Question

Writing a book from the beginning to the end is (so I heard) a very hard process. Planning a book is easier. This question is dual in a sense to the question "Books you would like to read (if somebody would just write them)". It is about a book that you feel you would like to write (if you just have the time). A book that will describe a topic not yet properly discribed or give a new angle to a subject that you can contribute.

The question is meant to refer to realistic or semi-realistic projects (related to mathematics). Answers about book projects based on existing survey articles or lecture notes can be especially useful.

Of course, If you had some progress in writing a book mentioned here please please update your answer!

Answer 1

Gosh, what a question, Gil. What is your answer?

I have written many books in my head, but I am much too lazy actually to write a book. I guess my first choice would be

Geometric nonlinear functional analysis, volume II

and my third choice

Geometric nonlinear functional analysis, volume III

neither of which will ever be written by the authors of volume I.

A less daunting topic, until you think about it a lot, is

The geometry of $L_p$ spaces.

Another one I toyed with was

Approximation properties of Banach spaces.

Answer 2

Book: The Differential Topology of Loop Spaces

Why: Because they are one of the first examples of spaces that are almost, but not quite, entirely unlike manifolds. They are relatively straightforward spaces which can be fairly conceptually grasped, but still contain enough intricacies to reveal some of the important differences between finite and infinite dimensions (though perhaps I should say between manifolds modelled on Banach spaces and more general manifolds). A book on their differential topology would thus be a gentle introduction to the topic than is (as far as I'm aware) currently available (in particular, although just about everything I'd want to say is covered in Kriegl and Michor's works, it's in such a context and with such generality that "daunting" doesn't quite cut the mustard).

Who For: Me, 10 years ago. That is, I'd try to write the book I wish I'd had when starting out in infinite dimensional differential topology so I wouldn't have made all the mistakes that I made.

Why Me: Because I work in that area and I think I've made just about every wrong assumption about loop spaces possible so I know lots of the traps for unwary differential topologists venturing out into the miasma that is infinite dimensional topology.

Will I Ever Actually Write It: Maybe, maybe not (vote for this answer if you want me to!). I made a start by writing up some seminar notes. I've started transferring them in to the nLab (but in the process I've been generalising them which slightly goes against the purpose of the project as I described it above). I'd certainly like to write it, if only to convince myself that I no longer have all those false assumptions, but whether or not I ever actually do it ... (hey, I've an idea, maybe all the time I put into MO and meta.MO could be reallocated to book-writing. Then it'll be finished next week.).

Answer 3

Question seems a little silly to me, unless it's meant as motivation. But for those who answer the question and then are motivated to go ahead with their book project, I can offer some personal experience on the process.

Step 1. Start with a detailed outline and 100+ pages of detailed notes from a course that you've taught on the subject.

Step 2. Estimate about how long you think it will take to turn those notes into a published book. (In my case, I figured that it couldn't take more than a year or so.)

Step 3. Triple the value in Step 2 to arrive at an accurate estimate.

Answer 4

While I find the question borderline, I succumb to the temptation to answer.

Knot Theory: Kawaii examples for topological machines.

Topology is full of big machines, which may seem rather daunting to the student. But knot theory is a wonderful playground for toy models of many of these machines, where you can see how they work and visualize what they are doing. And one can draw pictures.
I think that a collection of these examples would be useful to students (I would have loved to have had it) or to people who would like to teach topology. And I don't think anything like this exists, really. The machine itself would be introduced only briefly, refering to somewhere else for more detail, while the knot theory example would be fleshed out in full.
For example, curvature of knots is the perfect playground for the Gauss-Bonnet Theorem. Computations of homology in knot theory give perfect toy examples (with pictures you can draw) for Mayer-Vietoris, the snake lemma, and other homological arguments. Ideas such as localization and Brown representability come up naturally. And an Alexander module gives a perfect playground for commutative algebra over a UFD.
So the idea would be to give sophisticated proofs of simple facts, letting the topological machines play the lead role. The student of topological machine X might then read the book by looking up the relevant section, which would give a kawaii (cute?) example in knot theory, highlighting how exactly the machine is working, and shedding light on its nature.
How likely am I to write it? I've toyed with the idea for a long time. For the book to be useful, it needs to be very visual and pedagogical, to make it light fluffy reading for one who knows the machine, and educational reading for one who doesn't. And becaue I have high asprations for it, it may take a while. But I do have intentions of actually writing it at some point, even if I don't yet know when that might be.

