Plenty. Citation counts depend on matching algorithms. The algorithms try to pair an item in the reference list of an article with a known item in a database. Usually, you want to have matches or near matches on multiple points: author name, title, year of publication, page range, source (name of the journal). However, bibliographic styles are not consistent. And some authors make mistakes or take shortcuts, providing too little information. Some journals enforce telegraphic reference styles. Here is an example I chose at random from a respected physics journal:

- R. Yang and Z. Q. Wu,
*Earth Planet. Sci. Lett.***404**, 14 (2014). - J. C. Crowhurst, J. M. Brown, A. F. Goncharov, and S. D. Jacobsen,
*Science***319**, 451 (2008). - H. Marquardt, S. Speziale, H. J. Reichmann, D. J. Frost, and F. R. Schilling,
*Earth Planet. Sci. Lett.***287**, 345 (2009).

Note that there are no titles. Also, a page range isn’t given, just a starting page. This style makes it hard for the matching algorithm, but it is a standard style in the physics literature, not just this journal.

Some old-school citations are almost impossible for an algorithm to find. Here’s an old old-school example, from an old paper by Lefschetz in the Annals of Mathematics in 1920 .

• • • • •

The references are given almost as prose, but heavily abbreviated. My copy of Whittaker and Watson is full of citations as footnotes. (And I can only imagine how many times that book is referenced the way I just did, by the authors’ names, not by the title.¹) Such citations are rarer now, but they still occur.

Errors in citations can propagate. When I wrote my PhD thesis, an important result that I used was the Borel-Weil-Bott Theorem. Bott’s paper is

MR0089473

Bott, Raoul

Homogeneous vector bundles.

*Ann. of Math. (2)* **66** (1957), 203–248.

However, I found that many papers cited it incorrectly. Moreover, I could see that some authors copied the citation to Bott’s paper from the references in another paper. If one paper had it incorrect, then subsequent articles make the same mistake. I don’t remember exactly which ones I encountered way back in grad school, but examples are easy to find. For instance, one paper has the year and pages correct, but has the volume number as 56. Another puts the volume number at 60. (Getting warmer!) Kostant’s paper establishing his famous formula for the multiplicity of a weight gets everything right except the page range. In his paper on Lie algebra cohomology and the Borel-Weil-Bott Theorem (published two years later), Kostant has a complete and correct citation.

Citations to books can be troublesome. Often, the citation is spare, giving the author, title, and year. Here is a citation from a paper published in 2016 to a famous book by Dautray and Lions:

21. Dautray R, Lions JL. *Mathematical Analysis and Numerical Methods for Sciences and Technology*. Springer: Berlin, 1990.

We matched that to

**MR1036731**

Dautray, Robert(F-POLY); Lions, Jacques-Louis(F-CDF)

Mathematical analysis and numerical methods for science and technology. Vol. 1.

Physical origins and classical methods. With the collaboration of Philippe Bénilan, Michel Cessenat, André Gervat, Alain Kavenoky and Hélène Lanchon. Translated from the French by Ian N. Sneddon. With a preface by Jean Teillac. *Springer-Verlag, Berlin,* 1990. xviii+695 pp. ISBN: 3-540-50207-6; 3-540-66097-6.

But it could also have been Volume 3 or Volume 4, which were also published in 1990:

**MR1064315**

Dautray, Robert(F-POLY); Lions, Jacques-Louis(F-CDF)

Mathematical analysis and numerical methods for science and technology. Vol. 3.

Spectral theory and applications. With the collaboration of Michel Artola and Michel Cessenat. Translated from the French by John C. Amson. *Springer-Verlag, Berlin,* 1990. x+515 pp. ISBN: 3-540-50208-4; 3-540-66099-2

**MR1081946**

Dautray, Robert(F-POLY); Lions, Jacques-Louis(F-CDF)

Mathematical analysis and numerical methods for science and technology. Vol. 4.

Integral equations and numerical methods. With the collaboration of Michel Artola, Philippe Bénilan, Michel Bernadou, Michel Cessenat, Jean-Claude Nédélec, Jacques Planchard and Bruno Scheurer. Translated from the French by John C. Amson. *Springer-Verlag, Berlin,* 1990. x+465 pp. ISBN: 3-540-50209-2; 3-540-66100-X

Volume 2 was published in 1988. Volume 5 was 1992, and Volume 6 was 1993.

Formats vary greatly, with some including the city of publication, some including series information (such as *Ergebnisse* or maybe *Ergebnisse der Mathematik* or *Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge*). If the series information is given, a volume number might be included. In checking changes in citations after releasing the new features for MathSciNet, we realized that there were instances of a citation to a book in a series mixing up the volume number and the year of publication. (We fixed them.)

The propagation of the errors with citations to Bott’s paper described above was partly due to authors taking a shortcut. Few people wanted to figure out the appropriate abbreviation for the *Annals of Mathematics* or where to put the volume number versus the publication year. So many of us looked at the references in another paper to sort that out. With some of the tools built into MathSciNet, such shortcuts are no longer necessary. At Mathematical Reviews, we work very hard to make sure we have complete and accurate bibliographic information for the entries in MathSciNet. We also work hard to make it easy for you to use that information. My earlier post References and Citations tells you ways to do that, including obtaining the information in BibTeX format.

If you use the bibliographic data from MathSciNet in your references, then everybody’s matching algorithms will have a much easier time pairing those references with the paper in their databases. This helps people count. And you will have an easier time writing up the paper!

¹ This is related to the problem of *Alice’s Restaurant*. Sometimes what we call a thing is not the name of the thing. There is a song called *Alice’s Restaurant*, which is about Alice’s restaurant. But “Alice’s Restaurant” is not the name of the restaurant – it’s just the name of the song *about* the restaurant.

**We have given MathSciNet some upgrades.**

As of January 3rd, 2017, MathSciNet will be running on new software, which has allowed us to add some great new features, with more to come in February 2017. This post provides some highlights of how MathSciNet is bigger, better, faster, more.

And please visit the AMS Booth at the JMM in Atlanta to see demonstrations of the new features of MathSciNet, as well as to meet some of the Editors from Mathematical Reviews. There will be free access to MathSciNet at JMM!

Lists of search results will be sortable in a number of different ways, including chronologically, reverse chronologically, and by citations.

Here are the results of a search for publications with Anywhere=(fundamental lemma), sorted three different ways:

**Classic sort: Newest First**

**Oldest First**

**Sorted by Number of Citations**

Previously, searches in MathSciNet assumed you were looking for a phrase. Now, searches in MathSciNet insert Boolean **AND** between terms by default. Thus, the search

will return all items where the word **fundamental** and the word **lemma** both occur somewhere. They need not be adjacent.

The search

will search for the phrase **fundamental lemma** anywhere in our listing of the item. The two words will need to be adjacent to each other.

Here is what the results of the second search, for “fundamental lemma”, look like:

Notice that the number of matches has dropped from 2510 to 512, which is to be expected.

Here are the results sorted by citations:

Lists of search results will now include facets that allow users to filter and refine searches. These are the choices listed in the sidebar on the left. The possible refinements here are item type, author, institution, primary classification, journal, and year. Returning to the example just listed (searching for the phrase “fundamental lemma”), we can refine the search by choosing an author:

These matches are based on the author identification (author disambiguation) that goes into creating the Mathematical Reviews author database. (See Who Wrote That? for more on our author identification.) The numbers in parentheses tell how many matches there are: 14 matches for Báo Châu Ngô, 13 for Yuval Z. Flicker, 13 for Jean-Loup Waldspurger, and so on.

Clicking on **Laumon, Gérard** produces

Notice that the refined results are still sorted by number of citations. The choice of sorting follows along with the refining process.

In this screen shot, notice that the first two items have a light blue background:

That is because the two items are related. Indeed, they are Parts I and II. We gather them together, even though they were published as two separate books. Indeed, they have one joint review, written by Jonathan David Rogawski.

We can further refine the search by making a choice of Primary Classification.

We see that there are just two choices: Number Theory (MSC=11) with 6 items and Topological groups, Lie groups (MSC=22) with 4 items. Let’s look at the 4 items with primary classification Topological groups, Lie groups:

Notice that the results are still sorted by “Citations”.

