What are you reading now? (STEM only)

[jasonRF]

I picked up two gems for ~$25, Theory of Electromagnetic Waves by Kong and Variational Techniques in Electromagnetism by Cairo & Kahan.

Both look pretty good, Kong treats electromagnetic materials in a moving frame immediately, which gives insight into materials becoming bianisotropic immediately. I think it's going to take time for me to digest it but worth the investment.

I didn't realize Theory of Electromagnetic Waves was that different from the 1990 Electromagnetic Wave Theory. If you aren't familiar, the newer book starts with two chapters designed for a junior level course, then proceeds to enough advanced material for a full year graduate course. Waves in moving media is discussed starting on page 913. It sounds like you are in for some pretty interesting (and advanced!) reading.

jason

 

[vanhees71]

How does it answer the Abraham-Minkwski controversy?

 

[Dr Transport]

How does it answer the Abraham-Minkwski controversy?

No, it doesn't even mention it. Way at the end of the text, when discussing the Lagrangian formulation, he defines the momentum density, ala Minkowsky, as $\vec{G} = \vec{D}\times\vec{H}$.

 

[vanhees71]

? That I've never seen. It's either $\vec{E} \times \vec{H}/c^2$ (Abraham) or $\vec{D} \times \vec{B}$ (Minkowski).

The salomonic conclusion of this dilemma is that one momentum density is the kinetic momentum density (Abraham) or the canonical momentum density (Minkowski) of the em. field. Taking the sum of total the kinetic or canonical momenta of the medium and the em. field you get the same result and a conserved quantity for a closed system of a polarizable medium and the em. field.

Which momenta are to be used to describe the local effects in a polarizable medium depends on the physical situation. For a nice review, see (open access!)

https://doi.org/10.1098/rsta.2009.0207

 

[Dr Transport]

? That I've never seen. It's either $\vec{E} \times \vec{H}/c^2$ (Abraham) or $\vec{D} \times \vec{B}$ (Minkowski).

The salomonic conclusion of this dilemma is that one momentum density is the kinetic momentum density (Abraham) or the canonical momentum density (Minkowski) of the em. field. Taking the sum of total the kinetic or canonical momenta of the medium and the em. field you get the same result and a conserved quantity for a closed system of a polarizable medium and the em. field.

Which momenta are to be used to describe the local effects in a polarizable medium depends on the physical situation. For a nice review, see (open access!)

https://doi.org/10.1098/rsta.2009.0207

Oops, my bad, mis read it, $\vec{D} \times \vec{B}$, not what I mentioned previously.

 

[Demystifier]

K. Huang, Quantum Field Theory From Operators to Path Integrals
- One of better QFT textbooks that I have been reading (and I've been reading a lot of them). One of nice things about it is that it explains the essence of renormalization already at page 5, in a manner a'la Wilson that does not depend on quantization.

Speaking of Huang and the essence of renormalization, see also a short review https://arxiv.org/abs/1310.5533 .

 

[mathwonk]

Still reading Mumford's redbook of algebraic geometry, since 2017!

edit: July 20,2020. Got sidetracked again by the Coronavirus shutdown, I guess. Only up to page 153, but have had to read essentially an entire (small) book on commutative algebra, Undergraduate Commutative Algebra, by Miles Reid, which I highly recommend for learning from. The one problem in Mumford, on page 153, made me feel the need to learn the classification of finitely generated modules over a "Dedekind domain" (a domain, all of whose localizations at non zero primes are principal ideal domains). This generalizes naturally the classification of fin.gen. modules over a principal ideal domain, but is not taught in all books or courses that include the more standard result over p.i.d.'s; e.g. it is not in my own algebra course notes, nor in Lang, which I had thought to be pretty encyclopedic. I am consulting Dummitt and Foote, which is proving quite useful.

