Besides free MathSciNet at ICIAM in Valencia, it is possible to have demos of how to use MathSciNet.

Here I am talking with two mathematicians from Morocco about using the Mathematical Reviews Journal Database behind MathSciNet to help decide where to send a paper. We began by searching for all items in MSC 49 [Calculus of variations and optimal control; optimization] that were published in the last three years and that were published in journals (rather than in conference proceedings or as books). We then looked at the **Journals** facet in the sidebar to see what the most frequently occurring journals in the list were. The most frequent was *Journal of Optimization Theory and* Applications. The second most frequent was *SIAM Journal on Control and Optimization*. We clicked on the journal link from one of the items in* SIAM Journal on Control and Optimization*, which took us to the MR Journal Profile page for it. From there we could see all sorts of information about the journal: the number of articles published per year, the most commonly occurring subject areas, citation data, and a link to the journal’s website, where an author could find information about how to submit a paper. Next, we repeated the procedure for MSC 65 [Numerical analysis], which led us to look more closely at *Journal of Computational and Applied Mathematics* and *SIAM Journal on Scientific Computing*.

My earlier blog post, Journal Profile Pages, has more details about using the MR Journal Profile Pages. Take a look!

]]>**ICIAM Collatz Prize:**

Siddhartha Mishra

(ETH Zürich, Switzerland)

**ICIAM Lagrange Prize**

George Papanicolaou

(Stanford University, USA)

**ICIAM Maxwell Prize**

Claude Bardos

(Université Paris Denis Diderot, France)

**ICIAM Pioneer Prize **

Yvon Maday

(Sorbonne University, Paris, France)

**ICIAM Su Buchin Prize **

Giulia Di Nunno

(University of Oslo, Norway)

Mathematical Reviews will be at the ICIAM in Valencia next week, July 15-19. Naturally, we will be at the AMS booth, which is at locations 7, 8, and 9 in the Exhibit Area.

The AMS is sponsoring some extras at the coffee break on Tuesday, July 16, from 16:30 to 17:00 in the Exhibit Area. Stop by then – or any other time.

As with other important meetings, we have arranged for free access to MathSciNet at the conference sites and most of the affiliated hotels. You won’t need to do anything special, other than to connect using the internet or wifi service of the conference sites or hotels.

We hope to see you in Valencia next week!

]]>In what follows, I will highlight some of the new features. However, I also invite you to do some exploring of your own, since there are some new features that I will gloss over and seeing some of them on your own will be a lot more fun!

The main bibliographic information about the journal is in the first card, labeled **Journal Details**. Here is an example, using the *Proceedings of the London Mathematical Society*. I have chosen this journal because it has just enough features to make the example interesting without having so many features to make it complicated.

The title of the journal is at the top of the page. Some publisher information is at the top of the card, along with one or more links to the publisher’s website for the journal. The number after **Publications Listed** tells how many items (articles) from the journal have been included in the Mathematical Reviews Database and uploaded to MathSciNet. The number is actually a link that will take you to a listing of all 6,254 items, displayed as in the results of a MathSciNet publication search. The number for **Publications Cited** tells how many of the articles have been cited from the reference lists in MathSciNet. In this case, that number is 3,890, which is 62.2% of all the 6,254 articles we have for the journal in the database. The total number of citations in MathSciNet to *Proceedings of the London Mathematical Society* is 52,865. Apparently some of the items are receiving a lot of multiple citations! These 52,865 citations come from 42,917 items in the database. Since this number is smaller than 52,865, the pigeonhole principle tells us that some of these items are citing more than one article in *Proceedings of the London Mathematical Society**.*

The **Journal Details** card includes information about the **Latest Issue** we have, including a link to the contents of that issue in MathSciNet. It also includes information about the **Earliest Issue** that we have in the database, including a link to the contents of that issue in MathSciNet.

