Other What are you reading now? (STEM only)

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Current reading among participants focuses on various STEM books, including D. J. Tritton's "Physical Fluid Dynamics," which is appreciated for its structured approach to complex topics. J. MacCormick's "Nine Algorithms That Changed the Future" is noted for its accessibility in explaining computer algorithms. Others are exploring advanced texts like S. Weinberg's "Gravitation and Cosmologie" and Zee's "Gravitation," with mixed experiences regarding their difficulty. Additionally, books on machine learning, quantum mechanics, and mathematical foundations are being discussed, highlighting a diverse range of interests in the STEM field. Overall, the thread reflects a commitment to deepening understanding in science and mathematics through varied literature.
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What book are you reading now, or have been reading recently? Only STEM (science, technology, engineering and mathematics) books are counted.
 
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D. J. Tritton, Physical Fluid Dynamics. I never formally learned this topic, but I now need it for my teaching. I really like the way the book is structured, starting with phenomenology before delving into the equations.
 
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Recently I was reading J. MacCormick, Nine Algorithms That Changed the Future
https://www.amazon.com/dp/0691158193/?tag=pfamazon01-20
Some of the most widely used computer algorithms explained in a simple non-technical way. Very readable.
 
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Demystifier said:
Recently I was reading J. MacCormick, Nine Algorithms That Changed the Future
https://www.amazon.com/dp/0691158193/?tag=pfamazon01-20
Some of the most widely used computer algorithms explained in a simple non-technical way. Very readable.
I'll have to add this to my reading list.
 
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Computational Electromagnetics for RF and Microwave Engineering, David Davidson
 
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This semester, I've to create problems for a GR/cosmology lecture. So I'm right now reading a bit in the literature. Whenever there's something unclear, I turn (of course) to

S. Weinberg, Gravitation and Kosmologie, Wiley&Sons, Inc., New York, London, Sydney, Toronto, 1972.
 
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DrClaude said:
D. J. Tritton, Physical Fluid Dynamics. I never formally learned this topic, but I now need it for my teaching. I really like the way the book is structured, starting with phenomenology before delving into the equations.
I like this book for the same reason, along with the experimental results that are included throughout the book. Was easy to read as a student - much nicer than Landau and Lifshitz, the other book we used for the class.
 
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I've been reading "Mathematics for the physical sciences" by Laurent Schwartz, mostly to see how he presents distribution theory for an audience of non-mathematicians.
 
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Tom M. Apostol, Calculus I, II. I never had a chance to study rigorous Calculus, so back to the basics!
 
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  • #10
I am reading "Perfect Rigour" - Masha Gessen, I don't know if that counts.

vanhees71 said:
This semester, I've to create problems for a GR/cosmology lecture. So I'm right now reading a bit in the literature. Whenever there's something unclear, I turn (of course) to

S. Weinberg, Gravitation and Kosmologie, Wiley&Sons, Inc., New York, London, Sydney, Toronto, 1972.

If I had to make a list of books on this topic I would put that at the end. No, in fact I will not put it in the list.
 
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  • #11
I am reading Zee's Gravitation. I am going really slow as I find it is a relatively hard book to read but it is very rewarding... What I am really enjoying though is the video series on Mathematical Physics by Prof. Balakrishnan. I am also reading a bit on AP calculus topics (more like getting familiar with) as I will soon have to teach my daughter.
 
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  • #12
Rereading MTW Gravitation. Much prefer this canonical geometric GR approach to Weinberg's book which I read about a year ago.
 
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  • #13
I am now reading https://www.amazon.com/dp/3642372759/?tag=pfamazon01-20.
I consider it a really amazing textbook on the subject. If I remember well, Leonard Susskind somewhere in his Modern Physics Special Relativity lectures remarked that SR can be learned within days. This book doesn't support this opinion. Looks like I'm definitely not in Susskind's league :-)
 
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  • #14
I've been reading the 2nd edition of Sutton and Barto's Reinforcement Learning, trying to learn how the biology and machine learning ideas are related.
 
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  • #15
Explorations in Mathematical Physics by Don Koks. I want to see physics math done from a geometric algebra point of view. (But I am afraid that the physics will be too tough for me.)
 