Am also sidetracked by watching the "pseudo lectures" on scheme theory from Ravi Vakil of Stanford, available on youtube, and continuing through the summer of 2020, (originally recorded every saturday at 8am pacific time, made available later), as a way of "spitting in the face" of the virus. Also reading his online notes "The rising sea", the title being a reference to Grothendieck's description of his way of thinking about solving math problems. Amazingly, Ravi seems to have signed up almost 800 fairly active participants worldwide for his "pseudo course", people interested in schemes, but coming from many walks of scientific inquiry. If interested, see the links below:

https://math216.wordpress.com



http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf (these are the notes, but site is not secure)

 

[MathematicalPhysicist]

Still reading Mumford's redbook of algebraic geometry, since 2017!

edit: July 20,2020. Got sidetracked again by the Coronavirus shutdown, I guess. Only up to page 153, but have had to read essentially an entire (small) book on commutative algebra, Undergraduate Commutative Algebra, by Miles Reid, which I highly recommend for learning from. The one problem in Mumford, on page 153, made me feel the need to learn the classification of finitely generated modules over a "Dedekind domain" (a domain, all of whose localizations at non zero primes are principal ideal domains). This generalizes naturally the classification of fin.gen. modules over a principal ideal domain, but is not taught in all books or courses that include the more standard result over p.i.d.'s; e.g. it is not in my own algebra course notes, nor in Lang, which I had thought to be pretty encyclopedic. I am consulting Dummitt and Foote, which is proving quite useful.

Am also sidetracked by watching the "pseudo lectures" on scheme theory from Ravi Vakil of Stanford, available on youtube, and continuing through the summer of 2020, (originally recorded every saturday at 8am pacific time, made available later), as a way of "spitting in the face" of the virus. Also reading his online notes "The rising sea", the title being a reference to Grothendieck's description of his way of thinking about solving math problems. Amazingly, Ravi seems to have signed up almost 800 fairly active participants worldwide for his "pseudo course", people interested in schemes, but coming from many walks of scientific inquiry. If interested, see the links below:

https://math216.wordpress.com



http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf (these are the notes, but site is not secure)

If you are interested for an introduction to commutative algebra you can't go wrong by reading Atiyah's and Macdonald's textbook:
https://www.amazon.com/dp/0201407515/?tag=pfamazon01-20

Though I haven't finished reading it, I'll return to it someday.

 

[martinbn]

Still reading Mumford's redbook of algebraic geometry, since 2017!

When I was a student for algebraic geometry I studied Hartshorne's book. There was one exercise that I worked on and wasn't very confident if I was right. Purely by chance, browsing int the library, I opened the red book and the first proposition I saw was the statement of that same probelem with a detailed proof. Since then I think very highly of the book.

 

[mathwonk]

I have several works on commutative algebra, including Atiyah-Macdonald, Zariski - Samuel, Eisenbud, Northcott, Matsumura, Milne, and Dieudonne's Topics in local algebra, as well as chapters in general algebra books, such as Lang, Dummitt and Foote, Hungerford, Jacobson, Mike Artin, and Van der Waerden. Out of all these, I find Miles Reid's little book the most useful (although everything actually included in Mike Artin's book is helpful) in the sense of being easy to read, insightful, and limited in its goals. I find I benefit from reading books aimed at people with far less training them myself. I.e. as a postgraduate myself, I often benefit from an explanation that is aimed at graduates or even undergraduates. Atiyah - Macdonald is very authoritative, and the proofs are very efficient and slick and correct, but it is the sort of book whose explanations go "in one ear and out the other" at least for me. The exercises in A-M are also frequently hard for me, whereas the ones in Reid are not only easier, but also more instructive. I did consult A-M for a treatment of general valuation theory, which Miles omits.

To be fair, I think one reason Miles' book is preferable to me, is that he had A-M available for the proofs and only had to augment the insights, improve the readability, and create better exercises. I also have only the earlier work by Matsumura, his Commutative Algebra. His later work Commutative Ring Theory is widely considered to be easier to learn from, and perhaps benefited from its translation by Miles Reid. So while those other books are ones I have spent time in and then stopped, only to return to the same topic later having forgotten it, Miles' book seems to be one that I think I would enjoy reading all of, and then setting it aside, having actually learned it. So far I have read chapters 5,6,7,8, but benefited so much that I actually went back to chapter 1, and learned something. Hence I am tempted to read 2,3,4, even though they had seemed too elementary at first glimpse. I am also inclined to return afterwards to A-M to see if it then is more useful, and I thank you for the reminder of its quality.

remark: I have read chapter 1 of A-M and worked most of the exercises, but in general there are just too many exercises in there for me not to get bogged down. This book's text also goes too fast for me. The proofs come so fast and briefly I don't have time to understand their implications. So I would need to discipine myself to read this book very slowly, stopping to think about all the slick proofs.