The astute observer will have noted that the first issue listed is not from the Third Series of the *Proceedings of the London Mathematical Society**. *Indeed, we have collected all the incarnations of the journal into a journal group. If you want to see the main constituents of the group, they are in the **Concise History** card. If a journal changes publishers or changes its title (from Second Series to Third Series, for instance), we catalogue that information. More details about the constituents, including notes on the changes from one to another, can be found by clicking the **Journal Title History** button in the **Concise History** card. See the note at the end of this blog post for a little more information about what is in the **Journal Title History**.

The **Mathematical Citation Quotient (MCQ)** for a given year is defined as the number of times the items published in the journal in the previous five years were cited by items in reference list journals published in the given year, divided by the number of articles the journal published in that same five-year period. The **All MCQ** is the MCQ computed as if every item (every journal article, proceeding article, book, and thesis) had been published in one big journal. It provides a benchmark by which to understand the MCQ of a journal. In the new journal profile pages, the MCQ for each year can be seen graphically all at once, along with the **All MCQ** for each year. (Note that we don’t compute MCQs before the year 2000.) Here is the graphical representation of the MCQ data for the *Proceedings of the London Mathematical Society**.*

If you prefer numbers, clicking on the table tab provides the yearly MCQ data for the journal: the MCQ, the number of relevant citations, and the number of relevant publications in the journal. You can also see the table for the MCQ via its tab.

**Pro Tip**: In the bar graphs, clicking the small camera icon in the upper right-hand corner will download a PNG image of the graph to your desktop. For the tables, you can highlight the table to copy and paste to a text document or a spreadsheet.

This card presents data on the citations to the journal. The first tab shows a bar graph of the citations, broken down by the publication year of the cited papers. You can choose whether the citing articles have been published in a particular year or look at the data over all time. The default is to use all time. Change it by selecting a year from the **Citation Year** box and clicking **Update**. The horizontal axis shows the publication years for the particular journal. The default range of years is from 1990 to the present. However, if the journal started publishing after 1990, the graph is adjusted accordingly. There is a slider below the graph that allows you to adjust the left and right endpoints of the year range being displayed. Here is the bar graph for citations from articles published in 2017 to articles in the *Proceedings of the London Mathematical Society** *in the year range 1900 to 2018:

If you want to see the data in tabular form, just click **Table**. Here are the top few rows of the tabular form of the data in the above bar graph:

Year |
Citations by Year |
Self Citations |
Total Publications |
% of Pub. Cited |
---|---|---|---|---|

2017 | 30 | 0 | 67 | 77.6% |

2016 | 53 | 0 | 59 | 83.1% |

2015 | 141 | 3 | 89 | 93.3% |

2014 | 112 | 2 | 95 | 94.7% |

2013 | 116 | 2 | 86 | 96.5% |

2012 | 109 | 2 | 78 | 96.2% |

2011 | 89 | 1 | 64 | 98.4% |

2010 | 66 | 0 | 56 | 92.9% |

The last column tells you what percentage of the articles in that year of the journal have been cited in MathSciNet. You can have a year like 2010 where there are more citations than papers published in the journal, but still have 7.1% of the papers uncited. Clearly this means that some articles have multiple citations. The entries in the second and fourth columns are links. The links in the second column takes you to the citing papers as listed in MathSciNet. You may encounter a list of fewer items than the number of citations in the table, since it is possible for one paper to cite more than one article in the journal. In the results list, those papers are noted with the phrase “Multiple citations from this item to Proc. Lond. Math. Soc.” (using the abbreviated name of the journal in question). When you click through to the *Proceedings of the London Mathematical Society **Proc. *are boxed. Note: The link specifies a particular publication year for the journal – citations to a different year of the journal will not be boxed.