Answer 5

I would love to write something titled "Higher mathematics from engineer's perspective", which would consist of a few chapters each of which should be devoted to a single simple to state real engineering problem whose solution requires rather sophisticated mathematical tools. The main content of the chapter would be the shortest path to the full solution with all the relevant concepts explained, all relevant theorems fully proven, etc. For instance, one possible such chapter would be "How to shape an airplane wing and compute the lifting force?" with all that complex analysis etc. An easier one would be "How to shape the rollercoaster track?" about elementary space curve theory. A harder one would be "How to find defects in solids?" with PDE's, wave equation, etc. Something like that is certainly lacking though I doubt that the people who will read it need it and that the people who need it will read it.

Another book written by a mathematician that makes me really jealous is "Alice in Wonderland". Alas, I currently do not have any good idea of how to beat it though the perception of the surrounding reality by a mathematically inclined mind is much more subtle and "unusual" today than it was in Carroll's time. (I almost wrote "perverted" instead of "unusual" but it is a kind of "perversion" that is in the reality itself, not in its perception, so this word, if used, won't really be understood correctly without a long explanation).

Needless to say, I will write neither of the two. Still, somewhere in the Platonic domain both these books exist and occasionally I stumble upon an "excerpt" that is taken right from one of them (that "excerpt" is, of course, not necessarily in the form of a written text or a sound track, but I cannot find a better word (fragment?) now).

Answer 6

What I would really like to write is the new Da Vinci code, that is, a book that's an absolute piece of trash but sells 80 million copies. Purely for the sake of my bank account, of course.

Oh, but probably you were referring to mathematics books. Well, I don't think the question was very good anyway.

Answer 7

Book Title: An Introduction to Forcing (for people who don't care about foundations.)

Synopsis: Forcing is one of the most amazing techniques in use today, and it offers amazing insight into how objects in mathematics can be constructed. The aim of this book would be to focus on the tools and methods of Forcing, and provide examples of constructions which highlight the intrinsic beauty that can be found hiding under the surface of a forcing argument. Moreover, it would highlight the practical applications of, and sense of naturalness the "Forcing Perspective" brings to inductive mathematical constructions (which might be outside the domain of set-theoretic interest.)

Reason For Wanting to Write It: When I first learned about Forcing, the first thing that struck me was "Why the hell has no one ever told me about this? What the hell!? This is AWESOME!" That sense of awe has stayed with me throughout my very short "career." So the book would be a way for me to share this view with other mathematicians who don't really care all that much about "set theory", "category theory", or "foundations" (just like I did before I learned about independence proofs, etc.) Moreover, the aim would not be to convert them to some relativist view of mathematics, but to just show them how directly linking the logical structure of an object with its construction can open new doors, and add much needed perspective to any field.

When Would It Get Written: Honestly, not now, and not in the near future, maybe 10/20 years. The reason for this is, I just don't know enough yet, I'm still a student. That being said, I must admit, I am most likely not the first person anyone would pick to write such a book. However, if I was ever presented with the opportunity I would take it in a heartbeat. To me the importance of the ideas and perspective for mathematics as a whole out weigh the possible huge list of errors and corrections that would follow such a book (if written by me that is).

PS: if there are any spelling or grammar errors, feel free to fix them.

Answer 8

Categories for Computer Science Made Easy

Basically, I'd like to collect together the stuff I mentioned in this answer, as well as some more. The information is scattered through many books and papers right now. Many of these papers are very challenging, even though they often contain elementary parts that could stand on their own.