By the way, I could be wrong, but it seems to me the second to last sentence on page 31 of A-M is incorrect. They say there the A - algebra structure on the ring D is by means of the map a --> f(a) tensor g(a), whereas it seemed to me last time I read it was that it should be via a -->f(a) tensor 1 = 1 tensor g(a). Yes in fact this is forced by the very next sentence, giving the commutative diagram for the various given ring maps. The map they give is obviously not even additive, since f(a+b) tens g(a+b) does not equal f(a) tens g(a) + f(b) tens g(b). In my opinion that is the sort of thing that can happen when you go too fast and don't pause to explore the consequences of your statements, although these authors are so smart and knowledgeable, there seem to be remarkably few such errors.

I also have as introductory algebraic geometry books, Mumford's two books, yellow and "red", Hartshorne, Vakil, Miles Reid, James Milne, Mike Artin, Fulton, Walker, Shafarevich, Griffiths, Miranda, Griffiths and Harris, ACGH, Hassett, Bertram, Harris, Cox-Little-O'Shea, Semple and Roth, Fischer, Brieskorn and Knorrer, ... well I have a lot.

As a tip for reading Hartshorne's book, he himself wrote it after teaching several courses on the subject, some of which I sat in on. The first course was the basis for his chapter 4 on curves, and the second course was on surfaces, his chapter 5. Hence I recommend reading them in that order, i.e. start with chapters 4 and 5 and only then go back to 2 and 3 for background you may want to see developed in detail. Chapter 1 is independent of the others, a separate course on varieties and examples. In fact Hartshorne himself suggests starting in chapter 4, for "pedagogical" reasons, but only says so in the first paragraph of that chapter, which the reader may not have noticed until plunging haplessly into chapters 2 and 3.

I also celebrate the great effort Hartshorne has made to provide us a clear account of so many things, but his choice of just citing commutative algebra results without proof, does not work well for me. I prefer Shafarevich's model of actually proving the needed results as they are encountered, as he does especially in the first one - volume edition of his book, which I recommend highly. Mumford also tends to lose me in his red book on those occasions where he sends me to Zariski - Samuel for extensive background on fields, rather than just telling me the argument he needs. Zariski-Samuel is excellent, but the excursion means a big time sink for me.

Mumford is so knowledgeable and so succinct in his explanations that it is a great service for me when he just summarizes the proof of something, which he usually does. His redbook is the only place I know where one is told what is the relation between varieties over arbitrary fields, and the associated ones over their algebraic closure. After reading this, I was able to easily give a complete answer to a student question on stackexchange about what are the maximal ideals of R[X,Y], where R is the real numbers, whose full explanation had not been provided for some time (although correct answers to the more limited question actually asked had been given, and those people probably knew this as well).

Here is a tiny example of something I absorbed from Reid that I had not realized from any other source, although maybe I would have, had I read Bourbaki more fully. Namely, the primary decomposition theorem for noetherian rings says that every ideal in a noetherian ring, ( a ring in which every ideal has a finite number of ideal generators), can be written as an intersection of "primary" ideals. An ideal is prime, as you know, if when you mod out by it, you get a domain, i.e. a ring in which there are no zero divisors except zero. An ideal is primary if when you mod out by it, the only zero divisors are nilpotent. OK, the surprizing result is that even irredundant primary decompositions are not unique! I.e. the primary ideals involved are not always unique. BUT! those primary ideals that are minimal, ARE unique. Moreover, the prime radicals of both minimal and non minimal primary ideals are unique.

The geometric version of this says that every algebraic scheme in affine space, can be written as a union of irreducible algebraic schemes, and the maximal set theoretic components have a unique scheme structure, but those components that are contained in other larger components have a non unique structure. Nonetheless, the underlying sets of these component scheme are all unique, i.e. the radicals of the primary ideals are all unique. Now it had never dawned on me that the non uniquemess means that those primary ideals are not important. Namely it is the unique objects, namely the prime ideals occurring as their radicals that are important. Reid makes this clear by taking the Bourbaki approach to decomposition, showing that it is the "associated primes" of an ideal that should be focused on. I.e. one defines the associated primes, shows their uniqueness, and then proves that they are the same as the radicals of the primary ideals in an irredundant primary decomposition. Just a remark.

