Other Should I Become a Mathematician?

[mathwonk]

nonetheless, atiyah macdonald is considered very easy to read, and i recommend it. but i especially like zariski samuel which is much more detailed and hence extremely readable.

 

[mathwonk]

well i like Shafarevich, basic algebraic geometry. there are some mistakes, but overall it is one of the most geometrically intuitive and enjoyable introduction out there.

of course Hartshorne is also very highly recommnended, but i personally recommend Shafarevich first followed by Hartshorne.

oh yes, even before those two books, there are excellent books on curves, by Walker, and by Fulton, but Fulton is out of print and very hard to find. another very nice intro to complex curves is algebraic curves by Griffiths.

so here is a list of introductory books i myself read and liked, roughly in order of difficulty: walker: algebraic plane curves. fulton: algebraic curves; shafarevich: basic algebraic geometry; mumford: algebraic geometry I: projective varieties; mumford: red book of algebraic geometry; hartshorne: algebraic geometry.

a more recent one, written from experience teaching at brown and harvard, is algebraic geometry by joe harris. it is filled with good well explained examples. griffiths and harris: principles of algebraic geometry is also useful for accessible explanations of many important topics. I especially benefited from the chapter on riemann surfaces. this book has a number of technical errors and gaps in some proofs, but one still learns a lot from it.

then maybe arnaud beauville: complex algebraic surfaces; mumford: abelian varieties; mumford:lectures on algebraic surfaces, mukai: moduli theory? and mumford on theta functions, three volumes.

many people also like miles reids introductory book: undergraduate algebraic geometry. and also his undergraduate commutative algebra. if you find atiyah macdonald too hard, try reid.

another good introductory book is algebraic curves and riemann surfaces, by rick miranda. there are also lots of free notes on the web by people such as ravi vakil, igor dolgachev, james milne, miles reid, and others.

there are also many other books written since i was a student, and hence less familiar to me. the one listed are mostly the ones i read as a student. one recent book by a colleague that is well received is invitation to arithmetic geometry by dino lorenzini. another that may be useful for making the intuitive leap from varieties to schemes is geometry of schemes by eisenbud and harris.

i myself am currently reading varchenko et al on singularities, and looijenga on isolated complete intersection singularities. there are also many good more advanced books like the one by mori and kollar on classification of varieties.

 

[mathwonk]

it just dawned on me eastside, i think no one else here ever asked me the one thing i might possibly know something about!

 

[mathwonk]

hulek also has a new introduction to alg geom. and there is the classic book by semple and roth.

 

[eastside00_99]

it just dawned on me eastside, i think no one else here ever asked me the one thing i might possibly know something about!

I do what I can. I will look into these book. I have read Fulton's "Algebraic Curves" and that's basically it for the books on that list. So, they are welcomed additions. By the way, the new Oxford Graduate Texts in Mathematics series has two books out on Algebraic Geometry--one which introduces the study of Arithemtic Geometry and the other Algebraic Groups.

 

[eastside00_99]

Oh, Mathwonk, have you read "The Geometry of Szyzgies" by Eisenbud and Harris. I am thinking I will read this soon. Also, as you mention Principles of Algebraic Geometry, do you ever pick up EGA or SGA either as references or as intructive learning material. Does Serre have anything that you have found helpful. Anyway, I guess I should read Harthsorne soon. I think I will try and do this in the summer. Thanks again for you list.

 

[mathwonk]

i did pick up EGA but never got much out of it, except very locally as a reference. Mumford used to say the way to read EGA was not to start at the beginning, but to find then topic you wanted, then trace it back though all the prior references, understand it, then write it up yourself in 2 pages.

serre's algebraic groups and class fields is excellent for curves, and especially the duality theorem. He also has a classic paper Faisceaux algebrique coherent, which is standard for cohomology.

I have not heard of geometry of syzygies. if you have already read fulton on curves you already know quite a lot. there are three big resuits inn there: bezout, resolution of singularities, and riemann roch. and fulton is the master on bezout, well he is also a master on riemann roch.

i would suggest shafarevich next, or maybe even mumford's red book, then hartshorne or serre FAC.

and if you are tempted by advanced books like SGA, you will enjoy looking at Fulton's Intersection theory.

