Should I Become a Mathematician?

[mathwonk]

i see i ignored a question asking what the grad school experience is like, passing prelims, writing a thesis, etc...

so here goes a little on that. of course everyone enters grad school, in US anyway, with a different background. Since we have a shortage of PhD mathematicians in US, we are always scouting for talent and recruiting people to our programs. So many people get in who are less than ideally prepared.

Thus the beginning of the grad experience can be rough for the less prepared. At UGA we have recently begun a testing and placement program, and have courses designed to help people get up to speed on some crucial undergrad material they may not have been fully taught before. This is new, as it used to be more or less sink or swim.

Different schools are different on this matter of sink or swim, and it would be wise to find out whether your school just brings in and watches to see which ones survive, ot whether they try to help everyone make it. I suspect today most try to be helpful, but that may be less true where the school is very popular and the professors are very busy.

The most fortunate people are those who know all the basic stuff and are ready to begin work toward a thesis right away, the primary reason for being there.

At the other extreme, I entered not knowing what an ideal was and was immediately plunged into an algebra course on homological ring theory. I had also never had complex variable (in undergrad they said, "oh you'll learn that in grad school") and began in an analysis course that spent one month on reals and one on complex and moved on to Riemann surfaces!

So you need to come in knowing as much as possible, and also choose a school where the introduction is somewhat sensitive to what people know. (Brandeis was still a great place to be, and has no doubt also changed totally since then.)

So you must talk to people currently at the schools of interest, students as well as professors, to find out the department's expectations, and how those are viewed by students.

the first thing then is to get up to speed as quickly as possible. Writing a thesis will take much longer and be much harder usually than you could have imagined, so you need to get ready to do it and sart doing it as soon as possible.

so since for many students the first big hurdle is the prelims, i will post here the current prelim syllabi from UGA in a few subjects. The requirements may vary, and are constantly changing, but a pure mathematics aspirant should hope to be able to pass prelims in all 3 pure subjects, say topology, algebra, and at least one type of analysis, real or complex.

We have gradually made these syllabi less and less demanding over the years, continually removing material, to where they will probably read like undergraduate syllabi to students from abroad (or elite US schools) now.

Notice for example the algebra syllabus no longer covers noetherian rings and modules, nor tensor products, and the complex syllabus no longer covers elliptic functions or riemann mapping theorem. Still it is quite challenging for an average undergrad from the US to master all this in a short amount of time, i.e. a year or so of grad school.

Be sure to get these syllabi from your target school, as they may be very different at different places.

 

[mathwonk]

We have recently upgraded the syllabus by calling half the material "undergraduate" material and trying to offer it prior to the grad course. We still need to make this work in practice, as this is new. Thus we spend only one semester in grad school teaching the alg prep course now, as opposed to the old days when it was a year. This is one justification for deleting some material from the syllabus.

Study Guide for Algebra Qualifying Exam
Proposed: April 2006

UNDERGRADUATE MATERIAL

Group Theory: MATH 6010
subgroups
quotient groups
Lagrange's Theorem
fundamental homomorphism theorems
group actions with applications to the structure of groups such as
the Sylow Theorems
group constructions such as:
direct products
structures of special types of groups such as:
p-groups
dihedral, symmetric and alternating groups, cycle decompositions
the simplicity of An, for n ≥ 5

References: [1,3,5].

Linear Algebra: MATH 6050
determinants
eigenvalues and eigenvectors
Cayley-Hamilton Theorem
canonical forms for matrices
linear groups (GLn , SLn, On, Un)
dual spaces, dual bases, pull back, double duals
finite-dimensional spectral theorem

References: [1,2,5]GRADUATE MATERIAL (MATH 8000)

Foundations:
Zorn's Lemma and its uses in various existence theorems such as that of a basis for a vector space or existence of maximal ideals.

References: [1,4]

Group Theory:
Sylow Theorems
free groups, generators and relations
semi-direct products
solvable groups
References: [1,3,5].