Summary: the primary ideals of embedded components are not important since not unique, rather the support of embedded components matter more. I never realized this before reading Reid. I could still be wrong of course, but I feel I have learned something.

 

[Demystifier]

Why do mathematicians find algebraic geometry so sexy? Is it just because of its highest level of abstraction (schemes etc) or is there some other reason? To me, algebraic geometry does not look very fundamental for mathematics as a whole, in the sense that knowledge of abstract algebraic geometry does not help much in most other branches of math.

 

[Mondayman]

Does algebraic geometry show up in physics? I recall seeing Nima Arkani-Hamed reading Principles of Algebraic Geometry by Joe Harris in Particle Fever.

 

[member 587159]

Why do mathematicians find algebraic geometry so sexy? Is it just because of its highest level of abstraction (schemes etc) or is there some other reason? To me, algebraic geometry does not look very fundamental for mathematics as a whole, in the sense that knowledge of abstract algebraic geometry does not help much in most other branches of math.

That's a huge generalisation. I rather dislike algebraic geometry and I truly hate Hartshorne's book. Now you have an opinion from the opposite side of the spectrum :)

 

[Demystifier]

Does algebraic geometry show up in physics? I recall seeing Nima Arkani-Hamed reading Principles of Algebraic Geometry by Joe Harris in Particle Fever.

Algebraic geometry at some level shows up in physics, but I think this level is not what mathematicians find so sexy. I never seen in physics mentioning things like Grothendieck schemes or Zariski topology.

 

[Demystifier]

That's a huge generalisation. I rather dislike algebraic geometry and I truly hate Hartshorne's book. Now you have an opinion from the opposite side of the spectrum :)

Well, by mathematicians I obviously meant many mathematicians, not all mathematicians. Anyway, what are your favored branches of math?

 

[Dragon27]

Why do mathematicians find algebraic geometry so sexy? Is it just because of its highest level of abstraction (schemes etc) or is there some other reason? To me, algebraic geometry does not look very fundamental for mathematics as a whole, in the sense that knowledge of abstract algebraic geometry does not help much in most other branches of math.

This is from Eisenbud, The Geometry of Schemes

The theory of schemes is the foundation for algebraic geometry formulated by Alexandre Grothendieck and his many coworkers. It is the basis for a grand unification of number theory and algebraic geometry, dreamt of by number theorists and geometers for over a century. It has strengthened classical algebraic geometry by allowing flexible geometric arguments about infinitesimals and limits in a way that the classic theory could not handle. In both these ways it has made possible astonishing solutions of many concrete problems.
...
No one can doubt the success and potency of the scheme-theoretic methods. Unfortunately, the average mathematician, and indeed many a beginner in algebraic geometry, would consider our title, “The Geometry of Schemes”, an oxymoron akin to “civil war”. The theory of schemes is widely regarded as a horribly abstract algebraic tool that hides the appeal of geometry to promote an overwhelming and often unnecessary generality. By contrast, experts know that schemes make things simpler. The ideas behind the theory — often not told to the beginner — are directly related to those from the other great geometric theories, such as differential geometry, algebraic topology, and complex analysis. Understood from this perspective, the basic definitions of scheme theory appear as natural and necessary ways of dealing with a range of ordinary geometric phenomena, and the constructions in the theory take on an intuitive geometric content which makes them much easier to learn and work with.

Doesn't it just make your loins quiver a little bit?

Here are some more passages from Manin, Introduction to the Theory of Schemes

Meanwhile the elements of algebraic geometry became everyday language of working theoretical physicists and the need for concise accessible textbooks only increased.
...
The methods described in these lectures are currently working tools of theoretical physicists studying subjects that range from high-energy physics (see [Del]), where the Large Hadron Collider still (now is year 2016) struggles to confirm or disprove the supersymmetry of our world (or rather models of it), to solid-state physics, where supersymmetric models already work (see, e.g., the very lucid book [Ef] with a particularly catchy title).

