Should I Become a Mathematician?

[mathwonk]

Intro to alg geom 2

II. NEW METHODS FOR PLANE CURVES: TOPOLOGY and COMPLEX ANALYSIS
Riemann associated to a plane curve f(X,Y)=0 its set of complex solutions, compactified and desingularized. This is its "Riemann surface", a real topological 2 manifold with a complex structure obtained by a branched projection onto the complex line. For instance the curve y^2 = 1-X^3 becomes its own Riemann surface after adding one point at infinity, making it a topological torus. Projection on the X coordinate is a 2:1 cover of the extended X line, branched over infinity and the solutions of 1-X^3 = 0.
This association is a functor, i.e. a non constant rational map of plane curves yields an associated holomorphic map of their Riemann surfaces, in particular a topological branched cover. Riemann assigns to a real 2 manifold its "genus" (the number of handles), and calculates that branched covers cannot raise genus, and the only surface of genus zero is the sphere = the Riemann surface of the complex t line. Hence if the Riemann surface of a plane curve has positive genus, it cannot be the branched image of the sphere, hence the curve cannot be parametrized by the coordinate t.
Riemann also proved a smooth plane curve of degree d has genus g = (d-1)(d-2)/2, so smooth cubics and higher degree curves all have positive genus and hence cannot be parametrized. He proved conversely that any curve whose Riemann surface has genus zero can be parametrized, e.g. hyperbolas, circles, lines, parabolas, ellipses, or any curve of degree < 3. Moreover a singularity, i.e. a point where the curve has no tangent line, like (0,0) on Y^2 = X^3, lowers the genus during the desingularization process, and this is why such a "singular" cubic can be parametrized.

One also obtains a criterion for any two irreducible plane curves to be rationally isomorphic, namely their Riemann surfaces should be not just topologically, but holomorphically isomorphic. By representing a smooth plane cubic as a quotient of the complex line C by a lattice, using the Weierstrass P function, one can prove that many complex tori are not holomorphically equivalent, by studying the induced map of lattices. It follows that there is a one parameter family of smooth plane cubics which are rationally distinct from each other.

This shows briefly the power and flexibility of topological and holomorphic methods, which Riemann largely invented for this purpose, an amazing illustration of thinking outside traditional confines.

 

[mathwonk]

Intro to alg geom 3

III. RINGS and IDEALS
To go further in the direction of arithmetic questions, one would like more algebraic techniques, applicable to fields of characteristic p, algebraic numbers fields, rings of integers, power series rings,... One can pose the question of isomorphism of plane curves algebraically, using ring theory, as follows. Since all roots of multiples of the polynomial f vanish on the zero locus of f, it is natural to associate to the curve V:{f=0} in k^2, the ideal rad(f) = {g in k[X,Y]: some power of g is in (f)}. Then the quotient ring R = k[X,Y]/rad(f) is the ring of polynomial functions on V. Moreover if p is a point of V, evaluation at p is a k algebra homomorphism R-->k with kernel a maximal ideal of R. In case k is an algebraically closed field, like C or the algebraic numbers, this is a bijection between points of V and maximal ideals of R.
In fact everything about the plane curve V is mirrored in the ring R in this case, and two irreducible polynomials f,g, in k[X,Y], define isomorphic plane curves if and only if their associated rings R and S are isomorphic k algebras. Indeed the assignment of R to V is a "fully faithful functor", with algebraic morphisms of curves corresponding precisely to k algebra maps of their rings. To recover the points from the ring one takes the maximal ideals, and to recover a map on these points from a k algebra map, one pulls back maximal ideals. (Since these rings are finitely generated k algebras and k is algebraically closed, a maximal ideal pulls back to a maximal ideal.) Any pair of generators of the k algebra R defines an embedding of V in the plane.
Similarly, if f (irreducible) in k[X,Y,Z] defines a surface V:{f=0} in k^3, (k still an algebraically closed field), then not only do points of V correspond to maximal ideals of R = k[X,Y,Z]/(f), but irreducible algebraic curves lying on V correspond to non zero non maximal prime ideals in R. Again this is a fully faithful functor, with polynomial maps corresponding to k algebra maps. In particular the pullback of maximal ideals is maximal, but now the pullback of some non maximal ideals can also be maximal, i.e. some curves can collapse to points under a polynomial map.

