• Yesterday, Sean Carroll announced (and twittered) he’ll suspend blogging at Cosmic Variance in order to concentrate on his research.
• A couple of months ago, John Baez dropped a bom saying he will stop writing ‘This Week’s Finds in Mathematical Physics’ after week 300, due to a change of focus (“I’ve realized that our little planet needs my help a lot more than the abstract structure of the universe does”). Since then, the n-category cafe is grinding to a halt.
• At Coctail Party Physics there was a series of reposts because : “Life is currently kicking our collective asses, both professionally and (for some of us) personally, hence the eerie quiet of late at the cocktail party physics”.

These are no exceptions. More science- and math-blogs are struggling to maintain an illusion of activity. Sure, in these uncertain times one is more focussed on essentials (job, family) rather than peripherals (such as blogging). But, perhaps there’s more to it.

RSS-feeds became status-updates

Most people digest a newspaper by skimming the titles and actually read only a small selection of the articles in some detail.

A few years ago, every blog was its own newspaper. People checked their favorite blogs periodically by hand (the early adopters had a very small bloglist in their RSS-aggregator), read most of the new material and frequently bookmarked, commented on or linked to the post.

Today, the RSS-feed has become the newspaper itself. People subscribe to such a large collection of different feeds (news sites, blogs, status updates, forums etc.) that they can only skim the titles, rarely click through and push the ‘mark all read’ button in order to keep things somewhat under control. A bit like you manage your Twitter- or Facebook-status update.

As a result, one gets less interaction and feedback, the stats show decreasing on-site activity and one begins to question the futility of it all. However, there is no reason to despair.

GoogleAnalytics, a blogger’s best (only) friend

Trying to write a good post is still worthwhile, at least in the long run. If you’re on Google Analytics, check-out under ‘Content’ the posts having accumulated the largest number of hits and having the longest average on-page time.

Chances are these are your better posts. But, you will see that most of these hits didn’t take place via RSS-feeds the day it was posted, but over time, via referrals through sites like Wikipedia or MathOverflow, linking to your post.

That is, bloggers need to go for long term effects rather than instant gratification via comments or visitor-stats. But then, thinking-long-term is so last millennium!

Microblogging isn’t the alternative

Perhaps we can combine blogging with getting instant response? I’m two weeks on Twitter now and thanks to the wide variety of people I’m following I discovered a lot of material, quickly

• BreakingNews : yesterday, I first learned of Jose Saramago’s death via @thebookslut
• SillyThings : also yesterday, @divbyzero taugth me how to add vuvuzelas to #noncommutative
• GoodReads : @JenLucPiquant pointed me to several interesting posts, mostly on science writing

But, if you wonder about the futility of blogging, Twitter is the nec-plus-ultra in futility. A tweet, not picked up immediately, is forgotten and lost in the twitterverse by tomorrow.

A compromise? Tweet-blogging

Perhaps, we can have the best of both worlds, by writing better posts through immediate feedback on drafts via Twitter. Here’s the idea (refer to the excellent post New and Dirty : Tweet Blogging for more details)

• Aim for shorter, crispier posts. Your Twitter-experience of trying to capture an idea in 140 characters or less will help you in this.
• Condense the main points you want to make in your post into a couple of tweets and twitter them as ‘draft versions’ of your post.
• If you’ve collected a good mix of followers, you might expect excellent feedback on your draft, this is also known as Crowd sourcing. Edit : BoraZ corrects this as ‘mindcasting’.
• Edit your post by combining the key-points (the tweets) with the responses you obtained.
• Tweet the URL of your post, thanking your collaborators.

To see it in practice, @divbyzero got me involved in a question on quiver-terminology resulting in his post What’s in a name.

Noncommutative geometry and noncommutative algebra appear to have a problem when it comes to their visibility on social-media sites, compared to related topics such as representation theory, algebraic geometry, category theory or string theory.

At times when science journalists turn to tools like Twitter for crowdsourcing and more and more people start their information-search from within preferred social-media sites, the noncommutative message is under threat to become irrelevant, quickly. In future posts we’ll try out strategies to (hopefully) remedy this a bit, but first we need to get the sobering numbers straight.

Google Search : A quick way to get an estimate for the number of webpages and blogposts having ‘noncommutative geometry’ in the title is to use Google-search and type

allintitle:”noncommutative geometry”

and record the total number of results returned (search ‘Web’ resp. ‘more/Blogs’). Below the blogpost-ranking, together with (blog – web) data for our list of topics.

