Suppose you have just written your first paper in an area and now you want to figure out where to send it. Maybe it’s your first paper ever. (If so, congratulations!) Maybe you normally work in one area, but had a good idea in a neighboring subject that turned into a paper. How can you determine some suitable journals for your paper? A good idea is to ask colleagues. Here is something you can do with MathSciNet, as a supplement to colleagues’ advice or in case no one has any good suggestions.

I’m going to discuss this by using the example of representations of $p$-adic Lie groups, a subject that is on the boundary of what I know.

**Step 1. **Figure out the MSC class that represents the subject area. If you managed to get some results and write a paper on the subject, you probably know a few papers in that area, or maybe some authors who work in the area. For $p$-adic Lie groups, I know that Paul Sally worked extensively in the area. Here are three papers by Sally that have “$p$-adic” in the title.

**MR2798422**

Adler, Jeffrey D. (1-AMER-MS); DeBacker, Stephen (1-MI); Sally, Paul J., Jr. (1-CHI); Spice, Loren (1-TXC)

Supercuspidal characters of ${\rm SL}_2$ over a $p$-adic field . (English summary) *Harmonic analysis on reductive, p-adic groups, *19–69,

Contemp. Math., 543, *Amer. Math. Soc., Providence, RI,* 2011.

22E50

**MR1039842**

Sally, Paul J., Jr. (1-CHI)

Some remarks on discrete series characters for reductive $p$-adic groups. *Representations of Lie groups, Kyoto, Hiroshima, 1986, *337–348,

Adv. Stud. Pure Math., 14, *Academic Press, Boston, MA,* 1988.

22E50 (22E35)

**MR0744292**

Moy, Allen (1-YALE); Sally, Paul J., Jr. (1-CHI)

Supercuspidal representations of ${\rm SL}_n$ over a $p$-adic field: the tame case.

*Duke Math. J.* 51 (1984), no. 1, 149–161.

22E50

The class **22E50** is the primary classification for all three, with **22E35** as a secondary class on one of the papers. Looking up the descriptions of the classes, we find:

**22E35 **Analysis on $p$-adic Lie groups

**22E50 **Representations of Lie and linear algebraic groups over local fields [See also 20G05]

It helps to know a couple of things here. First, $p$-adic fields are local fields – indeed they are about the only examples of local fields I can think of without going to Wikipedia or MathWorld. Second, an important tool in studying representations of any sort of group is the analysis of the group’s characters, which are functions on the group related to the representations. So, I am confident that **22E50 **and

**Step 2.** Do a Publications Search for MSC **22E50**.

There are 1689 matches.

**Step 3**. In the search results, look in the sidebar under the heading “Journals”.

The journals are listed in decreasing order of the number of matches in the search results. Here are the journals that have at least 20 articles with **MSC 22E50** as the primary class.

Journal |
Count |

Pacific J. Math. | 78 |

Represent. Theory | 68 |

Canad. J. Math. | 55 |

Duke Math. J. | 52 |

J. Reine Angew. Math. | 48 |

Amer. J. Math. | 46 |

J. Algebra | 43 |

Compositio Math. | 37 |

J. Number Theory | 37 |

Trans. Amer. Math. Soc. | 37 |

Israel J. Math. | 36 |

Manuscripta Math. | 36 |

Invent. Math. | 33 |

Math. Ann. | 33 |

Ann. Sci. École Norm. Sup. (4) | 29 |

Compos. Math. | 29 |

Proc. Amer. Math. Soc. | 29 |

Astérisque | 27 |

Int. Math. Res. Not. IMRN | 27 |

J. Lie Theory | 24 |

J. Inst. Math. Jussieu | 23 |

Bull. Soc. Math. France | 20 |

C. R. Acad. Sci. Paris Sér. I Math. | 20 |

This gives you a list of journals to consider.

**Step 4.** These articles are all on representations of groups over local fields, which is going to be a fairly tight match due to the narrowness of the definition of the subject area. Some other classifications are broader, with more variations within the class. Consider, for instance, **94A08** *Image processing (compression, reconstruction, etc.) in information and communication theory*. This covers a lot of ground. Indeed, a search for **MSC Primary **=** 94A08** produces 11986 matches. It might be sufficient to find a journal that publishes papers in image processing, but how close to the specific topic of your paper are they? Close enough? To look at the papers on image processing showing up in a particular journal, click on it in the sidebar. The search results are now filtered to include only papers from that journal. You can now look to see if they are close to the topic of your paper. If so, click on the journal’s name or abbreviation in any of the matches, which brings you to the **MathSciNet** profile page for that journal. You can then see information about the most frequently occurring authors in the journal, the most frequent 2-digit subject areas, the journal’s Math Citation Quotient (our computation of the number of citations divided by number of papers), and so on. There will also be a link to the journal’s website, where you will find information about how to submit a paper to the journal.

This method isn’t perfect, and isn’t a substitute for consulting experts, but it does give you a good list that is based on what the journal has published.

**Some modifications.**

*Narrow the time-frame. *Some journals have been around a long time. For instance, *Pacific Journal of Mathematics* started in 1951. It has published a lot of papers on representations of $p$-adic Lie groups, but what has it done recently? In **Step 2**, you could look for papers after the year 2000.

This produces 1026 matches. Here are the journals with the most articles meeting the criteria:

Journal |
Count |

Represent. Theory | 63 |

Pacific J. Math. | 42 |

Canad. J. Math. | 40 |

J. Number Theory | 34 |

J. Algebra | 33 |

Compos. Math. | 29 |

Manuscripta Math. | 29 |

Int. Math. Res. Not. IMRN | 27 |

J. Reine Angew. Math. | 27 |

Amer. J. Math. | 26 |

Israel J. Math. | 23 |

J. Inst. Math. Jussieu | 23 |

Duke Math. J. | 22 |

J. Lie Theory | 22 |

Astérisque | 20 |

Invent. Math. | 18 |

Trans. Amer. Math. Soc. | 17 |

Math. Z. | 16 |

Proc. Amer. Math. Soc. | 16 |

Bull. Soc. Math. France | 13 |

*Search MSC Primary/Secondary*. Some subject areas have fewer papers than others. For instance, if I had searched for **MSC Primary = 22E35** and **Publication Year > 2000**, I would find only 132 matches. The journals with at least 3 papers matching the criteria are:

Journal |
Count |

Int. Math. Res. Not. IMRN | 6 |

Forum Math. | 5 |

Pacific J. Math. | 5 |

Trans. Amer. Math. Soc. | 5 |

Canad. J. Math. | 4 |

J. Funct. Anal. | 4 |

C. R. Math. Acad. Sci. Paris | 3 |

Canad. Math. Bull. | 3 |

Israel J. Math. | 3 |

J. Algebra | 3 |

J. Inst. Math. Jussieu | 3 |

Manuscripta Math. | 3 |

Proc. Amer. Math. Soc. | 3 |

Represent. Theory | 3 |

If I loosen the search to be **MSC Primary / Secondary = 22E35** and **Pub Year > 2000**, the number of matches jumps to 403.

*Combine classes*. For the image processing example, where there were too many matches, you might search for both **MSC Primary = 94A08** and something for an **MSC Primary/Secondary** class. For instance, wavelets are often used in image processing. You could add **MSC 42C40** = *Nontrigonometric harmonic analysis involving wavelets and other special systems* to the search.

This was effective in that it cut the number of results down to just 217. The journals with at least five articles matching the criteria are:

Journal |
Count |

IEEE Trans. Image Process. | 40 |

Int. J. Wavelets Multiresolut. Inf. Process. | 29 |

Appl. Comput. Harmon. Anal. | 16 |

SIAM J. Imaging Sci. | 8 |

J. Comput. Appl. Math. | 7 |

J. Math. Imaging Vision | 5 |

I hope these suggestions are helpful. There is a lot of information in the Math Reviews Database that powers MathSciNet. It can be used in many different ways to help mathematicians, not all of which are pre-programmed into MathSciNet.

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The successful applicant will have mathematical breadth with an interest in current developments, and will be willing to learn new topics in pure and applied mathematics. In particular, we are looking for an applicant with expertise in **probability** (MSC 60) and/or **statistics**.(MSC 62) The ability to write well in English is essential. The applicant should ideally have several years of relevant academic (or equivalent) experience beyond the Ph.D. Evidence of written scholarship in mathematics is expected. The twelve-month salary will be commensurate with the experience that the applicant brings to the position. Benefits include travel funds and the opportunity for study leave.

More information is available from the posting on MathJobs: https://www.mathjobs.org/jobs/list/18981

*Mathematical Reviews* is a great place to work. You get to do something important and useful. You would also be working with great people, including the other editors. A list of the current editors is here.

**Note**: several editors from Mathematical Reviews will be at the JMM in Seattle. Come by our section of the AMS booth to find out more.