Answer 9

I would like to write a book on Forcing and large cardinals.

The idea would be to give a complete account of the interaction of these two central set-theoretic concepts, aiming at their intersection, rather than at their union. How are large cardinals affected by forcing? What kinds of forcing can preserve which kinds of larger cardinals? To what extent do the standard forcing notions affect large cardinals? For example, to what extent can we preserve large cardinals while forcing GCH, V=HOD, or their negations, among other set-theoretic features commonly obtained by forcing? By what methods can we show that large cardinals are preserved? To what extent can large cardinals be made indestructible by (certain kinds of) forcing? What are the most general things that can be said about how the large cardinal embeddings of one model relate to the large cardinal embeddings of its forcing extensions and ground models? The topic has a fundamentally category-theoretic flavor, since it is at essence about how large cardinal embeddings are affected by forcing, ideas that can often be expressed by means of large commutative diagrams, involving lifts of embeddings from models to their forcing extensions.

Let me confess: the truth is that I have been working on writing such a book for the past ten years, and have about 320 pages completed, sitting on my computer; I have used drafts of this book when teaching graduate courses in set theory, and over the years I have allowed various versions of these drafts to become distributed to various other researchers. In fact, it appears that this "book" has already been cited a number of times by various authors in published articles, even though it does not yet exist as a book.

So I would like to complete it. But somehow I keep getting distracted by other interesting and worthwhile projects...

Answer 10

"Thinking with categories" a small introduction for the layman.
May be a more commercial title would be "Functorial Thinking".
A small book (circa 120 p.) with the goal of explaining basic category theory using plenty of examples but mostly non mathematical ones.
Intended for an audience of linguists, philosophers, computer designers and any curious intellectual.

The book presuppose a reader not adverse to a minimum of algebra, yet it should mostly contains basic defining algebraic equations for categories, functors , natural transformations and adjunctions.

The goal of this book: It should enable a philosopher (not necessarily specialized in logic) to grasp properly what an adjunction is in 2 to 4 hours.

The basic motivation: Find proper real-life examples (as in elementary set theory) for category theory.

To illustrate : A 5-subset of a football team can be made by picking some players randomly, but a sub-object is a set of 5 players that can play together! In fact common language would call it sub-team. So far when trying to design examples in real life you end up too often with groupoids and thin category(posets).

Any suggestions of places from which to draw material/inspiration would be most welcome.

Answer 11

Related to my question:

http://mathoverflow.net/questions/54250/references-for-constructible-sheaves-on-complex-analytic-stacks

I'd like to read/write a book on constructible sheaves and the six operation formalism on complex analytic stacks, as it seems there are not too many references in literature (I would apologize if there is one that I'm not aware of), and there are so many basic facts in étale cohomology that one expects (at least I expect, in my research) to be true for analytic stacks but I couldn't find any reference, and therefore had to prove them from the beginning.

Planned content: (all on complex analytic stacks) dualizing complexes and the six operations (with any coefficient ring; the analytic topology allows us to do so), adic theory (mimic Laszlo-Olsson's theory), various results in étale cohomology in this setting, including proper/smooth base change, purity, Artin's comparison (for analytifications of complex Artin stacks), Künneth formula. Also I hope I could discuss Hodge theory (already done in Deligne's Hodge III, in terms of simplicial schemes), perverse sheaves (say with $\mathbb Q$-coefficient, again mimic Laszlo-Olsson) and mixed Hodge modules.

Answer 12

Over the years I had a few ideas about books as well as the appealing idea of not writing a book. When I see books others have written I am usually quite amazed by them, and the amount of work involved seems rather alarming. (Being able to write unpolished things and to jump from one topic to another is an advantage of writing a blog.) In any case, I would prefer to write a book with an electronic version using the full possibilities of hyperlinks. Here are some specific ideas about books I would have liked to write had this been painless:

1) Face numbers, graphs and skeleta of polytopes and complexes. This is an area of combinatorial geometry which I find very exciting and it is related to various other areas of combinatorics and mathematics. (I am quite an expert in the area of the proposed book but not an expert in these related areas.) This topic is discusses in several books but I don't think there is a book devoted to this subject. My starting idea for this project is simple: To take Chapters 18 (by Billera and Bjorner) and Chapter 20 (by me) from the Handbook of discrete and computational geometry update them and add proofs.