1. string theory : (12000 – 71700)
2. algebraic geometry : (911 – 109000)
3. category theory : (441 – 32400)
4. representation theory : (405 – 86300)
5. noncommutative geometry : (244 – 24400)
6. noncommutative algebra : (12 – 3480)

Google Adwords : Surely (?) nobody’s going to place a bid on ‘noncommutative geometry’ but Google’s AdWords Traffic Estimator (click through to the new beta version) is an excellent free tool to estimate the number of monthly global searches for specific terms, and their estimated CPC (cost per click). Here the search-ranking and corresponding CPC-value

1. string theory : 110000 ($ 0.19)
2. algebraic geometry : 18000 ($ 0.52)
3. representation theory : 8100 ($ 0.42)
4. category theory : 5400 ($ 4.54)
5. noncommutative geometry : 1300 ($ 0.40)
6. noncommutative algebra : – ($ 0.05)

Note the hight CPC for ‘category theory’ and insufficient data for ‘noncommutative algebra’ to give a credible monthly search estimate.

The above data gives a rough indication of the relative visibility and popularity of these topics, at least when one restricts to people searching information via Google (or similar search-engines).

Today, more and more people are surfing the web via the search function of their preferred social-media site, such as Digg, Reddit, Facebook or Twitter (and about a 100 similar sites). Some of these have their own search engines, such as Twitter search, but they track only the most recent entries and don’t return quantifiable data needed to compare different search terms. Another useful twitter-tool is Trendistic to spot trends in Twitter. Unfortunately, for all our topics there is insufficient data to draw a full chart and the tool restricts to recent activity.

A tool giving back quantifiable data (of sorts) is Social Mention. Social Mention monitors 100+ social media properties directly including: Twitter, Facebook, FriendFeed, YouTube, Digg, Google etc. and attaches numbers to activity over the last few months mentioning the search term. The numbers include

• Strength is the likelihood that the term is being discussed in social media. A very simple calculation is used : search term mentions within the last 24 hours divided by total possible mentions.
• Sentiment is the ratio of mentions that are generally positive to those that are generally negative.
• Passion is a measure of the likelihood that individuals talking about the search term will do so repeatedly. For example, if you have a small group of very passionate advocates wo talk about the topic all the time, you will have a higher passion score. Conversely if every mention is written by a different author you will have a lower score.
• Reach is a measure of the range influence. It is the number of unique authors referencing the topic divided by the total number of mentions.

Below the ranking according to strength, with inclusion of the other data.

1. category theory : strength 24%, sentiment 7:1, passion 33%, reach 16%
2. string theory : strength 21%, sentiment 3:1, passion 24%, reach 18%
3. representation theory : strength 21%, sentiment 3:1, passion 31%, reach 16%
4. algebraic geometry : strength 4%, sentiment 1:1, passion 36%, reach 14%
5. noncommutative geometry : strength 0%, sentiment 3:1, passion 23%, reach 16%
6. noncommutative algebra : strength 0%, sentiment 19:1, passion 14%, reach 10%

An extremely valuable feature of Social Mention is the option to obtain similar data for all the different types of social media : blogs, microblogs, bookmarks, comments, etc. etc. .

For each of these, Social Mention gives the Top Keywords, Top Users, Top Hashtags and Top Sources. That is, one can quickly determine the most prominent and active voices on a specific topic and click through to their recent postings.

Similar tools (lacking unfortunately number-data) are Same Point Social Media Search (“a conversation search engine that lets you see what people are talking about”) and Who’s Talking?.

In this series of posts we will walk you through the main paradigm shifts that took place in algebraic geometry over the last century. The short story is : the use of (higher) categories led to (more) flexible geometric objects..

That is, if varieties are 0-categorical geometric objects, then schemes are 1-categorical and stacks 2-categorical objects. The longer story starts here.

Algebraic geometry addresses a very natural problem. Say, you have a (possibly infinite) system of polynomial equations in a number of complex variables

what can you say about its set of solutions? The solutions to just one of these equations form an hypersurface in n-dimensional affine space , so we want to describe the intersection of these (possibly infinite) hypersurfaces.

Linear algebra tells us the answer in the easy case when all the polynomials are linear forms,

and in this case we have to consider the intersections of the corresponding hyperplanes in . A quick recap of the main facts :

• (basis) : we can reduce to a finite set of linear forms. Take a basis for the subspace in the -dimensional space of all linear forms in n variables.
• (existence) : there is a solution to the linear system of equations iff the constant function . In fact, is the space of all linear forms vanishing on solutions.
• (size) : if solutions exist, they form an affine subspace of of dimension .