Here is a picture of us eagerly watching for the arrival of your application:

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On Monday, November 29, 2021, 12:00 am- 11:59 pm Eastern Standard Time, we honor our AMS members via “AMS Day”, a day of specials on AMS publications, membership, and more. Our exciting limited-time offerings include:

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]]>At the end of the article is some information about how you can donate to a fund that supports the program, which I am happy to repeat here. There are three ways you can donate to the MDC Fund: on the AMS website here, by sending a check to the AMS Development Department (address below) and instructing the AMS to direct the gift to the MDC Fund, or by calling the Development Office at +1 (401) 455-4126.

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This was not a random question on Trefethen’s part, he had written a letter to the editor of SIAM News on the topic, which was published in the July/August 2015 issue. The letter was based on data collected by members of the Numerical Analysis Group at Oxford, of which Trefethen is the head. The average length of papers in the three SIAM journals *SIAM Journal on Applied Mathematics* (SIAP), *SIAM Journal on Numerical Analysis* (SINUM), and *SIAM Journal on Scientific Computing* (SISC) had doubled from 1975 to 2020.

There is more on this topic on Trefethen’s blog, such as here.

The data for the *Transactions of the AMS *produce a graph very similar to the graphs for the SIAM journals, at least after things stabilized in the 1960s.

The average number of pages in the years from 1965 to 1969 ranges from 13.5 to 16.8. In the most recent five-year period, the average number of pages ranges from 28.8 to 31.7.

We can compute the average number of pages per article by decade, as in the following table, which shows that the average length of an article in *Transactions* has indeed doubled since the 1960s.

Transactions of the AMS |
|||

Decade | #pgs | #articles | pgs/art |

1920 | 4004 | 229 | 17.5 |

1930 | 8891 | 459 | 19.4 |

1940 | 9785 | 416 | 23.5 |

1950 | 14741 | 724 | 20.4 |

1960 | 28556 | 1735 | 16.5 |

1970 | 49678 | 2852 | 17.4 |

1980 | 46821 | 2500 | 18.7 |

1990 | 52421 | 2398 | 21.9 |

2000 | 56825 | 2422 | 23.5 |

2010 | 87778 | 3034 | 28.9 |

2020 | 9007 | 280 | 32.2 |

It is tempting to consider the question for the *Proceedings of the AMS,* but it has a page limit. Assuming that the upper bound is mostly obeyed, it is unlikely that we will see the phenomenon in the *Proceedings.* I am not aware of any strict page limit for the AMS journal *Mathematics of Computation*, however. Let’s check it out. Here is the plot for *Math. of Comp.* from 1960 to 2020:

Visually, the growth is rather clear. As before, we can compute the average number of pages per article by decade, as in the following table:

Mathematics of Computation |
|||

Decade | #pgs | #articles | pgs/art |

1960 | 6600 | 683 | 9.7 |

1970 | 11165 | 1092 | 10.2 |

1980 | 14412 | 1062 | 13.6 |

1990 | 17626 | 1082 | 16.3 |

2000 | 20883 | 1107 | 18.9 |

2010 | 28274 | 1180 | 24.0 |

2020 | 3071 | 111 | 27.7 |

As with the SIAM journals, the *Transactions of the AMS *and *Mathematics of Computation* has also seen the average length of an article at least double since the 1960s.

What about *Annals of Mathematics? *Well, here is the plot of the data for *Annals*:

For the *Annals*, the size of the articles prior to 1965 did not vary as much as we saw for *Transactions*. Again, the growth is quite visible from the plot. The table of the data for the *Annals *by decade shows consistent growth in the length of articles:

Annals of Mathematics |
|||

Decade | #pgs | #articles | pgs/art |

1930 | 8924 | 600 | 14.9 |

1940 | 9223 | 555 | 16.6 |

1950 | 12517 | 717 | 17.5 |

1960 | 12032 | 528 | 22.8 |

1970 | 11976 | 452 | 26.5 |

1980 | 12761 | 417 | 30.6 |

1990 | 14857 | 398 | 37.3 |

2000 | 22208 | 537 | 41.4 |

2010 | 27473 | 565 | 48.6 |

2020 | 3666 | 62 | 59.1 |

Going by decades, the average length of an article has more than doubled from the 1960s to the present.

So what is going on? There were no definitive answers during the discussion at the Committee on Publications meeting. In his letter to *SIAM News*, Trefethen suggested that this is an indication of the professionalization of mathematics. That is to say, 45 years ago, it was OK to write a paper that just presented an idea. Now, an article represents a “piece of work”, explaining connections with other work. The good thing about a piece of work is that it is less important for you to be part of the in-crowd to understand the paper. The bad thing about a piece of work is that, in our time-crunched lives, we are more likely just to skim it, rather than to read it fully.

The phenomenon is curious, though, because it contradicts the supposed trend of “salami slicing” your work into several articles rather than one big article, aiming for the LPU (“Least Publishable Unit” or “publon”). On the other hand, these six journals are not necessarily typical.

I am grateful to Nick Trefethen for asking the question, then encouraging my exploration of a very partial answer to it. Thank you, also, to SIAM and *SIAM News *for so readily granting permission to reprint Trefethen’s original plot.

Those of us at the AMS know and appreciate Andrés’s many talents. He is a wonderful colleague. It is great to see him recognized. Moreover, because Andrés moved from a university faculty position to working as an editor at Mathematical Reviews, the profile puts a spotlight on the many different ways in which Latinx and Hispanic mathematicians have careers that advance research and support mathematics as a profession. Bravo Andrés! Bravo Lathisms!

The prizes and prize winners are listed below. The citations can be found on the Breakthrough Prize website. After the list, I have copied over some reviews of the work of the winners. Congratulations to the Breakthrough Prize Winners!

**Takuro Mochizuki**, Kyoto University

**Aaron Brown**, Northwestern University

**Sebastian Hurtado Salazar**, University of Chicago**Jack Thorne**, University of Cambridge**Jacob Tsimerman**, University of Toronto

**Sarah Peluse**, Institute for Advanced Study and Princeton University (PhD Stanford University 2019)**Hong Wang**, University of California, Los Angeles (PhD MIT 2019)**Yilin Wang**, MIT (PhD ETH Zürich 2019)

**MR2919903**

Mochizuki, Takuro (J-KYOT-R)

Wild harmonic bundles and wild pure twistor D-modules. (English, French summary)

Astérisque No. 340 (2011), x+607 pp. ISBN: 978-2-85629-332-4

14J60 (14F10 32L99)

This monograph provides a systematic analysis of the asymptotic behaviour of wild harmonic bundles on complex analytic manifolds. Important applications are then given to the study of the structure of (possibly irregular) flat meromorphic connections and to some open questions about $\scr{D}$-modules: among these problems is the complete proof of a stimulating conjecture of M. Kashiwara [in Topological field theory, primitive forms and related topics (Kyoto, 1996), 267–271, Progr. Math., 160, Birkhäuser Boston, Boston, MA, 1998; MR1653028] about an extension of the Hard Lefschetz Theorem and other nice properties from pure sheaves to semisimple $\scr{D}$-modules, which has drawn the attention also of notable researchers not necessarily working in the close vicinity of the subject [see, e.g., V. G. Drinfeld, Math. Res. Lett. 8 (2001), no. 5-6, 713–728; MR1879815]. Actually, Kashiwara’s conjecture appears even to be the original motivation under which the author started ten years ago a deep analysis of tame harmonic bundles, corresponding to (Fuchsian) regular differential equations or more generally to regular flat meromorphic connections “à la Deligne” [see T. Mochizuki, Geom. Topol. 13 (2009), no. 1, 359–455; MR2469521 and the references therein], and in this sense the present work is the ultimate step of the successful generalization of these results from the tame to the wild case.

As we said above, the consequences for the study of the structure of flat meromorphic connections (in particular the existence of resolutions of the bundle around the problematic “turning points”) are remarkable: for a more concrete and accessible presentation of these results we refer the interested reader also to the paper by the author in [J. Inst. Math. Jussieu 10 (2011), no. 3, 675–712; MR2806465]. It must be mentioned that similar conclusions were reached later by K. S. Kedlaya through different methods [see Duke Math. J. 154 (2010), no. 2, 343–418; MR2682186; J. Amer. Math. Soc. 24 (2011), no. 1, 183–229; MR2726603].

The importance of this extensive work is widely acknowledged in the current research literature, e.g. in the works concerning the recent substantial progresses in the irregular Riemann-Hilbert correspondence [see C. Sabbah, Introduction to Stokes structures, Lecture Notes in Math., 2060, Springer, Heidelberg, 2013; MR2978128; A. D’Agnolo and M. Kashiwara, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 10, 178–183; MR3004235; “Riemann-Hilbert correspondence for holonomic ${\scr D}$-modules”, preprint, arXiv:1311.2374].