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  • #16
I'm preparing for uni in September by working through Newtonian Mechanics by French, and reviewing calculus from Lang/Kline. Occasionally I'll reference HRW if I find myself struggling with a problem.
 
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  • #17
I am reading Jackson's ED 3rd edition, Aitchinson's and Hey's Gauge book latest edition, also Peskin's, Brown's, Ryder's and Zuber's books and Ashcroft's book accompanied with a problem book by Han on Solid state physics.

A few months ago (November,December a bit of january), I was also reading books of Munkres on Analysis on Manifolds and a book on representation of finite groups by Liebeck's and Gordon's; I should really return to these book someday.
 
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  • #18
I'm working through the new 5th edition (2016) of Gilbert Strang's https://www.amazon.com/gp/product/0980232775/?tag=pfamazon01-20. I like Strang because he puts a lot of effort into showing you how to think of the subject on an intuitive level.

Also, I stumbled across this little gem... Kuldeep Singh's https://www.amazon.com/gp/product/0199654441/?tag=pfamazon01-20 I find it a great, light book for very quickly building up intuition and the big picture. Much of the book is devoted to Question/Answer dialog as if you were conversing with a prof and it has many fully solved problems. Sometimes I find it light enough that I just skim some pages but that's perfect because there are plenty of other books that are tough slogging. I could see folks who are self studying, finding this book very appealing as an appetizer before taking on something more meaty like Friedman or Treil.
 
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  • #19
FactChecker said:
Explorations in Mathematical Physics by Don Koks. I want to see physics math done from a geometric algebra point of view. (But I am afraid that the physics will be too tough for me.)
I have this book/ Looks great. Been meaning to read it for a while now...so much to read and so little time.
 
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  • #20
I was reading Shankar Quantum Mechanics but I had to take it back to the library.
Now I am browsing Whittaker, Analytical Dynamics, and also Torge, Geodesy.
 
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  • #21
Geometry and The Imagination: David Hilbert. Fascinating Stuff.
 
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  • #22
Foundations of Geometry, also by David Hilbert. I'm reading this because I've been working on automatic theorem proving as applied to Euclid. Hilbert filled in some logical gaps in Euclid.
 
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  • #23
Linear Alegebra and its applications - Gilbert Strang
Introduction to Mechanics - kleppner and kolenkow
Electricity and Magnetism - Edward Purcell

I borrowed these physics books but now I find them very difficult.
 
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  • #24
Be warned about Purcell. It's quite confusing and unnecessarily complicated in its attempt to be pedagogical. It's easier to use the mathematics of Minkowski space rather than handwaving pedagogics.
 
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  • #25
Buffu said:
Linear Alegebra and its applications - Gilbert Strang
Introduction to Mechanics - kleppner and kolenkow
Electricity and Magnetism - Edward Purcell

I borrowed these physics books but now I find them very difficult.
Buffu,

If you haven't already studied vector calculus and introductory calculus-based mechanics and electromagnetism (from a source such as Halliday and Resnick, or some other equivalent book) then those physics books will be quite difficult. I took a course out of Purcell, and even with access to very helpful Professor and TA it was brutal.

Strang should be fine - perhaps it just isn't your style. Have you looked at the mit open courseware site for the class that is based on that book?
 
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  • #26
"Structures (Or why things don't fall down)" ...by EJ Gordon
 
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  • #27
jasonRF said:
If you haven't already studied vector calculus and introductory calculus-based mechanics and electromagnetism (from a source such as Halliday and Resnick, or some other equivalent book) then those physics books will be quite difficult. I took a course out of Purcell, and even with access to very helpful Professor and TA it was brutal.

I have studied Electromagnetism and Mechanics in school. I think the maths is hitting me most but I think I will understand it after some time.
It is hard but I just love reading them. They are so very well written and formatted in Latex. No bullshit pictures and no hyperlinks to some "help" sites.
 
  • #28
martinbn said:
I am reading "Perfect Rigour" - Masha Gessen, I don't know if that counts.
It counts, popular STEM books are also STEM books.
 
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  • #30
https://www.amazon.com/dp/B00YSILNL0/?tag=pfamazon01-20 by Sebastian Raschka. Very good overview of the subject. After this, I want to go back and fill in much of the statistical background that I'm missing for machine learning. I've become the machine learning guy at work because of my Python skills, but I'm woefully lacking in the statistics background.
 
