with cone points at the zeros of $\omega$, i.e., a flat surface or translation surface [A. Zorich, in Frontiers in number theory, physics, and geometry. I, 437–583, Springer, Berlin, 2006; MR2261104]. The space $H(\alpha)$ admits an action of $\textrm{SL}(2,\Bbb R)$ which generalizes the action of $\textrm{SL}(2,\Bbb R)$ on the space $\textrm{GL}(2,\Bbb R)/\textrm{SL}(2,\Bbb Z)$ of flat tori. A “unit hyperboloid” $H_1(\alpha)$ is defined as a subset of translation surfaces in $H(\alpha)$ of area one: $\frac i2\int_M\omega\wedge\overline\omega=1$.

The aforementioned measure-classification result of [A. Eskin and M. Mirzakhani, op. cit.] is: A probability measure on $H_1(\alpha)$ that is invariant under $P:=\left\{\left(\begin{array}{cc}*&*\\ 0&*\end{array}\right)\right\}\subset\textrm{SL}(2,\Bbb R)$ is $\textrm{SL}(2,\Bbb R)$-invariant and affine (i.e., supported on an immersed submanifold and compatible with Lebesgue measure in a particular way; the submanifold is then also said to be affine).

With $a_t:=\textrm{diag}(e^t,e^{-t})$ and $r_\theta:=\left(\begin{array}{cc}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{array}\right)$ we can now state the main isolation property of this paper.

If $\varnothing\subseteqq M\subset H_1(\alpha)$ is an affine invariant submanifold, then there is an $\textrm{SO}(2)$-invariant $f\colon H_1(\alpha)\to[1,\infty]$ such that:

- $M=f^{-1}(\infty)$.
- $f$ is bounded on compact subsets of $H_1(\alpha)\smallsetminus M$.
- $\overline{f^{-1}([1,\ell])}$ is compact for all $\ell$.
- $\exists b$ (depending only on the “complexity” of $M$) $\forall c\in(0,1)$ $\exists T>0$ $$ (A_tf)(x):=\frac1{2\pi}\int_0^{2\pi}f(a_tr_\theta x)\,d\theta\le cf(x)+b $$ whenever $x\in H_1(\alpha)\smallsetminus M$ and $t>T$.
- There is $\sigma>1$ such that $\sigma^{-1}f(x)\le f(gx)\le\sigma f(x)$ for all $x\in H_1(\alpha)$ and $g\in\textrm{SL}(2,\Bbb R)$ near the identity.

Here is an overview of the many consequences derived here.

Orbit closures in $H_1(\alpha)$ are affine invariant submanifolds (the unipotent counterpart is in [M. Ratner, Duke Math. J. 63 (1991), no. 1, 235–280; MR1106945]) and any closed $P$-invariant subset of $H_1(\alpha)$ is a finite union of affine invariant manifolds.

The space of ergodic $P$-invariant probability measures on $H_1(\alpha)$ is weak*-compact (the unipotent counterpart is in [S. Mozes and N. A. Shah, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 149–159; MR1314973]).

Equidistribution for sectors, random walks and Følner sets (the first of which implies that for any $x\in H_1(\alpha)$ there is a unique affine invariant manifold of minimal dimension that contains $x$); uniform versions of the equidistribution results (Theorems 2.7, 2.9) are analogous to [S. G. Dani and G. A. Margulis, in I. M. Gelʹfand Seminar, 91–137, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993 (Theorem 3); MR1237827], which plays a key role in applications of the theory.

Orbit counting in rational billiards: Let $N(Q,T)$ denote the number of cylinders of periodic trajectories of length at most $T$ for the billiard flow on a rational polygon $Q$. It is known that this grows quadratically [H. A. Masur, Ergodic Theory Dynam. Systems 10 (1990), no. 1, 151–176; MR1053805; in Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), 215–228, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988; MR0955824]: $N(Q,e^t)e^{-2t}$ is bounded above and away from 0 for $t>1$. The uniform equidistribution result for sectors implies that

$$ \frac1t\int_0^tN(Q,e^s)e^{-2s}\,ds\underset{t\to\infty}\rightarrow c,

$$

where $c$ is the Siegel-Veech constant [W. A. Veech, Ann. of Math. (2) 148 (1998), no. 3, 895–944; MR1670061; A. Eskin, H. A. Masur and A. Zorich, Publ. Math. Inst. Hautes Études Sci. No. 97 (2003), 61–179; MR2010740] associated to the affine invariant submanifold $M = \textrm{SL}(2,\Bbb R)S$ with $S$ the flat surface obtained by unfolding $Q$. The authors find it natural to conjecture that in fact $N(Q,e^t)e^{-2t}\,ds\underset{t\to\infty}\rightarrow c$, but this seems beyond current methods (yet is known in special cases [A. Eskin, H. A. Masur and M. Schmoll, Duke Math. J. 118 (2003), no. 3, 427–463; MR1983037; A. Eskin, J. Marklof and D. W. Morris, Ergodic Theory Dynam. Systems 26 (2006), no. 1, 129–162; MR2201941; K. Calta and K. Wortman, Ergodic Theory Dynam. Systems 30 (2010), no. 2, 379–398; MR2599885; M. Bainbridge, Geom. Funct. Anal. 20 (2010), no. 2, 299–356; MR2671280]).