Reviewed by Corrado Marastoni

**MR3702679**

Brown, Aaron (1-CHI-NDM); Rodriguez Hertz, Federico (1-PAS-NDM); Wang, Zhiren (1-PAS-NDM)

Global smooth and topological rigidity of hyperbolic lattice actions. (English summary)

Ann. of Math. (2) 186 (2017), no. 3, 913–972.

37C85 (37D20)

Let $G$ be a connected semi-simple Lie group with finite centre, no compact factors, and all almost-simple factors having real rank at least 2 and let $\Gamma<G$ be a lattice. The superrigidity theorem of G. A. Margulis shows that any linear representation of $\Gamma$ into ${\rm{PSL}}_d(\Bbb{R})$ extends, up to a compact error, to a continuous representation of $G$. R. J. Zimmer subsequently put forward a series of conjectures and questions related to representations into the group ${\rm{Diff}}^{\infty}(M)$ for a compact manifold $M$, based on the analogy between linear groups and diffeomorphism groups. This initiated what is now called the Zimmer program for understanding and classifying smooth actions by lattices of higher rank. This paper is a significant contribution to one line of enquiry in this program, studying the global rigidity of actions of lattices of higher rank on nilmanifolds under the hypothesis of hyperbolic linear data. Under some mild hypotheses a rather complete picture emerges of global rigidity phenomena in this setting. Using these results, the authors establish $C^{\infty}$ global rigidity for Anosov actions by uniform lattices, for Anosov actions of ${\rm{SL}}_n(\Bbb{Z})$ on $\Bbb{T}^n$ for $n\geqslant5$, and for probability-preserving actions of lattices of higher rank on nilmanifolds.

Reviewed by Thomas Ward

**MR3375524**

Hurtado, Sebastian (1-CA)

Continuity of discrete homomorphisms of diffeomorphism groups. (English summary)

Geom. Topol. 19 (2015), no. 4, 2117–2154.

57S05

This work is about the continuity of certain (discrete) homomorphisms between groups of diffeomorphisms of smooth manifolds and the classification of such homomorphisms when the manifolds involved are of the same dimension.

Let $M$ be a $C^\infty$ manifold and denote by $\mathrm{Diff}_c(M)$ its group of $C^\infty$ compactly supported diffeomorphisms isotopic to the identity endowed with the (metrizable) weak topology [see M. W. Hirsch, Differential topology, Springer, New York, 1976; MR0448362]. Let $d_{C^\infty}$ be a metric compatible with the weak topology. For any compact set $K\subseteq M$, let $\mathrm{Diff}_K(M)$ denote the group of diffeomorphisms in $\mathrm{Diff}_c(M)$ supported in $K$ with the induced topology. Let $N$ be another smooth manifold. A group homomorphism $\Phi\:\mathrm{Diff}_c(M)\longrightarrow\mathrm{Diff}_c(N)$ is weakly continuous if for every compact set $K\subseteq M$, the restriction $\Phi_{\mathrm{Diff}_K(M)}$ of $\Phi$ to $\mathrm{Diff}_K(M)$ is continuous.

Based on a theorem by E. Militon [“Éléments de distorsion du groupe des difféomorphismes isotopes à l’identité d’une variété compacte”, preprint, arXiv:1005.1765] the author proves the following lemma:

Let $\Phi\:\mathrm{Diff}_c(M)\longrightarrow\mathrm{Diff}_c(N)$ be a (discrete) group homomorphism. Let $K\subseteq M$ be compact and suppose $\{h_n\}_n$ is a sequence in $\mathrm{Diff}_K(M)$ such that $\mathrm{\lim}_{n\rightarrow\infty}\ d_{C^\infty}(h_n,{\rm Id})=0$. Then $\{\Phi(h_n)\}_n$ contains a subsequence converging to a diffeomorphism $H$, which is an isometry for a $C^\infty$ Riemannian metric on $N$.

Using the above result, the main theorem of the present work is shown, which asserts that any discrete group homomorphism $\Phi\:\mathrm{Diff}_c(M)\longrightarrow\mathrm{Diff}_c(N)$ is weakly continuous.

Generalizing previous work of K. Mann [Ergodic Theory Dynam. Systems 35 (2015), no. 1, 192–214; MR3294298], the author proves that when $N$ is a closed manifold, $\dim(M)\geq\dim(N)$ and $\Phi \: \mathrm{Diff}_c(M)\longrightarrow\mathrm{Diff}_c(N)$ is a nontrivial homomorphism, then $M$ and $N$ are of the same dimension and $\Phi $ is “extended topologically diagonal”.

The paper includes some illuminating examples and finishes with some relevant questions and remarks.

Reviewed by Ricardo Berlanga Zubiaga

**MR3327536**

Thorne, Jack A. (4-CAMB-NDM)

Automorphy lifting for residually reducible $l$-adic Galois representations.

J. Amer. Math. Soc. 28 (2015), no. 3, 785–870.

11F80 (13D10)

This paper proves important modularity theorems for $n$-dimensional $\ell$-adic Galois representations over totally real fields, in some residually reducible cases.

Let $F$ be a CM field, and denote by $F^+$ its maximally totally real subfield. Define $G_F=\mathrm{Gal}(\overline F/F)$. If an automorphic representation $\pi$ of $\mathrm{GL}_n(\Bbb A_F)$ is regular, algebraic, essentially conjugate self-dual and cuspidal, then there exists a continuous semisimple representation $$ \rho(\pi)\:G_F\longrightarrow \mathrm{GL}_n(\overline{\Bbb Q}_\ell) $$ attached to $\pi$, uniquely characterized (up to isomorphism) by the collection of local data coming essentially from the local Langlands correspondence [see G. Chenevier and M. H. Harris, Camb. J. Math. 1 (2013), no. 1, 53–73; MR3272052; A. Caraiani, Algebra Number Theory 8 (2014), no. 7, 1597–1646; MR3272276; Duke Math. J. 161 (2012), no. 12, 2311–2413; MR2972460].

One says that a Galois representation $\rho\:G_F\rightarrow \mathrm{GL}_n(\overline{\Bbb Q}_\ell)$ is modular if it is isomorphic to $\rho(\pi)$ for some $\pi$.

Fix a continuous irreducible Galois representation $$ \rho\:G_F\longrightarrow \mathrm{GL}_n(\overline{\Bbb Q}_\ell). $$ Suppose that it is conjugate self-dual: if $c$ denotes the non-trivial element in $\mathrm{Gal}(F/F^+)$, this condition means that $\rho^c\simeq \rho^\vee\epsilon^{1-n}$ where $\epsilon$ is the $\ell$-adic cyclotomic character. We also assume that $\rho$ is de Rham with distinct Hodge-Tate weights. The paper addresses the modularity of such a representation.

One may find a Galois stable lattice in $\Bbb Q_\ell^n$, and realize $\rho$ as a continuous representation $$ \rho\:G_F\longrightarrow \mathrm{GL}_n(\mathcal O_K) $$ where $K$ is a finite extension of $\Bbb Q_\ell$ and $\mathcal O_K$ is its ring of algebraic integers. In particular, it makes sense to consider the residual representation $$ \overline\rho\:G_F\longrightarrow \mathrm{GL}_n(k) $$ where $k$ is the residue field of $K$.

A previous result of the author [J. Inst. Math. Jussieu 11 (2012), no. 4, 855–920; MR2979825] proves modularity results under the condition that the residual representation $\overline\rho$ is absolutely irreducible and adequate in the terminology of [J. A. Thorne, op. cit.]. The paper under review addresses the modularity problem in some cases when $\overline\rho$ is not absolutely irreducible. The main problem with non-irreducible residual representations is that the universal deformation ring may not exist, and therefore there is no hope to address $R=T$ theorems.

The cases treated in this paper are those when $\overline\rho$ is Schur. This technical condition is sufficient to guarantee the existence of universal deformation rings. This condition was first introduced in [L. Clozel, M. H. Harris and R. L. Taylor, Publ. Math. Inst. Hautes Études Sci. No. 108 (2008), 1–181; MR2470687]. The next problem is to apply the Taylor-Wiles method. Here a technical difficulty appears, since all relevant Galois cohomology computations require that the residual representation is absolutely irreducible. This problem is solved using an argument by C. M. Skinner and A. J. Wiles [Inst. Hautes Études Sci. Publ. Math. No. 89 (1999), 5–126 (2000); MR1793414]. The idea is to use Hida families and move from the residual representation $\overline\rho$ to an irreducible representation with coefficients in a one-dimensional quotient of the Iwasawa algebra, and then apply the usual arguments to a localization of the universal deformation ring $R$ at the dimension one prime corresponding to this representation. Note that to develop this method the author is forced to assume that $\rho$ is ordinary at primes dividing $\ell$. A third technical problem arises when one needs to show that the codimension of reducible Galois representations inside $\mathrm{Spec}(R)$ is large. In this case, there is no reason to expect such a property. For this, the author is led to assume another additional hypothesis to guarantee that the locus of reducible deformations is small. The condition he requires in the main result is that $\overline\rho$ admits a place $v$ at which the associated Weil-Deligne representation of the restriction of $\rho$ at $G_{F_v}$ corresponds under the local Langlands correspondence to a twist of the Steinberg representation.