To give the algebraic notion full flexibility, in particular to embrace non Jacobson rings with too few maximal ideals to carry all the desired structure, Grothendieck understood one should discard the restriction to rings without radical and expand the concept of a "point", to include irreducible subvarieties, i.e. consider all prime ideals as points, as follows.

 

[mathwonk]

Intro to alg geom 4

IV. AFFINE SCHEMES
If R is any commutative ring with 1, let X (= "specR") be the set of all prime ideals of R, with a topological closure operator where the closure of a set of prime ideals is the set of all prime ideals containing the intersection of the given set of primes. (Intuitively, each prime ideal contains the functions vanishing at the corresponding point, so their intersection is all functions vanishing at all the points of the set, and the prime ideals containing this intersection hence are all points on which that same set of functions vanishes. So the closure of a set is the smallest algebraically defined locus containing the set.) This closure operator defines the "Zariski topology" on X.
Now any ring map defines a morphism of their spectra by puling back prime ideals, and in particular a morphism is continuous, although this alone says little since the Zariski topology is so coarse. Notice now maximal ideals may pull back to non maximal ones, e.g. under the inclusion map Z-->Q of integers to the rationals, taking the unique point of specQ to a dense point of specZ. Maximal ideals now correspond to closed points, and in particular there are usually plenty of non closed points. Intuitively, every irreducible subvariety has a dense point, and together these "points", one for each irreducible subvariety, give all the points of specX.
If K is a ring, a "K valued point" of X is given by a ring homomorphism R-->K, not necessarily surjective. E.g. if K is a field, the pullback of the unique maximal ideal of K is a not necessarily maximal, prime ideal P of R, the K valued point. Even if the point is closed, i.e. if P is maximal, we get information on which maximal ideals correspond to points with coefficients in different fields. If say k = the real field, and f is a polynomial over k, then a k algebra map g:k[X,Y]/(f)-->k has as kernel a maximal ideal corresponding to a point of {f=0} in k^2, i.e. a point of {f=0} in the usual sense, with real coefficients. The coordinates of this point are given by the pair of images (g(X),g(Y)) in k^2 of the variables X,Y, under the algebra map g, which after all is evaluation of functions at our point. But if say f = Y-X^2, the map from k[X,Y]/(f) -->C taking X to i, and Y to -1, corresponds to the C (complex) - valued point (i,-1), in C^2 rather than k^2.
More generally, if I is any ideal in Z[X1,...,Xn] generated by integral polynomials f1,...fr, and A is a ring, a ring homomorphism Z[X1,...,Xn]/I -->A takes the variables Xj to elements aj of A such that all the polynomials fi vanish at the point of A^n with cordinates (a1,...an). I.e. the map defines an "A valued point " of the locus defined by I. E.g. if M is a maximal ideal of R,we can always view the coordinates of the corresponding point in the residue field R/M, i.e. the point M of specR is "R/M valued".
This approach let's us recover tangent vectors too, in case say of a variety V with ring R = k[X1,...,Xn]/I, where radI = I, and k is an algebraically closed field. Consider the ring S = k[T]/(T^2), with unique maximal ideal (t) generated by the nilpotent element t. Then we claim tangent vectors to V correspond to S valued points (over spec(k)), i.e. to k algebra maps R-->S. E.g. if R = k[X], and we map R-->S by sending X to a+bt, then the inverse image of the maximal ideal (t) is the maximal ideal (X-a), and two elements of (X-a) have the same image in S if and only if they have the same derivative at X=a. Thus S valued points of V are points of the "tangent bundle" of V.

 

[mathwonk]