 

[mathwonk]

apparently the book on syzygies is solo by eisenbud, and i think i have seen it. it should be nice, but is mostly very much commutative algebraic, and i am personally not very knowledgeable in that direction, although it is beautiful and fascinating.

oddly, it is easiest to forget to mention some books one is most familiar with, as if one takes them for granted. an outstanding book on complex curves is geometry of algebraic curves, vol. I, by arbarello, cornalba, griffiths, and harris.

there actually is a volume II, written and polished over the past 30 years, and intended for release soon, probably in the next year, (I have seen a pre - release copy). The exercises alone in vol.I contain an extensive education on curves.

another one of my favorite books is abelian integrals, by george kempf, available from the university autonoma de mexico.

these books deal with the so called brill noether problem, or what i sometimes call beyond riemann roch. the problem is to determine for each line bundle how many sections it has, or for each divisor how many meromorphic functions have poles bounded by that divisor.

the answer is given in a sense by riemann roch, but there are 2 unknowns in that formula, h^0 and h^1, and all RRT tells you is their difference. so the picard or jacobian variety provides a space that parametrizes all line bundles with the same numerical invariants, i.e. the same difference h^0-h^1, and it is filtered by the different possible values of h^0. to determine the dimensions and geometric properties of these strata is the brill - noether problem.

there was a fundamental conference on the topic at UGA in 1979, with Griffiths headlining it, and the ACGH book is the outgrowth of that meeting.

 

[mathwonk]

now that you have read fulton on curves, including RRT from the algebraic point of view, you might take a look at my notes on RRT on my website, from riemann's, i.e. the complex analytic, point of view. this is the original approach and is well explained in the chapter on riemann surfaces in Principles of algebraic geometry.

i am getting the impression however that your point of view is perhaps more algebraic than mine. If this is correct, i especially recommend serre's book mentioned above, and probably also eisenbud's syzygies book.

there are other purely commutative algebra books out there that people recommend too, by matsumura, namely commutative algebra, and commutative ring theory (a later version). these books are pretty austere, but i have been told the second one is more readable.

i also recommend reading some original papers, like say zariski's paper on the concept of a simple point, in his collected works. he was a prime mover in putting commutative algebra in place firmly as a foundational tool for algebraic geometry.

 

[mathwonk]

i cannot yet find a book on algebraic groups in that series, but liu's book on arithmetic geometry and curves looks very ambitious and probably hard to read for a beginner.

he takes what is to me the least accessible approach, familiar to students of the 60's, namely the most heavy abstract machinery first, before any even basic results on the simplest objects, curves. if this appeals to you, you might like EGA.

actually mumford once told us that even grothendieck used to start out with these little charming and specific results, but he was never satisfied to leave them that way, and would then go back and think about them in more and more abstract terms until eventually they were unrecognizable.

plus his written account in EGA is due to Dieudonne, who writes in a very dry way. You should look at SGA or something Grothendieck actually wrote to get more of an idea of his own style, still very very abstract.

as a tiny example of EGA style, if you have studied sheaves at all, you know how abstract they can be. well in the beginning of EGA, they point out that topological spaces are really too special for the topic and the right setting is sheaves on partially ordered sets,...

this kind of thing is a bit off putting to young persons, unless they are blinded by the prophetic zeal of their leader. for grothendieck's own treatment of sheaf cohomology, not filtered through anyone else, look at his paper in Tohoku journal, "sur quelque points d'algebre homologique".

for algebraic versus analytic cohomology there is also serre's great paper GAGA. but we are getting a little off the deep end here for starters.

i recommend one learn something about curves, surfaces (e.g. del pezzo surfaces and scrolls), abelian varieties, and cohomology. then go in any direction you want, maybe much sooner.

 

[mathwonk]

for aspiring algebraic geometers, i will post another brief sketch of riemann's theory of curves, hopefully not one i have posted before; it should soon be on my webpage as well.