Reviewed by Matteo Longo

**MR3744855**

Tsimerman, Jacob (3-TRNT-NDM)

The André-Oort conjecture for $\mathcal A_g$. (English summary)

Ann. of Math. (2) 187 (2018), no. 2, 379–390.

11G15 (11G18 14G35)

The author of this paper proves the following theorem: There exists $\delta_g > 0$ such that if $\Phi$ is a primitive CM type for a CM field $E$, and if $A$ is any $g$-dimensional abelian variety over $\overline{\Bbb{Q}}$ with endomorphism ring equal to the full ring of integers $\mathcal{O}_E$ and CM type $\Phi$, then the field of moduli $\Bbb{Q}(A)$ of $A$ satisfies $[\Bbb{Q}(A): \Bbb{Q}] > |{\rm Disc}(E)|^{\delta_g}$.

By a result of J. S. Pila and the author [Ann. of Math. (2) 179 (2014), no. 2, 659–681; MR3152943], this theorem implies the André-Oort conjecture for the coarse moduli space $\mathcal{A}_g$ of principally polarized abelian varieties of fixed dimension $g \ge 1$. The André-Oort conjecture, as stated in this paper, asserts that an irreducible closed algebraic subvariety $V$ of a Shimura variety $S$ contains only finitely many maximal special subvarieties. Alternatively, every irreducible component of the Zariski closure of any set $\Sigma$ of special points in $S$ is a special subvariety of $S$ [see E. Ullmo and A. Yafaev, Ann. of Math. (2) 180 (2014), no. 3, 823–865; MR3245008]. In the latter cited paper and its sequel [B. Klingler and A. Yafaev, Ann. of Math. (2) 180 (2014), no. 3, 867–925; MR3245009], a conditional proof of the general conjecture was given which depends on the generalized Riemann hypothesis (GRH) for zeta functions of CM fields. The results of the paper under review are unconditional.

Reviewed by Patrick Morton

**MR4199235**

Peluse, Sarah (1-IASP-SM)

Bounds for sets with no polynomial progressions. (English summary)

Forum Math. Pi 8 (2020), e16, 55 pp.

11B30 (11B25)

This is a seriously impressive paper obtaining the first quantitative bounds for a large number of cases of the celebrated Bergelson-Leibman theorem [V. Bergelson and A. Leibman, J. Amer. Math. Soc. 9 (1996), no. 3, 725–753; MR1325795].

Recall that the Bergelson-Leibman theorem states the following: Let ${P_1,\dots,P_m\in\Bbb{Z}[X]}$ be polynomials with $P_i(0)=0$ for all $i$. Let $\alpha>0$. Then, provided that $N$ is sufficiently large in terms of $\alpha,P_1,\dots,P_m$, any set $A\subset\{1,\dots,N\}$ of cardinality at least $\alpha N$ contains a nontrivial configuration ${(x,x+P_1(d),\dots,x+P_m(d))}$. Note that the case $P_i(X)=iX$ is already Szemerédi’s theorem.

Bergelson and Leibman’s proof uses ergodic theory and does not lead to effective bounds (of any kind, even in principle). Finding such bounds is a major open problem in additive combinatorics. In the paper under review, the author finds the first “reasonable” bounds in the case that the $P_i$ have distinct degrees. She shows that in this case $A$ contains a configuration of the stated type provided that $|A|\ll N/(\log\log N)^c$, where $c=c_{P_1,\dots,P_m}$. For comparison, we remark that this is a bound of the same strength as that obtained by W. T. Gowers [Geom. Funct. Anal. 11 (2001), no. 3, 465–588; MR1844079] in his famous work on Szemerédi’s theorem, although it should be noted that the work under review does not directly extend that work since, in the Szemerédi case, the degrees of the $P_i$ are not distinct.

Previously, bounds of the strength the author obtains were known only when $m=1$ (where, in fact, better bounds are known, the state of the art being work of T. F. Bloom and J. Maynard [“A new upper bound for sets with no square difference”, preprint, arXiv:2011.13266]) and the case $P_1(X)=X$, $P_2(X)=X^2$, which was handled by the author and S. Prendiville [“Quantitative bounds in the nonlinear Roth theorem”, preprint, arXiv:1903.02592]. The present work, while it builds from that case, contains some substantial new innovations and the overall scheme of argument is vastly more complicated, though at its heart it remains a density-increment argument as with almost all quantitative bounds for problems of this kind over the integers. A key feature, critical in obtaining good bounds, is the use of comparatively “soft” arguments to avoid any need to invoke the inverse theory of Gowers norms or to introduce any discussion of nilsequences. For this to be possible the distinct degree condition is essential.

There seems little point in trying to sketch the argument here, not least because Section 3 of the paper does just that, providing in addition a stylish diagram illustrating the dependencies between various arguments. From Section 4 onwards, the technical details are at times quite formidable.

A number of intermediate results in the paper could be of independent interest. Foremost in this category is probably Theorem 3.5, a “quantitative concatenation” result for Gowers norms, which roughly speaking asserts that certain averages of box norms are controlled by Gowers norms. The author also highlights Lemma 5.1, a more technical result of a similar flavour, as being potentially portable elsewhere.

Whilst the bounds obtained in this paper are impressive compared to what went before, it is plausibly true that the true bound in all of the cases considered is polynomial! I am not aware of any counterexamples to such a possibility.

Reviewed by Ben Joseph Green

**MR4055179**

Guth, Larry (1-MIT); Iosevich, Alex (1-RCT); Ou, Yumeng (1-CUNY2); Wang, Hong (1-MIT)

On Falconer’s distance set problem in the plane. (English summary)

Invent. Math. 219 (2020), no. 3, 779–830.

42B20 (28A80)

Falconer’s distance problem is a famous and difficult problem in geometric measure theory. Roughly speaking it asks for the relationships between the dimensions of a Borel set $F \subseteq \Bbb{R}^d$ and the distance set of $F$, defined by $$ D(F) = \{ |x-y| : x,y \in F\}. $$ From now on all sets $F$ are Borel. There are various ways to formulate the problem precisely. One conjecture is that if $\dim_H F >d/2$, then $D(F)$ should have positive Lebesgue measure. Here $\dim_H$ denotes the Hausdorff dimension.

In his original paper [Mathematika 32 (1985), no. 2, 206–212 (1986); MR0834490], K. J. Falconer proved that $\dim_H F >d/2+1/2$ ensures that $D(F)$ has positive Lebesgue measure. T. H. Wolff [Internat. Math. Res. Notices 1999, no. 10, 547–567; MR1692851; addendum, J. Anal. Math. 88 (2002), 35–39; MR1979770] improved this in the plane, proving that $\dim_H F >1+1/3$ ensures that $D(F)$ has positive Lebesgue measure. The main result of the paper under review is a further improvement on this. Specifically, if $F$ is a planar Borel set with $\dim_H F >1+1/4$, then $D(F)$ has positive Lebesgue measure. The proof is long and technical with many new insights and techniques coming from harmonic analysis and geometric measure theory. The results also hold for pinned distance sets and distance sets where the distances are taken with respect to a norm which has a unit ball with a smooth boundary of non-vanishing Gaussian curvature.

The distance set problem has seen a lot of activity in the last few years—for example [T. Orponen, Adv. Math. 307 (2017), 1029–1045; MR3590535; T. Keleti and P. S. Shmerkin, Geom. Funct. Anal. 29 (2019), no. 6, 1886–1948; MR4034924] and various other works.