Perhaps surprisingly, David Hilbert and Emmy Noether
were able to extend these facts to arbitrary systems of polynomial equations!

Their main idea was to translate the problem into algebra, whence algebraic geometry. In the polynomial algebra we can consider the ideal generated by all the polynomials . Clearly, if in a point all the polynomials evaluate to zero, then so does every polynomial contained in the ideal . That is, all zero sets are in fact zero-sets of ideals in .

The Hilbert basis theorem asserts that the polynomial algebra is Noetherian, that is, every ideal is finitely generated. In particular, our ideal is of the form for a finite set of polynomials . Phrased differently, all zero sets are zero sets of finite systems of polynomial equations, similar to the situation of linear equations. The difference between the two is that the new polynomials cannot necessarily be chosen among the original poynomials .

An existence criterium for solutions is given by Hilbert’s weak Nullstellensatz. It states that there exist solutions to the system of polynomial equations whenever the associated ideal is proper, that is, does not contain the constant function . Compare this to the existence criterium for systems of linear equations.

The easy part of the proof is that whenever is a proper ideal, it is contained in a maximal ideal determining a quotient map to a field

If we could now show that , we would be done as then the point would be a zero for all polynomials in and hence a common zero to all polynomials .

However, showing that turns out to be somewhat harder than expected and is a consequence of the Noether Normalization lemma. This says that an affine commutative -algebra, that any quotient algebra of the form for an ideal , is always a finitely generated module over a polynomial subalgebra . The proof can be viewed as the polynomial version of solving a linear system by substitution of variables.

The number of remaining variables in the formulation of the Noether normalization lemma is then the dimension or the size of the zero set of , that is, the number of free parameters on which it depends. The ‘size’ refers here to the biggest chunk of the solution set. In general, a zero set can be decomposed in its irreducible components, that is zero-sets which cannot be written as the union of two strictly smaller sets of zeroes.

In our dictionary between the geometry of zero-sets and the algebra of ideals in the polynomial algebra, irreducible zero-sets correspond to prime ideals, that is ideals for which imples that or belongs to . The geometric picture of writing the zero-set of as the finite union of its irreducible components corresponds to the algebraic picture of determining the finite number of prime ideals, minimal over the ideal .

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Proofs of these results can be found in the book Undergraduate Algebraic Geometry by Miles Reid. More details (and, at times, more accurate proofs) are to be found in his other Student Textbook Undergraduate Commutative Algebra.

Summarizing, Hilbert and Noether gave polynomial versions of basic facts from linear algebra : we can reduce to finite systems of equations, we can calculate whether the system has a solution and can determine the size of the set of all solutions. Further, we have translated the geometric problems on zero-sets into algebraic problems concerning ideals of polynomial algebras.

However, this dictionary is not perfect as there may be distinct ideals in having the same set of zeroes. For example, it is clear that zeroes of a polynomial are also zeroes of any power . The full-blown version of the Hilbert Nullstellensatz assures us that this is the only problem that remains, that is, two ideals and have the same set of zeroes precisely when their radicals coincide . Recall that for some and that it is the intersection of all prime ideals containing , which fits nicely with the above decomposition of zero-sets as the union of their irreducible components.

So, taking as our geometric objects the zero sets in we are only considering radical ideals in . Next time we will extend our class of geometric objects beyond zero-sets so that we can attach distinct objects to distinct ideals, and we will explain why this may be of use.

MathOverflow was started by Berkeley graduate students and postdocs Anton Geraschenko, David Brown, and Scott Morrison in October 2009. Half a year later it has become an essential online research tool for mathematicians, possibly surpassed only by the arXiv.
Why should you be on MathO, or is it a pure waste of time?

5 reasons to be on MathOverflow

1. CivicDuty : You should answer Math0-questions in your specific field of expertise. It doesn’t cost you much effort to explain what is already evident to you, and you’ll make MathO a more valuable tool.

2. ResearchBooster : On occasion, a MathO-question may be the hook for you to finally write that paper you’ve postponed so long. Sometimes, there is that question you feel you should know the answer to, but you simply don’t.

3. MeetingGround : Following MathO via your favorite tag, say noncommutative geometry, you’ll get to know new people in your field and the questions they’re interested in.