Riemann’s view of plane curves: 1.

Riemann’s goal was to classify all complex holomorphic functions of one variable.

1) The fundamental equivalence relation on power series: Consider a convergent power series as representing a holomorphic function in an open disc, and consider two power series as representing the same function if one is an analytic continuation of the other.

2) The monodromy problem: Two power series may be analytic continuations of each other and yet not determine the same function on the same open disc in the complex plane, so a family of such power series does not actually define a function.

Riemann’s solution: Construct the (“Riemann”) surface S on which they do give a well defined holomorphic function, by considering all pairs (U,f) where U is an open disc, f is a convergent power series in U, and f is an analytic continuation of some fixed power series f0. Then take the disjoint union of all the discs U, subject to the identification that on their overlaps the discs are identified if and only if the (overlap is non empty and the) functions they define agree there.

Then S is a connected real 2 manifold, with a holomorphic structure and a holomorphic projection S-->C mapping S to the union (not disjoint union) of the discs U, and f is a well defined holomorphic function on S.

3) Completing the Riemann surface: If we include also points where f is meromorphic, and allow discs U which are open neighborhoods of the point at infinity on the complex line, then we get a holomorphic projection S-->P^1 = C union {p}, and f is also a holomorphic function
S-->P^1.

4) Classifying functions by means of their Riemann surfaces:
This poses a new 2 part problem:
(i) Classify all the holomorphic surfaces S.
(ii) Given a surface S, classify all the meromorphic functions on S.

 

[mathwonk]

riemann on curves 2.

5) The fundamental example
Given a polynomial F(z,w) of two complex variables, for each solution pair F(a,b) = 0, such that ∂F/∂w (a,b) ≠ 0, there is by the implicit function theorem, a neighborhood U of a, and a nbhd V of b, and a holomorphic function w = f(z) defined in U such that for all z in U, we have f(z) = w if and only if w is in V and F(z,w) = 0. I.e. we say F determines w = f(z) as an “implicit” function. If F is irreducible, then any two different implicit functions determined by F are analytic continuations of each other. For instance if F(z,w) = z-w^2, then there are for each a ≠ 0, two holomorphic functions w(z) defined near a, the two square roots of z.

In this example, the surface S determined by F is essentially equal to the closure of the plane curve X: {F(z,w) = 0}, in the projective plane P^2. More precisely, S is constructed by removing and then adding back a finite number of points to X as follows.

Consider the open set of X where either ∂F/∂w (a,b) ≠ 0 or
∂F/∂z (a,b) ≠ 0. These are the non singular points of X. To these we wish to add some points in place of the singular points of X. I.e. the set of non singular points is a non compact manifold and we wish to compactify it.

Consider an omitted i.e. a singular point p of X. These are always isolated, and projection of X onto an axis, either the z or w axis, is in the neighborhood of p, a finite covering space of the punctured disc U* centered at the z or w coordinate of p. All such connected covering spaces are of form t-->t^r for some r ≥ 1, and hence the domain of the covering map, which need not be connected, is a finite disjoint union of copies of U*. Then we can enlarge this space by simply adding in a separate center for each disc, making a larger 2 manifold.

Doing this on an open cover of X in P^2, by copies of the plane C2, we eventually get the surface S, which is in fact compact, and comes equipped with a holomorphic map S-->X, which is an isomorphism over the non singular points of X. S is thus a “desingularization” of X. For example if X crosses itself with two transverse branches at p, then S has two points lying over p, one for each branch or direction. If X has a cusp, or pinch point at p, but a punctured neighborhood of p is still connected, there is only one point of S over p, but it is not pinched.Theorem: (i) The Riemann surface S constructed above from an irreducible polynomial F is compact and connected, and conversely, any compact connected Riemann surface arises in this way.
(ii) The field of meromorphic functions M(S) on S is isomorphic to the field of rational functions k(C) on the plane curve C, i.e. to the field generated by the rational functions z and w on C.

I.e. this example precisely exhausts all the compact examples of Riemann surfaces.

Corollary: The study of compact Riemann surfaces and meromorphic functions on them is equivalent to the study of algebraic plane curves and rational functions on them.