Reviewed by Jonathan MacDonald Fraser

**MR3959854**

Wang, Yilin (CH-ETHZ)

The energy of a deterministic Loewner chain: reversibility and interpretation via ${\rm SLE}_{0+}$. (English summary)

J. Eur. Math. Soc. (JEMS) 21 (2019), no. 7, 1915–1941.

30C55 (30C62 60J67)

The author studies some features of the energy of a deterministic Loewner chain. According to the chordal Loewner description, a simple curve $\gamma$ from 0 to infinity in the upper half-plane $\Bbb H=\{z\in\Bbb C\:{\rm Im}\,z>0\}$ is parameterized so that the conformal map $g_t$ from $\Bbb H\setminus\gamma[0,t]$ onto $\Bbb H$ with $g_t(z)=z+o(1)$ as $z\to\infty$ satisfies in fact $g_t(z)=z+2t/z+o(1/z)$ as $z\to\infty$. Extend $g_t$ continuously to the boundary point $\gamma_t$. The real-valued driving function $W_t := g_t(\gamma_t)$ of $\gamma$ is continuous. The Loewner energy is the Dirichlet energy of $W_t$ given by $$ I(\gamma):= \frac{1}{2}\int_0^{\infty}\left(\frac{dW_t}{dt}\right)^2 dt=\frac{1}{2}\int_0^{\infty}\left(\frac{d(g_t(\gamma_t))}{dt}\right)^2dt. $$ The scale invariance $I(u\gamma)=I(\gamma)$ for $u>0$ makes it possible to define the energy $I_{D,a,b}(\eta)$ of any simple curve $\eta$ from a boundary point $a$ of a simply connected domain $D$ to another boundary point $b$ to be the energy of the conformal image of $\eta$ via any uniformizing map $\Psi$ from $(D,a,b)$ to $\Bbb (H,0,\infty)$. For such $\eta$, define its time-reversal $\widehat\eta$ that has the same trace as $\eta$, but which is viewed as going from $b$ to $a$. The first main contribution of the paper is presented in the following theorem.

Main Theorem 1.1. The Loewner energy of the time-reversal $\widehat\eta$ of a simple curve $\eta$ from $a$ to $b$ in $D$ is equal to the Loewner energy of $\eta$: $I_{D,a,b}(\eta)=I_{D,b,a}(\widehat\eta)$.

When $\lambda\in C([0,\infty))$, consider the Loewner differential equation $$ \partial_tg_t(z)=\frac{2}{g_t(z)-\lambda_t}$$ with the initial condition $$ g_0(z)=z. $$ The chordal Loewner chain in $\Bbb H$ driven by $\lambda$ (or the Loewner transform of $\lambda$) is the increasing family $(K_t)_{t>0}$ defined by $K_t=\{z\in\Bbb H\:\tau(z)\leq t\}$, where $\tau(z)$ is the maximum survival time of the solution $g_t(z)$. Let $H\subset C[0,\infty)$ be the set of finite $I$ energy functions. The author proves the following statement.

Proposition 2.1. For every $\lambda\in H$, there exists $K=K(I(\lambda))$, depending only on $I(\lambda)$, such that the trace of the Loewner transform $\gamma$ of $\lambda$ is a $K$-quasiconformal curve.

The deterministic results in the paper are closely linked with the Schramm-Loewner Evolutions (SLE) theory. In the final section, the author establishes some connections with ideas from SLE restriction properties and SLE commutation relations.

Reviewed by Dmitri Valentinović Prokhorov

]]>

The work was done at the Zentrum für Data Analytics, Visualization and Simulation (DAViS). Prof. Dr. Heiko Rölke is the leader of DAViS. Thomas Keller is the project leader overseeing the calculations. The computation took 108 days and 9 hours on a supercomputer. The new record was in the news this week. The announcement from their institution is here (in German). Here is a short NPR (audio) story. *Popular Mechanics* had a nice article, and contacted Keller about their methods, as well as about the meaning of the calculation. A key element to the DAViS group’s computation is the Chudnovsky algorithm, which has been used by other teams computing digits of $\pi$.

There is a long tradition of computing digits of $\pi$. There is even an MSC2020 class for the family of such computations: **11Y60** *Evaluation of number-theoretic constants*. Long ago, such computations were done by hand. Wikipedia has a nice page with the history of the computations, and some related facts. Early mathematicians in Egypt, Babylon, and China had approximations for $\pi$, usually as fractions. As a formalized system, continued fractions provide an accessible method for generating good rational approximations to irrational numbers, including $\pi$. The first few convergents in the continued fraction for $\pi$ are $\frac{3}{1}$, $\frac{22}{7}$, $\frac{333}{106}$, $\frac{355}{113}$. The sequence A001203 in the OEIS is the continued fraction representation of $\pi$, in the standard continued fraction shorthand. Once we switched to using the decimal system, the digits became the target. Various mathematicians, some unknown, some known, some famous, worked on computations of the digits of $\pi$. Now, of course, we rely on computers, but we still need good algorithms.

As a non-expert in this area, one result that I find particularly fascinating is

MR1415794

Bailey, David (1-NASA9); Borwein, Peter (3-SFR); Plouffe, Simon (3-SFR)

On the rapid computation of various polylogarithmic constants. (English summary)

Math. Comp. 66 (1997), no. 218, 903–913.

11Y60

which provides a way of skipping intermediate digits and computing the $N$th hexadecimal digit of $\pi$ – also for certain other number-theoretic constants. $N$ can be as large as you want, as the authors demonstrate by computing the ten billionth hexadecimal digit of $\pi$.

For the news stories, the journalists usually ask “What’s this good for?” The obvious answer is the tautological answer, but they are looking for applications outside number theory and outside mathematics. Inside mathematics, methods of computing digits of $\pi$ and other mathematical constants are connected with developments of independent areas of mathematics. One example is the theory of continued fractions, mentioned earlier. Some of the efficient formulas (or series) for computing $\pi$ and other number-theoretic constants have connections with modular forms. The work of David Bailey, Peter Borwein, Simon Plouffe involves the evaluation of special polylogarithms. Peter Borwein and Jonathan Borwein wrote a nice book about the connection between $\pi$ and the algebraic-geometric mean (AGM).

MR0877728

Borwein, Jonathan M. (3-DLHS); Borwein, Peter B. (3-DLHS)

Pi and the AGM.

A study in analytic number theory and computational complexity. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1987. xvi+414 pp. ISBN: 0-471-83138-7

11Y60 (68Q30)

Doing insanely advanced versions of standard computations is also a nice way to test your algorithms and your hardware. Keller is quoted in the *Popular Mechanics* article saying, “For us, the record is a byproduct of tuning our system for future computation tasks.” An older example of this idea was a somewhat unintentional test and wasn’t computing digits of $\pi$, but Thomas Nicely‘s computations of primes, twin primes, and the like led to the discovery of hardware flaws in an Intel Pentium chip design in 1994.

MR1415794

Bailey, David (1-NASA9); Borwein, Peter (3-SFR); Plouffe, Simon (3-SFR)

On the rapid computation of various polylogarithmic constants. (English summary)

Math. Comp. 66 (1997), no. 218, 903–913.

11Y60

The authors give algorithms for the computation of the $n{\rm th}$ digit of certain transcendental constants in (essentially) linear time and logarithmic space. The complexity class considered is denoted by ${\rm SC}^*$, which means ${\rm space}=\log^{O(1)}(n)$ and ${\rm time}=O(n\log^{O(1)}(n))$. As a typical example, the authors show how to compute, say, just the billionth binary digit of $\log(2)$, using single precision, within a few hours.

The existence of such an algorithm, which appears to be quite surprising at first, is based on the following idea. Suppose a constant $C$ can be represented as $C=\sum_{k=0}^\infty1/(b^{ck}q(k))$, where $b\ge2$ and $c$ are positive integers, and $q$ is a polynomial with integer coefficients $(q(k)\ne0)$. The task is to compute the $n{\rm th}$ digit of $C$ in base $b$. First observe that it is sufficient to compute $b^nC$ modulo 1. Clearly,

$$

b^nC\bmod1=\sum_{k=0}^\infty\frac{b^{n-ck}}{q(k)}\bmod1= \

\sum_{k=0}^{[n/c]}\frac{b^{n-ck}\bmod q(k)}{q(k)}\bmod1+\sum_{k=1+[n/c]}^\infty\frac{b^{n-ck}}{q(k)} \bmod1.

$$

In each term of the first sum, $b^{n-ck}\bmod q(k)$ is computed using the well-known fast exponentiation algorithm modulo the integer $q(k)$. Division by $q(k)$ and summation are performed using ordinary floating-point arithmetic. Concerning the infinite sum, note that the exponent in the numerator is negative. Thus, floating-point arithmetic can again be used to compute its value with sufficient accuracy. The final result, a fraction between 0 and 1, is then converted to the desired base $b$. With certain minor modifications, this scheme can be extended to numbers of the form $C=\sum_{k=0}^\infty p(k)/(b^{ck}q(k))$, where $p$ is a polynomial with integer coefficients.

It now happens that a large number of interesting transcendentals are of the form described. Many of the formulas depend on various polylogarithmic identities. Thus, define the $m{\rm th}$ polylogarithm $L_m$ by $L_m(z)=\sum_{j=1}^\infty z^j/j^m,\ |z|<1$. Then, for example, $-\log(1-2^{-n})=L_1(1/2^n)$, or $\pi^2=36L_2(\frac12)-36L_2(\frac14)-12L_2(\frac18)+6L_2(\frac1{64})$. One of the most striking identities is, however,$$\pi=\sum_{j=0}^\infty\frac1{16^j} \bigg(\frac4{8j+1}-\frac2{8j+4}-\frac1{8j+5}-\frac1{8j+6}\bigg).$$Using these formulas, it is easily shown that $-\log(1-2^{-n}),\ \pi$ or $\pi^2$ are in ${\rm SC}^*$. The authors demonstrate their technique by computing the ten billionth hexadecimal digit of $\pi,$ as well as the billionth hexadecimal digit of $\pi^2$ and $\log(2)$.