4. CelebrityHugging : Browsing through the first few pages of MathO users you’ll recognize a lot of top-level mathematicians. Click through to their profiles and then to the questions they answer, and you’ll learn a lot of new facts quickly.

5. TimeEfficient : Following MathO efficiently doesn’t cost time. Subscribe to the RSS-feeds of your specific selection of tags, and you’ll get 3 to 5 new questions a day, top. Browse through them, and if you happen to know the answer, login to MathO via your OpenId, answer, and disappear again in the background. If a specific discussion is of interest, subscribe to that question’s RSS-feed.

5 reasons NOT to be on MathOverflow

1. OverTheTop : Arguably there was a better signal/noise ratio a couple of months ago at MathO. Perhaps it’ll only go downhill from here. See also the meta-mathO- thead ‘is mathO becoming less fun?’.

2. ReputationStress : If you take your virtual mathO-reputation seriously you shouldn’t get involved, purely on medical grounds.

3. CorruptVoting : At times you’ll get frustrated because your ‘perfect’ answer wasn’t approved while another was, or you’ll spot groups of MathO-ers consistently upvoting group-member answers. See the meta-mathO threads on perceived bullying, constructive downvoting etc.

4. KansasKiddies : Sometimes, young peoples’ enthusiasm for their ph.d. topic is only surpassed by idolatry for their advisers. They use mathO to promote their advisers’ great works and, if needed, to rewrite history. Try to ignore them, sectarians are lost cases.

5. TimeWaster : If you happen to check mathO on an hourly basis or if you’re into ‘playing the system’ to increase your rating, shut down your computer and try to get some genuine work done.

So, you’ve decided to spend the summer-break finding out (finally) what noncommutative geometry is all about? How would you go about achieving this goal?

Chances are you’d love to find a readable, entry-level introductory book on the subject. Searching for “noncommutative geometry” under Books, Amazon.com spits back a list of no fewer than 212 items!

The list contains genuine text-books but also a fair number of conference proceedings, and in addition, it is not necessarily ordered in terms of accessibility … to put is mildly.
We propose a list of the Top 10 entry-level introductions to noncommutative geometry, and we will suggest the chapters in them most relevant to obtain a feel for the subject quickly!

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1. Basic Noncommutative Geometry by Masoud Khalkhali. (NDG)

Blurb : “This book provides an introduction to noncommutative geometry and some of its applications. It can be used either as a textbook for a graduate course on the subject or for self-study. It will be useful for graduate students and researchers in mathematics and theoretical physics and all those who are interested in gaining an understanding of the subject. One feature of this book is the wealth of examples and exercises that help the reader to navigate through the subject. While background material is provided in the text and in several appendices, some familiarity with basic notions of functional analysis, algebraic topology, differential geometry, and homological algebra at a first-year graduate level is helpful.”

This is a marvelous, carefully written book and anyone should read the first two chapters (1. Examples of algebra-geometry correspondences and 2. Noncommutative quotients), whether you are into NDG (that is, noncommutative differential geometry) or NAG (noncommutative algebraic geometry). They contain enough material familiar to algebraists (affine varieties, affine schemes, Morita equivalence etc.) and relate this to the corresponding notions in the realm of -algebras. I haven’t seen a more successful attempt to bridge the divide between the two flavors of noncommutative geometry! The remaining chapters (3. Cyclic cohomology and 4. Connes-Chern character) are more technical but, even if you decide to skip those for the time being, you should have a look at section 4.4 on pages 184-185 : ‘A final word: basic noncommutative geometry in a nutshell’. Here, the whole book is summarized in one diagram identifying the topological index with the analytic index.

2. Non-commutative Algebraic Geometry by S. Paul Smith. (NAG)

This isn’t a book (yet?), but rather a forgotten (?) book-project. Fortunately, you can still download an unfinished version from Paul’s website. The reason for including it in this textbook-list being that it is the best elementary introduction to the categorical approach of NAG. One might start with chapter 3 (‘Non-commutative spaces’), explaining the main idea of viewing Abelian or Grothendieck categories as noncommutative spaces using lots of concrete examples, and consult the two first chapters for categorical details. As soon as you grasp the key facts spend as much time as you need on the marvelous chapter 4 (‘Some non-commutative surfaces’) containing many very detailed examples. After that you may want to browse through chapter 5 (‘Non-commutative projective spaces’) to get a readable introduction to NPG (noncommutative projective geometry), you know the stuff with point- and line-modules to study interesting graded algebras such as Sklyanin algebras. An indispensable read if you want to understand NAG!