You can also see the journals that are citing this journal most frequently, either in a bar graph or in a table. Clicking on the name of the journal takes you to its Journal Profile Page. Here is the table for the *Proceedings of the London Mathematical Society* for the citing year 2017:

Publication |
Citations |
---|---|

J. Algebra | 102 |

Adv. Math. | 100 |

J. Math. Anal. Appl. | 59 |

Trans. Amer. Math. Soc. | 53 |

Proc. Amer. Math. Soc. | 49 |

Comm. Algebra | 48 |

Int. Math. Res. Not. IMRN | 41 |

Math. Z. | 38 |

J. Number Theory | 37 |

J. Funct. Anal. | 31 |

Classifications

It is possible to see the distribution of MSCs (Mathematics Subject Classifications) in the journal, using either a three-year window or over all time. By default, just the top ten classes are displayed, but clicking **Show All** displays all of them. Here are the top ten for the *Proceedings of the London Mathematical Society* over all time:

Notice that **Other** is the most common class that occurs. The Mathematics Subject Classification has not existed for as long as the *Proceedings of the London Mathematical Society* has been publishing. Indeed, the journal began publishing papers in 1865, before there was an American Mathematical Society and long before there was a Mathematical Reviews! We received bibliographic data from the journal that allowed us to back fill those years. The items that come from the contributed data do not have classes attached to them and are marked as **Other**. Clicking on the two-digit MSC brings you to a full description of the subject class. Clicking on the number in the **Count** column takes you to a listing of all the items in the journal with that class as primary class from the time period selected.

The last feature I want to point out is the table of the authors most frequently appearing in the journal. As with the MSCs, you can look either at the most recent three years or over all time. Here is the table for the *Proceedings of the London Mathematical Society** *over all time.

Clicking on the author’s name takes you to his or her Author Profile Page in MathSciNet. Clicking on the number in the **Papers** column takes you to the MathSciNet listing of the author’s papers in the journal.

I have skipped over a few things, but I hope they will be sufficiently self-evident. Besides, I don’t want to steal your enjoyment of discovering what’s new with the MathSciNet Journal Profile Pages!

For cataloging and bibliographic reasons, different incarnations of the same journal are distinguished in the database. For instance, a journal may change publishers, as the *Proceedings of the London Mathematical Society* has done a few times, starting off with Francis Hodgson, moving to Cambridge University Press, and now being published by Wiley. It is also possible that a journal changes its title, say from *Bulletin of the American Mathematical Society* to *American Mathematical Society. Bulletin. New Series*. Or, a journal may split, as *Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences. Séries A et B* did, creating three journals, including *Comptes Rendus des Séances de l’Académie des Sciences. Série I. Mathématique.* * *The new journal profile pages provide information on these transitions in the **Journal Title History** area. For foreign-language journals, the profile page has information about translations that are also covered by Mathematical Reviews.

The new Journal Profile Pages were created by **Travis Smith** and **Erol Ozil**. They were guided by the tremendous staff of Mathematical Reviews, who provided great suggestions and feedback. They had particular help from **Kathy Wolcott** and **Norman Richert**.

We had a prototype of the Journal Profile Pages at the Joint Mathematics Meetings in Baltimore (January 2019). We received valuable suggestions from quite a few mathematicians, librarians, and publishers at the JMM. Some of the librarians took extraordinarily close looks at the prototype and provided insightful comments. We are especially grateful for the time they devoted to helping improve the pages.

]]>To find the list, login as a reviewer via the MR Reviewer page: https://mathscinet.ams.org/mresubs/. After logging in, you are at your reviewer home page. Here is a screenshot from mine:

The arrow points to the new link. Let’s follow it! We come to my reviewer history page. Here is a screenshot of the top of that page:

Now I can see a list of all my reviews. Moreover, we have set it up so that you can copy and paste the list into a text file. **Note**: if you copy into a WORD file, you may want to use the “Copy and Match Formatting” option in WORD.

The list in your Review History is **not** set up for easy pasting into a BiBTeX file. However, getting such a BibTeX list has been possible for a while now, but *requires that you do it while connected via a subscription to MathSciNet*. Here is how.