Intro to alg geom 5

V. SCHEMES
One next defines a scheme as a space with an open cover by affine schemes, by analogy with topological manifolds, which have an open cover by affine spaces. For this we need to be able to glue affine schemes along open subsets, so we need to understand the induced structure on an open subset of V = specR. A basis for the Zariski topology on specR is given by the open sets of form V(f) = {primes P in specR with f not in P}. Intuitively this is the set of points where f does not vanish. (The analogy is with a "completely regular" topological space whose closed sets are all cut out by continuous real valued functions.)
On the set V(f), the most natural ring is R(f) = {g/f^n: g in R, n a non negative integer}/{identification of two fractions if their cross product is annihilated by a non neg. power of f}. I.e. since powers of f are now units, anything annihilated by a unit must become zero, so g/f^n = h/f^m if for some s, f^s[gf^m - hf^n] = 0 in R. Intuitively these are rational functions on V which are regular in V(f). This construction defines an assignment of a ring to each basic open set V(f) in V, i.e. it defines a sheaf of rings on a basis for V, and hence on all of V, by a standard extension device. This sheaf is called O, perhaps in honor of the great Japanese mathematician Oka, who proved much of the foundational theory for analytic sheaves.
Then one develops a number of technical analogues of properties of manifolds, in particular of compactness, and Hausdorffness, now called properness and separation conditions. Since the Zariski topology is very coarse, the usual version of Hausdorffness almost always fails but there is a better analogue of separation which usually holds. The point is that Hausdorffnes has a descriptiion in terms of products, and algebraic or scheme theoretic products also differ from their topological versions.
In making these constructions, mapping properties come to the fore, and are crucial even for finding the right definitions, so categorical thinking is essential. It is also useful to keep in mind, that some technically valuable varieties are not separated even in the generalized sense. I.e. sometimes one can prove a theorem by relaxing the requirement of algebraic separation.

 

[mathwonk]

Intro to alg geom 6

VI. COHOMOLOGY
To really take advantage of methods of topology one wants to define invariants which help distinguish between different varieties, i.e. to measure when they are isomorphic, or when they embed in projective space, and if so then with what degree and in what dimension. One wants to recover within algebra all the rich structure that Riemann gave to plane curves using classical topology and complex analysis. Since the Zariski topology is so coarse, again one must use fresh imagination, applied to the information in the structure sheaf, to extract useful definitions of basic concepts like the genus, the cotangent bundle, differential forms, vector bundles, all in a purely algebraic sense. This means one looks at "sheaf cohomology", i.e. cohomology theories in which more of the information is contained in the rings of coefficients than in the topology. This is only natural since here the topology is coarse, but the rings are richly structured. Computing the genus of a smooth plane curve V over any algebraically closed field for instance, is equivalent to calculating H^1(V,O), where O is the structure sheaf.
The first theory of sheaf cohomology for algebraic varieties was given by Serre in the great paper Faiseaux Algebriques Coherent, where he used Cech cohomology with coefficients in "coherent" sheaves, a slight generalization of vector bundles. (They include also cokernels of vector bundle maps, which are not always locally free where the bundle map drops rank. This is needed to have short exact sequences, a crucial aspect of cohomology.) Cech cohomology is analogous to simplicial or cellular homology, in that it is calculable in an elementary sense using the Cech simplices in the nerve of a suitable cover, but can also become cumbersome for complicated varieties. Worse, for non coherent sheaves which also arise, the Cech cohomology sequence is no longer exact.
Other constructions of cohomology theories by resolutions ("derived functors"), e.g. by flabby sheaves or injective ones, have been given by Grothendieck and Godement, which always have exact cohomology sequences, but they necessarily differ from the Cech groups, hence computing them poses new challenges. (Just as one computes the topological homology of a manifold from a cover by cells which are themselves contractible, hence are "acyclic" or have no homology, one also computes sheaf cohomology from a resolution by any acyclic sheaves - sheaves which themselves have trivial cohomology. This is the key property of flabby and injective sheaves.)
As in classical algebraic topology, no matter how abstract the definition of cohomology, it becomes somewhat computable, at least for experts, once a few basic exactness and vanishing properties are derived. A fundamental result is that affine schemes have trivial cohomology for all coherent sheaves. This makes it possible to calculate coherent Cech cohomology on any affine cover, without passing to the limit, e.g. to calculate the cohomology H*(O(d)) of all line bundles on projective space. But once the affine vanishing property is proved for derived functor cohomology, it too allows computation of the groups H*(O(d)).

 

[mathwonk]