2) Analysis of Boolean functions. This is a fairly new research area which again I find very exciting. It has connections to various areas of combinatorics and computer science, to probability and to harmonic analysis. Yet it is a sufficiently young field that a book is possible. How to go about it? Muli Safra and I wrote a related survey article about thresold phenomenon what seems to be missing is an additional survey on Fourier analysis of Boolean functions and then adding-proofs transformation as part of what is required to make them into a book.

Update There are two recent very nice related books: Lectures on noise sensitivity and percolation is a new beautiful monograph by Christophe Garban and Jeff Steif. Ryan O’Donnell is writing a book about Fourier analysis of Boolean functions and he serializes it on a blog entiled Analysis of Boolean Function.

3) A different idea that Gunter Ziegler and I played with was to write " The book of examples" (mimmicking perhaps the style of Aigner and Ziegler's "Proofs from the book") The mathoverflow question on fundamental examples is very much related to this idea. So given the many answers all that is "left to do" is to select some of the examles, to divide them into chapters, to ellaborate more on each selected example and indicate important connections. (This can also be done collectively.)

4) A different direction would be to transform the posts from my blog "Combinatorics and More" into a book (like Terry Tao and Dick Lipton have done for their blogs.)

5) (ADDED: AUG 2011) I forgot to mention that I did write an Internet Book Entitled "Gina Says: Adventures in the Blogosphere String War" , which contains all sort of things and also some mathematics. I would like to edit it further to make it suitable to a larger audience and possibly publish it via a commercial publisher.

6) (ADDED: Nov 2012) The content of my debate with Aram Harrow on quantum fault-tolerance that started in this post and concluded in this post over the blog "Godel lost letter and P=NP" can be the basis for an interesting book.

Answer 13

It seems that the point of most calculus books currently existing is to: 1. Provide homework problems 2. Provide sample solutions to such problems so that students can pattern match 3. provide formulas in little boxes.

I would like to write a calculus book which really forces students to think about calculus. This means that they will have to develop the calculus themselves. The book will assist in this task by asking very leading questions, and asking students to work out examples which contain the essence of each new idea. A course based on such a book would consist of students working through the relevant section the night before, and the "lecture" is a group discussion aimed at clarifying the ideas developed. Of course, this must be supplemented with plenty of calculations, but these must always be accompanied by written explanations of the thought process behind each calculation.

Of course, these thoughts apply equally well to any other book about mathematics, especially those aimed at undergraduates. The ones aimed at graduate students or researchers could also benefit from this, but by that time most students have learned how to do this kind of thing for themselves.

Answer 14

In an ideal world where I would have unlimited time for nice book projects, I would like to write an update and english translation of my book on Poisson geometry and deformation quantization, which is unfortunately in german (I was young, needed the money...)

In course of such an update and translation I would like to incorporate some new topics (any suggestions?) and include in particular a treatment of symmetries, Morita theory, and existence&classification of star products also for the Poisson case (based on formality and gloalization a la Dolgushev...), and perhaps, also some more details on reduction. On the other hand, I would try to make the symplectic and Poisson geometry part much shorter, perhaps even in form of an appendix, to focus on the DQ part. I would like to keep the balance between mathematical presentation of the material with additional motivation section from mathematical physics.

But the world is far from being ideal, so I can not promise when I will find time for doing so... ;)

Answer 15

I would like to write a book related to my question Singular semi-Riemannian Geometry: usefulness and state of the art.

In fact, I have some 120 pages of drafts, but I would like to detail more some parts which I wrote too quickly, and to explain more between the equations.