 

[mathwonk]

reiemann on curves 3.

6) Analyzing the meromorphic function field M(S).

If S is any compact R.S. then M(S) = C(f,g) is a finitely generated field extension of C of transcendence degree one, hence by the primitive element theorem, can be generated by two elements, and any two such elements define a holomorphic map S-->X in P^2 of degree one onto an irreducible plane algebraic curve, such that k(X) = M(S).

Question: (i) Is it possible to embed S isomorphically onto an algebraic curve, either one in P^2 or in some larger space P^n?
(ii) More generally, try to classify all holomorphic mappings S-->P^n and decide which ones are embeddings.

Riemann’s intrinsic approach:
Given a holomorphic map ƒ:S-->Pn, with homogeneous coordinates z0,...,zn on P^n, the fractions zi/z0 pull back to meromorphic functions ƒ1,...,ƒn on S, which are holomorphic on S0 = ƒ-1(z0≠0), and these ƒi determine back the map ƒ. Indeed the ƒi determine the holomorphic map S0-->Cn = {z0≠0} in P^n.

Analyzing ƒ by the poles of the ƒi
Note that since the ƒi are holomorphic in ƒ-1(z0≠0), their poles are contained in the finite set ƒ-1(z0=0),and on that set the pole order cannot exceed the order of the zeroes of the function z0 at these points. I.e. the hyperplane divisor {z0 = 0}0 in P^n pulls back to a “divisor” ∑ njpj on S, and if ƒi = zi/z0 then the meromorphic function ƒi has divisor div(ƒi) = div(zi/z0) = div(zi) - div(z0) = ƒ*(Hi)-ƒ*(H0).
Hence div(ƒi) + ƒ*(H0) = ƒ*(Hi) ≥ 0, and this is also true for every linear combination of these functions.

I.e. the pole divisor of every ƒi is dominated by ƒ*(H0) = D0. Let's give a name to these functions whose pole divisor is dominated by D0.

Definition: L(D0) = {f in M(S): f = 0 or div(f) +D0 ≥ 0}.

Thus we see that a holomorphic map ƒ:S-->Pn is determined by a subspace of L(D0) where D0 = ƒ*(H0) is the divisor of the hyperplane section H0.

Theorem(Riemann): For any divisor D on S, the space L(D) is finite dimensional over C. Moreover, if g = genus(S) as a toplogical surface,
(i) deg(D) + 1 ≥ dimL(D) ≥ deg(D) +1 -g.
(ii) If there is a positive divisor D with dimL(D) = deg(D)+1, then S ≈ P^1.
(iii) If deg(D) > 2g-2, then dimL(D) = deg(D)+1-g.

Corollary of (i): If deg(D) ≥ g then dim(L(D)) ≥ 1, and deg(D)≥g+1 implies dimL(D) ≥ 2, hence, there always exists a holomorphic branched cover S-->P1 of degree ≤ g+1.

Q: When does there exist such a cover of lower degree?

Definition: S is called hyperelliptic if there is such a cover of degree 2, if and only if M(S) is a quadratic extension of C(z).

Corollary of (iii): If deg(D) ≥ 2g+1, then L(D) defines an embedding S-->P^(d-g), in particular S always embeds in P^(g+1).

In fact S always embeds in P^3.
Question: Which S embed in P^2?

Remark: The stronger Riemann Roch theorem implies that if K is the divisor of zeroes of a holomorphic differential on S, then L(K) defines an embedding in P^(g-1), the “canonical embedding”, if and only if S is not hyperelliptic.

 

[mathwonk]

riemann on curves 4.

7) Classifying projective mappings
To classify all algebraic curves with Riemann surface S, we need to classify all holomorphic mappings S-->X in P^n to curves in projective space. We have asociated to each map ƒ:S-->P^n a divisor Do that determines ƒ, but the association is not a natural one, being an arbitrary choice of the hyperplane section by H0. We want to consider all hyperplane sections and ask what they have in common. If h: ∑cjz^j is any linear polynomial defining a hyperplane H, then h/z0 is a rational function f with div(f) = ƒ*(H)-ƒ*(H0) = D-D0, so we say

Definition: two divisors D,D0 on S are linearly equivalent and write D ≈ D0, if and only if there is a meromorphic function f on S with D-D0 = div(f), iff D = div(f)+D0.