Intro to alg geom 7-8

VII. SPECIAL TOPICS
It is hard to prove many deep theorems in great generality. So having introduced the most general and flexible language, one often returns to the realm of more familiar varieties and tries to study them with the new tools. E.g. one may ask to classify all smooth irreducible curves over the complex numbers, or all surfaces. Or one can study the interplay between topology and algebra as Riemann did with curves, and ask in higher dimensions what restrictions exist on the topology of an algebraic variety. Hodge theory, i.e. the study of harmonic forms, plays a role here.
Instead of global questions, one can focus instead on singularities, the special collapsing behavior of varieties near points where they do not look like manifolds. Brieskorn says there are three key topics here: resolution, deformation, and monodromy. Resolution means removing singularities by a sort of surgery while staying in the same rational isomorphism class. Deformation means changing the complex structure by a different sort of topological surgery which allows the singular object to be the central fiber in a family of varieties whose union has a nice structure itself. This leaves the algebraic invariants more nearly constant than does resolution. Monodromy means studying what happens to topological or other subvarieties of a smooth fiber in a family, as we "go around" a singular fiber and return to the same smooth fiber.
E.g. if a given homology cycle on a smooth fiber is deformed onto other nearby smooth fibers, when it goes around the singular fiber and comes back to the original smooth fiber, it may have become a different cycle! I.e. if we view the homology groups on the smooth fibers as a vector bundle on the base space, sections of this bundle are multivalued and change values when we go around a singularity, just as a logarithm changes its value when we go around its singularity at the origin.

People who like to study particular algebraic varieties may look for ones that are somewhat more amenable to computation that very general ones, e.g. curves, special surfaces, group varieties like abelian varieties. The latter is my area of specialization, especially abelian varieties arising from curves either as jacobians, or as components of a splitting of jacobians induced by an involution of a curve (Prym varieties).

Others study curves, surfaces and threefolds which occur in low degree in projective space such as curves in projective 3 space, or as double covers of the projective plane or of projective 3 space branched over hypersurfaces of low degree such as quadrics. Dual to varieties of low dimension are those of low codimension, e.g. the study of general projective hypersurfaces, varieties defined by one homogeneous polynomial. Some study vector bundles on curves, or on projective space.
Some examine how varieties can vary in families. One beautiful and favorite object of study are called "moduli" varieties, which are a candidate for base spaces of "universal" families of varieties of a particular kind, the guiding case always being curves. A very active area is the computation of the fundamental invariants of the moduli spaces M(g) of curves of genus g, and of their enhanced versions M(g,n), moduli of genus g curves with n marked points.
Another very rich source of accessible varieties is the class of "toric" varieties, ones constructed from combinatorial data linked to the exponents of monomials in the defining ideal.

VIII. PRERECQUISITES
To do algebraic geometry it obviously helps to know algebraic topology, complex analysis, number theory, commutative algebra, categories and functors, sheaf cohomology, harmonic analysis, group representations, differential manifolds,... even graphs, combinatorics, and coding theory! But one can start on the most special example that one finds attractive, and use its study to motivate learning some tools. This is a commonly recommended way to begin.

 

[mathwonk]

the pure number theorists at uga whose work was of interest to cryptography did not themselves work in cryptography, but provided factoring algorithms, and estimated their speed.

implementing those algorithms within cryptography was left to other applied mathematicians. still that application provided notoriety and funding opportunities also for the pure guys.

 

[tehno]

the pure number theorists at uga whose work was of interest to cryptography did not themselves work in cryptography, but provided factoring algorithms, and estimated their speed.

implementing those algorithms within cryptography was left to other applied mathematicians. still that application provided notoriety and funding opportunities also for the pure guys.

BIG HURRAY for pure number theorists!

 

[J77]

Jobs for pure mathematicians

I'm not a recruiter for them but, as an example,... http://www.gchq.gov.uk/recruitment/careers/math_videosmall.html

 

[tehno]

I'm not a recruiter for them but, as an example,... http://www.gchq.gov.uk/recruitment/careers/math_videosmall.html

Good one.. ..
althought my feeling watching it is the baldy is trying to talk that chick into group sex,not a collaborative math work

 

[mathwonk]

please, this is a family thread.

 

[pivoxa15]

Mathwonk, you said that you haven't heard of a Phd not being able to find post doc work but the woman in that ad said academic jobs in universities are hard to find. Does she have a Phd? If not than obviously it will be hard to find an academic job such as a research job. She has the option of being a teaching assistant at univeristy which is less hours than a school teacher. If she does have a Phd in maths than it would be 'low' teaching at school wouldn't you say? Do you know of any maths Phds teaching in a school?

 

[mathwonk]

no. but in the old days i heard that top private schools like andover, exeter, may have had science profs that were very well trained, possibly phd.

high, low, if you enjoy teaching good students, then top high school or prep school teaching might ring your bell.

i once taught high school students for free, for a year or so, 2 days a week, and a month in summer. although it was lower level maths than some of my uni teaching and my research work, i enjoyed it greatly because the students were more responsive.

two of my high school students later went to ivies and obtained phds in physics and maths. one of them is now a full prof at an ivy himself.