 

[member 587159]

Well, by mathematicians I obviously meant many mathematicians, not all mathematicians. Anyway, what are your favored branches of math?

I'm currently doing a lot of functional analysis with a special focus on $C^*$-algebras.

These are truly fascinating objects. The interplay between algebra and topology is really fascinating. For example, every abelian $C^*$-algebra can be realized as $C_0(X)$ where $X$ is a locally compact Hausdorff space and the Gelfand-Naimark theorem actually says that the study of $C^*$-algebras is equivalent with the study of adjointly closed complete subalgebras of $B(\mathcal{H})$ (bounded operators on the Hilbert space $\mathcal{H}$).

 

[Demystifier]

Manin, Introduction to the Theory of Schemes

I like the book covers.

But more to the point, can you give some examples of "astonishing solutions of many concrete problems" that Eisenbud and Harris refer to?

 

[Dragon27]

But more to the point, can you give some examples of "astonishing solutions of many concrete problems" that Eisenbud and Harris refer to?

Oh, I just omitted that from the quote:

On the number-theoretic side one may cite the proof of the Weil conjectures, Grothendieck’s original goal (Deligne [1974]) and the proof of the Mordell Conjecture (Faltings [1984]). In classical algebraic geometry one has the development of the theory of moduli of curves, including the resolution of the Brill–Noether–Petri problems, by Deligne, Mumford, Griffiths, and their coworkers (see Harris and Morrison [1998] for an account), leading to new insights even in such basic areas as the theory of plane curves; the firm footing given to the classification of algebraic surfaces in all characteristics (see Bombieri and Mumford [1976]); and the development of higher-dimensional classification theory by Mori and his coworkers (see Kollár [1987]).

They're rather mathy, of course. For me, it's the abstract (and categorical) language and concepts from the modern theory (like schemes) that are of interest. I've come to the conclusion (from my experience of self-studying), that abstractions make the theory look cleaner, more conceptual and transparent, less messy. And the language of modern algebraic geometry has apparently spread around into many other areas of math and physics (mathematical and theoretical), like category theory (which was first created to help solve certain problems in algebraic topology and then just went out of control and spread all over the math). And that's probably for a reason.

 

[Demystifier]

And the language of modern algebraic geometry has apparently spread around into many other areas of math and physics (mathematical and theoretical), like category theory (which first was created to help solve certain problems in algebraic topology and then just went out of control and spread all over the math). And that's probably for a reason.

So can it be said that's it's not the subject of algebraic geometry itself (namely algebraic varieties) that is so sexy, but rather the general language that is used in the modern abstract formulation of it?

 

[martinbn]

That's a huge generalisation. I rather dislike algebraic geometry and I truly hate Hartshorne's book. Now you have an opinion from the opposite side of the spectrum :)

That I never understood. I can understand if some likes some area more that others, or if he doesn't have an interest in some areas. But to actually dislike any part of mathematics is strange to me!

I'm currently doing a lot of functional analysis with a special focus on $C^*$-algebras.

These are truly fascinating objects. The interplay between algebra and topology is really fascinating. For example, every abelian $C^*$-algebra can be realized as $C_0(X)$ where $X$ is a locally compact Hausdorff space and the Gelfand-Naimark theorem actually says that the study of $C^*$-algebras is equivalent with the study of adjointly closed complete subalgebras of $B(\mathcal{H})$ (bounded operators on the Hilbert space $\mathcal{H}$).

That is even more puzzling, because conceptually the abstract algebraic geometry is very much in this spirit. I might be wrong but I think some of the Grothendick's ideas were motivated by Gelfand's work.

 

[martinbn]

Why do mathematicians find algebraic geometry so sexy? Is it just because of its highest level of abstraction (schemes etc) or is there some other reason? To me, algebraic geometry does not look very fundamental for mathematics as a whole, in the sense that knowledge of abstract algebraic geometry does not help much in most other branches of math.

I think this is true for algebraicly minded mathematicians. The analysts are not that keen on algebraic geometry.

 

[member 587159]

That I never understood. I can understand if some likes some area more that others, or if he doesn't have an interest in some areas. But to actually dislike any part of mathematics is strange to me!

That is even more puzzling, because conceptually the abstract algebraic geometry is very much in this spirit. I might be wrong but I think some of the Grothendick's ideas were motivated by Gelfand's work.