Reviewed by Boris Hasselblatt

]]>You can trace Saint-Raymond’s work in MathSciNet, with its persistent themes, such as the Boltzmann equation and limiting phenomena in statistical mechanics. Her work has inspired several long reviews in MathSciNet, including MR2683475, MR1952079, and MR3157048. In an earlier post, I mentioned that Cédric Villani’s review of her paper with François Golse on the Navier-Stokes limit of the Boltzmann equation “verges on being a short course.” The text of Nader Masmoudi’s long review of her volume in *Lecture Notes in Mathematics* is below.

Meanwhile, IMPA has videos of Saint-Raymond giving a mini-course on some of her work: Class 1, Class 2, and Class 3.

MR2683475

Saint-Raymond, Laure (F-ENS-DAM)

Hydrodynamic limits of the Boltzmann equation.

Lecture Notes in Mathematics, 1971. Springer-Verlag, Berlin, 2009. xii+188 pp. ISBN: 978-3-540-92846-1

From a physical point of view, we expect that a gas can be described by a fluid equation when the mean free path (Knudsen number) goes to zero. In his sixth problem, on the occasion of the International Congress of Mathematicians held in Paris in 1900, Hilbert asked for a full mathematical justification of these derivations. During the last two decades this problem has attracted a lot of interest.

Let us first give some background about this problem (see Chapters 1 and 2 in the book). The molecules of a gas can be modeled by spheres that move according to the laws of classical mechanics. However, due to the enormous number of molecules to be considered, it is hopeless to describe the state of the gas by giving the position and velocity of each individual particle. Hence, we must use some statistics and instead of giving the position and velocity of each particle, we specify the density of particles $F(x,v)$ at each point $x$ and velocity $v$. Under some assumptions (rarefied gas, etc.), it was proved by Boltzmann (and Lanford for a rigorous proof in the hard sphere case) that this density is governed by the Boltzmann equation (B): $$ \partial_t F + v\cdot \nabla_{x} F = B(F,F). $$ To derive fluid equations from the Boltzmann equation, one has to introduce several dimensionless parameters: the Knudsen number ${\rm Kn}$ (which is related to the mean free path), the Mach number ${\rm Ma}$ and the Strouhal number ${\rm St}$ (which is a time rescaling). With these parameters, one can rewrite the Boltzmann equation as $$ {\rm St}\cdot \partial_t F + v\cdot \nabla_{x} F = \frac1{\rm Kn} B(F,F) $$ with $F = M (1 + {\rm Ma}\cdot f)$ where $M$ is a fixed Maxwellian. It is worth noting that the Reynolds number ${\rm Re}$ is completely determined by the relation ${\rm Ma} = {\rm Kn}\cdot {\rm Re}$. Several fluid equations can be derived that depend on these dimensionless parameters: Compressible Euler system, acoustic waves, Incompressible Navier-Stokes-Fourier system, Stokes-Fourier system, Incompressible Euler system, etc. There are several approaches to deal with this problem: the weak compactness method initiated by C. Bardos, F. Golse and C. D. Levermore, asymptotic expansions [see A. De Masi, R. Esposito and J. L. Lebowitz, Comm. Pure Appl. Math. 42 (1989), no. 8, 1189–1214; MR1029125], the energy method [Y. Guo, Comm. Pure Appl. Math. 59 (2006), no. 5, 626–687; MR2172804; erratum, Comm. Pure Appl. Math. 60 (2007), no. 2, 291–293; MR2275331], etc.

This book gives an overview of some of these results and mainly the derivation of the Incompressible Navier-Stokes [F. Golse and L. Saint-Raymond, Invent. Math. 155 (2004), no. 1, 81–161; MR2025302] and Incompressible Euler [L. Saint-Raymond, Arch. Ration. Mech. Anal. 166 (2003), no. 1, 47–80; MR1952079] systems.