On your reviewer home page is a link to your Author Profile Page on MathSciNet, indicated by the red arrow in the screenshot below:

Clicking the link takes you to your Author Profile Page on MathSciNet. Below the basic information on the Author Profile Page, is a box with some links, including a link to a list of all of your reviews in MathSciNet:

The results list in MathSciNet looks like this:

To continue, click the link **Mark All**. The default format for batch downloads is the full review in HTML. However, we want to list all the items in BibTeX format. The arrow points to the button that allows you to change the format for batch downloading the results. Choose **Citations (BibTeX)**. Then click **Retrieve Marked**. The result is a list of all your reviews (well, in this case, all *my* reviews) in standard BibTeX format. Here is the top of that list:

@article {MR3185209, AUTHOR = {Olshanski, Grigori}, TITLE = {Projections of orbital measures, {G}elfand-{T}setlin polytopes, and splines}, JOURNAL = {J. Lie Theory}, FJOURNAL = {Journal of Lie Theory}, VOLUME = {23}, YEAR = {2013}, NUMBER = {4}, PAGES = {1011--1022}, ISSN = {0949-5932}, MRCLASS = {22E30 (41A15)}, MRNUMBER = {3185209}, MRREVIEWER = {Edward G. Dunne}, }

@article {MR3072155, AUTHOR = {Boyer, Adrien}, TITLE = {Semisimple {L}ie groups satisfy property {RD}, a short proof}, JOURNAL = {C. R. Math. Acad. Sci. Paris}, FJOURNAL = {Comptes Rendus Math\'{e}matique. Acad\'{e}mie des Sciences. Paris}, VOLUME = {351}, YEAR = {2013}, NUMBER = {9-10}, PAGES = {335--338}, ISSN = {1631-073X}, MRCLASS = {22E45}, MRNUMBER = {3072155}, MRREVIEWER = {Edward G. Dunne}, DOI = {10.1016/j.crma.2013.05.007}, URL = {https://doi.org/10.1016/j.crma.2013.05.007}, } etc.

You can now copy them and paste them into a BibTeX file!

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by Kevin Knudson and Evelyn Lamb features Ursula Whitcher,

who is an Associate Editor at Mathematical Reviews.

Listen to the interview here.

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For an insightful description of some of Uhlenbeck’s work, the recent article by Simon Donaldson in the *Notices of the AMS* is tremendous. Below are the texts of some reviews of her work in MathSciNet.

**MR0264714**

Uhlenbeck, K.

Harmonic maps; a direct method in the calculus of variations.

*Bull. Amer. Math. Soc.* **76 **1970 1082–1087.

It was shown by J. H. Sampson and the reviewer [Amer. J. Math. 86 (1964), 109–160; MR0164306] that in every homotopy class of maps of one compact Riemannian manifold M into another N of negative curvature, there is a harmonic map. Furthermore, S. I. Alʹber [Dokl. Akad. Nauk SSSR 178 (1968), 13–16; MR0230254] and P. Hartman [Canad. J. Math. 19 (1967), 673–687;MR0214004] have established certain uniqueness results if N has strictly negative curvature. Our technique was to follow gradient lines (of the tension field of the energy function E) in a suitable space of maps, since limit points are harmonic maps; that involved a rather delicate and explicit study of the appropriate elliptic and parabolic systems. We went on [the reviewer and Sampson, Proc. U.S.-Japan Sem. Differential Geometry (Kyoto, 1965), pp. 22–33, Nippon Hyoronsha, Tokyo, 1966; MR0216519] to show that without any curvature restrictions, there exists in every homotopy class a polyharmonic map, where the degree of polyharmonicity depends on the dimension of M. The proof in that case was based on Morse theory on Hilbertian manifolds of maps, and in particular on verification of the Palais-Smale condition (C) for the poly-energy function.