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3. An Introduction to Noncommutative Spaces and Their Geometries by Giovanni Landi. (NDG)

Blurb : “These notes arose from a series of introductory seminars on noncommutative geometry the author gave at the University of Trieste. They are mainly an introduction to Connes’ noncommutative geometry and could serve as a “first aid kit” before one ventures into the beautiful but bewildering landscape of Connes’ theory. The main difference with other available introductions to Connes’ work is the emphasis on noncommutative spaces seen as concrete spaces.”

Whereas Khalkhli brought you as far as spectral triples, to understand them you should read this book. Unfortunately, it seems no longer to be in print, but you can always download the arXiv paper on which the book is based! By now you may already be familiar with most of chapter 2 (‘Noncommutative spaces and algebras of functions’), but I would suggest going through chapter 3 (‘Noncommutative lattices’) in some detail if you’ve never heard of AF-algebras or Bratelli diagrams. Then, skip to chapter 5 (‘The spectral calculus’) and try to understand 5.5 (The Canonical Triple over a Manifold) giving the connection between spectral triples and classical (commutative) differential geometry. The finite point spaces (5.8) you may remember from Khalkhali’s book and combining both cases one gets the application of NDG to the standard model in physics in chapter 8.

[amazonify]0821838334[/amazonify]
4. Arithmetic Noncommutative Geometry by Matilde Marcolli. (NDG)

Blurb : “Arithmetic noncommutative geometry uses ideas and tools from noncommutative geometry to address questions in a new way and to reinterpret results and constructions from number theory and arithmetic algebraic geometry. This general philosophy is applied to the geometry and arithmetic of modular curves and to the fibers at Archimedean places of arithmetic surfaces and varieties. Noncommutative geometry can be expected to say something about topics of arithmetic interest because it provides the right framework for which the tools of geometry continue to make sense on spaces that are very singular and apparently very far from the world of algebraic varieties. This provides a way of refining the boundary structure of certain classes of spaces that arise in the context of arithmetic geometry.”

From now on the material quickly becomes more advanced and I’ll stop pointing you to ‘must see’ chapters as this choice may depend on your skills and interest. Personally, I like this book because it is terribly well written. For example, you may read through chapter 1 (‘Ouverture’) to test what you’ve learned before from Khalkhali and Landi. Pointers I would like to suggest are section 2.2 (The noncommutative boundary of modular curves) and most of chapter 3 on Bost-Connes like systems (if you’re interested in possible applications of NDG to the Riemann hypothesis). The book is freely available from Matilde’s website as a pdf-file.

[amazonify]9048145775:right[/amazonify]
5. Noncommutative Algebraic Geometry and Representations of Quantized Algebras by Alexander L. Rosenberg. (NAG)

Blurb : “This book contains an introduction to the recently developed spectral theory of associative rings and Abelian categories, and its applications to the study of irreducible representations of classes of algebras which play an important part in modern mathematical physics. Audience: A self-contained volume for researchers and graduate students interested in new geometric ideas in algebra, and in the spectral theory of noncommutative rings, currently invading mathematical physics. Valuable reading for mathematicians working on representation theory, quantum groups and related topics, noncommutative algebra, algebraic geometry, and algebraic K-theory.”

I’ve included this book, more because it appears to have influenced people working in the categorical approach to NAG, rather than finding it terribly well-written (imho). So, mainly for historical reasons, you might look at the stuff on the spectrum of an Abelian category (chpt. 2) and his use of ‘monads’ in chpt 5.

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6. An Introduction to Noncommutative Geometry by Joseph C. Varilly. (NDG)

Blurb : “Noncommutative geometry, inspired by quantum physics, describes singular spaces by their noncommutative coordinate algebras and metric structures by Dirac-like operators. Such metric geometries are described mathematically by Connes’ theory of spectral triples. These lectures, delivered at an EMS Summer School on noncommutative geometry and its applications, provide an overview of spectral triples based on examples. This introduction is aimed at graduate students of both mathematics and theoretical physics. It deals with Dirac operators on spin manifolds, noncommutative tori, Moyal quantization and tangent groupoids, action functionals, and isospectral deformations. The structural framework is the concept of a noncommutative spin geometry; the conditions on spectral triples which determine this concept are developed in detail. The emphasis throughout is on gaining understanding by computing the details of specific examples.”