Reviewed by Andreas Guthmann

**MR0877728**

Borwein, Jonathan M. (3-DLHS); Borwein, Peter B. (3-DLHS)

Pi and the AGM.

A study in analytic number theory and computational complexity. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. *John Wiley & Sons, Inc., New York,* 1987. xvi+414 pp. ISBN: 0-471-83138-7

11Y60 (68Q30)

This book reveals the close relationship between the algebraic-geometric mean iteration and the calculation of $\pi$.

The topic of algebraic-geometric mean iteration leads to a discussion on the theory of elliptic integrals and functions, theta functions and modular functions. The calculation of $\pi$ leads into the area of calculating algebraic functions, elementary functions and constants plus the transcendence of $\pi$ and $e$. The calculation of $\pi$ advanced with a frenzy with the advent of modern computers. Briefly, in 1706 the first 100 digits of $\pi$ were calculated. By 1844 the first 205 digits were known. In 1947 the first 808 digits of $\pi$ were computed using a desk calculator.

Then the modern computer came on the scene. Now the known first digits changed rapidly, which Table A partially illustrates:

Table A: | Year | Number of known first digits of $\pi$ |
Required Time to Calculate |
---|---|---|---|

1949 | 2 037 | 70 hours | |

1961 | 100 000 | 9 hours | |

1973 | 1 000 000 | 24 hours | |

1983 | 16 000 000 | 30 hours | |

1986 | 29 360 000 | 28 hours | |

1986 | $2^{25}$=33 554 432 | 96 minutes |

These figures reveal, and Table B shows, how much the speed of computing has increased in a very short time.

Table B: | Year | Approximate Digits/Hour |
---|---|---|

1949 | 29.1 | |

1961 | 11 111.1 | |

1973 | 41 666.7 | |

1983 | 533 333.3 | |

1986 | 1 048 571.4 | |

1986 | 20 971 520.0 |

Reviewed by H. London

**MR1021452**

Chudnovsky, D. V. (1-CLMB); Chudnovsky, G. V. (1-CLMB)

The computation of classical constants.

*Proc. Nat. Acad. Sci. U.S.A.* 86 (1989), no. 21, 8178–8182.

11Y60 (11-04 11Y35 33A99)

In this very interesting paper the authors make a large number of valuable comments on mathematics and algorithmics in connection with their computation of $\pi$ up to one billion digits. They give a short history of the computation of $\pi$ and some remarks on the evaluation of values of the hypergeometric functions. They explain how the Legendre relations for elliptic curves with complex multiplication give rise to Ramanujan’s series which are now used to compute $\pi$. Finally, some remarks on computer implementations are made.

Reviewed by F. Beukers

**MR1222488**

Borwein, J. M. (3-WTRL-B); Borwein, P. B. (3-DLHS)

Class number three Ramanujan type series for $1/\pi$. (English summary)

Computational complex analysis.

*J. Comput. Appl. Math.* 46 (1993), no. 1-2, 281–290.

11Y60 (11R11)

S. Ramanujan [Quart. J. Math. 45 (1914), 350–372; Jbuch 45, 186] exhibited certain series which converge very rapidly to $1/\pi$, of the form $\sum_{n\geq 0}(-1)^n((A+nB)/C^{3(n+1/2)})((6n)!/(3n)!n!^3)$. It turns out that for each squarefree integer $d$, the numbers $A$, $B$, $C$ are determined by the field $\mathbf{Q}(\sqrt{-d})$, and are actually $h$th degree algebraic integers, where $h$ is the class number of the field. The Chudnovskys have used such a series (with $h(-427)=2$), which adds 25 digits per term, to compute $\pi$ to a record two billion digits. Herein, the Borweins continue their fraternistic rivalry with the Chudnovskys by finding an example with $h(-1555)=4$ which adds about 50 digits per term. However, they have to deal with a quartic rather than quadratic irrational.

They also explicitly give the series approximating $1/\pi$ corresponding to each imaginary quadratic field of class number $3$ (surprisingly though, they provide no reference to tell us how they know that they have the complete list).

{For the collection containing this paper see MR1222468.}

Reviewed by Andrew Granville

]]>I first learned of Yoshimura crush patterns some years ago, when my friends David Wright and Lisa Mantini visited my family and gave us a set of tumblers that were made in the distinctive shape. Two of the tumblers are in the photo at the start of this post.

The history of the crush patterns (also called “crease patterns”) demonstrates the interplay between theory and experiment, as described below. It also demonstrates how dissemination of scientific information has changed over the last eighty years. Yoshimaru Yoshimura studied the patterns that bear his name in a paper published in Japan in 1951. An article in English was published in 1955. The patterns were analyzed even earlier, though, by Theodore von Kármán and Hsue-Shen Tsien in 1941. Yoshimura, however, was unaware of their work due to the war. Over the decades, *Mathematical Reviews* has worked to provide information about published research to researchers everywhere. This was especially important during the Cold War. Today, of course, the internet provides many ways for people to discover research done by researchers halfway around the world: via MathSciNet, the arXiv, or simply via a Google search. It seems that the greatest impediment to keeping up on the literature now is the immense quantity of what is published.

Both studies were motivated by the gap between experiment and theory. As von Kármán and Tsien write, they had discussed in two previous papers “the inadequacy of the classical theory of thin shells in explaining the buckling phenomenon of cylindrical and spherical shells. It was shown that not only the calculated buckling load is 3 to 5 times higher than that found by experiments, but the observed wave pattern of the buckled shell is also different from that predicted.” Their 1941 paper begins with the buckling patterns observed in experiment and provides an analysis that explains them. The starting point is the observation that the bending energy is related to the curvature of the surface, as previous investigators, such as L.H. Donnell, had shown. The analysis by von Kármán and Tsien seeks to minimize the energy computed using curvature terms arising only via the radial displacement of the shell. The resulting PDEs are nonlinear, which they solve by considering products of cosine functions and the method of undetermined coefficients. In their conclusion, they point out limitations of their approach. “However, due to the complexity of the problem, the results given in this paper can be only considered as a rough approximation and most of the statements made are qualitative rather than quantitative. To put the new theory on a solid footing, a more accurate solution of the differential equations of equilibrium is necessary.” Von Kármán included his work with Tsien in his Gibbs Lecture on the topic “the engineer grapples with nonlinear problems”, delivered at the December 1939 meeting of the American Mathematical Society in Columbus, Ohio. The lecture was published in the *Bulletin of the AMS*.

Yoshimura’s investigation also acknowledges the gap between experiment and theory. As compared to von Kármán and Tsien, though, he takes a broader view, and makes intrinsic use of the geometry of the buckling. He writes, “The state which may be actually realized after buckling must be determined by minimizing the energy, not only with respect to the magnitude of deflection, but also to the aspect ratio and the circumferential number of buckling waves. The actual buckling load will be given by a comparison of energy levels before and after buckling and the energy barrier to be jumped over in buckling. Based on such a concept, the general buckling and the local buckling of a cylindrical shell are considered to be quite different phenomena from the energy viewpoint, though they are equivalent with respect to the load.” An important observation by Yoshimura is that while both a flat sheet and a cylindrical shell are geometrically flat (zero Gauss curvature), the cylindrical shell has nearby deformations that are also developable surfaces, i.e., deformations that also have zero Gauss curvature. These are his eponymous crush patterns.

Crumpling paper has been in the news recently. Siobhan Roberts has published two articles about crumpling paper in the *New York Times, *one in 2018 and the other in March 2021. The first article describes the doctoral work of Omer Gottesman, along with Jovana Andrejevic, Chris H. Rycroft, and Shmuel M. Rubinstein, who studied the growth of crease length in repeatedly crumpled sheets of paper. The primary finding is that, despite complications and variations, the total length of the creases grows logarithmically. The second article describes work by Jovana Andrejevic, Lisa M. Lee, Shmuel M. Rubinstein, and Chris H. Rycroft on a two-dimensional aspect of the work: analysis of the facets that develop upon repeated crumpling. Their work relates paper crumpling to fragmentation distributions, including work by Kolmogorov from around the time the von Kármán and Tsien were doing their work on buckling of thin cylinders.