In particular, D≈D0 implies that L(D) isom. L(D0) via multiplication by f. and L(D) defines an embedding iff L(D0) does so. Indeed from the isomorphism taking g to fg, we see that a basis in one space corresponds to a basis of the other defining the same map to P^n, i.e. (ƒ0,...,ƒn) and (fƒ0,...,fƒn) define the same map.

Thus to classify projective mappings of S, it suffices to classify divisors on S up to linear equivalence.

Definition: Pic(S) = set of linear equivalence classes of divisors on S.

Fact: The divisor of a meromorphic function on S has degree zero.

Corollary: Pic(S) = ∑ Pic^d(S) where d is the degree of the divisors classes in Pic^d(S).

Definition: Pic^0(S) = Jac(S) is called the Jacobian variety of S.

Definition: S^(d) = (Sx..xS)/Symd = dth symmetric product of S
= set of positive divisors of degree d on S.

Then there is a natural map S^(d)-->Pic^d(S), taking a positive divisor D to its linear equivalence class O(D), called the Abel map. [Actually the notation O(D) usually denotes another equivalent notion the locally free rank one sheaf determined by D.]

Remark: If L is a point of Pic^d(S) with d > 0, L = O(D) for some D>0 if and only if dimL(D) > 0.
Proof: If D > 0, then C is contained in L(D). And if dimL(D)>0, then there is an f ≠ 0 in L(D) hence D+div(f) ≥ 0, hence > 0.QED.

Corollary: The map S^(g)-->Pic^g (S) is surjective.
Proof: Riemann’s theorem showed that dimL(D)>0 if deg(D) ≥ g. QED.

It can be shown that Pic^g hence every Pic^d can be given the structure of algebraic variety of dimension g. In fact.
Theorem: (i) Pic^d(S) isom C^g/L, where L is a rank 2g lattice subgroup of C^g.
(ii) The image of the map S^(g-1)-->Pic^(g-1)(S) is a subvariety “theta” of codimension one, i.e. dimension g-1, called the “theta divisor”.
(iii) There is an embedding Pic^(g-1)-->P^N such that 3.theta is a hyperplane section divisor.
(iv) If O(D) = L in Pic^(g-1)(S) is any point, then dimL(D) = multL(theta).
(v) If g(S) ≥ 4, then g-3 ≥ dim(sing(theta)) ≥ g-4, and dim(sing(theta)) = g-3 iff S is hyperelliptic.
(vi) If g(S) ≥ 5 and S is not hyperelliptic, then rank 4 double points are dense in sing(theta), and the intersection in P(T0Pic^(g-1)(S)) isom P^(g-1), of the quadric tangent cones to theta at all such points, equals the canonically embedded model of S.
(vii) Given g,r,d ≥0, every S of genus g has a divisor D of degree d with dimL(D) ≥ r+1 iff g-(r+1)(g-d+r) ≥ 0.

 

[mathwonk]

having discussed riemann's plan for classifying functions on a fixed curve, the next chapter should be his idea for classifying all curves, but that is not written. see some survey of moduli of curves.

the basic fact is that the set of isomorphism classes of smooth connected algebraic curves of genus g has the structure of an algebraic variety of dimensions 1 if g=1, and dimension 3g-3 if g>1.

it is irreducible, and not compact, but has a nice compactification obtained by adding curves with ordinary double points which still have only a finite number of automorphisms if g > 1. there is only one curve to add if g=1, the unique curve obtained by identifying two distinct points of P^1.

the compactification is due to mumford and mayer and deligne. its detailed properties are still a subject of intense study.

 

[pivoxa15]

Mathswonk, this may seem neive but why did you choose algebraic geometry rather than algebraic topology?

What are the differences between the two? I get the feeling that topology is a more global study of things so more abstract? Or both as abstract as each other?