Another way to refine these results further is to enter a new search term in the “Search within results” box. This adds the new search term and looks for it anywhere in our record for the item: Author, Title, Journal, Review Text, etc. This example is left as an exercise to the reader…

The results of an Author Search now display with more information. Let’s use the Fields Medalist Wendelin Werner as an example, pretending that we don’t remember his first name. Navigating to the **Authors** tab, we enter “Werner” in the search field.

The results look vastly different now:

There is plenty of extra information to help choose the Werner we were actually looking for. In this case, there are 503 matches. My screen shot only allows us to see the first 9, sorted by Profile Name. **Note** that the search is looking for “Werner” either as a first name or as a family name. By default, the results are listed alphabetically by Profile Name (the name that appears in the Author Profile). We can re-sort by Citations:

There are five matches with over 1000 citations. Only one of them has last name Werner. We can also see that this Werner started publishing at around the right time to have recently won a Fields Medal. Clicking on Werner, Wendelin brings us to his Author Profile page on MathSciNet

An alternative way to have found Wendelin Werner, assuming we knew something of his mathematics, would have been to refine by Primary Classification. That is to say, to select the classification in which he has published most frequently. In Werner’s case, that is Probability Theory and Stochastic Processes (MSC=60). Note that 38 of these authors have this class as their most frequently occurring MSC.

Selecting Probability Theory and Stochastic Processes yields

Note that, again, the sorting by Citations has followed us as we have refined the search. Wendelin Werner, the Fields Medalist, is now the top result. Clicking on his name leads us to his Author Profile page

These are just a few highlights of what can be done with the new features of MathSciNet. You can watch a demo video of some of the sorting and faceting improvements. (The video was made when the new features were still prototypes – so some things will look slightly different.) But, the best thing is to try them out — after January 3rd!

And keep watch for more new features that will come later in February 2017:

**Search Result Alerts:**Users will be able to log in and create email alerts that will send them any new results for queries that they create. This will allow users to be notified when, for example, an author’s citations count changes, or a new issue is added to a journal.**Autosuggest for Journals and Author Search Boxes:**MathSciNet’s journals and author search boxes will now suggest useful search strings for journal titles and author names.

I would be remiss if I didn’t thank **Erol Ozil**. He is the head of IT at Mathematical Reviews and is the mastermind behind these new features. To begin with, he set the stage for the upgrades by initiating the change in our search software. He then did an amazing job of getting the changes coded in the new environment. As database experts know, this requires restructuring the underlying databases to take advantage of the new code. There were a thousand issues, both large and small, that came up along the way. Erol dealt with them all.

Thank you, Erol.

]]>

Here is a list of the prizewinners that have been announced so far, with links to their Author Profiles on MathSciNet and to the news announcements of the prizes. (A list of all AMS Prizes and Awards is available here.) From the Author Profiles, you can explore the award-winning work of these mathematicians.

- James G. Arthur to receive 2017 AMS Steele Prize for Lifetime Achievement
- Leon Simon to receive 2017 AMS Steele Prize for Seminal Contribution to Research
- Dusa McDuff and Dietmar Salamon to receive 2017 AMS Steele Prize for Exposition
- András Vasy to receive 2017 AMS Bôcher Prize
- Henri Darmon to receive 2017 AMS Cole Prize in Number Theory
- Laura DeMarco to receive 2017 AMS Satter Prize
- László Erdős and Horng-Tzer Yau to receive 2017 AMS Eisenbud Prize
- John Friedlander and Henryk Iwaniec to receive 2017 AMS Doob Prize
- David Bailey, Jonathan Borwein, Andrew Mattingly, and Glenn Wightwick to receive 2017 AMS Conant Prize
- David H. Yang to Receive 2017 AMS-MAA-SIAM Morgan Prize.

Congratulations!

]]>You can see other statistics about us at MathSciNet by the Numbers.

]]>John O’Connor found 1706 mathematicians with biographies in their archive who have author profiles in MathSciNet. For each of these mathematicians, they have inserted a link from their biography to our Author Profile Page. (It’s at the end of the biography, with the other links.) And reciprocally, we have a link from the MathSciNet Author Profile page to the biography page at St. Andrews.

Here are 37 examples:

- Lars Ahlfors: MathSciNet, MacTutor
- Vladimir Arnold: MathSciNet, MacTutor
- Sergei Bernstein: MathSciNet, MacTutor
- Lipman Bers: MathSciNet, MacTutor
- Raoul Bott: MathSciNet, MacTutor
- Alberto Calderón: MathSciNet, MacTutor
- Constantin Carathéodory: MathSciNet, MacTutor
- Henri Cartan: MathSciNet, MacTutor
- S.S. Chern: MathSciNet, MacTutor
- Richard Courant: MathSciNet, MacTutor
- H.S.M. Coxeter: MathSciNet, MacTutor
- Jean Dieudonné: MathSciNet, MacTutor
- Ferdinand Gotthold Max Eisenstein: MathSciNet, MacTutor
- Arthur Erdélyi: MathSciNet, MacTutor
- Paul Erdős: MathSciNet, MacTutor
- Friedrich Ludwig Gottlob Frege: MathSciNet, MacTutor
- Israel Gelfand: MathSciNet, MacTutor
- Josiah Willard Gibbs: MathSciNet, MacTutor
- Boris Vladimirovich Gnedenko: MathSciNet, MacTutor
- Daniel Gorenstein: MathSciNet, MacTutor
- Hans Grauert: MathSciNet, MacTutor
- Godfrey Harold Hardy: MathSciNet, MacTutor
- Olga Alexandrovna Ladyzhenskaya: MathSciNet, MacTutor
- Anneli Cahn Lax: MathSciNet, MacTutor
- Peter Lax: MathSciNet, MacTutor
- André Lichnerowicz: MathSciNet, MacTutor
- Otto Neugebauer: MathSciNet, MacTutor
- Ilya Piatetski-Shapiro: MathSciNet, MacTutor
- Marina Ratner: MathSciNet, MacTutor
- Julia Hall Bowman Robinson: MathSciNet, MacTutor
- James Burton Serrin: MathSciNet, MacTutor
- Claude Elwood Shannon: MathSciNet, MacTutor
- John Robert Stallings: MathSciNet, MacTutor
- William Paul Thurston: MathSciNet, MacTutor
- Hermann Klaus Hugo Weyl: MathSciNet, MacTutor
- Jean-Christophe Yoccoz: MathSciNet, MacTutor
- Oscar Zariski: MathSciNet, MacTutor

I am very grateful to John O’Connor who did the matching between the two sites.

]]>Here are some examples with links:

Henri Cartan

Claude Chevalley

Georges de Rham

Jean Dieudonné

Wolfgang Doeblin

Charles Ehresmann

Gaston Julia

Pierre Lelong

André Lichnerowicz

Raphaël Salem

Laurent Schwartz

André Weil

To find the listing for an item that comes from contributed data for a thesis, you should go to the main search page of MathSciNet and enter “Thesis” in the “MSC Primary”. If you want to narrow the results to just those theses coming from NUMDAM, enter “NUMDAM” in the “Anywhere” field. The biggest source of data and links for theses we have is from ProQuest. As of this morning, we have metadata for 79,758 ProQuest theses. (More information about theses in MathSciNet was given in an earlier post.)

We are very grateful to NUMDAM and to ProQuest for their help in providing these data and links to users of MathSciNet.

]]>

Fall Central Sectional Meeting

University of St. Thomas, Minneapolis, MN

October 28-30, 2016 (Friday – Sunday)

Mike Jones will be giving the demos and answering questions.

Fall Southeastern Sectional Meeting

North Carolina State University, Raleigh, NC

November 12-13, 2016 (Saturday – Sunday)

Edward Dunne will be giving demos and answering questions. (Yes, that’s me!)

]]>

Lots of information is on the card. Note that before the annotation, nothing is labeled. There are accepted rules that tell a librarian (or a patron) what each piece of data is. For most pieces of this data, a non-librarian would be likely to figure out what everything meant.

In an online catalog (such as from the Library of Congress), you might see:

Personal name: |
Malinowski, Bronislaw, 1884-1942. |

Title: |
Magic, science and religion and other essays |

Published/Created: |
Boston, Beacon Press, 1948. |

LCCN Permalink: |
https://lccn.loc.gov/48006987 |

Description: |
xii, 327 p. front. 22 cm. |

LC classification (full): |
GN8 .M286 |

LC classification (partial): |
GN8 |

Related names: |
Redfield, Robert, 1897-ed. |

Contents: |
Magic, science and religion.–Myth in primitive psychology.–Baloma: the spirits of the dead in the Trobriand Islands.–The problem of meaning in primitive language.–An anthropological analysis of war. |

Subjects: |
Anthropology. |

Notes: |
Bibliographic footnotes. |

LCCN: |
48006987 |

Dewey class no. |
572.04 |

Type of material: |
Book |

Notice that now the data fields are all labeled.