The planned content is pretty much the same as it is now, but more elaborated, and with more examples:

I. Singular Semi-Riemannian Manifolds with Variable Signature Metric

1. Tensor Operations on Degenerate Inner Product Spaces

2. Tensor Operations in Singular Semi-Riemannian Geometry

3. Differential Operations on Singular Semi-Riemannian Manifolds

4. Curvature of Semi-Regular Semi-Riemannian Manifolds

5. Warped Products of Semi-Regular Semi-Riemannian Manifolds

II. Applications to the Singularities in General Relativity

1. Einstein Equation on Spacetime with Degenerate Metric

2. Time Evolution in Singular General Relativity

3. Black Hole Information Paradox in Singular General Relativity

I have some additional directions in which I intend to develop the subject, and I want to add them to the book. Also I would like to make sure that I was not reinventing the wheel.

Answer 16

What this country needs is a successor to Courant/Robbins' "What is mathematics?", first published in 1941. Gowers' wonderful "Princeton companion to mathematics" cannot serve as a modern replacement of this volume, insofar as it addresses a group which is already deeply interested in mathematics and definitely knows what mathematics is all about. Not unlike Gowers' compendium the book I'm dreaming of would be the work of a devoted collective of authors, but in addition it would need a unifying editorship to make it the landmark in the field for decennia to come, as it was the case with Courant/Robbins' book seventy years ago.

Answer 17

Vertex Algebras (for Beginners)$^2$ (=Vertex Algebras for Beginners for Beginners)

because I've spent the past ~3 years carrying around Vertex Algebras for Beginners and only in the past few weeks have I understood what on earth is really going on.

Answer 18

After reflexion, I think I will reduce my contribution to this: Don't think too much about the book you want to write, just write it down. Don't wait that everything is perfect, just begin. Anyway, it will take years.

Answer 19

I'm writing a "Theory of category" personal Book (in latex) in Bourbaki style..

but is in Italian at now, and in working progressing

Answer 20

Note  this answer is superseded by a more detailed answer to which the reader is referred.

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I would very much like to write the book (or much easier, read someone else's book) Geometric Dynamics with Practical Applications in Classical and Quantum Systems Engineering.

The envisioned book would encompass, via geometric methods, many of the dynamical and informatic themes that Michael Nielsen and Isaac Chuang so excellently cover via algebraic methods, in their Quantum Computation and Quantum Information (2000).

It would be the book that—in an alternative history of 20th century quantum physics—would have grown from Saunders Mac Lane's 1970 Chauvenet Lecture Hamiltonian mechanics and geometry, had Mac Lane first read Nielsen and Chuang's now-classic text, and then read Abhay Ashtekar and Troy Schilling's Geometrical formulation of quantum mechanics (1999), and finally read Carlton Caves' (on-line) note Completely positive maps, positive maps, and the Lindblad form (2000, revised 2002 and 2008).

As for notation, this book would embrace the notation of Jack Lee's admirably clear Introduction to Smooth Manifolds (2003) ... no need for "bras" and "kets"!

Our UW Quantum Systems Engineering (QSE) Group first explored these ideas in an article Practical recipes for the model order reduction, dynamical simulation and compressive sampling of large-scale open quantum systems. This article is sufficiently lengthy (at 96 pages) as to provide much of the material for a short textbook. However, an all-recipe textbook on quantum simulation would lack an overall ordering perspective.

Thus, a void in the existing quantum dynamics literature is an integrative synthesis of the above references by Mac Lane, Nielsen and Chuang, Ashtekar and Schilling, Caves; moreover many other authors—Arnold, Carmichael, Abraham and Marsden, Thurston, etc.—could be added to this list. It takes quite a bit of reading to appreciate that these authors' ideas and formalisms are naturally congruent.

The arxiv preprint Elements of naturality in dynamical simulation frameworks for Hamiltonian, thermostatic, and Lindbladian flows on classical and quantum state-spaces (arXiv:1007.1958) is a first-draft summary of those results needed to establish a coherent set of geometric ideas associated to practical large-scale quantum simulation (fortunately, not very many new results are needed).