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  • #31
Demystifier said:
What book are you reading now, or have been reading recently? Only STEM (science, technology, engineering and mathematics) books are counted.
Richard Dawkins - The Greatest Show On Earth
One of his better, more science focused book. He tends often go on about his anti-religious antics from time to time, that even shows up sometimes in his books. Which I don't care about in my opinion, I don't practice a belief system. But for those, like me who just wants to gobble up in science and/or biology in general. This is a great book.
 
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  • #32
I started to read '' The Mathematical Foundations of Quantum Mechanics'' by David A.Edwards
 
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  • #33
Ssnow said:
I started to read '' The Mathematical Foundations of Quantum Mechanics'' by David A.Edwards
This is a long review paper, so I think we can count it as a "book". :smile:
 
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  • #34
Well, then one has to define, how a long review paper becomes a book. I'd say that some review articles well deserve the status of a book (e.g., Abers and Ben Lee's review article about gauge theories, which is among the best presentations of the subject I know:

E. Abers and B. Lee, Gauge Theories, Phys. Rept., 9 (1973), p. 1–141.
http://dx.doi.org/10.1016/0370-1573(73)90027-6
 
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  • #35
I'm reading one of the "Very Short Introduction" books from Oxford University Press: Philosophy of Science: A Very Short Introduction.

I've noticed in a couple of threads on PF that some folks get offended or disdainful when they hear the phrase "philosophy of science"; and will even scoff that "science needs no philosophy." I've never agreed with this attitude as it makes little sense. Philosophy of science touches on not just the history of science, but the evolution of scientific communities, standards, and methods; all of which is relevant to not just the doing of science, but the understanding of it by the public (I count myself as a member of the public).

So far this particular book seems well done & I'm learning things as I go along that seem very relevant to science today, including topics I read about on this forum as well as in the mainstream media. E.g. the chapter on scientific inference has a primer on causation that is basic, but still useful for lay readers; the chapter then precedes to explain the importance of probability to inference, the distinction between objective and subjective probabilities, the rules of conditionalization, etc. I know a little about this because I studied the math of classical probability some years back; however conditional probability is something I need to learn more about, and this offers a what seems a decent conceptual introduction.
 
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  • #36
vanhees71 said:
I'd say that some review articles well deserve the status of a book
Here are some of my candidates:

F. Gieres, Mathematical surprises and Dirac's formalism in quantum mechanics, quant-ph/9907069.
It is not so long (56 pages), but the style of presentation is such that it looks like a book chapter.

R. Slansky, Group theory for unified model building, Phys. Rep. 79 (1981) 1-128.
A classic.

T. Eguchi, P.B. Gilkey, A.J. Hanson, Gravitation, gauge theories and differential geometry, Phys. Rep. 66 (1980) 213-393.
Another classic.
 
  • #37
The first one is indeed a masterpiece. The other two I don't know (yet) :-).
 
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  • #38
I'm reading through Griffith's Introduction to Elementary Particles. I just got done with my school's equivalent of Modern Physics (though I wouldn't call it simply Modern Physics) and after our few weeks on atomic, nuclear, and particle, I had to know more.
 
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  • #39
Karadra said:
Richard Dawkins - The Greatest Show On Earth
One of his better, more science focused book. He tends often go on about his anti-religious antics from time to time, that even shows up sometimes in his books. Which I don't care about in my opinion, I don't practice a belief system. But for those, like me who just wants to gobble up in science and/or biology in general. This is a great book.

There is an album by Nightwish of the same name inspired by this book of Dawkins. They had Dawkins on stage in Wembley, obviously he did not sing but gave some kind of "prologue".
 
  • #40
Markushevich, Theory of Functions of Complex Variable, 2nd edition
Van Trees, Detection, Estimation, and Modulation Theory, Part I, 1st edition
 
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  • #41
deskswirl said:
Van Trees, Detection, Estimation, and Modulation Theory, Part I, 1st edition
Cool - you are going old-school (but insightful!).
 
  • #42
jasonRF said:
Cool - you are going old-school (but insightful!).

I thinking about reading Kay's book next. I'm not aware of any other estimation books. Are you?
 
  • #43
deskswirl said:
I thinking about reading Kay's book next. I'm not aware of any other estimation books. Are you?