 

[mathwonk]

well my first love was indeed algebraic topology. but i was frustrated at the time in that pursuit for some reason. then i was just very captivated by the lectures of a brilliant young algebraic geometer, alan mayer, and turned to that subject. herb clemens cemented my decision with his course on riemann surfaces and became my advisor.

actually i was blessed by great courses in several subjects, topology from ed brown jr,. alg number theory by paul monsky, algebra by maurice auslander, complex and real analysis from hugo rossi and robert seeley, and algebraic geometry by alan mayer, (and there were excellent courses from others: tom sherman, ronnie wells, alphonse vasquez, ron stern, ...). so i had the great opportunity to choose the direction that really appealed to me.

one reason i made my decision, crazy as it may be in truth, was that i felt a real intuition for topology and so topology seemed too easy to me. algebraic geometry mixed the subject i had an affinity for, geometry, with one i always felt very difficult, algebra. i liked the challenge of combining the two.

 

[mathwonk]

the difference between algebraic topology and algebraic geometry is that all curves of genus g are equivalent in algebraic topology, but they have a 3g-3 dimensional space of different possible complex structures in algebraic geometry. i could not picture that and so found that subject more fascinating.

 

[eastside00_99]

i cannot yet find a book on algebraic groups in that's eries, but liu's book on arithmetic geometry and curves looks very ambitious and probably hard to read for a beginner.

he takes what is to me the least accessible approach, familiar to students of the 60's, namely the most heavy abstract machinery first, before any even basic results on the simplest objects, curves. if this appeals to you, you might like EGA.

actually mumford once told us that even grothendieck used to start out with these little charming and specific results, but he was never satisfied to leave them that way, and would then go back and think about them in more and more abstract terms until eventually they were unrecognizable.

plus his written account in EGA is due to Dieudonne, who writes in a very dry way. You should look at SGA or something Grothendieck actually wrote to get more of an idea of his own style, still very very abstract.

as a tiny example of EGA style, if you have studied sheaves at all, you know how abstract they can be. well in the beginning of EGA, they point out that topological spaces are really too special for the topic and the right setting is sheaves on partially ordered sets,...

this kind of thing is a bit off putting to young persons, unless they are blinded by the prophetic zeal of their leader. for grothendiecks own treatment of sheaf cohomology, not filtered through anyone else, look at his paper in Tohoku journal, "sur quelque points d'algebre homologique".

for algebraic versus analytic cohomology there is also serre's great paper GAGA. but we are getting a little off the deep end here for starters.

i recommend one learn something about curves, surfaces (e.g. del pezzo surfaces and scrolls), abelian varieties, and cohomology. then go in any direction you want, maybe much sooner.

Haha, it seems a minute detail between sheafs on topologies and posets whereas I can't imagine where one would use sheafs for posets? I think I do lean more toward learning the abstract machinary although I found your exposition on the Riemann Surfaces refreshing. Part of the problem though is that I have had little complex analysis, just a few definitions. I can never bare to bring myself to study the material when I have free time but next year the first thing I will do is take a year of complex analysis. I have had other classes: like manifold theory, functional analysis, algebra, and a course already on algebraic geometry from a computational perspective that the idea of sheaves and schemes are not that hard to grasp. The only problem is that they are beasts and it takes time to work many of the problems that are worth working. I have read, or mostly read, "Algebraic Varieties" by Kempf and I found that extremely useful in helping get me to schemes. But, still, I have just started out learning this stuff so I am not really all that well-versed.

You see the idea of sheaves in what you have posted about the interplay between Algebraic Curves and Riemann Surfaces. It seems to me that this is partly the construction above when considering discs U and holomorphic functions on U. As to what you have outlined, I felt I should it in an informal way:

Every compact Reimann Surfaces is isomorphic to an Algebraic Curve in some affine space as P^n is isomorphic to a subvariety of some A^m.