However, in modern libraries, there is more information than (normally) meets the eye. Here is a more detailed view of the digital record, as provided by the MARC record:

000 01142cam a22002891 4500

001 8352402

005 20050721194408.0

008 750617s1948 mau b 000 0 eng

035 __ |9 (DLC) 48006987

906 __ |a 7 |b cbc |c oclcrpl |d u |e ncip |f 19 |g y-gencatlg

010 __ |a 48006987

035 __ |a (OCoLC)1395532

040 __ |a DLC |c FTS |d OCoLC |d DLC

042 __ |a premarc

050 00 |a GN8 |b .M286

082 __ |a 572.04

100 1_ |a Malinowski, Bronislaw, |d 1884-1942.

245 10 |a Magic, science and religion, and other essays; |c selected and with an introd. by Robert Redfield.

260 __ |a Boston, |b Beacon Press, |c 1948.

300 __ |a xii, 327 p. |b front. |c 22 cm.

504 __ |a Bibliographical footnotes.

505 0_ |a Magic, science and religion.--Myth in primitive psychology.--Baloma: the spirits of the dead in the Trobriand Islands.--The problem of meaning in primitive language.--An anthropological analysis of war.

650 _0 |a Anthropology.

700 1_ |a Redfield, Robert, |d 1897- |e ed.

985 __ |e OCLC REPLACEMENT cdsdistr

991 __ |b c-GenColl |h GN8 |i .M286 |t Copy 1 |w OCLCREP

991 __ |b c-GenColl |h GN8 |i .M286 |p 00017792015 |t Copy 2 |w CCF

Wow! Lots more information! Moreover, I can’t understand most of it. It is all labeled, but using a secret code. How is this helpful? Well, MARC stands for “MAchine Readable Cataloging”. So I’m not supposed to be able to understand this, but a computer parses the information easily. This is an example of metadata for the digital world. (The machine code is available here as an XML file, in case you want to be unable to read it in another format.)

In the right hands, metadata are pieces of information that is developed, structured, and maintained to describe materials in ways that meet the particular needs of a group of users. At Mathematical Reviews, metadata for each bibliographic entry in MathSciNet are created to serve the research needs of mathematicians, librarians, and others who work with the mathematics literature. Metadata describe each item listed in the database in terms of its type of publication, creators and publisher, length, edition, online availability, subject area (using MSCs), and other identifying characteristics.

MathSciNet metadata provide consistent and well-structured information about what has been published in mathematics over time and how publications are related to each other based on elements such as authors, publication date, subject matter, references, and editions. Staff members at Mathematical Reviews work to assure the consistency and accuracy of the metadata created for roughly 120,000 items each year. High quality metadata are essential to the many features of MathSciNet including author and publications searching, the Citation Database, Author Profile pages, and the rich and growing set of links within MathSciNet that enable efficient and accurate exploration of the mathematical literature.

The creation of bibliographic metadata at Mathematical Reviews takes place at two levels. The first level involves the bibliographic description of books, journals and issues. At this level, the work resembles the cataloging at a library. The second level focuses on the description of individual papers within journal issues and book collections. At this second level, the work can be quite different, working in finer detail with a narrower range of data. Cataloging principles and standards guide the bibliographic description of materials at Mathematical Reviews. One important set of principles is found in the Resource Description and Access (RDA) standards. (See also this page from the Library of Congress.) These and other principles and standards must be interpreted and incorporated into a cataloging framework designed to meet the information needs of the mathematical community. Adhering to principles and standards allows us to maintain continuity and consistency in MathSciNet across all the literature we cover, and across the wider bibliographic world (e.g., your library).

Publishers create metadata for their publications. One of the earliest resources was Bowker’s *Books in Print*. (See also this.) Years ago, this was produced annually as a bound volume that you could find in your library or bookstore. It attempted to list the bibliographic information of every book printed in a given year. It was a book about books – metadata! For scholarly and academic journals, there was Ulrich’s *Periodicals Directory*. Bowker’s and Ulrich’s only had the information from the publishers who gave it to them. Both resources were widely used in libraries, for purchasing as well as for identification and awareness purposes. Bowker’s was also heavily used by bookstores and book distributors, particularly for making purchases. Needless to say, it was a good idea for publishers to provide data to Bowker’s and to Ulrich’s. Both are now solely online.

For books, Bowker’s established a format for the electronic delivery of metadata. Data in this format was frequently used by both online bookstores and brick-and-mortar bookstores. Other formats exist, such as what amazon.com uses. The Library of Congress records described above are important. When readying a book, the publisher applies for “CIP data” – cataloging in publishing data. The record returned to the publisher from the Library of Congress becomes an important component of the metadata attached to the publication.

Mathematical Reviews receives metadata from publishers through our Preliminary Data program. Publisher metadata received in this program are used to create preliminary entries in MathSciNet, while editorial decisions, cataloging, and classification are completed. These items are marked in MathSciNet with the icon . Preliminary MathSciNet entries speed up the availability of information about publications by several weeks.

While preliminary data accelerate the posting of items to MathSciNet, it is not the case that the data arrive and we can just blithely post away! Data arrive in all sorts of formats. And with various extras. Upon arrival, preliminary data go through an initial check, during which materials such as cover images, front and back matter, and other non-article information are removed. Duplicate materials are also removed and issue and journal level information is verified and corrected as needed. The papers are then sent to our editors, who select those for which permanent listings in MathSciNet will be made.

The selected papers move through the departments at Mathematical Reviews and the bibliographic data are checked, edited, and enhanced at each point. Discrepancies between the data received and the published online version are addressed and additional information about the paper may be added. Author disambiguation is done, links to Author Profile pages are made, institutional codes are added, and classifications assigned. When these steps are completed, the preliminary entry is replaced with its permanent and complete MathSciNet entry. Articles from approximately 800 journals now arrive through our Preliminary Data program. We continue to add journals to the program and look forward to working with additional publishers in the future.

Accurate, timely, and consistent metadata are integral to the information and services we provide to the mathematical community. MathSciNet metadata development is an ongoing process accomplished by experienced and dedicated staff throughout Mathematical Reviews.

I am very grateful to Kathy Wolcott, Librarian and Manager of the Mathematical Reviews Acquisitions Department, for help with this post. Large swathes of this post are ~~plagiarized~~ verbatim from a document written by her.

Is “data” singular or plural? For that question, I defer to the wisdom of xkcd.

]]>Sargsyan’s review is exceptional because he takes the time to explain the context of the paper as well as the paper’s contents. He has a few criticisms, too, but presents his opinions in a constructive way. Sargsyan starts by taking four paragraphs to set the stage for the subject. At the end of the fourth paragraph, we discover the objective of inner model theory (showing that certain minimal and canonical models exist for all large cardinals). A few more paragraphs about what you can do with inner model theory leads Sargsyan up to the declaration that “inner model theory is simply great”, but “learning it is tough”. Sargsyan describes Steel as “one of the most experienced and accomplished inner model theorists”, and thinks the task of learning the theory is unimaginable without Steel’s outline. But… Steel has set himself a big task, which is to “present the theory in the greatest possible generality known at the time”. To accomplish this, the reader either needs to know what inner model theory is generally about or be ambitious (i.e., willing to take things on faith to forge ahead). Steel recommends agains the second method. And he even suggests how to construct your own crash course in getting to read this 90-page Outline of inner model theory. (I would add one item to that list: read this review!) Sargsyan encourages the readers to slog through all this because at the end you get a prize: enough understanding to do research in a beautiful, deep area of mathematics.

MR2768698

Steel, John R.(1-CA)

An outline of inner model theory. Handbook of set theory. Vols. 1, 2, 3, 1595–1684, Springer, Dordrecht, 2010.

03E45 (03E15 03E35 03E55 03E60)

It is a well-documented phenomenon that the set-theoretic universe becomes more complex as one assumes stronger and stronger large cardinal axioms. For instance, the existence of a measurable cardinal implies that the universe is far away from the constructible universe $L$, and this can be measured via objective means. If there is a measurable cardinal then the constructible universe computes no successor cardinal correctly and, in particular, the set of reals of the constructible universe is countable.