After the construction of the renormalized solutions to the Boltzmann equation by R. J. DiPerna and P.-L. Lions [Ann. of Math. (2) 130 (1989), no. 2, 321–366; MR1014927], there was a program initiated by Bardos, Golse and Levermore [J. Statist. Phys. 63 (1991), no. 1-2, 323–344; MR1115587; Comm. Pure Appl. Math. 46 (1993), no. 5, 667–753; MR1213991] to derive incompressible models from the Boltzmann equation. In particular the main objective was to recover the Leray [J. Leray, Acta Math. 63 (1934), no. 1, 193–248; MR1555394; JFM 60.0726.05] global weak solutions of the incompressible Navier-Stokes system starting from the DiPerna-Lions solutions.

There were five main assumptions in their first work:

(1) Because of a problem coming from the rapid time-oscillations of acoustic waves, only the time independent case was considered.

(2) Local conservation laws were assumed even though these are not known to hold for the renormalized solutions.

(3) The lack of high-order moment estimates required the restriction of the discussion to the momentum equation and no heat equation was derived.

(4) A key equi-integrability estimate was assumed on the solutions of the Boltzmann equation. This is due to the fact that the natural space for the Boltzmann equation is $L\log L$ whereas for the Navier-Stokes system the natural space is $L^2$.

(5) Due to a technical estimate for the inverse of the linearized Boltzmann kernel, only very particular collision kernels were considered.

These five assumptions have been removed one by one in the past two decades:

(1) In [P.-L. Lions and N. Masmoudi, Arch. Ration. Mech. Anal. 158 (2001), no. 3, 173–193, 195–211; MR1842343] the time-oscillating acoustic waves were treated using a compensated compactness type argument coming from the compressible-incompressible limit [P.-L. Lions and N. Masmoudi, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 5, 387–392; MR1710123].

(2)–(3) In [P.-L. Lions and N. Masmoudi, op. cit., MR1842343 (pp. 195–211)], the assumption on the local conservation in the momentum equation was removed, and in [Comm. Pure Appl. Math. 55 (2002), no. 3, 336–393; MR1866367], Golse and Levermore were able to derive the Stokes-Fourier system. The main idea is to recover the moment conservation laws at the limit.

(4) The main breakthrough of [F. Golse and L. Saint-Raymond, op. cit.; MR2025302] was a new $L^1$ averaging lemma that allows one to prove the key equi-integrability estimate.

(5) In [F. Golse and L. Saint-Raymond, J. Math. Pures Appl. (9) 91 (2009), no. 5, 508–552; MR2517786] the result was extended to hard cutoff potentials satisfying Grad’s cutoff assumption and in [C. D. Levermore and N. Masmoudi, Arch. Ration. Mech. Anal. 196 (2010), no. 3, 753–809; MR2644440] it was also extended to both hard and soft potentials. Another important extension was done by D. Arsénio [“From Boltzmann’s equation to the incompressible Navier-Stokes-Fourier system with long-range interactions”, Arch. Ration. Mech. Anal., to appear], who treated the non-cutoff case.

We also note that the case where the problem is considered in a bounded domain was treated in [N. Masmoudi and L. Saint-Raymond, Comm. Pure Appl. Math. 56 (2003), no. 9, 1263–1293; MR1980855] where Navier and Dirichlet boundary conditions were derived starting from the Maxwell boundary condition.

Chapter 3 of this book presents the main mathematical tools used in dealing with the hydrodynamic limit. In particular several estimates coming from the entropy, the entropy dissipation and Darrozès-Guiraud information are presented. Also the new $L^1$averaging lemma is proved.

Chapter 4 deals with the incompressible Navier-Stokes limit using the weak compactness method. In particular the author shows how to combine the ideas from [N. Masmoudi and L. Saint-Raymond, op. cit.; MR1980855] to treat the case of a bounded domain with Maxwell boundary conditions.

Chapter 5 deals with the incompressible Euler limit using the relative entropy method [L. Saint-Raymond, op. cit.; MR1952079].

Finally, Chapter 6 gives a survey of the known results about the compressible Euler limit. It is worth noting that if we are interested in starting from the renormalized solutions then none of the methods used in the incompressible case can be adapted. The author gives some open problems and perspectives.

Reviewed by Nader Masmoudi

Yves Meyer has been selected to win the 2017 Abel Prize. The citation is “for his pivotal role in the development of the mathematical theory of wavelets”. His work is certainly well known within mathematics, especially within harmonic analysis and in its important applications in image processing, data compression, signal analysis, and many other modern settings.