- Andrejevic, J., Lee, L.M., Rubinstein, S.M.
*et al.*A model for the fragmentation kinetics of crumpled thin sheets.*Nat Commun***12,**1470 (2021). https://doi.org/10.1038/s41467-021-21625-2. - Donnell, L. H., Stability of Thin-Walled Tubes Under Torsion,
*N.A.C.A. Technical Report No. 479*, 1934. - Gottesman, O., Andrejevic, J., Rycroft, C.H.
*et al.*A state variable for crumpled thin sheets.*Commun Phys***1,**70 (2018). https://doi.org/10.1038/s42005-018-0072-x. - von Kármán, Theodore; Tsien, Hsue-Shen, The buckling of thin cylindrical shells under axial compression.
*J. Aeronaut. Sci.*8 (1941), 303–312. - von Kármán, Theodore, The engineer grapples with non-linear problems.

Bull. Amer. Math. Soc. 46 (1940), 615–683. MR0003131 - Kolmogoroff, A. N., Über das logarithmisch normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstückelung. C. R. (Doklady) Acad. Sci. URSS (N. S.) 31, (1941). 99–101. MR0004415
- Yoshimura, Y.: Theory of Thin Shells with Finite Deformation.
*Rep. Inst. Sci. and Tech.*, Tokyo Univ., 2, 1948, P. 167; 3, 1949,P. 19. - Yoshimura, Y.: Local Buckling of Circular Cylindrical Shells and Scale Effects.
*Proc. of the 1st Japan National Congress for Appl. Mech.*, 1951. - Yoshimura, Yoshimaru: On the mechanism of buckling of a circular cylindrical shell under axial compression.
*Technical Memorandum 1390*. National Advisory Committee for Aeronautics, July 1955.

Note: this paper had been available from the NASA site: https://ntrs.nasa.gov/citations/19930093840. At the time of this writing (18 July 2021), I can no longer find it on the NASA server. A copy is available from the University of North Texas: https://digital.library.unt.edu/ark:/67531/metadc62872/m1/1/.

**MR0006926**

von Kármán, Theodore; Tsien, Hsue-Shen

The buckling of thin cylindrical shells under axial compression.

*J. Aeronaut. Sci.* 8 (1941), 303–312.

73.2X

This paper is devoted to the solution of the problem stated in the title under the same general assumptions as were made in an earlier paper by the same authors on buckling of the spherical shell [J. Aeronaut. Sci. 7, 43–50 (1939); MR0003177]. [See also a paper by v. Kármán, Dunn and Tsien [J. Aeronaut. Sci. 7, 276–289 (1940); MR0003178] and a paper by K. Friedrichs [Theodore von Kármán Anniversary Volume, California Institute of Technology, Pasadena, 1941, pp. 258–272; MR0004599].] The essentially new idea in this as in the earlier papers is that it is possible to explain why thin shells buckle at a much lower pressure than that predicted by the linear theory of buckling by considering the effect of certain nonlinear terms; even the quantitative results of the nonlinear theory for the spherical shell were found to be in quite good accord with experiment. The cylindrical shell is a much more difficult case than that of the spherical shell (because of a lack of symmetry in the buckled state) so that the authors restrict themselves in the main to qualitative rather than quantitative comparison with experiment. Solutions are obtained by an energy method.

Reviewed by J. J. Stoker

**MR0003131** **(2,167d)** Reviewed

von Kármán, Theodore

The engineer grapples with non-linear problems.

*Bull. Amer. Math. Soc.* 46 (1940), 615–683.

71.0X

von Kármán appeals to pure mathematicians for cooperation with engineers who are struggling with a great variety of mathematical problems. He summarizes the intention of his Gibbs lecture as follows: “An attempt is made to show the application of analytical methods available for the solution of certain nonlinear problems in which the engineer is interested. Some gaps are shown and frontiers indicated beyond which the safe guidance of the mathematical analysis is for the time being lacking.” After contrasting linear with nonlinear problems von Kármán states: “in most nonlinear problems physical reasoning is not sufficient or fully convincing, so that in these cases the questions of existence and uniqueness represent a real challenge to the mathematician.” The whole field of nonlinear mathematical engineering problems is then discussed in a rather detailed survey.

1. Nonlinear vibrations result from differential equations of the type: $$ \ddot x+\omega^2x=f(x,\dot x), $$ where $x$ represents a deflection and $\omega$ the natural frequency of the system. Periodic solutions, corresponding to Poincaré’s limit cycles, represent “self-excited” vibrations. In some limit cases there are sudden transitions between deflections of opposite sign, “relaxation vibrations” according to van der Pol. If a periodic force is applied to the system, the phenomenon of “subharmonic resonance” may result; it is important in radio-technique but also occurs in airplane vibrations; it is here treated by a perturbation method analogous to a procedure developed by Poincaré for celestial mechanics.

2. Nonlinear differential equations occur in the theory of elasticity when deflections are permitted to be large. Bending and buckling of thin rods (“elastica” problem) and plates are discussed. The differential equations for plates, derived by von Kármán in 1910, are $$ \Delta\Delta F=E(w_{xy}^2-w_{xx}w_{yy}),\quad C\Delta\Delta w=F_{yy}w_{xx}+F_{xx}w_{yy}-2F_{xy}w_{xy}, $$ where $E$ and $C$ are constants. The most interesting phenomenon is to be found in the case of very thin plates $(C\rightarrow 0)$. The solutions, which recently have been treated by asymptotic integration, will be constants except in narrow strips near the boundary where sudden changes occur. For curved arches and shells the presence of nonlinear terms provides a solution which is attained in reality even before the linear theory comes into play; in this case, therefore, the linear theory fails entirely to give account of the actual situation. A brief review is given of the rather recent general theories of nonlinear elasticity due to Murnaghan and Biot. Plastic deformations are characterized by quite a different type of nonlinear problems, namely, boundary value problems for hyperbolic differential equations of the type $$ (f_{xx}+f_{yy})^2-C(f_{xx}f_{yy}-f_{xy}{}^2)=0. $$

3. In the theory of fluid flow a variety of problems occur in which the differential equation (the potential equation) is linear while the boundary conditions are nonlinear. Many problems in this field have been thoroughly treated, for example, the flow around obstacles and through nozzles, also waves with large amplitudes; many other problems remain unsolved, for example, that of heavy jets over spill-ways or the meteorological problem of the progressing cold front; a main difficulty seems to be the determination of the proper singularity of the analytic function which represents the flow.

4. The flow of viscous fluids is governed by a nonlinear differential equation (essentially of the fourth order) for which exact solutions are known only in few instances. From an engineering point of view the limiting case that the viscosity approaches zero is more important; it had defied the powers of analysis until it was made accessible to methods of asymptotic integration through Prandtl’s ingenious boundary layer theory. There remain, however, unanswered purely mathematical questions; for example, what is the flow pattern around a submerged body?

5. The interesting feature of the nonlinear differential equations of compressible fluids is that they are elliptic or hyperbolic depending on whether the velocity is below or above that of sound. For both cases some solutions are known. For problems, however, where the equation is elliptic in one part and hyperbolic in another part (a condition encountered, for example, with aerial bombs dropped from large heights), no methods are as yet available. The invention of such methods “would be an achievement both from a practical and mathematical point of view.”

The paper is furnished with a great many instructive figures and supplemented by an extensive bibliography.

Reviewed by K. Friedrichs

**MR0004415**

Kolmogoroff, A. N.

Über das logarithmisch normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstückelung. **(German)**

*C. R. (Doklady) Acad. Sci. URSS (N. S.)* **31, **(1941). 99–101.

60.0X

It is stated that observations show that the logarithms of the sizes of particles such as mineral grains are frequently normally distributed [Rasumovski, in the same C. R. 28, 814–816 (1940)]. The purpose of the present paper is to explain this phenomenon by a plausible probabilistic scheme. Consider a random process in which the number of particles at time $t\ (=0,1,2,\cdots)$ is $N(t)$, and the number of particles of dimension not greater than $r$ is $N(r,t)$ (it is irrelevant how this “dimension” is defined). It is supposed that the probability that a particle of size $r$ splits, during $(t,t+1)$, into $n$ particles of sizes $x_1r,x_2r,\cdots,x_nr$ is independent of $t$ and $r$. Let then $Q(x)$ be the mean value of the number of particles of size not greater than $xr$ originated during $(t,t+1)$ from a particle of size $r$. It is shown that under some slight additional assumptions $N(e^x,t)/N(t)$ tends to a Gaussian distribution with mean value $mt=t\int_0^1\log ydQ(y)/Q(1)$ and variance $t\int_0^1(\log y-m)^2dQ(y)/Q(1)$.

Reviewed by W. Feller

]]>The cicadas of Brood X have emerged throughout much of the eastern United States. In certain areas of Ann Arbor, Michigan, where Mathematical Reviews is physically located, they have become quite loud. They belong to the genus *Magicicada *of periodical cicadas that emerge either in 13-year cycles or in 17-year cycles. The cycle lengths are prime numbers, which makes mathematicians wonder why.