In the paper under review (which is an announcement, with sketch of proofs. Let us hope that a full account will appear in the near future, as these ideas deserve) a new method of proof is described for the existence of harmonic maps. It involves verification of (C) for perturbations (e.g., adding pth powers of the differential, or of the Laplacian) of the energy E, in the context of Morse theory on Finsler manifolds of maps. (A special case of these perturbations was discovered, for the same purpose, by H. I. Eliasson [Global analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), pp. 67–89, Amer. Math. Soc., Providence, R.I., 1970;MR0267605].) The method displays how the curvature restriction on N forces convergence of critical points of the perturbed energies to critical points of the energy itself. In addition to establishing existence, the method provides unique results (in the spirit of the proof of Alʹber). It would be interesting to see whether these methods permit extension to the Plateau problem (where M has a boundary and harmonic maps have prescribed image on the boundary).

Reviewed by J. Eells

**MR0464332** ** **

Uhlenbeck, K.

Generic properties of eigenfunctions.

*Amer. J. Math.* **98 **(1976), no. 4, 1059–1078.

Let $M^n$ be a compact $n$-manifold and let $L_b$ be a family of self-adjoint elliptic operators on $M^n$ with the parameter $b\in U$ an open subset of a Banach space $B$. The author shows that under reasonable hypotheses the following properties are generic with respect to $B$, i.e., for almost all $b\in U$, (a) $L_b$ has one-dimensional eigenspaces; (b) zero is not a critical value of the eigenfunctions, restricted to the interior of the domain of the operator; (c) the eigenfunctions are Morse functions on the interior of $M$; (d) if $\partial M\neq\varnothing$ and Dirichlet boundary conditions have been imposed, then the normal derivative of the eigenfunctions has zero as a regular value. The author gives several applications. For example, let $\Delta_g$ be the Laplace operator for a metric $g\in\scr M_k=${$C^k$-metrics on $M^n$} for $k>n+3$. Then {$g\in\scr M_k\colon\Delta_g$satisfies (a), (b), (c) and (d) on nonconstant eigenfunctions} is residual in $\scr M_k$.

Reviewed by A. J. Tromba

**MR0604040** ** **

Sacks, J.; Uhlenbeck, K.

The existence of minimal immersions of $2$-spheres.

*Ann. of Math. (2)* **113 **(1981), no. 1, 1–24.

Let $M_p$ be a closed Riemann surface of genus $p\geq 0$ and $N$ a compact Riemannian manifold. In this interesting paper, the authors establish the existence of harmonic maps in three cases: (i) If $\pi_2(N)=0$, every homotopy class of maps $M_p\rightarrow N$ contains a harmonic map of minimum energy—this was established by L. Lemaire [J. Differential Geom. 13(1978), no. 1, 51–78; MR0520601]; see also the article by R. Schoen and S. T. Yau [Ann. of Math. (2) 110 (1979), no. 1, 127–142; MR0541332]. (ii) If $\pi_2(N)\neq 0$, then a generating set for $\pi_2(N)$ modulo the action of $\pi_1(N)$ can be represented by harmonic maps $M_0\rightarrow N$ of minimum energy—such a map is automatically a conformal branched immersion of minimum area. (iii) If $N$ has noncontractible universal covering space, there exists a nonconstant conformal minimal branched immersion $M_0\rightarrow N$ (which may be a saddle point for the energy). The method is to perturb the energy functional $E$ to give a functional $E_\alpha$ ($\alpha\geq 1,E_1=E+$ constant) which satisfies the condition (C) of Palais and Smale and then to study the complicated convergence of critical maps of $E_\alpha$ as $\alpha\rightarrow 1$. Note that a nonexistence theorem contrasting with (ii) was given by A. Futaki [Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 6, 291–293;MR0581474].

Reviewed by John C. Wood

**MR0664498** ** **

Schoen, Richard; Uhlenbeck, Karen

A regularity theory for harmonic maps.

*J. Differential Geom.* **17 **(1982), no. 2, 307–335.