There are plenty of mathematicians that dislike some areas of mathematics. I could give plenty of examples but the pure versus applied mathematics debate already says enough. However, maybe I should add some nuance and admit that the 'dislike' is put too strong. The ideas from algebraic geometry feel unnatural to me and the subject just doesn't click as other subjects do. Maybe the problem is that I self-studied from the (awful$^{(*)}$) book of Hartshorne and I should go look for a better book. But for now, I have no need for algebraic geometry anyway so I feel I can spend my time on other topics instead.

Also, comparing functional analysis and algebraic geometry is a no-go for me. These fields do not have much overlap.

$(*)$ For self-study. Might be a good reference book.

 

[Dragon27]

So can it be said that's it's not the subject of algebraic geometry itself (namely algebraic varieties) that is so sexy, but rather the general language that is used in the modern abstract formulation of it?

Well, modern algebraic geometry has certainly outgrown its classical roots, but I'm not entirely sure what is still considered its subject. The notion of algebraic variety has been generalized as well (from its classical "the set of solutions of a system of polynomial equations"). And the algebraic varieties appear in some problems related to mathematical physics, interplay with differential geometry, etc (don't ask me to provide examples, though :) ).
With regards to sexiness specifically, probably something like that. This kind of intriguing conceptual way it uses to look at the classical problems in geometry and algebra, and how it intertwines them. And the proof of Fermat's Last Theorem...edit:
An interesting article/blog post
https://johncarlosbaez.wordpress.com/2019/03/15/algebraic-geometry/

 

[martinbn]

Also, comparing functional analysis and algebraic geometry is a no-go for me. These fields do not have much overlap.

I didn't mean to compare functional analysis with algebraic geometry. I meant it in a specific way. Gelfand's idea to represent curtain commutative algebras as algebras of functions on some space. To points of the space are the maximal ideals of the algebra and so on. This is also what happens in algebraic geometry. A commutative ring is the ring of regular functions on an affine variety, whose points are the prime ideals of the ring and so on.

 

[mathwonk]

I suspect one reason algebraic geometry comes across as sexy is that so many fields medals went there in recent decades. But in my own experience, it seems to draw people in by its "sweep". When I went to UGA in 1977 I was almost the only person in algebraic geometry. Some years later there were so many people claiming to be in algebraic geometry in some form, people who previously announced themselves as specialists in some other area, that as a joke, my friend, an operator theorist, announced himself on the day we introduced ourselves to each other, as an algebraic geometer.

In practical terms, it turned out that algebraic geometers at UGA were so broadly trained and interested, that they could work together with many other people. Collaborative efforts developed between algebraic geometers and algebraists, number theorists, differential geometers, complex analysts, geometric analysts, even applied mathematicians. Algebraic geometers, at least the ones I know, also know at least something about algebra, algebraic topology, representation theory (not me), differential geometry, several complex variables and complex manifold theory, number theory, and some also are experts in logic. People with questions often showed up in the offices of algebraic geometers to have them answered, even if sometimes they closed the doors first to conceal that they were asking.

I once had a colleague in functional analysis come to me excited about his recent results on something he apologetically said was some abstruse concept called "Fredholm operators". I was puzzled that he thought I would not know what this was, as it was very familiar to me, namely an operator on a Banach space with finite dimensional kernel and cokernel. As is well known, certainly to "all" algebraic geometers, these are basic examples of operators with a well defined concept of "index", namely the difference of the dimensions of those two subspaces, as well as the fact that this index is constant on connected components of the space of such operators. Basic theorems in algebraic geometry and global analysis, (Atiyah-Singer Index theorem), concern giving topological formulas for such indices, and the Riemann - Roch theorem is a classical precursor of these results. Indeed the famous topic of K-theory, developed by Grothendieck in connection with his generalized Riemann - Roch theorem, involves both Fredholm operators and the space B* of units of the Banach algebra B of all bounded operators on a separable complex Hilbert space; i.e. both those spaces, Fred and B*, are "classifying spaces" for K theory, (see K-theory, appendix, by Michael Atiyah). So algebraic geometers tend to know something about Banach algebras and Fredholm operators even if (some) functional analysts do not know what K theory is.