There are other ways in which the universe becomes complex under the assumption that it contains large cardinals. For example, the existence of infinitely many Woodin cardinals implies that there is no projective well-ordering of the reals. Or the existence of a supercompact cardinal implies the failure of the square principle above it.

On the other hand, it is known that many large cardinals can exist in universes that are well understood and permit detailed analysis, universes that after being analyzed and thoroughly studied are not so complex anymore. For example, it is a well-known fact that, assuming the existence of a measurable cardinal, there is a minimal transitive model that contains all ordinals and has a measurable cardinal. It is important to note that by minimal we do not mean “minimal under inclusion” but rather that it generates all the others, or is embedded into all the others. This minimal model has the form $L[\mu]$ where $L[\mu]\models “\mu$ is a normal $\kappa$-complete ultrafilter on $\kappa$ for some $\kappa$”. It was shown by K. Kunen that any other model of this form is obtained from the minimal one by iterating the ultrapower construction. The modern representation of $L[\mu]$ as a mouse is somewhat more involved but has the additional advantage that it permits a complete analysis similar to the fine structural analysis of $L$ carried out by R. B. Jensen in his seminal paper “The fine structure of the constructible hierarchy” [Ann. Math. Logic 4 (1972), 229–308; erratum, ibid. 4 (1972), 443; MR0309729].

Similar facts are known for large cardinals up to the level of a Woodin cardinal that is itself a limit of Woodin cardinals and somewhat past it. Showing that such minimal and canonical models exist for all large cardinals is the goal of inner model theory.

Any substantial theory needs test problems and unexplained phenomena that lead the way to further developments. Resolving these test problems and explaining the unresolved phenomena is what keeps us, the practitioners and builders of the theory, honest in our pursuit. Otherwise, in the case of inner model theory, why not just declare $V$ as canonical and be done with it? Inner model theory has both test questions that it tries to resolve and unexplained phenomena that it tries to elucidate.

The inner model-theoretic approach to building minimal and canonical models for large cardinals has been used to calibrate lower bounds of set-theoretic statements. For instance, it is the most successful method for deriving strength from the Proper Forcing Axiom, which has been conjectured to have large cardinal strength—that of a supercompact cardinal. The problem of whether the consistency of the Proper Forcing Axiom implies the consistency of ${\sf{ZFC}}\,+\,$“there is a supercompact cardinal” is one of the main test questions of inner model theory.

The existence of canonical inner models for large cardinals explains our experience with determinacy. For instance, projective determinacy is true because for each $n$ there is a canonical inner model, a mouse, with $n$ Woodin cardinals. In fact the two theories, projective determinacy and the theory ${\sf{ZFC}}\,+\,$” for each $n$ there is a mouse with $n$ Woodin cardinals”, are equiconsistent. Similar results hold for more complex forms of determinacy and more complex canonical inner models. In a sense, the goal of inner model theory is to show that all determinacy is a consequence of the existence of mice with large cardinals and vice versa. That this is true is the content of the Mouse Set Conjecture.

In short, inner model theory is simply great. However, anyone who has any experience with it can say that learning it is tough. The chapter under review (in what follows, the Outline), written by John R. Steel, one of the most experienced and accomplished inner model theorists, makes the experience of learning it significantly easier. I don’t know how students who came before me, before the time of the Outline, learned inner model theory. For me, learning this subject simply means developing a thorough understanding of every section of the Outline. It is not so clear to me what the meaning of “learning inner model theory” would be in a world that is not blessed with this chapter.

The goal of the author is clear; he states it in the second paragraph. He is setting out to “present the theory in the greatest possible generality known at the time”. This goal is too grand, yet the author accomplishes it with as few shortcomings as is possible. It is the best exposition of inner model theory available, and I highly recommend it to both students and researchers.

Since we have been glorifying inner model theory and the Outline, it is fair that we start with some of the Outline’s shortcomings. There is simply no quick fix, no magic potion that we take and after that we know inner model theory. Anyone who aims to present the theory in the “greatest possible generality” must make some concessions.

The first concession made is the amount of background material that is assumed. There is a lack of standard graduate textbooks in set theory, and so it is hard to say what the minimum required level is for reading the Outline with a relative ease. To read the later sections, one certainly has to be familiar with forcing. While notions such as extenders, elementary embeddings and Woodin cardinals are introduced in the chapter, I do think one has to have a prior knowledge of these topics. The introduction to extenders is too quick and there are no exercises, so while theoretically it is possible to just learn extenders from the chapter and ambitiously continue, I think those who have worked out some exercises on extenders and have read the relevant chapters of the books by T. J. Jech [Set theory, the third millennium edition, revised and expanded, Springer Monogr. Math., Springer, Berlin, 2003; MR1940513] and A. Kanamori [The higher infinite, second edition, Springer Monogr. Math., Springer, Berlin, 2003; MR1994835] will have a better understanding of the material.

The next basic shortcoming of the chapter is the lack of a historical introduction. In the modern world of mathematical publishing it is taboo to write papers without an appropriate introduction. The author does so and he gets away with it perfectly, as what he presents in the chapter has a far greater value than conforming to certain unjustified taboos. Nevertheless, he pays a relatively high cost for this.

The chapter is simply not accessible to those who don’t already know what inner model theory is about. The author, however, knows that his intended audience has this knowledge. After all, what follows a one-page Introduction is a section that introduces premice, the main object studied by inner model theorists. The reasons behind our endless search for canonical models for large cardinals among the models of the form $L[\vec{E}]$ are not given. Fine structural notions are not motivated, and the motivation behind the definition of fine extender sequence is not given (Definition 2.4). The non-sophisticated reader, one who has not acquired the proper background for reading the chapter, might have the feeling that one sunny day these notions just fell down from the sky.

If our ambitious reader, the one who ignored the warning to become more sophisticated elsewhere, made it past the initial hurdles, it is hard to see how they could get past Definition 2.7. Here three types of premice are introduced without any prior warning. The author makes an attempt to encourage our ambitious reader to venture on by saying that these distinctions are not important in what follows and only appear in technical details suppressed by the Outline. This is indeed true, so if you are the ambitious reader with no or limited prior knowledge of inner model theory don’t give up yet; however, there will be more of this. For example, wait until you reach the point where you are suddenly asked to go and read about the $r\Sigma_n$ hierarchy somewhere else (prior to Definition 2.20). Let me stop this. I simply do not recommend the Outline to anyone who wants to acquire practical knowledge of inner model theory and wants to use the Outline as their first source. In all fairness to the author, it is indeed the case that if you work out the courage to ignore some of the technicalities, then you can work through the Outline. But there is a better, more productive way to use the chapter.

The following is a reasonable way of approaching the Outline. First one should master forcing and basic large cardinal theory. The excellent books by Kunen [Set theory, reprint of the 1980 original, Stud. Logic Found. Math., 102, North-Holland, Amsterdam, 1983; MR0756630; Set theory, Stud. Log. (Lond.), 34, Coll. Publ., London, 2011; MR2905394], Jech [op. cit.] and Kanamori [op. cit.] can be used for this goal. One also needs to have an understanding of the theory of $0^\sharp$ and the theory of $L[\mu]$. Again, the texts of Jech and Kanamori can be used for this. A reader of the Outline who expects to understand the material should in addition to the aforementioned topics also work through Jensen’s “Fine structure of the constructible universe” [op. cit.]. One also needs to be motivated. The Introduction of the Outline contains excellent references that deal with history and motivations. I highly recommend Jensen’s “Inner models and large cardinals” [Bull. Symbolic Logic 1 (1995), no. 4, 393–407; MR1369169]. At this point one can start reading the Outline with relative ease.

I would also recommend fragments of “Fine structure and iteration trees” by W. J. Mitchell and Steel [Lecture Notes Logic, 3, Springer, Berlin, 1994; MR1300637] and “A proof of projective determinacy” [J. Amer. Math. Soc. 2 (1989), no. 1, 71–125; MR0955605] and “Iteration trees” [J. Amer. Math. Soc. 7 (1994), no. 1, 1–73;MR1224594] by D. A. Martin and Steel. One can also read more modern expositions, for instance “Fine structure” by R.-D. Schindler and M. Zeman [in Handbook of set theory. Vols. 1, 2, 3, 605–656, Springer, Dordrecht, 2010; MR2768688] or “Determinacy in $L({\Bbb R})$” by I. Neeman [in Handbook of set theory. Vols. 1, 2, 3, 1877–1950, Springer, Dordrecht, 2010; MR2768701]. See also “A recommended roadmap into inner models”, a question by Asaf Karagila at MathOverflow (http://mathoverflow.net/questions/73075/). Then everything in the Outline will make perfect and clear sense.