Let $M^n$ and $N^k$ be Riemannian manifolds, with $M$ compact. Let $u\colon M\rightarrow N$ be an $L_1^2$-map minimizing the energy functional $E(u)=\int\langle du(x),du(x)\rangle dV$, possibly modified to include lower order terms. Theorem: If $u(M)$is in a compact subset of $N$ a.e., then $u$ is smooth on $M-\scr S_u$ for a suitable closed set $\scr S_u$ of Hausdorff dimension $\leq n-3$. If $n=3$, then $\scr S_u$ is discrete. In fact, $\scr S_u=\{a\in M\colon\liminf_{l\rightarrow 0}E_{D_l(a)}(n)/l^{n-2}>0\}$. (An important special case of that result was obtained simultaneously by M. Giaquinta and E. Giusti [MR0648066 above; Analysis, to appear]; there $u(M)$ is required to lie in a coordinate chart.) The case $n=2$ is due to C. B. Morrey; for arbitrary $n$a key idea from potential theory is Morrey’s Dirichlet growth lemma [C. B. Morrey, Multiple integrals in the calculus of variations, Springer, New York, 1966; MR0202511]. Another basic idea (used also by Giaquinta-Giusti), this time from geometric measure theory, is H. Federer’s reduction theorem [Bull. Amer. Math. Soc. 76 (1970), 767–771; MR0260981]. A major technical difficulty overcome in this paper is to find comparison maps (as used by Morrey) which have images in $N$. By refining their arguments the authors obtain the theorem: In addition to the hypotheses of the preceding theorem, suppose that every harmonic map $\theta\colon S^j\rightarrow N$ is constant for $2\leq j\leq n-1$, where $S^j$ is the Euclidean $j$-sphere; (more generally and importantly; if $p\colon{\bf R}^{j+1}-\{0\}\rightarrow S^j$ denotes radial projection and $\theta\colon S^j\rightarrow N$ is a harmonic map such that $\theta\circ p$ minimizes energy on compact subsets of ${\bf R}^{j+1}$, then $\theta\circ p$ is constant $(2\leq j\leq n-1)$). Then $\scr S_u=\varnothing$; i.e., $u$ is smooth on $M$. This is a very important contribution to the theory of harmonic maps. For instance, these hypotheses are satisfied if the universal cover of $N$ supports a strictly convex smooth function. That case provides a new proof and generalization of the existence theorem of the reviewer and J. H. Sampson [Amer. J. Math. 86 (1964), 109–160; MR0164306] —and much else, as well.

{In the correction, it is noted that “harmonic” should be deleted from the statement of Lemma 2.5. The other results are not affected by the change.}

Reviewed by J. Eells

**MR0710054** ** **

Schoen, Richard; Uhlenbeck, Karen

Boundary regularity and the Dirichlet problem for harmonic maps.

*J. Differential Geom.* 18 (1983), no. 2, 253–268.

This paper follows an earlier one by the authors [same journal 17 (1982), no. 2, 307–335; MR0664498]. Both papers are based on the facts that a harmonic map $u\;(u\in L^2_{1\,{\rm loc}}({\bf R}^n,N))$ which is constant along the rays from 0 a.e. defines a new map $w:S^{n-1} \rightarrow N$ such that $u(x)=w(x/\vert x\vert )$ which is also harmonic and conversely; furthermore $u$ has a singularity at 0 if and only if $w$ is not a constant map (Theorems III and IV). This result supplies an estimate of the Hausdorff dimension of the set of singularities. The first paper dealt with inner regularity for harmonic maps, this one deals with boundary regularity for harmonic maps satisfying a Dirichlet problem. The boundary regularity is actually stronger. This is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres $S^{n-j}_+$ which map the boundary $S^{n-j-1}$to a point $(1\leq j\leq n-2)$. In fact, the Euler-Lagrange equation is deduced by minimizing a functional $\tilde E$ slightly more general than the energy $E,\tilde E(u)=E(u) +V(u)$, where $V(u)$ is the integral over $M$ of $\Sigma_i\Sigma_\alpha\,\gamma ^\alpha_i(x,u(x))\partial u^i/\partial x^\alpha+\Gamma(x,u(x))$. The main regularity theorem is the following: Let $M$ be a compact manifold with $C^{2,\alpha}$ boundary. Suppose $u\in L^2_1(M,N)$ is $E$-minimizing and satisfies $u(x)\in N_0$ a.e. for a compact subset $N_0\subset N$. Suppose $v\in C^{2,\alpha}(\partial M,N_0)$ and $u=v$ on $\partial M$. Then the singular set $S$ of $u$ is a compact subset of the interior of $M$; in particular, $u$ is $C^{2,\alpha}$ in a full neighborhood of $\partial M$.