Current research interests focus on verification, validation, and runtime estimation (VVR), with a view toward establishing consonance between practical VVR and various "no go" results that complexity theory provides. These investigations are still in an early stage; they largely motivate our TCS StackExchange questions "Are runtime bounds in P decidable? (answer: no)" and "Do runtimes for P require EXP resources to upper-bound? … are concrete examples known? (answer: yes and yes)".

We are working on a narrative that integrates these ideas, our MathOverflow answer to the question "What a Geometer Should Know" is our working outline for that narrative.

The PNAS article "Spin Microscopy's Heritage, Achievements, and Prospects" describes our technical objectives, which originate in roadmaps set forth by von Neumann, Wiener, and Feynman in the era 1946-59; most of my questions and answers here on MathOverflow, and also on TCS StackExchange, are conditioned upon these concrete (and predominantly medical) objectives.

And finally, this first book Geometric Dynamics with Practical Applications in Classical and Quantum Systems Engineering is envisioned as the first of two books ... the second book will be Applications of Quantum Spin Microscopy in Regenerative Medicine. That second book will someday be a topic for a second post.

Answer 21

"The Laws of Relations" the title echoing famous "The Laws of Thought" By George Boole but in spirit closer to "Logic of Relatives" by Charles Sanders Peirce. The subject is algebra of relations with named attributes of arbitrary finite arity. It is predicate calculus without quantifiers where the predicates are identified by their names only (not the names, nor positions of predicate attributes) with syntax reminiscent of Peirce-Tarski Relation algebra.

Admittedly, the number of publications on the subject is less than dozen, which makes it pie-in-the-sky sort of wish.

Answer 22

I would like to write a book about algebraic shifting. Your survey is too compact and (in my opinion) is not user-friendly. On the other hand rewriting all proofs with algebraic machineries kills the beauty of this theory (again, in my opinion). So I don't like Herzog and Hibi's book. I think it is necessary to show the concrete combinatorial nature of algebraic shifting.

Answer 23

Undergraduate Deformation Theory And Quantum Groups

It would be based on a course I took 2 years ago at the CUNY Graduate Center given by John Terilla and Tom Tradler and would focus on the basic concepts of the Gerstenhaber bracket, deformations of associative algebras,operads and quantum groups.It would differ from the usual texts in that would have very minimal prequisites:strong undergraduate backgrounds in algebra and topology (an algebra course based on Herstien and a topology course based on John Lee's book would suffice).

It would focus mainly on the material in "classical" deformation theory i.e. Gerstenhaber's original papers as well as the work done by Stasheff and Markl on the theory. It would end concievably with a glimpse at the modern theory on sheaves and prepare the student for Harshorne's book and the current literature.

Answer 24

Acknowledgments   The following three-volume answer to Gil Kalai's MathOverflow question “A Book You Would Like to Write” is chiefly inspired by Tim Gowers' thoughtful comments on Gödel's Lost Letter and P=NP, which he wrote in response to Dick Lipton's question “Make your own world: what would you do if you could do anything?”

Prologue   Here is an excerpt from Gowers' comment on Gödel's Lost Letter:

Gowers' choice  “If on the other hand P!=NP, then the price I ask … is that we come to understand far better the subclass of mathematical statements and proofs we are actually interested in. … I would like a world where exactly one of the statements ‘P=NP’ and ‘mathematical creativity can be automated’ is true.”

Let us regard Gowers' choice (as we will call it) not as a wish, but as an engineering directive whose fulfillment requires a bespoke mathematical toolset such that “creativity can be automated.” Specifically, we regard Gowers' choice as a path toward Bill Thurston's goal:

Thurston's goal  “The goal of mathematics is to develop enhanced ways for humans to see and think about the world. Mathematics is a transforming journey, and progress in it can better be measured by changes in how we think than by the external truths we discover.

