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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Here are a couple of photos from the meetings.

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**Complimentary MathSciNet at the meeting**. By special arrangement, free MathSciNet will be available when using the wifi at the conference site. This year, access to MathSciNet should be available again in the hotels! All you have to do is point your browser to https://mathscinet.ams.org/mathscinet and we will take care of the rest! And you can pair your mobile device (laptop, tablet, phone) with the JMM MathSciNet to have 30 days of free MathSciNet after the Meetings are over. If you don’t know how to pair a device with a MathSciNet description, either look up this old post or stop by the AMS booth for an explanation.

**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! If you can’t come to one of the scheduled demos, stop by the booth any time and we will be happy to give you an impromptu demo.

**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 the jobs. My earlier post explains the open positions, or you can look them up on MathJobs.org. 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 the reviewers, who play a key role in creating the Mathematical Reviews database. Refreshments will be served. The location is the Promenade Room, 1st Floor, Marriott Inner Harbor.

AMS members can sign up for **email notifications of newly added items by subject area**. You can select up to three 2-digit Mathematics Subject Classifications. Then, about once a month, we will send you a list of all the items that have been added to the Mathematical Reviews database in those areas in the last month.

This service has existed for years, but many people don’t seem to know about it. It is the electronic version of *Current Mathematical Publications (CMP)*, hence is known as e-CMP. The *CMP* was a printed publication that was meant to give a quick listing of the newest material that had been received at Mathematical Reviews. When the *CMP* was started (in 1965), *Mathematical Reviews* existed only in print form. Items were only added to the printed *Mathematical Reviews *volumes once the work was done on them, including the review. There could be some delay. So, we started offering a publication of quicker notifications of published items. As email caught on, we started offering electronic notifications to AMS members. When *Mathematical Reviews* went online, it became possible to load bibliographic information about the items without waiting for a review. Nevertheless, the same email notification service has continued as a member-only benefit. And it is quite handy.

**How to activate / modify e-CMP notifications**. Go to the e-CMP information page. Click on the “Go to e-CMP” link, which will take you to a sign-in page. Login with the credentials for your AMS membership, and you will come to a screen that looks like this:

Pick some subject classes. Make the changes. And check your Inbox the first week of every month.

I hope you find this as useful as I do.

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For each of the positions, the successful applicant will have mathematical breadth with an interest in current developments, and will be keen to learn new topics in pure and applied mathematics. Candidates are expected to have a doctorate, plus several years of relevant experience post-Ph.D.

- Job #13067: [To start as soon as possible in 2019.] The new editor should have expertise in
**combinatorics**and related areas. - Job #13068: [To start in late spring or summer 2019.] The new editor should have expertise in
**mathematical physics and areas of analysis**. - Job #13069:[To start in late spring or summer 2019.] The new editor should have expertise in
**number theory**and related areas of algebra, analysis, and geometry.

*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.

]]>The ceremony is a big production, with celebrities from television, film, and music present, as well as the scientists. The prizes come with significant cash awards, currently $3,000,000 (US). The first five winners of the Breakthrough Prizes in Mathematics started a tradition of using a part of their awards to fund the IMU Breakout Graduate Fellowships. And all the subsequent prize winners have followed this great tradition of helping mathematics graduate students from and in the developing world.

The trophy, designed by Olafur Eliasson, is a wireframe toroidal shape, which simultaneously hints at mathematics, life sciences, and fundamental physics, the three areas where prizes are awarded. There is a nice photograph of it here.

The citations for the winners are given below, along with links to their author profiles on MathSciNet. Congratulations Breakthrough Laureates!

**Vincent Lafforgue** – *CNRS (National Center for Scientific Research, France)* and *Institut Fourier, Université Grenoble Alpes.*

**Citation:** For ground-breaking contributions to several areas of mathematics, in particular to the Langlands program in the function field case.

**Chenyang Xu**–*Massachusetts Institute of Technology*and*Beijing International Center for Mathematical Research*

**Citation:**For major advances in the minimal model program and applications to the moduli of algebraic varieties.**Karim Adiprasito**and**June Huh**–*Hebrew University of Jerusalem*and*Institute for Advanced Study*, respectively

**Citation:**For the development, with Eric Katz, of combinatorial Hodge theory leading to the resolution of the log-concavity conjecture of Rota.**Kaisa Matomäki**and**Maksym Radziwill**–*University of Turku*and*California Institute of Technology*, respectively

**Citation:**For fundamental breakthroughs in the understanding of local correlations of values of multiplicative functions.

Don’t forget the Open House at Mathematical Reviews Saturday, October 20^{th}!

Thank you to Don McClure for some help on the details of the Breakout Graduate Fellowships.

]]>Mathematical Reviews is hosting an **Open** **House** as part of the AMS Fall Central Sectional Meeting at the University of Michigan in Ann Arbor. The open house will take place Saturday, October 20 from 12:30 to 2:00pm at the Mathematical Reviews building, 416 S. Fourth Street, Ann Arbor, MI. Come see where the magic happens!

Here is a map with directions for how to walk from the site of the sectional meeting to Mathematical Reviews. The walk should take 15 to 20 minutes.

It’s lunch time, so we will have sandwiches. We work in a former brewery, so we will have samples of some local beers. [See this earlier post for more about the history of our building.]

All attendees of the Sectional Meeting and their guests are welcome to attend. We hope to see you on Saturday!

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