Sections 1–6 of the Outline are taken from various papers. Nevertheless, the Outline offers a short and succinct presentation of these topics, and often it communicates ideas that cannot be found elsewhere. For instance, the Outline contains a sketch of the proof of Condensation that has all the main ideas clearly presented. In Section 6, the proof of the Branch Existence Theorem is left out and instead it is explained how it leads to iterability via unique branches. This is also where the all too useful zipper argument is presented (Theorem 6.10). Also, the idea of $\mathcal{Q}$-structures is introduced and it is shown how it leads to uniqueness of branches (Corollary 6.14). I am unaware of any other publication prior to the Outline that makes a strong case for the idea that the good branches must be identified via good $\mathcal{Q}$-structures. This is one of the ideas that invites descriptive set theory into inner model theory. One starts looking for such canonical $\mathcal{Q}$-structures inside the largest countable set operators associated with nice pointclasses.

Section 2 introduces premice and fine structure. Section 3 introduces iteration trees and proves the Comparison Lemma (Theorem 3.11). Section 4 contains technical yet important lemmas that are used in showing that the various constructions of mice converge. Here one can learn about the Dodd-Jensen lemma and the weak Dodd-Jensen lemma. These are important results used in the proofs of various fine structural facts such as universality and solidity, which are the topic of Section 5. Section 6 deals with the constructions producing mice, the $K^c$ constructions. Here everything introduced up to this point comes together. The convergence of the $K^c$constructions is reduced to the iterability conjecture (Conjecture 6.5), which says that the countable substructures of the models appearing in a $K^c$ construction are iterable.

One shortcoming is that there is no discussion of the different background conditions that can be used in $K^c$ constructions. Varying such conditions produces different models, and in many applications the countably certified $K^c$ defined in the Outline is not good enough. Another shortcoming is that there is no review of core model theory (for a quick introduction to this topic, the reader may consult the paper by B. Löwe and Steel [in Sets and proofs (Leeds, 1997), 103–157, London Math. Soc. Lecture Note Ser., 258, Cambridge Univ. Press, Cambridge, 1999; MR1720574]). Perhaps the biggest disappointment is that there are no proofs of iterability anywhere in these sections. However, the Outline should be viewed as a section of the Handbook of set theory edited by M. Foreman and Kanamori. Here it is published alongside Neeman’s “Determinacy in $L({\Bbb R})$” [op. cit.], E. Schimmerling’s “A core model toolbox and guide” [in Handbook of set theory. Vols. 1, 2, 3, 1685–1751, Springer, Dordrecht, 2010; MR2768699], Schindler and Zeman’s “Fine structure” [op. cit.] , P. D. Welch’s “$\Sigma^*$ fine structure” [in Handbook of set theory. Vols. 1, 2, 3, 657–736, Springer, Dordrecht, 2010; MR2768689], and W. J. Mitchell’s “Beginning inner model theory” [in Handbook of set theory. Vols. 1, 2, 3, 1449–1495, Springer, Dordrecht, 2010; MR2768696] and “The covering lemma” [in Handbook of set theory. Vols. 1, 2, 3, 1497–1594, Springer, Dordrecht, 2010;MR2768697]. These works together contain most of what is left out by the Outline. In particular, Schimmerling’s chapter contains a guide to core model theory and Neeman’s chapter contains a proof of iterability.

The highlight of the Outline is its last two sections, sections 7 and 8, which can be viewed as an introduction to descriptive inner model theory [see G. Sargsyan, Bull. Symbolic Logic 19 (2013), no. 1, 1–55; MR3087400]. This is a subject that lies in the crossroads of descriptive set theory and inner model theory. The two subjects meet in several ways. Often descriptive set-theoretic methods are used to show that the $\omega_1$-iteration strategies that we build are universally Baire, and hence, generically absolute. The aforementioned $\mathcal{Q}$-structure idea is one way of showing that $\omega_1$-iteration strategies are universally Baire. For example, if $V$ is closed under the sharp function, $x\mapsto x^\sharp$, and an iteration strategy is guided by $\mathcal{Q}$-structures whose descriptive set-theoretic complexity is below the $\sharp$ operator, then the strategy in question is universally Baire.

Often inner model theory is used to analyze the structure of the universe under the assumption of ${\sf{AD}}$ or ${\sf{AD}}^+$. A typical example is the computation of the inner model ${\rm HOD}$ of models of determinacy. It is a long-standing open problem whether the ${\rm HOD}$ of models of ${\sf{AD}}^++V=L({\wp}({\Bbb R}))$ satisfies ${\sf{GCH}}$. It is known that it satisfies ${\sf{CH}}$. There has been a great deal of progress on this question using methods from inner model theory.

The goal of the last two sections of the Outline is to introduce two important topics of descriptive inner model theory. The first one is mouse capturing. This is the statement that under ${\sf{AD}}^++V=L({\wp}({\Bbb R}))\,+\,$ “there is no mouse with a superstrong cardinal”, for $x, y\in {\Bbb R}$, $x$ is ordinal definable from $y$ if and only if $x$ is in some mouse over $y$.

The Outline only deals with mouse capturing in $L({\Bbb R})$ under the additional assumption that $\mathcal{M}_\omega$ exists. This is not the optimal hypothesis. W. H. Woodin has shown that mouse capturing in $L({\Bbb R})$ is a consequence of ${\sf{AD}}^{L({\Bbb R})}$, and also that the set-theoretic strength of ${\sf{AD}}^{L({\Bbb R})}$ is weaker than that of the existence of $\mathcal{M}_\omega$. The proof of this result can be found, for instance, in “A theorem of Woodin on mouse sets” by Steel [in Ordinal definability and recursion theory: The Cabal Seminar, Volume III, 243–256, Cambridge Univ. Press, 2016].

The first part of Section 7 contains the proof of the first half of mouse capturing in $L({\Bbb R})$. In general, this portion of mouse capturing is just a consequence of comparison, but the author wants to prove more, namely that the ordinal definable reals are exactly those that are in $\mathcal{M}_\omega$. For this one needs techniques that track fragments of the iteration strategy of $\mathcal{M}_\omega$ inside $L({\Bbb R})$. Definition 7.7 introduces weak iterability, and after some discussion a conclusion is drawn that weak iterability is absolute between $V$ and $L({\Bbb R})$ (Theorem 7.8). It is then shown that under $V=L({\Bbb R})+{\sf{AD}}$, weak iterability is equivalent to iterability. This then naturally leads to the fact that every initial segment of $\mathcal{M}_\omega|\omega_1^{\mathcal{M}_\omega}$ is iterable in $L({\Bbb R})$.

The second part of Section 7 contains the proof of the second part of mouse capturing, namely that the ordinal definable reals are only those that are in $\mathcal{M}_\omega$ (Corollary 7.18). This part of the proof uses a great deal of sophisticated machinery. First, Woodin’s extender algebra and genericity iterations are introduced (Theorem 7.14). Genericity iterations are one of the most important techniques of descriptive inner model theory. Given an iterable transitive model $M$with a Woodin cardinal $\delta$ and a real $x$, it is possible to iterate $M$ to make $x$ generic over the iterate.

Such iterations can be dovetailed together to obtain a representation of $L({\Bbb R})$ as a symmetric extension of some iterate of $\mathcal{M}_\omega$, a very useful fact (Theorem 7.15). This fact, coupled with homogeneity of the Lévy collapse, easily implies that every ordinal definable real is in $\mathcal{M}_\omega$. At this point it would have been nice to explain Woodin’s derived model theorem but the author refers to his paper “A stationary-tower-free proof of the derived model theorem” [in Advances in logic, 1–8, Contemp. Math., 425, Amer. Math. Soc., Providence, RI, 2007; MR2322359] for a complete proof of this result that only uses inner model-theoretic techniques.