Brood X cicadas have been in the news in the US because they are have emerged after 17 years as larvae living underground. Many of the stories (such as this and this) mention that the cycles are prime numbers: Magicicada septendecim having a 17-year cycle and Magicicada tredecim a 13-year cycle. Intuitively, this sounds interesting, but just what are the details? The point that is generally made is that these cycle lengths help keep the cicada broods out of sync with potential predators, who instead rely on some other prey.

In 1979, Robert M. May wrote about the periodicity of cicadas in *Nature*. He nicely lays out the issues and the possible explanations. He mentions that 13 and 17 are prime, but doesn’t dwell on that, instead focusing on the relatively long duration of the nymph stage. May is interpreting the cicadas’ life cycle as a dynamical system with some parameters. His hypothesis is that for these particular species, the important parameters are in some special intermediate range that prevents the cicadas from being wiped out by predators or falling into an annual life cycle. May also lays out one of the difficulties in studying cicadas with such life cycles, writing, “Whatever their significance in the world of pedators and parasitoids, 13 and 17 years are much longer than the time scale for most research grants and tenure decisions.”

Here are some mathematical results related to cicadas that can be found using MathSciNet. One theme is using discrete dynamical systems based on nonlinear Leslie matrix models for semelparous populations, with magical cicadas being a motivating example.

After the excerpts from MathSciNet, there is a link to an engaging video, followed by an observation about being careful when searching “anywhere” in MathSciNet.

**MR2501473** ** **

Cushing, J. M. (1-AZ)

Three stage semelparous Leslie models. (English summary)

*J. Math. Biol.* 59 (2009), no. 1, 75–104.

92D25 (37N25 39A10 92D40)

The author considers nonlinear Leslie matrix models for the dynamics of semelparous populations. Synchronous cycles describe temporally synchronized collections of age cohorts that appear in periodic outbreaks of periodical insects (such as cicadas). In mathematical models, destabilization of an extinction equilibrium and the subsequent occurrence of a global continuum branch of positive equilibria represent synchronous cycles as a bifurcation scenario. The inherent net reproductive number $R_0$ plays an important role in the bifurcation which occurs at $R_0=1$. Although bifurcation behaviors at $R_0=1$ have been well researched for two-dimensional semelparous Leslie models, not much is known for semelparous models in three or higher dimensions due to complexity in analyses. In this paper, a complete description of the bifurcation at $R_0=1$ is presented for the three-dimensional case under some monotonicity assumptions on the nonlinear interaction terms.

Following the introduction of preliminary results, the author investigates the dynamics on the boundary of $\Bbb{R}_3^{+}$. For the classification of dynamics, symmetry/asymmetry as well as strength in competition among inter-age classes play an important role. A single-class three-cycle is a fixed point on a positive coordinate axis $A_+^0$ which is one possible representation of synchronous cycles. The other type of synchronous cycles are called two-class three cycles, which correspond to a fixed point on a positive coordinate plane $P_+^0$. It is shown that under the monotonicity assumptions, single-class three-cycles are globally asymptotically stable (GAS) on $A_+^0$ if $R_0>1$. However, several alternative results can be obtained which depend on parameter constraints if $R_0>1$ but near to $1$: two-class three-cycles can occur on $P_+^0$. Interestingly, the dynamics in the interior of $\Bbb{R}_3^{+}$ is affected by the dynamics on boundaries. A heteroclinic cycle can occur. As summarized in the abstract, strong inter-age class competitive interactions promote oscillations with separated life-cycle stages, while weak interactions promote stable equilibrations with overlapping life-cycle stages. The methods used in this paper include the theory of planar monotone maps, average Lyapunov functions, and bifurcation theory techniques. Several figures depicted by numerical computations are shown to illustrate the possible outcomes.

Reviewed by Shinji Nakaoka

**MR2350059** ** **

Diekmann, Odo (NL-UTRE); Wang, Yi (PRC-HEF); Yan, Ping (FIN-HELS-MS)

Carrying simplices in discrete competitive systems and age-structured semelparous populations. (English summary)

*Discrete Contin. Dyn. Syst.* 20 (2008), no. 1, 37–52.

39A11 (37N25 92D25)

The authors consider a nonlinear Leslie (discrete time matrix) model for a semelparous population under the assumption that density dependence (through competition among age classes) is a monotone (specifically Beverton-Holt) function of a single weighted total population size. The main result of the paper concerning this model is the existence of a carrying simplex, which is then exploited to prove the existence of a heteroclinic cycle lying on the coordinate axes of the positive cone. Ecologically this cycle represents a dynamic in which generations are separated in time and each erupts periodically as cohorts age (periodical insects such as cicadas provide specific biological examples). To obtain this result, the others use a generalization of the well-known work of M. W. Hirsch on the existence of carrying simplexes in competitive systems. The authors modify the hypotheses of this generalized theory in order to obtain a new theorem that is more amenable to the Leslie model application.

Reviewed by J. M. Cushing

**MR1761531** ** **

Behncke, Horst (D-OSNB-MI)

Periodical cicadas. (English summary)

*J. Math. Biol.* 40 (2000), no. 5, 413–431.

92D25 (39A10)

A few mathematical models are developed to study the evolution of periodicity and synchronicity of magical cicadas. The key features include “underground habitat limitation, stochastic variations in predation and habitat, competition and the influence of the fungus”. The models are represented either in recursion form or with a Leslie matrix. For these models, the author shows convergence to stable generation distributions. Conditions for synchronous or periodic solution are also derived. “In addition, the intermediate evolutionary models are studied in order to show that cicada populations are capacity limited, which is tacitly assumed in the standard model.”

Reviewed by Anthony Leung

**MR2523298**

Gourley, Stephen A. (4-SUR); Kuang, Yang (1-AZS)

Dynamics of a neutral delay equation for an insect population with long larval and short adult phases. (English summary)

*J. Differential Equations* 246 (2009), no. 12, 4653–4669.

34K20 (34K40 92D40)

The authors study the stability of equilibria in a nonlinear autonomous neutral delay differential population model recently formulated by Bocharov and Hadeler via the reduction of a standard structured population model [G. A. Bocharov and K.-P. Hadeler, J. Differential Equations 168 (2000), no. 1, 212–237; MR1801352; K.-P. Hadeler, in The 8th Colloquium on the Qualitative Theory of Differential Equations, No. 11, 18 pp., Electron. J. Qual. Theory Differ. Equ., Szeged, 2008; MR2509170]. This model describes the intriguing dynamics of an insect population, such as periodical cicadas and flightless marine midges, with long larval and short adult phases. In the present paper, the study of the stability for the nonlinear neutral delay differential model is transformed into an appropriate non-neutral nonautonomous delay differential equation of unbounded delay. It is shown that the biologically meaningful solutions are always positive and bounded, provided that the time adjusted instantaneous birth rate at the time of maturation is less than 1. Global stability of the extinction and positive equilibria is also obtained by the method of iteration. In addition, the analysis in the paper reveals the fact that if the time adjusted instantaneous birth rate at the time of maturation is greater than 1, then the population will grow unboundedly regardless of the population death process.

Reviewed by Si Ning Zheng

MR1849824

Webb, G. F. (1-VDB)

The prime number periodical cicada problem. (English summary)

Discrete Contin. Dyn. Syst. Ser. B 1 (2001), no. 3, 387–399.

92D25

Summary: “Mathematical models are presented to argue for the significance of prime number emergences of 13 year and 17 year periodical cicadas (Magicicada spp.). The prime number values arise as resonances of emergences with 2 and 3 year quasi-cycling predators. Predators with 2 and 3 year quasi-cycles are present due to their age dependent fecundity and mortality rates. Their quasi-cycles are enhanced by the predation of cicadas during emergences and thus exert significant influence on the cicada periodic life cycles.”

Here is a short video about cicadas, featuring David Attenborough:

Searching for “cicada” using the **Anywhere** field in a publication search brings up some curious matches.

One such curious match is:

**MR2720210 **

Ferguson, Andrew (4-WARW-MI); Jordan, Thomas (4-BRST); Shmerkin, Pablo (4-MANC-CDA)

The Hausdorff dimension of the projections of self-affine carpets. (English summary)

Fund. Math. 209 (2010), no. 3, 193–213.

28A80 (28A78)

If you check out the item in MathSciNet, you will be hard-pressed to find the word “cicada” in the title of the paper, the review text, or any of the normal places. Publication searches in MathSciNet include a variety of possible fields from the database. One such field is the **institution**. It turns out that the address of the third author on this paper is:

**4-MANC-CDA**

Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA), School of Mathematics

University of Manchester

(merged with University of Science and Technology in Manchester UMIST)

Manchester M13 9PL

ENGLAND

The “cicada” that was found is the acronym for the author’s institution! The moral: “Anywhere” really means *anywhere* when doing a publication search.

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