An application is an amusing proof of a theorem of Sacks and Uhlenbeck on the existence of minimal 2-spheres representing the second homotopy group of a manifold: “If $N$ is compact with convex or empty boundary, any smooth map $v:S^2\rightarrow N$which does not extend continuously to $B^3$ is homotopic to a sum of smooth harmonic (hence minimal) maps$u_j:S^2\rightarrow N$, $j=1,2,\cdots,p$”. The last section deals with approximation of $L^2_1$ maps by smooth maps. The authors give a simple example of a map $u\in L^2_1(B^3_1,S^2)$ such that $u(x)=x/\vert x\vert $ cannot be an $L^2_1$ limit of continuous maps; this result is still true for $L^2_1(M,N)$ with $\dim M\geq 3$. On the contrary if $\dim M=2$, an $L^2_1$ map is a limit of smooth maps.

Reviewed by Liane Valere Bouche

**MR0861491** ** **

Uhlenbeck, K.(1-CHI); Yau, S.-T.

On the existence of Hermitian-Yang-Mills connections in stable vector bundles.

Frontiers of the mathematical sciences: 1985 (New York, 1985).

*Comm. Pure Appl. Math.* 39 (1986), no. S, suppl., S257–S293.

The Yang-Mills equations, which arise in particle physics, have been applied with great success to study the differential topology of $4$-manifolds. Of particular interest are complex surfaces, in which case the Yang-Mills equations have a holomorphic interpretation. Let $X$ be a compact Kähler manifold (of any dimension) and $E$ a holomorphic bundle over $X$. A Hermitian metric on $E$ determines a canonical unitary connection whose curvature is a $(1,1)$-form $F$ which is a skew-Hermitian transformation of $E$. The inner product of $F$ with the Kähler form is then an endomorphism of $E$, and the connection satisfies the Yang-Mills equations if and only if this endomorphism is a multiple of the identity. Metrics which give rise to such connections are called Hermitian-Einstein metrics, and the resulting connection is termed Hermitian-Yang-Mills. It is natural to ask which bundles admit such metrics. Recall first a definition from algebraic geometry. The slope of a bundle is the ratio of its degree to its rank, and a bundle $E$ is said to be stable if the slope of any coherent subsheaf of lower rank is strictly less than the slope of $E$. The main theorem of the present paper asserts that a stable holomorphic bundle over a compact Kähler manifold admits a unique Hermitian-Einstein metric. For complex curves this is an old result of Narasimhan and Seshadri. S. K. Donaldson [Proc. London Math. Soc. (3) 50 (1985), no. 1, 1–26; MR0765366] gave a proof for projective algebraic surfaces, and (subsequent to the work of the authors) extended his work to cover projective complex manifolds of any dimension [Duke Math. J. 54 (1987), no. 1, 231–247;MR0885784].

The authors make a direct study of the partial differential equation arising from the Hermitian-Yang-Mills condition. The continuity method is used to demonstrate the existence of solutions to a perturbed equation. Of course, this involves a priori estimates for the solutions. Let $\varepsilon$ be the perturbation parameter, and $h_\varepsilon$ the solution to the perturbed equation. Then as $\varepsilon\rightarrow 0$ either the $h_\varepsilon$ converge to a solution, or (after appropriate renormalization) the limit represents an $L_1^2$ holomorphic projection onto a subbundle. A major step in the proof consists in showing that this projection is smooth outside a subvariety of codimension at least $2$ and that the image is a coherent subsheaf of $E$. Then the previous estimates are used, via the Chern-Weil theory, to show that this subsheaf is destabilizing for $E$.