Section 7 ends with the Mouse Set Conjecture (see the discussion after Definition 7.23), one of the modern driving forces of inner model theory. The Mouse Set Conjecture states that mouse capturing holds. The last sentence of Section 7 is rather dramatic. It expresses the author’s belief that it is impossible to build mice with superstrong cardinals without first proving the Mouse Set Conjecture. This is a belief that has greatly influenced a generation of inner model theorists that grew up reading the Outline.

The second important topic is the computation of ${\rm HOD}$ of models of ${\sf{AD}}^++V=L({\wp}({\Bbb R}))$. Section 8 is devoted to it. Assume $\mathcal{M}_\omega$ exists. Let $\mathcal{H}={\rm HOD}^{L({\Bbb R})}$. It was known to the Cabal group that ${\sf{CH}}$ holds in $\mathcal{H}$, and whether ${\sf{GCH}}$ holds in $\mathcal{H}$ was left open. Motivated by the result that the reals of $\mathcal{M}_\omega$ are exactly those that are in $\mathcal{H}$, Steel and Woodin started investigating the structure of $\mathcal{H}$. Soon Steel showed that, assuming $\mathcal{M}_\omega$ exists, $V_\Theta^{\mathcal{H}}$ is a mouse, and Woodin showed that it is in fact an initial segment of an iterate of $\mathcal{M}_\omega$. Here $\Theta$ is the successor of the continuum. Woodin also computed the full $\mathcal{H}$ by showing that it has the form $L(V_\Theta^{\mathcal{H}}, \Sigma)$, where $\Sigma$ is some fragment of the iteration strategy of $V_\Theta^{\mathcal{H}}$. It is then easy to conclude that $\mathcal{H}\models {\sf{GCH}}$.

Let $\delta$ be the least cardinal such that $L_\delta({\Bbb R})$ is a $\Sigma_1$-elementary substructure of $L({\Bbb R})$. Section 8 contains a proof of the fact that $\mathcal{H}\cap V_{\delta}^{L({\Bbb R})}$ is a mouse (Theorem 8.20). A discussion of other results mentioned above follows but the proofs of these results are not presented. Recently Steel and Woodin wrote a long paper [in Ordinal definability and recursion theory: The Cabal Seminar, Volume III, 257–346, Cambridge Univ. Press, 2016] on ${\rm HOD}$ of models of ${\sf{AD}}^+ + V=L({\wp}({\Bbb R}))$ that contains the proofs of all these facts. Section 8 ends with the result due to the author that in $L({\Bbb R})$, every regular cardinal below $\Theta$ is a measurable cardinal (Theorem 8.27).

There are several important techniques and notions that are used in modern inner model theory that are not covered by the Outline. For example, the theory of homogenously Suslin sets and universally Baire sets is not present, and there is no in-depth discussion of the derived model theorem [see J. R. Steel, in Logic Colloquium 2006, 280–327, Lect. Notes Log., 32, Assoc. Symbol. Logic, Chicago, IL, 2009; MR2562557]. Certainly these are topics that a student of inner model theory must master. Topics that were still under development at the time the Outline was written are also, naturally, not in there. For instance, the reader will not find much on the core model induction or hod mice in the Outline. However, all of these topics have received a fair treatment in other publications [see R.-D. Schindler and J. R. Steel, “The core model induction”, unpublished manuscript, available at wwwmath.uni-muenster.de/u/rds; J. R. Steel, J. Symbolic Logic 70 (2005), no. 4, 1255–1296; MR2194247; G. Sargsyan, Mem. Amer. Math. Soc. 236 (2015), no. 1111, viii+172 pp.; MR3362806], and the Outline does its job at informing the reader of the existence of a world beyond what it covers.

At the beginning of this review, I made the claim that learning inner model theory means mastering the topics covered by the Outline. Then it seemed that I self-contradicted by saying that many details or several topics are omitted. In the Outline what the reader has is exactly what the title promises, an outline of a subject and its main components up to the time it was written. Undoubtedly the reader will have to consult other sources to develop a deeper understanding of inner model-theoretic techniques and notions, but it is absolutely invaluable information to know what exactly to learn, and this is the job of the Outline. A subject that is as spread out through various publications as inner model theory would be close to impossible to learn if we were not blessed with this chapter. The Outline is a torch that guides our way through the labyrinth whose walls are made of extenders, iteration trees, universally Baire representations, mice, ${\rm HOD}$, Woodin cardinals and other unworldly creatures. The prize we get for going through the labyrinth is the knowledge of topics needed to do research in one of the most beautiful and deep areas of set theory.

{For the collection containing this paper see MR2768678.}

Reviewed by Grigor Sargsyan

]]>Before I get going, I should point out that I was involved in the discussion with Tao about publishing some of his blog material as books, a point whose relevance will become clear below. Helfgott’s review has several exceptional characteristics. To begin with, it is thorough – as you might guess at a glance at the length. But it is not just that he says a lot, his discussion is quite thoughtful. As I see in many excellent reviews, Helfgott establishes the context of the work early on in the review. He begins, though, by giving a three-sentence *précis* of the book, followed by a quick three-sentence assessment of the book. So before you get going, you have a rough idea of what you will be reading. The main body of the review is a mix of telling us what is in the book – the main topics as well as the major theorems – and the reviewer’s comments on them. In the discussion of Chapter 1, Helfgott seems surprised that Tao proves a weak form of the Cheeger-Alon-Milman inequality, when with less fuss a shorter and more natural proof (which he sketches) gives an optimal form of the result. As we continue reading the review, we see several more times where Helfgott wonders about Tao’s choices. There are other instances where the book offers up a weaker form of a result, when the stronger form could be offered without a lot of extra fuss. Helfgott also points out how Tao’s great breadth introduces some shorthands that we mere mortals may need a bit more explanation. One example is in Chapter 3 where Tao quickly skips ahead writing “after regularizing $f_n$ in a standard fashion to make it smooth rather than merely Lipschitz”. An analyst (which is one of Tao’s métiers) will be fine with this, but lots of people interested in expanders in finite groups are not analysts. My experience with Tao’s writing is that he is extremely good at conveying how he thinks about particular problems or entire subjects. This can be great when you are able to keep up. If not, you need to do some extra work – though it is invariably worth the effort. Helfgott concludes his review with two general comments: an excellent blog does not automatically translate into an excellent book, but, at the same time, this book is a very valuable resource.

MR3309986

Tao, Terence

Expansion in finite simple groups of Lie type.

Graduate Studies in Mathematics, 164. American Mathematical Society, Providence, RI, 2015. xiv+303 pp. ISBN: 978-1-4704-2196-0

20D06 (05C25 05C81 11B30 17B20 20F65 20G30 22D10)

This is a book based on the author’s blog. The first half consists of lecture notes for a course taught by the author a few years ago on expansion in Cayley graphs. The second half is much more loosely organized; it is essentially a collection of blog posts having a more or less close link to the subject of the book.

The reviewer has just taught a course using this book as one of his main sources. The first thing to say is that it is undoubtedly a very useful resource. However, whoever uses the book for teaching will be well advised to collect other sources, and to take some time to reflect while reading each chapter, pondering whether matters can be taught in a shorter or cleaner way.

In part because the book’s choice of topics is actually quite good, we shall go over the text in some detail. Before starting, let us recall the different definitions of expansion, the book’s main subject. Let $\Gamma$ be a regular graph, i.e., one in which every vertex has the same degree (“valence”) $d$. A regular graph is an $\epsilon$-expander if it has spectral gap of size $\geq \epsilon$, i.e., if the second largest value of the adjacency matrix is $\leq (1 – \epsilon) d$, or, what is the same, if the second smallest value of the discrete Laplacian is $\geq \epsilon$. There are two slightly different definitions of combinatorial expansion, namely, vertex and edge expansion. A regular graph is an $\epsilon$-edge expander if, for every subset $S$ of the set of vertices $V$ with $|S|\leq |V|/2$, there are $\geq \epsilon d |S|$edges with their tails in $S$ and their heads outside it.

A family of graphs is an expander family if there is an $\epsilon>0$ for which they are all $\epsilon$-expanders. This is of particular interest if the degree of the graphs in the family is constant.