No doubt, a lot of people will prefer this book to Landi’s. It covers roughly the same material, is more recent and is very well written. So, switch 3. and 6. if you so prefer and look for similar material, that is chapters 2,3 and 8.

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7. Blowing Up of Non-Commutative Smooth Surfaces by Michel Van den Bergh. (NAG)

Blurb : “This text looks at topics that include: preliminaries on category theory; non-commutative geometry; pseudo-compact rings; Cohen-Macaulay curves embedded in quasi-schemes; derived categories; quantum plane geometry; and non-commutative cubic surfaces.”

As the series suggests, this is really a longer research article rather than a book. Still, it is one of the better texts to get an idea of what NPG is all about and why the categorical point of view may be useful. I would certainly urge you to read the introduction carefully and, if you’re into NAG and NPG, section 3 (‘Non-commutative geometry’) is also a must-read!
The book is freely available from Michel’s website as a ps-file.

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8. An Introduction to Noncommutative Differential Geometry and its Physical Applications by J. Madore. (NDG)

Blurb : “This is an introduction to noncommutative geometry, with special emphasis on those cases where the structure algebra, which defines the geometry, is an algebra of matrices over the complex numbers. Applications to elementary particle physics are also discussed. This second edition is thoroughly revised and includes new material on reality conditions and linear connections plus examples from Jordanian deformations and quantum Euclidean spaces. Only some familiarity with ordinary differential geometry and the theory of fiber bundles is assumed, making this book accessible to graduate students and newcomers to this field.”

A ‘must-read’ if you’re interested in possible applications of NDG to the standard model in physics. At least anyone should read the introduction, starting off with a nice quote from P.A.M. Dirac on Einstein’s aversion to quantum physics being entirely based on its non-commutativity…

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9. Noncommutative Geometry, Quantum Fields and Motives by Alain Connes and Matilde Marcolli. (NDG)

Blurb : “The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory. The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adele class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions. The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.”

Yeah well, after such a blurb what is there left to say? If Connes’ first book ‘Noncommutative Geometry’ is generally referred to as the ‘bible’ of NDG, then this book should be thought of as Connes’ ‘new testament’. Anyone aspiring to be a noncommutative-geometer-of-sorts-one-day must acquire a hard-copy of this book. First of all it is good value for money. Secondly, its function is similar to James Joyce’s ‘Finnegan’s Wake’ in English Literature circles : when visiting a colleague’s office and spotting a copy in the bookcase one tries to have a peek at the number of pages actually read… Anyway, my personal copy is pretty worn-out as I really like most of the introductory sections presenting an inspiring account of a wide variety of topics, ranging from number theory to physics. In good tradition, this book is freely available from Alain’s website as a pdf-file.

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10. Virtual Topology and Functor Geometry by Fred Van Oystaeyen. (NAG)

Blurb : “Intrinsically noncommutative spaces today are considered from the perspective of several branches of modern physics, including quantum gravity, string theory, and statistical physics. From this point of view, it is ideal to devise a concept of space and its geometry that is fundamentally noncommutative. Providing a clear introduction to noncommutative topology, Virtual Topology and Functor Geometry explores new aspects of these areas as well as more established facets of noncommutative algebra. By presenting new ideas for the development of an intrinsically noncommutative geometry, this book fosters the further unification of different kinds of noncommutative geometry and the expression of observations that involve natural phenomena.”

Fred’s books contain ideas which, in retrospect, turn out to be important. For example, in 1981 he was co-author of the first book having ‘noncommutative geometry’ in the title, at a time when the consensus was that one could investigate almost anything about noncommutative algebras, but their geometry. His latest book is on noncommutative topologies, that is ‘spaces’ in which the intersection of two opens may very well differ from . No doubt this will become important, one day… This book is included in the list for those who believe, after having mastered the previous ones, to know all about noncommutative geometry and the direction it will take in the future. Well, you simply don’t…

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It should be clear from the above that the ordering of the list is rather arbitrary from place 3 or 4 down, and, some people will object to some manifest omissions.

Judging from reactions and reviews on my own book, it may very well deserve a place in the list somewhere, but I’d rather leave that to others to determine. Most people find the introductory chapter helpful and you can download it on the arXiv as 3 lectures on noncommutative geometry@n. Also the full book is freely available as a pdf-file.

If this whetted your appetite, Matilde Marcolli has a Listmania Noncommutative Geometry list, and, preparing for this post I did compile a slightly larger list, too.