The regularity theorem for the projection is stated in general terms. Let $M$ be an algebraic manifold, which we assume is embedded in some projective space. A map $F$ from the unit ball in $\mathbf{C}$ to $M$ is said to be weakly holomorphic if it is in $L_1^2$ and if its differential maps the holomorphic tangent space of the ball into the holomorphic tangent space of $M$ almost everywhere. For balls of higher dimension a map is weakly holomorphic if for every linear coordinate system $\{z_1,\cdots,z_n\}$ and for almost every value of $z_2,\cdots,z_n$ it is weakly holomorphic as a function of $z_1$. Then the authors prove that any weakly holomorphic map into an algebraic manifold is meromorphic.

Both the result and the techniques of this paper are important. They have already found use in the theses of K. Corlette [“Flat $G$-bundles with canonical metrics”, J. Differential Geom., to appear] and C. Simpson [“Systems of Hodge bundles and uniformization”, Ph.D. Thesis, Harvard Univ., Cambridge, Mass., 1987; per revr.].

Reviewed by Daniel S. Freed

]]>There are two small new changes to MathSciNet^{®}: Search by DOI and Sort by Number of Authors.

The first tweak is the ability to search by DOI in a publications search.

In a Publication search, you have the following fourteen options for fielded search:

Author, Author Related, Title, Review Text, Journal, Institution, Series,

MSC Primary/Secondary, MSC Primary, MR Number, DOI, Reviewer,

Anywhere, References.

The newest of these is DOI. You can now search MathSciNet by entering the DOI of a paper. For example, if you have the DOI 10.1090/S0273-0979-1992-00266-5, entering it in the field

Make sure to put in the full DOI. A complete DOI contains a **prefix** and a **suffix**, separated by a “**/**“. Since people will often be copying and pasting DOIs into the field, we made is so that you can even put in the URL of the DOI resolver:

https://doi.org/10.1090/S0273-0979-1992-00266-5 or https://dx.doi.org/10.1090/S0273-0979-1992-00266-5

and the search will work. (It also works with **http** instead of **https** in the URL.)

**Note**: Sometimes publishers include a “**/**” in the suffix, as in https://doi.org/10.1093/imrn/rns214.

The result of the search in the example is the paper by Crandall, Ishii, and Lions on viscosity solutions:

**MR1118699**

Crandall, Michael G.(1-UCSB); Ishii, Hitoshi(J-CHUO); Lions, Pierre-Louis(F-PARIS9-A)

User’s guide to viscosity solutions of second order partial differential equations.

*Bull. Amer. Math. Soc. (N.S.)* 27 (1992), no. 1, 1–67.

35J60 (35B05 35D05 35G20)

A DOI search is a specialized search, but in certain circumstances, it is very useful.

The second tweak has to do with sorting. In a list of publications resulting from a search, you can sort the results in a variety of ways: Newest, Oldest, Citations, and now also by the number of authors on each item. For instance, a search for items authored byPaul Erdős results in 1445 matches, which, by default, are sorted by newest first. Clicking on the “Sort by” button near the top of the left-hand column allows you to resort the results by *oldest first*, by *decreasing number of citations*, or by *increasing number of authors* – labeled as “**#Authors**“. In the list of Erdős publications, clicking on “Show first 100 results” allows us to display 100 matches per page. If we resort by number of authors, we see 100 papers written by Erdős alone. Going to the next page, we see another 100 solo papers by Erdős. It is not until the fifth page that we see co-authored papers, starting with one written by Erdős and Diaconis (MR2126886).

We have had requests for this feature, and now it is available!

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