You probably know that Grothendieck, the most impactful algebraic geometer in a century, started out in functional analysis. Of course as noted just above, it is well known that in both subjects, one recovers a space from the algebra of functions on that space by taking the space of maximal ideals in that algebra, or more generally in scheme theory, prime ideals.

elementary exercises: there is a one-one correspondence between the maximal ideals of the ring of continuous functions on the closed interval [0,1] and the points of that interval.
there is a one-one correspondence between the maximal ideals of the polynomial ring C[X], and the space C, where C is the complex numbers.

less elementary: these correspondences hold also for continuous functions on compact hausdorff spaces, and polynomials on affine spaces C^n of any finite dimension. In both cases they are given by sending a point of the space to the maximal ideal of functions vanishing on that point.

In fact compactifications of a locally compact Hausfdorff space X correspond to constant - containing, point - separating, uniformly - closed, subalgebras of C(X).
(I hope I have this right, it has been over 50 years since I did the functional analysis exercises. I remember thinking it was fun to imagine which sub algebra compactifies an open disc as a closed disc, or as a sphere, or as projective 2-space.)

Remark: As to the influence of functional analysis on abstract algebraic geometry, Hilbert proved the algebraic geometry result above (Hilbert's nullstellensatz) in 1893, (Mathematische Annalen, 42 Band, 1 Heft, p.320), 20 years before the birth of Gelfand, who is often associated with its functional analysis counterpart.

In my case, before coming to algebraic geometry, I studied functional analysis, differential topology, algebraic topology, commutative algebra and (derived) functors, and several complex variables; none of it was wasted in the end. I wound up working in complex algebraic geometry, and am now trying to learn scheme theory, in retirement.

So , maybe today algebraic geometry is just seen as a very big tent, and lots of people shelter under it.

Speaking of a big tent, I was thinking one topic I knew nothing about was physics, and then remembered it depends what you consider as physics. I was once invited to deliver a series of lectures on Riemann surfaces to a conference of string theorists, who had decided that a Riemann surface should be considered an elementary particle! I also think of pde as foreign territory, but recall that the key result of Hodge theory, which I have studied (in the context of presenting Kodaira's proof of his "vanishing" theorem), is the representability of deRham cohomology classes on complex manifolds by "harmonic" forms, i.e. ones satisfying the Laplace equation. And the key ingredient of the theory of Jacobian varieties of complex curves is Riemann's theta function, a fundamental solution of the (several variable) heat equation. So the only basic one I have not consciously run across is the "wave equation".

By the way, if you think you don't like algebraic geometry, you might take a look at Semple and Roth, or Milkes Reid's Undergraduate algebraic geometry. I myself find my eyes glaze over when I peruse derived functor cohomology of sheaves, but am fascinated by exploring the structure of the 27 lilnes which lie on any smooth cubic surface in complex projective 3 space. I am even more magnetized by constructions like trying to see how those lines specialize when the cubic surface degenerates to three planes. I.e. If S is a smooth cubic surface and F is the union of three planes, consider the limit of the lines on S in the family F+tS as t-->0. Note that S meets each of the 3 lines where pairs of the planes of F meet, in 3 points. See if you can see why, as S approaches F, a line L of S must come to lie in one of the 3 planes of F, say ∏, and since the other two planes of F meet ∏ in two lines, M and N, the limit of L must join one of the three marked points of M to one of the three marked points of N. This gives all 27 limiting lines, 9 in each plane. For help, consult the book on lines on the cubic surface, by Beniamino Segre. I.e. to care about the modern formalization of algebraic geometry, it helps (me) to know some of the beautiful results that one wants to make precise and rigorous.

Here's another example: for a complete intersection curve C in P^3 of smooth surfaces S and T of degrees d and e, the canonical sheaf on C is O(de(d+e-4)), so 2g-2 = de(d+e-4), where g = genus(C). Hence if d = e = 2, we get 2g-2 = 0, hence C is a genus one curve, i.e. a torus. This result is found in Hartshorne, p. 352, i.e. after hundreds of pages of dense theory.