Kays book on estimation is good. It focuses on discrete problems (so no integral equations like in Van Trees) and is more straightforward to a non-expert like me.

There are other books:
Scharf, Statistical signal processing (good, but I haven't spent much time with it)
Poor, An introduction to signal detection and estimation (very mathy - you will see measures and integral equations. I'm not a fan but it is popular among some university professors)

Some books on random processes for engineers have some chapters/sections on estimation, depending on what you are looking for: Papoulis (especially fourth edition), Stark and Woods

Jason
 
  • #44
I'm currently reading Introduction to Quantum Mechanics by D. J. Griffiths. I'm a huge fan of QFT, and to master that, I need QM, so I'm reading that now. And seeing working with operators rather than functions can be clumsy. :confused:
 
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  • #45
i am reading Mumford's "red book" on algebraic geometry. i first perused it 50 years ago but did not really grok it thoroughly. now retired and with time on my hands i want to understand the basics as explained by the great experts.

[seven months later, I am up to page 50, but i have split a lot of firewood!]

[after another 15 months, april 2019, I am now up to page 120. I had no idea there was this much basic information about my subject that I did not know. Makes it somewhat embarrassing in hindsight to think back on all those conversations with people who actually knew the subject. But even now I see questions asked online whose answers are in this book, so not everyone has mastered this source.]

Now that I know how long it takes me to actually read a real book, I am daydreaming about a world wherein all of those prime years in jr high and high school would not have been wasted learning nothing. When I got to grad school I was forced to try to read such books as Spivak's Differential Geometry, and Kodaira and Morrow's Complex Manifolds, in a few days! Someone has to show you which books to read much earlier. Here is a moving article about a man who devoted his career to reaching out to young talents:
\
I got very bogged down in chapter II. section 8, specializations, from all the algebra, but am moving along again now, some months later. Nov. 18, 2019 and I am on page 133 and getting some feel for the ideas of the section. It involves considering a field k and a subring R which is a local ring and maximal for the relation of "dominance" of local rings. If m is the maximal ideal of R, this section discusses how to pass from an algebraic variety over k to one over R and then one over the quotient field L = R/m. One thinks of R as the ring of an infinitesimal curve C with a "fat" dense point a, and a small closed point b. Via the ring maps R-->k, and R-->L, one has maps in the other direction of projective spaces P^n(k)-->P^n(R), and P^n(L)-->P^n(R). There is also a map P^n(R)-->{a,b} so that P^n(k) is the fiber over a and P^n(L) is the fiber over b. Then one specializes an algebraic subvariety Z in P^(k) by mapping it first into P^n(R) and then intersecting it with the closed subspace P^n(L).

By example, one considers a variety over k = Q, defined by integer equations, and the associated integer points of the variety over Z defined by the same equations. Then one reduces it mod p for some prime p, and considers it over Z/pZ = L. The only general result so far is that given an irreducible (hence connected) variety over k, its specialization over L is still connected and of the same dimension.

Ok I finally finished chapter II, ending on p. 136, on December 6, 2019. It was slow but I learned a lot of algebra, including facts about torsion free, flat, and free modules, e.g. although these are consecutively more restrictive conditions in general, they are all equivalent for finitely generated modules over a valuation ring, since every finitely generated ideal in such a ring is actually principal. This gives an idea of the kind of specialized commutative algebra knowledge one needs for this chapter. So I am into my third year of reading this basic book in my specialty, and still enjoying and benefiting from it. Looking forward to the third chapter, on local properties of varieties, which promises to be more geometric. This current chapter was also challenge for me to see the geometry behjind the relentlessly algebraic description, but I learned a few things such as: saying a map X-->Y makes local rings of X torsion free modules over the local rings of Y, means e.g. that no component of X can map into a proper closed subset of Y, since then a function in Y vanishing on that closed subset would pull back to a function that equals zero on X when multiplied by a function vanishing on the other components of X. So the algebraic condition "torsion free" implies the geometric property of density of images of every component. stuff like that takes me time to absorb.

So reading chapter II required several excursions into algebra books, especially in section 4, fields of definition, for bolstering my knowledge of field theory, things like free joins and linear disjointness, and in section 8, for more on module theory, which is also being called on in chapter III.1, as well as localization. So I am taking another hiatus and reviewing bourbaki commutative algebra, chapters 1 and 2 on flat modules and localization.