 

[verbasel]

I have a question about "math fatigue". I've been questioning whether or not I'm really cut out to be a Mathematician.

Back in my second year, for various unwise reasons, I binged on honours Math courses. I thought I was going to do a specialist degree, so I took 4 honours math & stats credits and overall I was taking 6 full credits, which is the maximum course load at my uni. My year was an academic disaster, resulting in problems with anxiety and depression and the only decent marks I got were in the non-math/related courses.

After burning myself that way & seeing counselors both academic and otherwise, I opted for a double major in Math and Economics instead (adding another year to my degree). I've basically taken only Economics courses since then, and have completed the courses required for my Economics major. While the so-called 'math' in Economics infuriates me and the pure math courses I took way back when interested me greatly, I'm really apprehensive about taking math courses again.

I've further downgraded Math to a minor and plan to take the easiest courses in order to finish my degree without any further mishaps, but I know I would eventually like to return to the more rigorous math that fascinated and confounded me back in the early days.

How do I regain my confidence? Was I ever a mathematician to begin with? What's a good way to ease back into it?

 

[symbolipoint]

Verbasel,
Maybe you are not cut out to be a mathematician. A minor concentration in Mathematics might still be reasonable. What do you study between semesters? How much time (hours per week, and how many months) are you willing to dedicate to Mathmeatics? Are you willing to restudy courses which you already studied and earned credit in?

 

[J77]

Mathwonk, you said that you haven't heard of a Phd not being able to find post doc work but the woman in that ad said academic jobs in universities are hard to find. Does she have a Phd? If not than obviously it will be hard to find an academic job such as a research job. She has the option of being a teaching assistant at univeristy which is less hours than a school teacher. If she does have a Phd in maths than it would be 'low' teaching at school wouldn't you say? Do you know of any maths Phds teaching in a school?

Bear in mind, it was an advert for GCHQ -- therefore, she would likely say that academic jobs in universities are hard to find. However, there never seems any shortage of jobs available in the UK.

I would feel like teaching high school at some point -- my gf is one -- however, I would only like to teach kids who would be into it; ie. not there for "crwod control" -- which seems to be the norm in a lot of schools.

 

[pivoxa15]

I have a question about "math fatigue". I've been questioning whether or not I'm really cut out to be a Mathematician.

Back in my second year, for various unwise reasons, I binged on honours Math courses. I thought I was going to do a specialist degree, so I took 4 honours math & stats credits and overall I was taking 6 full credits, which is the maximum course load at my uni. My year was an academic disaster, resulting in problems with anxiety and depression and the only decent marks I got were in the non-math/related courses.

After burning myself that way & seeing counselors both academic and otherwise, I opted for a double major in Math and Economics instead (adding another year to my degree). I've basically taken only Economics courses since then, and have completed the courses required for my Economics major. While the so-called 'math' in Economics infuriates me and the pure math courses I took way back when interested me greatly, I'm really apprehensive about taking math courses again.

I've further downgraded Math to a minor and plan to take the easiest courses in order to finish my degree without any further mishaps, but I know I would eventually like to return to the more rigorous math that fascinated and confounded me back in the early days.

How do I regain my confidence? Was I ever a mathematician to begin with? What's a good way to ease back into it?

Interesting case. I took very much an opposite route to you in many ways than one. I didn't have a solid science maths background going into uni and enrolled in a commerce degree at first. But took some maths subjects in the first year. I loved it very much although found it extremely difficult especially the purer ones. I switched to a BSc in second year although decided only to take one maths subject and some other science and philosphy subjects. Looking back this may not have been a wise choice as I could be a better maths student had I done more maths in that year but at the time offcourse I wanted to explore other subjects and took physics for the first time so didn't know what to expect. I also felt that I didn't have enough mathematical maturity to do 2nd year maths even though I got 70 and 80 for 1st year pure and applied maths repectively. Now in my 4th year at uni, I am taking an overload (one extra subject) of 3rd year maths and physics subjects and although I also found it extremely challenging, have found that I enjoy it more than ever and can't wait to do higher maths in the future. But first thing is first, hopefully I complete this year successfully.