Now consider this 19th century quick calculation: degenerate one quadric surface to 2 planes, which thus meet the other quadric surface in 2 plane conics, both of genus zero (we assume this for the moment), and the two conics meet each other in two points (where the common line of the two planes meets the quadric surface). Since each conic is topologically a sphere, the union of two spheres with two common points is obviously the result of degenerating a torus by pinching two circles. So before degeneration we had a torus, i.e. a curve of genus one. To see why a plane conic has genus zero, project it from a point of the conic bijectively onto the (projective) x axis.

It is of course important to know why these calculations are rigorously correct, but it is also bad form to deprive students of powerful computational tools that were known and available well before the advent of rigorous methods.

 

[haushofer]

I just ordered Cosmology for the Curious by Vilenkin; this year I'll be covering some cosmology with my (pre-university) high school students.

 

[haushofer]

In between feeding and satisfying a baby I started with the book, and it's actually really nice. It has some nice historical notes, and also covers more advanced topics in a very clear and conceptual way. The conceptual explanation e.g. of the BGV-theorem, by on of the authors, is very nice. I also like that every now and then they don't mind making some more philosophical or even religious remarks (e.g. the role of BGV in discussions with William Laine Craig).

Excellent book, and for just 32 euro (hard cover) highly recommended for everyone with an interest in cosmology.

 

[Demystifier]

J. Schwichtenberg, Teach Yourself Physics (2020)
https://www.amazon.com/dp/3948763003/?tag=pfamazon01-20

It's not so much a book on physics, as it is a book on how to learn physics, especially if your goal is not to pass exams but to get a deep understanding. It's full of psychological, strategical and other tips useful in the process of self-learning. And it's very entertaining.

 

[Quarkman1]

J. Schwichtenberg, Teach Yourself Physics (2020)
https://www.amazon.com/dp/3948763003/?tag=pfamazon01-20

It's not so much a book on physics, as it is a book on how to learn physics, especially if your goal is not to pass exams but to get a deep understanding. It's full of psychological, strategical and other tips useful in the process of self-learning. And it's very entertaining.

I just finished that and found it quite a good read. It also encouraged me to pick up Feynman's Lectures on Physics - I just started Volume I. I always found Feynman to be a fascinating mind and excellent teacher. If my physics teachers were even half as good as he (and many others like Hawking, Neil deGrasse Tyson, Kip Thorne, etc.) was I might be a physicist today. I like teachers that encourage and drive curiosity in those they teach, not just giving them a book and a pencil and say "start memorizing theorems and equations." Feynman himself was not a fan of rote memorization and actively discouraged its practice. Swichtenberg writes good stuff. I also have No Nonsense Classical Mechanics which I am reading more to try to understand some of the math involved. My calculus skills are pretty rusty now.

 

[vanhees71]

Neither Physics nor math are about rote learning at all. To reach or even top Feynman as a teacher is impossible though. I think that of all famous physicists only Sommerfeld was an even better physics teacher.

 

[haushofer]

J. Schwichtenberg, Teach Yourself Physics (2020)
https://www.amazon.com/dp/3948763003/?tag=pfamazon01-20

It's not so much a book on physics, as it is a book on how to learn physics, especially if your goal is not to pass exams but to get a deep understanding. It's full of psychological, strategical and other tips useful in the process of self-learning. And it's very entertaining.

Is it also useful for physics teachers? I wrote a short manual for my students how to "learn/do" physics and am interested in this question.

 

[haushofer]

I just bought Luke Barnes' book on finetuning,"a fortunate universe". Maybe he can convince me :P

 

[Demystifier]

Is it also useful for physics teachers? I wrote a short manual for my students how to "learn/do" physics and am interested in this question.

It's not written for teachers, but I think that teachers who can read between the lines can find a lot of ways to improve their teaching.

 

[Quarkman1]

It's not written for teachers, but I think that teachers who can read between the lines can find a lot of ways to improve their teaching.

I would agree with this assessment. I think it is really geared to teaching you how to think and approach learning what can be a daunting subject. The goal, IMHO, is to encourage an admiration of the science of physics and the desire to want to learn. As Schwichtenberg writes (paraphrasing slightly), "my goal is not to teach you physics, my goal is to teach you what I wished I knew years ago, so you will want to learn physics." It certainly helped rekindle my fascination with physics and math, and encouraged me to pick up Feynman's QED and now Lectures on Physics.