I am getting a little better feel for tensor products, which have always seemed mysterious. The key properties are that the module MtensN is generated by elements of form mtensn, i.e. consists of linear combinations of them. Then the other key fact is to understand when two such linear combinations are equal, and for that the best thing to keep in mind is the mapping property, that linear maps out of MtensN correspond uniquely to bilinear maps out of MxN. I.e. the whole difficulty is that we like to deal with concrete elements, but it is very challenging in a tensor product to know just when a linear combination is actually equivalent to zero. Moreover it depends on which tensor product the element is considered as belonging to! I.e. in ZtensZ, the element 2tens3 is non zero, but in Ztens(Z/2), it equals zero! Also, even though Z and 2Z are isomorphic, and the element 2tens3 is zero in Ztens(Z/2), it is not zero in (2Z)tens(Z/2) ! The reason of course is that the isomorphism between them takes 2tens3 to 4tens3, which is zero in (2Z)tens(Z/2).

By the way, many people disparage Bourbaki as a text, but just today I found it to be the only adequate resource on my shelf for the algebra facts I needed on flatness. It was not covered in Atiyah Macdonald for instance, and when I turned to Hartshorne for a reference, his first one, Matsumura, dismissed the proof as follows: "the equivalence of properties 1-5 are well known". Thanks a lot. I also did not easily find what I wanted in Eisenbud, so I am gaining an appreciation for Bourbaki, which I also recall was a standard reference even for my great algebra teacher, Maurice Auslander. Eisenbud did have an enlightening remark about the proof of right exactness of tensoring however, which illuminated the somewhat more direct proof given in all other sources. No sources however gave an entirely direct proof, with elements, due to the difficulty above of dealing directly with linear combinations in a tensor product, and knowing just when one is equivalent to zero.

The more Bourbaki I read the more I like it. It gives complete coverage and complete proofs, very clearly exposed with no hand waving or steps left to the reader. This should recommend it to the people here who have said they want detailed explanations that do not leave big gaps for the reader. It also has exercises and even historical commentary. It seems that someone whom prepres in a subject from this source knows everything there is to know about it. Although the authors are anonymous, we know by now that they were all very famous top level mathematicians and this shines through in the quality of the coverage. it should suffice to mention e.g. Weil, Serre, Cartan, Chevalley, Dieudonne', Tate, Eilenberg, Borel, Grothendieck, Lang, Beauville, Raynaud, Samuel...

I wish I had the english translation but the french is also very clear and very easy french for someone with even a basic knowledge of the language. e.g. "commutative algebra" is "alg'ebre commutative", and "flat modules" is "modules plat". "localization" is "localisation". "ring" is "anneau". (think of annulus?)
 
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  • #46
mathwonk said:
i am reading Mumford's "red book" on algebraic geometry. i first perused it 50 years ago but did not really grok it thoroughly. now retired and with time on my hands i want to understand the basics as explained by the great experts.

Is algebraic geometry same as analytical geometry ?
 
  • #47
analytic geometry mostly studies geometric figures defined by linear and quadratic equations, in 2 or 3 dimensional affine space over the real numbers. algebraic geometry studies geometric loci defined by polynomials in any number of variables in affine or projective space of any dimension, over any field, as well as abstract versions of these loci defined analogously to manifolds by covering "charts", which themelves can be isomorphic to any affine locus. In particular "singular" points are welcomed, which are points where the locus is not like a manifold but can cross it self or have kinks and folds. In all these cases the functions acting on the loci are polynomials, or derived from them. In abstract algebraic geometry, an attempt is made to further include as rings of "functions" not just polynomials over a field, but any commutative ring with identity whatsoever. In this theory, one starts from such a ring A, and forms the set spec(A) consisting of all prime ideals of A. This is then given a topology in which the "closed" points are the maximal ideals, and prime ideals of coheight r are thought of as subloci of dimension r.