To associate Gowers' choice to Thurston's goal, we embrace Abraham Lincoln's view that our broad objective should be:

Lincoln's objective  “To make mutual exchange [of] discovery, information, and knowledge; so that, at the end, all may know every thing, which may have been known to but one, or to but a few [and] to stimulate that discovery and invention into extraordinary activity.”

Balancing these various ideas, we design the mathematical formalisms of the Gowers-Thurston world with a view toward providing “enhanced ways for humans to see and think about” their individual participatory roles in the emerging “extraordinary activities” of the 21st century.

To concretely specify this world's mathematical toolset, we apply a template that physicist Julian Schwinger's students affectionately distilled from his lectures:

Schwinger's template  “Although ‘1’ is not perfectly ‘0’ we can effectively regard …”

Applying Schwinger's template, we “effectively regard” the mathematical toolset of Gowers' choice and Thurston's goal as arising from this ansatz:

The Gowers-Thurston-Schwinger (GTS) Ansatz  “Although ‘NP’ is not known to be formally separable from ‘P’ we can effectively regard it as such whenever our main purpose is mathematical understanding. Similarly, although ‘Hilbert space’ is not known to be perfectly a ‘low-dimension secant variety of a Segre variety’ we can effectively regard it as such whenever our main purpose is dynamical understanding.”

The first part of the GTS ansatz restricts NP (and thus P) to those algorithms whose runtime attributes are decidable and whose outputs (including random samples) are verifiable; Juris Hartmanis has suggested that this restriction (suitably formalized) might render P and NP provably separable. In effect, the ansatz restricts P and NP to those algorithms that are humanly understandable in the Gowers-Thurston sense. The second part of the GTS ansatz focuses upon systems (both classical and quantum) whose trajectories are dynamically compressed onto low-dimension algebraic manifolds. In effect, the ansatz restricts computational simulations to the noisy and/or low-energy and/or highly symmetric dynamical trajectories that are commonly encountered in nature, in technology, and in the laboratory.

These considerations lead us to envision the Gowers-Thurston-Schwinger world as becoming a concrete 21st century reality via a 10-year path that (if we are lucky) will be described retrospectively by the following three MathSTEMnet reviews. In answer to Gil Kalai's question, the math of Volume I exists today; the math of Volume II foreseeably will exist within the next 36 months or so; the math of Volume III will be the work of many decades.

Needless to say, the MathSTEMnet reviews are entirely imaginary; in particular, the review of Volume III seeks to retell a classic Robert Heinlein medical narrative from 1958 in the dryly arch mathematical voice of Joseph Doob's 1948 review of Claude Shannon's Mathematical theory of communication (MR0026286).

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MR2739833
Lane, Alice; Lane, Bob
Elements of Naturality in Simulation and Sensing
Volume I of Surveys of Engineering for Enterprise
Constancy Press, Seattle, 2015. xviii+475 pp.
58-01 (53-01 57-01)

This volume aims to provide solid foundations for classical and quantum simulation. In the first of its three parts students learn the basics of differential and algebraic geometry at the same time that they learn the basics of Hamiltonian dynamics, first in the context of classical molecular dynamics, then in the context of classical interacting spins. From the beginning all state-spaces are treated as algebraic varieties (specifically, secant varieties of Segre varieties) that are endowed with symplectic and metric structure. The second of three parts treats (classical) thermostats and (quantum) Lindbladian processes within a mathematically natural Hamiltonian/Stratonovich formalism. In the final part, classical and quantum tools are merged in the practical context of quantum spin biomicroscopy, viewed both as a Shannon communication channel and as a target for simulation and sensing in synthetic biology.

The resulting volume reads as though Saunders Mac Lane, Vladimir Arnold, and Joe Harris teamed up to cover in one volume the dynamical elements of three classic texts: (1) Charlie Slichter's Principles of Magnetic Resonance, (2) Nielsen and Chuang's Quantum Computation and Quantum Information and (3) Frenkel and Smit's Understanding Molecular Simulation: from Algorithms to Applications — all in the flowing example-filled style of Jack Lee's Introduction to Smooth Manifolds. It is suitable for a senior undergraduate or first-year graduate course (that requires students to unlearn some of what they previously have been taught).