Here is the book <a href="http://www.oup.com/us/catalog/general/subject/Mathematics/PureMathematics/?view=usa&ci=9780198528319">An Introduction to Algebraic Geometry and Algebraic Groups</a>

As for abelian varieties, I wrote a brief paper on them for an Algebra class last semester--one in which the j-invariant is introduced. In the process, I had my hands on mumford's book abelian varieties for a little while. I could not read it for the most part. It is something in my future though; I know that. If I take your advice though and without the complex analysis, what book on curves do you recomend. Maybe something on space curves?

 

[mathwonk]

if you have read fulton, try serre next. groupes algebriques et corps de classes, i.e. algebraic groups and class fields.

 

[pivoxa15]

one reason i made my decision, crazy as it may be in truth, was that i felt a real intuition for topology and so topology seemed too easy to me.

SUrely there are extremely hard unsolved problems in that area. THe Poincare conjecture was still open at that time! Ever thought of cracking that one?!

 

[mathwonk]

well i guess i did not appreciate the global nature of the difficult problems in topology. i.e. all manifolds are locally the same, but not all algebraic varieties.

 

[Werg22]

Mathwonk, are mathematicians generally visual? I mean do most approach an idea visually whenever possible? I've been debating with myself: a strong mathematician does not need to resort to visual representations; to me it seems like "phony" mathematics.

 

[mathwonk]

well i am myself almost entirely visual, at least mentally. that is why algebra is so hard for me. but no, not all are. i have mentioned chatting with peskine once about a problem, and he only really got going when the dimensions moved up beyond where I could imagine them well.

so maybe the best ones are not as visual. i don't know. but they also say that ramanujam used to claim his ideas were revealed to him in dreams by goddesses. so people think in many different ways.

you might enjoy "the psychology of invention in the mathematical field", by hadamard. he talks there i believe about mozart claiming to see his symphonies in patterns in his brain before they all come together, and einstein as well, saying he thought in visual patterns.

i used to think using a calculator for research was inappropriate, but not as much after being shown what insights it could stimulate by a bright student.

in mathematics we are not ashamed of using whatever tools we can get our hands on. some of us enjoy visual thinking. the idea is to have fun, and make progress, not disparage each others thought processes.

 

[mathwonk]

here is quote from a letter by Einstein to Hadamard, in answer to the following question:

"It would be very helpful... to know what internal or mental images, what kind of 'internal word' mathematicians make use of; whetehr they are motor, auditory, visual, or mixed, depending on the subject which they are studying."

partial answer:

"the words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The physical entities which seem to serve as elements of thought are certain signs and more or less clear images which can be 'voluntarily' reproduced and combined...

The above mentioned elements are, in my case, of visual and some of muscular type."

...

Albert Einstein.

 

[Cincinnatus]

well my first love was indeed algebraic topology. but i was frustrated at the time in that pursuit for some reason. then i was just very captivated by the lectures of a brilliant young algebraic geometer, alan mayer, and turned to that subject. herb clemens cemented my decision with his course on riemann surfaces and became my advisor.

I find it hilarious to see Alan Mayer described as young.

 

[mathwonk]

well that was in 1967. i too was young. indeed the world was...

paul monsky, now retired, was so young i thought he was a student.

 

[waht]

Is the subject of non-commutative geometry just a continuation of algebraic geometry with emphasis on non-commutativity? If not, what are its prerequisites?

 

[pivoxa15]

it could stimulate by a bright student.

How can you tell a bright student from an ordinary one?

It seems at the undergraduate level , strong background knowledge (i.e knowledge of prereq) and willingness to work are the keys to success. What do you think?

 

[mathwonk]

i don't know much about non commutative geomnetry, but i think it has more to do with operator theory. i think alain connes is the guy to look up.

bright students just impress you. you are always impressed if someone teaches you something you don't know, or notices something you do not.

hard work is always the key to success. brains we get from our parents, there is nothing we can do to improve them except not waste them. the hard work is the only thing we ourselves control, hence is the main ingredient to focus on.

try not to be afraid to try, there are always much smarter people around, even fields medalist rene thom felt that when seeing grothendieck, but we may still do something they do not.

do not be discouraged if someone is smarter, you may still outwork them in a small area. or you may even collaborate with them!