Over the complex number field, the study of geometric loci of dimension one in the "plane" i.e. C^2, or the projective plane and polynomials and rational functions defined on them, is essentially equivalent to the study of one dimensional complex manifolds and holomorphic and meromorphic fuunctions defined on them.

so yes, it starts out a little like analytic geometry, but then you raise the degree and the dimension, and you generalize to more abstract fields and even rings. and you tend not to entertain transcendental functions like e^x, or sin and cos. and although you can imitate differential calculus, it is harder to do integral calculus, although i suppose the complex analytic theory of residue, which you can imitate, gives you a hand in that direction.

as example, the ring R[X] where R = reals, gives a space spec(R[X]) consisting of all prime ideals of R[X], i.e. zero, and all ideals generated by irreducible linear or quadratic real poynomials. If C = complexes, then spec(C[X]) is zero and all ideals generated by linear polynomials X-z where z is a complex number. The ring inclusion R[X]-->C[X] induces by pullback a geometric map spec(C[X])-->spec(R[X]) that is generically 2 to 1, roughly with each pair of conjugate complex numbers mapping to the irreducible real quadratic with those roots, and branched over the "real line" consisting of the maximal ideals of R[X] with linear generators. So from this point of view, the space spec(R[X]) has more information than just the real solutions of real polynomials, it also incorporates Galois orbits of complex solutions. Thus the theory lends itself also to study of number theory.

There are some general analogies with linear algebra, but geared up. Just as one linear equation on k^n defines a linear subspace of codimension one, so (if we assume k algebraically closed) does one polynomial equation on k^n define an algebraic variety of codimension one. More generally, the codimension of the locus in k^n defined by r equations cannot be more than r, in the general case as well the linear case. A surjective linear map from k^m to k^n has all fibers as linear spaces of dimension m-n, while a surjective polynomial map k^m-->k^n has all fibers of dimension at least m-n, and the general one of exactly that dimension.

if you want to begin reading about algebraic geometry, and are really a beginner, a good book is Algebraic Curves, by Robert Walker, or maybe with a bit more algebraic background, Undergraduate algebraic geometry, by Miles Reid. A fantastic book is the huge, scholarly tome: Plane algebraic curves, by Brieskorn and Knorrer. Oh another excellent one is Riemann surfaces and algebraic curves, by Rick Miranda. Bill Fulton has made his lovely 1969 book on curves available for free:

http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf

Basic algebraic geometry, by Shafarevich, is an excellent introduction to higher dimensional algebraic geometry, i.e. not just curves. All these books are more introductory than Mumford's red book. Mumford's book is of course wonderful, but you will appreciate it more with some background from some of these other books, which have more examples and exercises, and are less abstract.

as a measure of the difference in analytic geometry and algebraic geometry, even in dimension one, note that every (projective) plane curve, over the complex numbers, is a compact surface. those studied in analytic geometry, namely circles, parabolas and hyperbolas, are all (over the complexes) just spheres, whereas those of higher degree are compact surfaces of arbitrary genus g ≥ 0. E.g. plane cubics have genus 1 and smooth plane quartics have genus 3. Indeed defining the genus was a primary contribution by Riemann to the study of plane curves.

The three main theorems about plane curves are the bezout theorem on the number of intersections of two plane curves, the resolution of singularities saying that every plane curve with singularities is the image by a degree one map of a curve having no singularities, and the riemann roch theorem which computes the number of rational functions on a given curve with a given set of poles. all three of these theorems are proved in Walker and Fulton.

generalizing these theorems to higher dimensions have been a primary focus of research for a 150 years or more. The general riemann roch theorem was proved by hirzebruch in the 1950's i think and generalized further by grothendieck in the 1960's. the bezout theorem has been beautifully generalized by fulton in his book Intersection theory, and the resolution of singularities was published by hironaka in 1964 in characteristic zero, and announced by him this year(!) in characteristic p.

http://www.math.harvard.edu/~hironaka/pRes.pdf
 
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  • #48
I just realized I may have misunderstood the question about the relation of algebraic to analytic geometry. I understood you to be asking about the elementary subject of "analytic geometry" that is covered say in or before a beginning calculus course of one variable, say the material in George Thomas' Calculus and analytic geometry, or the chapter titled analytic geometry in the book Principles of mathematics by Allendoerfer and Oakley. But today, to a professional geometer, the term analytic geometry means the study of geometric loci in complex space of arbitrary dimension, which are defined by analytic, i.e. holomorphic functions. These also have abstract analogs as complex manifolds, and more generally complex analytic varieties. This is the subject covered for instance in the excellent book Complex analytic varieties, by Hassler Whitney.