• reviewed by Caradoc Dearborn

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MR2739833
Lane, Carla; Lane, David
Elements of Naturality in Surveys and Enterprises
Volume II of Surveys of Engineering for Enterprise
Constancy Press, Seattle, 2020. xxi+560 pp.
58-01 (53-01 57-01)

Volume II in this series takes up where Volume I leaves off: with the description of the molecular dynamics and quantum spin imaging of biological molecules. The first of three parts surveys the quantum theory of spin polarization transport, with an emphasis on transport-based techniques for generating order-unity dynamic nuclear polarization (T-DNP). Substantial emphasis is placed on efficient iterative evaluation of “musical” isomorphisms in trajectory integrations. The second part discusses 3D imaging methods that are enabled by the coherent polarization so achieved. The third part discusses the “crossover region”of imaging at 0.5 nm resolution, below which molecular dynamical simulations carry more information than direct imaging. Each chapter is accompanied by two-part design exercises, the first consisting of a pencil-and-paper (or SymPy) symbolic analysis, the second consisting of a large-scale (SAGE/PyQSE) numerical simulation; working code is provided for most exercises.

The concluding chapter requires students to design an enterprise for spin-imaging the entire nucleus of a eukaryotic cell (via quantum spin microscopy) at 0.5 nm resolution, then refining that imaging information (via molecular simulation) to sub-Angstrom scales. Present rapid developments in quantum spin microscopy, sample hyperpolarization, and molecular dynamic simulation ensure that this section will be outdated within a very few years …and yet no book better conveys the mathematical toolset that is so greatly in-demand to support the burgeoning global enterprise of observational synthetic biology.

• reviewed by Dilys Derwent

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MR2739833
Pomfrey, Ella; Longbottom, Finn
Elements of Naturality in Healing and Regeneration
Volume III of Surveys of Engineering for Enterprise
Constancy Press, Seattle, 2025. xxix+870 pp.
58-01 (53-01 57-01)

It is now ten years since Volume I of this series appeared, heralding a new era of comprehensive quantum spin imaging of biomolecular structure, and comprehensive simulation of the the molecular dynamics of these structures. It is now five years since Volume II heralded a new era of synoptic information regarding the workings of “every atom in its place”, very much as von Neumann and Feynman foresaw last century. Now Volume III has appeared, and the authors promise to provide a mathematical “natural” toolset for applying these capabilities in healing and regeneration.

Authors Ella Pomfrey and Finn Longbotton are members of the new breed of physician that are comfortable with symplectic structure and with bone structure, with individual molecules and with individual patients, with genetic and epigenetic variation, with complexity theory and with the evolving cognition of healing brains. They have mastered, both abstractly and in practice, the geometrically, algebraically, combinatorically, and informatically natural tools that previous generation of mathematicians brought to bear in the microscopic theory of healing and regeneration. Now in this volume, Pomfrey and Longbottom seek to bring this same natural toolset to bear on macroscopic healing processes. The emphasis throughout is upon practical clinical verification and validation procedures that ensure that bone, nerves, and minds all cleave to a path that leads to a satisfactory healing.

This reviewer entertains some doubt as to whether our understanding of healing and regeneration, in particular their epigenetic aspects, can ever match the naturality of our microscopic understanding … but no-one is better qualified than the authors, who have a distinguished record in the regenerative treatment of battle trauma, to meet the 21st century's grand challenge of healing, by evolving a mathematically natural understanding of it.

• reviewed by Mungo Bonham

Answer 25

I am writing the book "Algebraic General Topology. Volume 1" about my research in general topology (funcoids, reloids, and their generalizations, also the topic of filters and their generalizations).

Some draft fragments of it are available at my site.