 

[mathwonk]

well life choices are not so easy. i suggest gradual movements. stay at least partially with what is working, and go gradually in the direction of what you hope will work.

you are young and strong, and smart, so there are lots of openings.

but temporary fears and insecurities are common, at least in my experience.

the key is to persist with what you love.

if you are working at it, you are a mathematician, regardless of your success rate.

 

[mathwonk]

one thing that relieves math fatigue is contact with other mathematicians. i am now enjoying my birthday conference at uga and am extremely grateful to the visiting speakers and others who came to provide stimulus to those of us here. but guess what? at least one speaker said he himself was feeling the same lift from being here that we are feeling from having him here!

so try to get together with people who enjoy discussing together, and they will stimulate you and each other.

 

[pivoxa15]

Good point. I relieve maths fatigue by hanging around here.

 

[J77]

one thing that relieves math fatigue is contact with other mathemticians. i am now enjoying my birthday conference at uga and am extremely grateful to the visiting speakers and others who came to provide stimulus to those of us here. but guess what? at least one speaker said he himself was feeling the same lift from being here that we are feeling from having him here!

so try to get together with people who enjoy discussing together, and they will stimulate you and each other.

Yeah -- conferences certainly give you a lift. eg. I've just come back from a physics conference -- explaining your (mathematical) work to physicists really gives you new insight/avenues to explore.

 

[mathwonk]

what did you talk about?

 

[pivoxa15]

How common are mathematicians who scarifice their 'life' to do maths? i.e live alone without a partner or children and maintain minimum personal social interactions? How productive are they in the long term? I know Newton was one and the Russian who solved Poincare's problem but they have extroordinary abilities. How do people with lesser abilities do?

 

[mathwonk]

most mathematicians i know are pretty ordinary, and have families, friends, children, etc. those people you mentioned are very unusual, and not usually better mathematicians then the ordinary ones in my opinion.

 

[J77]

what did you talk about?

I pretty much fall under the category of nonlinear optics.

(ie. as opposed to GR/SR, HEP, Nano... etc.)

Response to above, also -- most mathematicians can usually be found occupying the local drinking spot at one time or the other. Communication and social skills are way up there if you want to succeed, imo.

 

[mathwonk]

feel like summarizing what you said about non linear optics? even if it way over my head, someone will enjoy it.

i will tell you for example what my friend asked me atmy conference.

he asked about generalizing riemanns proof of "jacobi inversion" for a single curve, to the analogous result for a pair of curves, one doubly covering the other.

in algebra its like generalizing a result about one field to the case of a quadratic extension of fields.

riemann showed that you could parametrize the jacobian of a genus g curve, which is a complex torus of dimension g, almost one to one, by a map from the product of g copies of the curve, via some "abelian integrals".

his argument can be given in two ways, one by using his theta function, (a fundamental solution for the heat equation), but there is another way, more geometric and almost tautological, using the dual torus called the picard variety of the curve.

of course this uses riemann's and abel's proof that the two tori are in fact isomorphic.

anyway my friend had done the analytic proof in the relative case and I then did the geometric proof the other night, between blogs here.

 

[mathwonk]

Now that my conference has ended and I am still exhilarated by the experience of meeting again so many mathematical friends and hearing so much interesting math that it has literally jump started my math research thinking again, I wanted to extend my earlier advice on becoming a mathematician to include strong advice to attend conferences.

Then I ran across edgardo's link to terence tao's advice, which contains everything i would have said and much more, but said more clearly and succinctly. Plus it has the stamp of approval of a Fields medalist. In fact I myself just reread Tao's advice for my own benefit.

Everyone here should read the advice of Terence Tao, and try to heed it. This is the best article i have seen on how to become a mathematician.

I am going to send it to our grad students at UGA for their benefit too.

For reference again, (and with thanks to edgardo):

http://www.math.ucla.edu/~tao/advice.html

 

[pivoxa15]

most mathematicians i know are pretty normal, and have families, friends, children, etc. those people you mentioned are very unusual, and not usually better mathematicians then the normal ones in my opinion.

But do you think had them or you not had a family etc would have enabled you to go further with your maths? In other words you would have had less distractions. Or do you think these things are necessary to make a good mathematician or at least keeping a balanced life is necessary to becoming a successful pure mathematian?

 

[pivoxa15]

Response to above, also -- most mathematicians can usually be found occupying the local drinking spot at one time or the other. Communication and social skills are way up there if you want to succeed, imo.

But what if it's pure maths?