The Hirzebruch Riemann Roch theorem mentioned above was proved in the context of complex manifolds, using tools from topology such as cobordism, while that of Grothendieck was in the context of algebraic varieties. Grothendieck had to give an algebraic version of chern classes for his work I believe.

Since polynomials in several variables are a particular type of analyic functions, this means that in a sense, algebraic geometry over the complex numbers is a special case of this broader notion of analytic geometry. Indeed the two subjects overlap significantly, and it was in the 19th century when Riemann introduced complex analysis and topology into the study of algebraic plane curves that algebraic geometry really deepened and started to become the vast subject it is today. Indeed until Riemann introduced topology into the subject, the concept of the genus of a "curve" was unknown. After his work this concept was algebraicized and introduced abstractly in terms of the dimension of the vector space of algebraic differential forms.

I.e. every algebraic plane curve in C^2 inherits a complex analytic structure from its embedding, and Riemann even showed how to remove the singularities from any plane curve and render it into a one dimensional complex manifold, the "Riemann surface" as we call it today, of that curve. A basic theorem is that the field of meromorpic functions on the riemann surface of a plane curve is isomorphic to the field of rational functions of the curve. He also gave an abstract definition of a one dimensional complex manifold and showed that when it is compact, it must arise from an algebraic plane curve, i.e. he gave a way to embed the complex manifold into complex projective space as the locus defined actually by polynomials, from which it could be projected into the plane.

In higher dimensions, even compact complex manifolds need not be algebraic however, since there exist compact complex surfaces, even tori, homeomorphic to the product of 4 circles, that carry no global meromorphic functions at all. For compact complex manifodls that can be embedded complex analytically into the projective space, it can be proved that the image of the embedding is always cut out by polynomials, so that analytically embedded compact complex manifold is actually an algebraic variety, algebrically embedded. It was Kodaira who generalized Riemann's algebraizability result to characterize exactly which compact complex manifolds have such embeddings, they are the ones that carry a sufficiently positive "line bundle", and since such a line bundle is detected by its chern form, it suffices for there to exist a positive integral 2 form of type (1,1), as i recall from distant memory. This is a certain type of cohomology class in H^(1,1);C intersect H^2;Z.

There is a famous paper of Serre referred to as GAGA, Geometrie analytique et geometrie algebrique, in which he shows that for complex projective varieties there is an equivalence of categories between their complex analytic coherent sheaf cohomology theory and the algebraic version defined by their algebraic structure. Sheaf theory was introduced in the mid 20th century as a tool in several complex variables as i recall, and Serre greatly enhanced algebraic geometry by giving an algebraic version of sheaf cohomology in his great paper FAC (faisceau algebrique coherent). Grothendieck then generalized sheaf cohomology further with a more general definition having better exact sequence properties (Serre had used Cech cohomology while Grothendieck used derived functor cohomology).

Having studied several complex variables myself in grad school, and having considered being a complex analyst, (and earlier having studied and contemplated doing algebraic and differential topology), when I returned to algebraic geometry, I brought with me and continued to use the complex analytic and topological tools I had available. So even though I call myself an algebraic geometer, in a significant sense I was really a more of a complex analytic geometer.
 
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  • #49
@mathwonk I was actually mentioning about the analytical geometry that is taught with calculus but your second post was also worth mentioning. Is differential geometry a subset of algebraic geometry ? My geometry knowledge approximately zero. I know plane geometry taught in school and a bit of conic sections.

I will read the book Reid's book after learning a bit of linear algebra.
Thank you for all the books.
 
  • #50
differential geometry includes a notion of length which is not part of algebraic geometry. the concept of curvature however seems to coexist in both in some form. Some of my friends did some work on the behavior of curvature on plane curves I believe, in particular Linda Ness.

Apparently there is a natural metric one can use on algebraic curves in affine or projective space, the :"Fubini - Study" metric, and one can then study differential geometric properties of algebraic varieties. Here is a part of Linda's thesis done under the direction of the famous algebraic geometer David Mumford.

http://www.numdam.org/article/CM_1977__35_1_57_0.pdf
 
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