Chapter 1 defines expander graphs in the two senses above. The relation between the two is established, in that the text proves a weak form of the discrete Cheeger(-Alon-Milman) inequality: for every $\epsilon>0$ and every $d\geq 1$ there is a $\delta>0$ such that if a graph of degree $d$ is an $\epsilon$-edge expander, then it is a $\delta$-expander. (The other direction, viz., that an $\epsilon$-expander is an $\epsilon/2$-edge expander, is easy.) Random walks are also mentioned; it is an exercise to show that expansion is equivalent to a very strong form of $\ell_2$ mixing in logarithmic time. Finally, it is shown that, for $d$ large, there is an$\epsilon>0$ such that random graphs of degree $d$ with $n$ vertices are $\epsilon$-expanders with probability tending to $1$ as $n\to \infty$, where a “random graph” is taken with respect to a slightly nonuniform distribution.

A few remarks are in order. The text proves a weak form of the Cheeger-Alon-Milman inequality; as it happens, one can do a little better with less trouble than in the book. To obtain a qualitatively optimal form of the inequality (namely, that an $\epsilon$-edge expander has a spectral gap $\delta \gg \epsilon^2$), one just needs to treat an arbitrary function $\phi\:V\to \Bbb{R}$ by first subtracting its median, and then applying what is essentially edge-expansion plus summation by parts (p. 12) to the part of $\phi$ above the median and the part below the median (as one can, given that each part has support of size $\leq |V|/2$, by the definition of “median”). This is shorter and arguably more natural than the proof in the book.

The proof that random graphs are expanders is also needlessly complicated. If one is going to take a nonuniform distribution anyhow, one can allow one’s graphs to have a few doubled edges or self-loops; one can then fix these without breaking the edge-expansion property.

Chapter 2 starts with the definitions of Cayley and Schreier graphs. It proceeds quite quickly, first defining Kazhdan’s property $(\rm T)$ and using induced representations to show that, given a locally compact group $G$ and a locally compact subgroup $H$ of finite covolume, $G$ has property $(\rm T)$ if and only if $H$ has. The idea of the ping-pong lemma is used to establish that if a probability measure on $\Bbb{R}^2$ is almost-invariant under the action of a compact neighborhood in ${\rm SL}_2(\Bbb{R})$, then it is almost entirely concentrated at the origin; this gives us that the pair $({\rm SL}_2(\Bbb{R}) \ltimes \Bbb{R}^2, \Bbb{R}^2)$ has relative property $(\rm T)$, and a construction based on the Mautner phenomenon (via Moore) is then used to show that this implies that ${\rm SL}_3(\Bbb{R})$ has property $(\rm T)$ (Kazhdan). Since ${\rm SL}_3(\Bbb{Z})$ is of finite covolume in ${\rm SL}_3(\Bbb{R})$, we conclude that ${\rm SL}_3(\Bbb{Z})$ has property $(\rm T)$, and so, given any set of generators $S$ of ${\rm SL}_3(\Bbb{Z})$ and any sequence of finite-index normal subgroups $N_n \triangleleft {\rm SL}_3(\Bbb{Z})$, the Cayley graphs $\{\Gamma({\rm SL}_3(\Bbb{Z})/N_n,S \bmod N_n)\}_n$ of ${\rm SL}_3(\Bbb{Z})/N_n$ with respect to $S \bmod N_n$ form an expander family [G. A. Margulis, Problemy Peredači Informacii 9 (1973), no. 4, 71–80; MR0484767]. In general, while the treatment is standard, this is one of the best parts of the book; the presentation is tight, and explains non-trivial concepts clearly.

The chapter ends with a brief sketch of a proof (following [O. Gabber and Z. Galil, J. Comput. System Sci. 22 (1981), no. 3, 407–420; MR0633542] and [S. Jimbo and A. Maruoka, Combinatorica 7 (1987), no. 4, 343–355; MR0931192], and [S. Hoory, N. Linial and A. Wigderson, Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 4, 439–561; MR2247919]) that Schreier graphs associated with ${\rm SL}_2(\Bbb{Z}/n\Bbb{Z}) \ltimes (\Bbb{Z}/ n\Bbb{Z})^2$ form an expander family. This proof skips property $(\rm T)$ and the Mautner phenomenon, and reduces the argument to its essence, viz., the use of the ping-pong lemma as above (applied this time to probability measures on $\Bbb{Z}$) combined with a little Fourier analysis over $(\Bbb{Z}/n\Bbb{Z})^2$. Pedagogically, it might have made more sense to start with this and then introduce property $(\rm T)$ (or just property ($\tau$)) to build matters up to ${\rm SL}_3(\Bbb{Z})$. This can be done quite naturally; one then ends with a proof of property $(\rm T)$ for ${\rm SL}_3(\Bbb{Z})$ that does not go through ${\rm SL}_3(\Bbb{R})$ ([M. Burger, J. Reine Angew. Math. 413(1991), 36–67; MR1089795; Y. Shalom, Inst. Hautes Études Sci. Publ. Math. No. 90 (1999), 145–168 (2001); MR1813225]; see also the expositions in [M. E. B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s property (T), New Math. Monogr., 11, Cambridge Univ. Press, Cambridge, 2008 (§4.2); MR2415834] or [E. Kowalski, The large sieve and its applications, Cambridge Tracts in Math., 175, Cambridge Univ. Press, Cambridge, 2008 (§D.4); MR2426239]). Of course, property$(\rm T)$ for ${\rm SL}_3(\Bbb{R})$ is of interest in itself.

Chapter 3 introduces “quasi-random groups”, meaning groups $G$ that have no non-trivial complex representations of small dimension. (This is, for instance, the case for $G = {\rm SL}_2(\Bbb{F}_p)$, which has no non-trivial complex representations of dimension $<(p-1)/2$ (Frobenius).) In a quasi-random group, the eigenspace of the Laplacian corresponding to a non-trivial eigenvalue cannot have small dimension, since $G$ acts non-trivially on it. This fact is used in two ways. First, it implies that large subsets of $G$ mix very quickly under multiplication (§3.1; see [W. T. Gowers, Combin. Probab. Comput. 17 (2008), no. 3, 363–387;MR2410393; L. Babai, N. V. Nikolov and L. Pyber, in Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 248–257, ACM, New York, 2008; MR2485310]). Second, it can be used, as in [P. C. Sarnak and X. X. Xue, Duke Math. J. 64 (1991), no. 1, 207–227; MR1131400], to prove a version of A. Selberg’s spectral gap [in Proc. Sympos. Pure Math., Vol. VIII, 1–15, Amer. Math. Soc., Providence, RI, 1965; MR0182610] (i.e., the fact that the second smallest eigenvalue of the Laplacian on the surface $\Gamma(N)/\Bbb{H}$ is bounded below by a constant). This spectral gap is then used to show (§3.2) that, for any set of generators $S$ of ${\rm SL}_2(\Bbb{Z})$, the Cayley graphs $\Gamma({\rm SL}_2(\Bbb{F}_p), S \bmod p)$ form an expander family (Lubotzky-Phillips-Sarnak, though the reference [A. Lubotzky, R. S. Phillips and P. C. Sarnak, Combinatorica 8 (1988), no. 3, 261–277; MR0963118] given here really treats other groups).

A note on this last implication: The natural thing would be to define a function $f(z)=\eta(d(z,A_i F_1))$, where $\eta$ is a smooth function, $d$ is the hyperbolic distance, $F_1$ is a fundamental domain for ${\rm SL}_2(\Bbb{Z})$, and $A_i$ is a hypothetical set of vertices with small boundary in the Cayley graph. (We want to show that $A_i$ does not exist, i.e., establish edge-expansion.) Here, what is used is essentially the characteristic function of a union of balls around points (this is fine, but does not show the underlying idea most clearly) and the smoothing is left essentially implicit. This is an example of a more general issue: phrases such as “after regularizing $\tilde{f}_n$ in a standard fashion to make it smooth rather than merely Lipschitz” will be clear to analysts (who barely need such an explanation), but other students will find them less helpful; there is also the issue that it could lead them to learn to hand-wave when they do not yet have the necessary acumen to do so safely. It is better to show them how simply a proof like this can be done in full.

The bound on the spectral gap given here is weaker than the one in [P. C. Sarnak and X. X. Xue, op. cit.] or [A. Gamburd, Israel J. Math. 127 (2002), 157–200;MR1900698]. The reason is that the author replaces the kernel used there by the heat kernel. Now, the original kernel is perfectly fine—it is just the convolution of the characteristic function of a (hyperbolic) disc with itself; this is the natural thing to do if one wants positivity on the spectral side. Exchanging this kernel for the heat kernel weakens and arguably also complicates matters slightly for everybody outside the class of readers who are deeply familiar with it.