 

[mathwonk]

well i find myself thinking sometimes that if I had no family obligations, then I could work more. There is a joke that a mathematician needs both a mistress and a wife because then when he is not with the mistress she thinks he is with the wife, and vice versa, so then he can skip out on both of them and go to the office and get some work done.

But in truth I never found it possible to complete my own grad studies and become a mathematician until i got married and had a normal family life. The birth of my children energized me also in my math.

Hironaka, the fields medalist once told me a joke about mathematicians who found they proved good theorems on getting married would sometimes get married several times to have this experience over again.

It is reminiscent of a remark made to me by an advisor at Harvard college on students who wanted get away from Cambridge and all its distractions to study more, but when they returned they found that the students who had stayed, somehow had accomplished more, even with all the distraction.

I personally cannot bear to stay longer than one week alone at a meeting or summer session. I love my work, but not exclusively, and I work better in a normal environment.

Life is not easy or simple. As my yoga teacher said, one has a spiritual self, a physical self, an intellectual self, an emotional self, etc...

The task is to keep them all functioning in harmony.

good wishes.

 

[Beeza]

Mathwonk,
At any point did you ever doubt your ability to succeed in advanced math courses? I just began self studying Apostol's Mathematical Analysis with a professor of mine (whom offered to continue working with me over the summer when the spring semester is over), and find even the beginning exercises very fun, but often time consuming and difficult. I sometimes worry I won't live up to my own expectations, or even my professors'.

 

[pivoxa15]

well i find myself thinking sometimes that if I had no family obligations, then I could work more. There is a joke that a mathematician needs both a mistress and a wife because then when he is not with the mistress she thinks he is with the wife, and vice versa, so then he can skip out on both of them and go to the office and get some work done.

But in truth I never found it possible to complete my own grad studies and become a mathematician until i got married and had a normal family life. The birth of my children energized me also in my math.

Hironaka, the fields medalist once told me a joke about mathematicians who found they proved good theorems on getting married would sometimes get married several times to have this experience over again.

It is reminiscent of a remark made to me by an advisor at Harvard college on students who wanted get away from Cambridge and all its distractions to study more, but when they returned they found that the students who had stayed, somehow had accomplished more, even with all the distraction.

I personally cannot bear to stay longer than one week alone at a meeting or summer session, because I enjoy being around my wife too much. I love my work, but not exclusively, and I work better in a normal environment.

Life is not easy or simple. As my yoga teacher said, one has a spiritual self, a physical self, an intellectual self, an emotional self, a sexual self, etc...

The task is to keep them all functioning in harmony.

good wishes.

What kind of distractions exist in Cambridge?

Your point about having balanced life is extremely important I think because we have evolved evolutionary and people who do a wide range of things are rewarded psychologically as a way of our body thinking us for what we have done to prolong its existence. Having children is one of those things I think. And when we don't do these things, our body punish us by making us feel depressed.

From your wide observations, what kind of wife is best suited to an academic mathematician? i.e another mathematician, school teacher, etc. OR is it too wide ranging to say?

 

[morphism]

pivoxa15, from all your posts I gather that you have some weird, disturbing idea about what a "pure mathematician" is. Mathematicians are humans, not machines that do mathematics...

 

[tehno]

Now that my conference has ended and I am still exhilarated by the experience of meeting again so many mathematical friends and hearing so much interesting math that it has literally jump started my math research thinking again, I wanted to extend my earlier advice on becoming a mathematician to include strong advice to attend conferences.

How many conferences,on average,you attend per year?Just being curious.

 

[symbolipoint]

Mathwonk, I just checked your initiating message on this topic and found what you said regarding foreign languages:

learn to struggle along in French and German, maybe Russian, if those are foreign to you, as not all papers are translated, but if English is your language you are lucky since many things are in English (Gauss), but oddly not Galois and only recently Riemann.

What more can you tell us about the usefulness of knowing Russian for the purposes of reading articles written in Russian about any Mathematics? How valuable? Do significant articles exist which have not yet been translated which Mathematical specialists might want to read and understand? In other words, is there still significant Mathematics work written in Russian which have not been translated? Would knowing Russian then be a special qualification for gaining admission to even an undergraduate Mathematics program (AS A STUDENT)? Were Russian Mathematicians known for any significant contributions to field of Mathematics (in other words, what were Russian Mathematicians famous for creating/discovering?)

symbolipoint