There are announcements of the prize in various places:

I will defer to these other sources for general information about the prize and about Meyer’s work. Here I would like to bring out a few aspects of his work with the help of *Mathematical Reviews*. First of all, at the time of this writing, there are 8448 items in MathSciNet with the word “wavelet” or “wavelets” in the title. Looking for either “wavelet” or “wavelets” anywhere in our records for items produces 13705 matches. The earliest is

MR0001894 Baker, Bevan B.; Copson, E. T. *The Mathematical Theory of Huygens’ Principle*. Oxford University Press, New York, (1939). vii+155 pp.

where “spherical wavelets” are mentioned in the review. (I suspect that these are not the same thing we normally think of as wavelets.)

Secondly, since citations are a big thing these days, let me point out that in MathSciNet, Meyer is cited 4834 times by 3262 authors. His most highly cited work is, not surprisingly, about wavelets:

MR1228209 Meyer, Yves. *Wavelets and operators*. Translated from the 1990 French original by D. H. Salinger. Cambridge Studies in Advanced Mathematics, 37. Cambridge University Press, Cambridge, 1992. xvi+224 pp. ISBN: 0-521-42000-8; 0-521-45869-2

This is the first part of a multi-part book. The second part was published as

MR1456993 Meyer, Yves; Coifman, Ronald. *Wavelets. Calderón-Zygmund and multilinear operators*. Translated from the 1990 and 1991 French originals by David Salinger. Cambridge Studies in Advanced Mathematics, 48. Cambridge University Press, Cambridge, 1997. xx+315 pp. ISBN: 0-521-42001-6; 0-521-79473-0

The book was published in two parts in English, but in three parts in French:

MR1085487 Meyer, Yves. *Ondelettes et opérateurs. I. Ondelettes*. Actualités Mathématiques. Hermann, Paris, 1990.

MR1085488 Meyer, Yves. *Ondelettes et opérateurs. II. Opérateurs de Calderón-Zygmund*. Actualités Mathématiques. Hermann, Paris, 1990.

MR1160989 Meyer, Yves; Coifman, R. R. *Ondelettes et opérateurs. III. Opérateurs multilinéaires*. Actualités Mathématiques. Hermann, Paris, 1991

Meyer and Coifman have 37 joint publications listed in MathSciNet. The earliest is a paper on singular integrals published in the *Transactions of the AMS*. Their most frequently cited joint work is on pseudodifferential operators, and was published as a volume in the esteemed series *Astérisque* from the Société Mathématique de France*.*

Meyer has published three books with the American Mathematical Society:

MR1342019 Jaffard, Stéphane; Meyer, Yves. *Wavelet methods for pointwise regularity and local oscillations of functions*. Mem. Amer. Math. Soc. 123 (1996), no. 587, x+110 pp.

MR1483896 Meyer, Yves. *Wavelets, vibrations and scalings*. With a preface in French by the author. CRM Monograph Series, 9. American Mathematical Society, Providence, RI, 1998. x+133 pp. ISBN: 0-8218-0685-8

MR1852741 Meyer, Yves. *Oscillating patterns in image processing and nonlinear evolution equations*. The fifteenth Dean Jacqueline B. Lewis memorial lectures. University Lecture Series, 22. American Mathematical Society, Providence, RI, 2001. x+122 pp. ISBN: 0-8218-2920-3

There was a story last year about a recent result of Meyer’s. Meyer was looking at variations on the Poisson formula as given in the work of Nir Lev and Alexander Olevskii, Quasicrystals with discrete support and spectrum. *Rev. Mat. Iberoam.* 32 (2016), no. 4, 1341–1352 [MR3593527]. After lecturing several times on the result, Meyer came up with a simpler proof of the result. Like any good researcher, before sending off the paper to a journal, he checked the existing literature. In the references to the Lev and Olevskii paper, he found a paper by Guinand:

MR0107784 Guinand, A. P. Concordance and the harmonic analysis of sequences. *Acta Math*. 101 1959 235–271.

Meyer dug up a copy of the paper and was surprised to find that Guinand had the same solution as his own. But no one had noticed this. Lev and Olevskii had not. Nor, apparently, had Salomon Bochner, who made no mention of it in his review of the paper in Mathematical Reviews. Meyer adjusted his paper accordingly, giving priority to Guinand, and it was published in the Proceedings of the National Academy of Science USA:

MR3482845 Meyer, Yves F. Measures with locally finite support and spectrum. *Proc. Natl. Acad. Sci. USA* 113 (2016), no. 12, 3152–3158.

In the section labeled *Significance*, Meyer wrote, “Our new Poisson’s formulas were hidden inside an old and almost forgotten paper published in 1959 by A. P. Guinand.”

Wavelet image by JonMcLoone – Own work, CC BY-SA 3.0, Link

]]>