Chapter 4 covers the Balog-Szemerédi-Gowers lemma and its consequences. This is really a lemma in graph theory, with a very useful implication on groups; as Tao himself showed, this implication (if framed appropriately) is valid for all groups, not just for abelian groups. We also see the result by J. Bourgain and A. Gamburd on how the lemma implies that a group in which sets grow rapidly under multiplication is a group in which measures become rapidly flatter under convolution by themselves. The version of the proof given in the book is particularly brief and elegant.

Chapter 5 is devoted precisely to the proof that, in $G={\rm SL}_n(K)$, $K$ a field, any set $A\subset G$ that generates $G$ grows rapidly: $|A A A|\geq |A|^{1+\delta}$, with $\delta>0$ a constant depending only on $n$, unless $A$ is already almost as large as $G$ itself. First, the sum-product theorem (Bourgain-Katz-Tao, first completed for small sets by Konyagin) is treated. This makes sense: even though it is not actually used in the treatment used here, it was used in the first proof of a result of this kind (due to the reviewer); moreover, some of the ideas in proofs of the sum-product theorem were abstracted from it (a task that started in [H. A. Helfgott, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 3, 761–851; MR2781932], as the author duly notes) and were crucial later. This is, in particular, the case for the “pivoting” argument.

Some special cases of the Larsen-Pink dimension-based estimates on intersections of finite subgroups of ${\rm SL}_2(K)$ with other subgroups of ${\rm SL}_2(K)$ are then proved. As the text again notes, what turned out to be analogues of these estimates were proved in [H. A. Helfgott, Ann. of Math. (2) 167 (2008), no. 2, 601–623; MR2415382] for intersections of generating sets $A$ of ${\rm SL}_2(K)$ with subgroups of ${\rm SL}_2(K)$; it is these analogues that are used in what follows. Section 5.3 contains what is called here a variant of the original argument to prove the growth result in [H. A. Helfgott, op. cit.; MR2415382] for ${\rm SL}_2$. It is indeed a very nice and easily generalizable variant, using the pivoting argument directly.

Sections 5.4 and 5.5 generalize the result to ${\rm SL}_n$, following [L. Pyber and E. Szabó, J. Amer. Math. Soc. 29 (2016), no. 1, 95–146; MR3402696] and [E. Breuillard, B. Green and T. C. Tao, Geom. Funct. Anal. 21 (2011), no. 4, 774–819; MR2827010]. To simply make clear the potential reach of each method, and not as any claim on “credit”: one can prove all the dimensional estimates here using the approach in [H. A. Helfgott, op. cit.; MR2781932], though it is best to systematize it further; as became clear only later, when the Lie algebra is simple, this can be done very cleanly, without algebraic manipulations that can look like ad hoc arguments.

Section 5.5 proves the dimensional estimates in the needed generality. Here the author chooses to use nonstandard analysis, through ultraproducts. This is unnecessary.

Section 6 sketches a proof of a non-concentration estimate needed by Bourgain and Gamburd’s proof [Ann. of Math. (2) 167 (2008), no. 2, 625–642; MR2415383] of expansion for ${\rm SL}_2(K)$ (which they later generalized to other groups). This proof is based on $|A A A|\leq |A|^{1+\delta}$ via flattening of measures, as said above. It is also proved that random generators give a Cayley graph with large girth; large girth is enough for Bourgain and Gamburd’s proof to apply.

Section 7 starts with a concise sketch of the Rosser-Iwaniec $\beta$-sieve (which, strangely, is uncredited here). After a discussion of strong approximation, the text gives a brief treatment of the work by Bourgain, Gamburd and Sarnak (“affine sieve”) [Invent. Math. 179 (2010), no. 3, 559–644; MR2587341], which uses the expansion property to sieve a set by the action of ${\rm SL}_2(\Bbb{Z})$, or other non-abelian groups. (In this framework, a classical sieve is seen to sieve by the action of $\Bbb{Z}$.) This is the end of the first part.

It may be regretted that permutation groups are barely touched upon. This is understandable: while there are many points in common in the study of growth over linear algebraic groups and over permutation groups, the study of the latter has arguably not yet reached quite the same stage of development as the former.

It would also have been desirable for the book to contain more material on random walks. There is only passing mention of mixing times—and short mixing times are one of the main applications of expansion. At the very least, one might have wished for a brief exposition on how diameters, mixing times and spectral gaps relate, in all directions of implication. Whoever uses the book for teaching or self-study would be well advised to supplement it with, say, [P. W. Diaconis and L. Saloff-Coste, Ann. Appl. Probab. 3 (1993), no. 3, 696–730; MR1233621], [L. Babai and M. Szegedy, Combin. Probab. Comput. 1 (1992), no. 1, 1–11; MR1167291], [L. Babai, in Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, 822–831, ACM, New York, 2006; MR2368881] and some sections of [D. A. Levin, Y. Peres and E. L. Wilmer, Markov chains and mixing times, Amer. Math. Soc., Providence, RI, 2009; MR2466937].

We come to the second part of the book. The structure here is less tight; the chapters, except for the last two, are not really connected to each other.

Chapter 8 uses Cayley graphs (drawn in many colors) to illustrate group cohomology. This is interesting, but not germane to the topic. The chapter could have easily been omitted.

Chapter 9 proves the Lang-Weil bound on rational points on varieties. This is essentially a consequence of the Hasse-Weil bound on rational points on curves. (In contrast, Deligne’s bounds, which do not seem to be mentioned, are the “true”, and harder, analogue of Hasse-Weil for varieties.) The text gives the Stepanov-Bombieri proof of the Hasse-Weil bound for the special case of a field whose number of elements is a perfect square. The Hasse-Weil bound for arbitrary finite fields can be deduced from this, but the text does not give a proof; the reader is referred instead to another book by the author.

Chapter 10 discusses the spectral theorem. Both the beginning and the end feel a little rushed: there is neither a proof for bounded operators (it is simply mentioned that this is the easier case) nor a full proof for the Laplace-Beltrami operator, even over $\Bbb{H}$. It would have been possible to give a full proof along the lines of, say, [H. Iwaniec, Spectral methods of automorphic forms, second edition, Grad. Stud. Math., 53, Amer. Math. Soc., Providence, RI, 2002 (Chapters 4–5);MR1942691], framing it in terms of the general theory.

Chapters 11 and 12 are entitled “Notes on Lie algebras” and “Notes on Lie type”. The first of these chapters gives a self-contained proof of the classification of simple Lie algebras over $\Bbb{C}$. It is nice to have an exposition of this standard topic written with the author’s usual clarity, but it isn’t needed here. (Moreover, given what the book is about, would it not make sense to at least mention the case of positive characteristic?) Chapter 12 starts by showing that a complex Lie group is simple (in the sense of having only $0$-dimensional proper normal subgroups) if and only if its Lie algebra is simple. It is a pity that the positive-characteristic analogue (which admittedly has exceptions for small characteristic; this was overlooked in [H. A. Helfgott, Bull. Amer. Math. Soc. (N.S.) 52 (2015), no. 3, 357–413 (§5.2); MR3348442]) was not treated here. That analogue is precisely what is needed to give a treatment in full generality of the dimensional estimates from the point of view of Lie algebras. In particular, what Chapter 5 calls “skewness” ([H. A. Helfgott, op. cit.; MR2781932] called this “sticking in different directions”) becomes clear as day in this way.

Chapter 12 concludes with an exposition of Chevalley groups, $(B,N)$ pairs and the Bruhat decomposition. All of this serves to go from the classification of Lie algebras to a classification of Lie groups. The end (§12.3) feels closer to a discussion than to a full treatment, though the case of finite characteristic is taken into account.

{Reviewer remarks: Unfortunately, an excellent blog does not necessarily translate into a uniformly first-rate book, especially when the path from the former to the latter has clearly been very short. A blog has an element of journalism to it, and the standards of journalism and permanent literature are not the same. It is useful to have the present text in a permanent format, but one must ask oneself whether the hard cover is worth the loss (of hyperlinks, say) incurred in the change in format, particularly when little else is added in the process.

{At the same time, this is a very valuable resource, though, as we have seen, there are some particulars in which it could be improved. We can hope that both the book and this review will prove useful to whoever writes a definitive work on the subject.}

Reviewed by Harald A. Helfgott

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