Theory of Rings and Modules:
basic properties of ideals and quotient rings
fundamental homomorphism theorems for rings and modules
characterizations and properties of special domains such as:
Euclidean implies PID implies UFD
classification of finitely generated modules over Euclidean domains
applications to the structure of:
finitely generated abelian groups and
canonical forms of matrices

References: [1,3,4,5].

Field Theory:
algebraic extensions of fields
fundamental theorem of Galois theory
properties of finite fields
separable extensions
computations of Galois groups of polynomials of small degree and cyclotomic polynomials
solvability of polynomials by radicals

References: [1,3,5]As a general rule, students are responsible for knowing both the theory (proofs) and practical applications (e.g. how to find the Jordan or rational canonical form of a given matrix, or the Galois group of a given polynomial) of the topics mentioned.References [Need to be updated; e.g. [3] and [4] are out of print.]

[1] Thomas W. Hungerford, Algebra, Springer, New York, 1974.
[2] Kenneth Hoffman and Ray Kunze, Linear Algebra, Prentice-Hall, 1961.
[3] Nathan Jacobson, Basic Algebra 1, W.H. Freeman, San Francisco, 1974.
[4] Nathan Jacobson, Basic Algebra 2, W. H. Freeman, San Francisco, 1980.
[5] Serge Lang, Algebra, Addison Wesley, Reading Mass., 1970

 

[mathwonk]

Here is the analysis syllabus, recently divided into reals and complex.Study Guide for Real Analysis Exam

I. Calculus and Undergraduate Analysis, 

Continuity and differentiation in one and several variables, 
Compactness and connectedness in analysis
, Sequences and series, 
Uniform convergence and uniform continuity
Taylor's Theorem, 
Riemann integrals
 References: [2]


II. Measure and Integration, 

Measurability:
*** Measures in Rn and on ơ-algebras
*** Borel and Lebesgue measures
*** Measurable functions
Integrability:
*** Integrable functions
*** Convergence theorems (Fatou’s lemma, monotone* and dominated
*** convergence theorems)
*** Characterization of Riemann integrable functions, 
Fubini and Tonelli theorems
Lebesgue differentiation theorem and Lebesgue sets
References: [1] Chapter 1, 2, 3.
[3] Chapter 3, 4, 5, 11, 12.
[4] Chapter 1, 2, 3, 6.

III. Lp and Hilbert Spaces

Lp space: Holder and Minkowski inequalities, completeness, and the dual of Lp
Hilbert space and L2 spaces: orthonormal basis, Bessel’s inequality, Parseval’s identity,
Linear functionals and the Riesz representation theorem.
References: [1] Section 5.5, Chapter 6.
[3] Chapter 6.
[4] Chapter 4.

[1]*G. Folland, Real Analysis, 2nd edition, John Wiley & Sons, Inc.
[2] W. Rudin, Principle of Mathematical Analysis, 3rd edition
[3]* H. Royden, Real Analysis, 3rd edition

[4] E. M. Stein and R. Shakarchi, Real Analysis, Princeton University Press.
Study Guide for Complex Analysis Exam

I. Calculus and Undergraduate Analysis

Continuity and differentiation in one and several real variables
Inverse and implicit function theorems
Compactness and connectedness in analysis
Uniform convergence and uniform continuity
Riemann integrals
Contour integrals and Green’s theorem
 References: [3].


II. Preliminary Topics in Complex Analysis

Complex arithmetic
Analyticity, harmonic functions, and the Cauchy-Riemann equations
Contour Integration in C
References: [1] Chapter 1, 2;
[2] Chapter 1, 2, 4;
[4] Chapter 1.

III. Cauchy's Theorem and its consequences

Cauchy's theorem and integral formula, Morera’s theorem
Uniform convergence of analytic functions
Taylor and Laurent expansions
Maximum modulus principle and Schwarz’s lemma
Liouville's theorem and the Fundamental theorem of algebra
Residue theorem and applications
Singularities and meromorphic functions, including the Casorati-Weierstrass theorem
Rouche’s theorem, the argument principle, and the open mapping theorem
References: [1] Chapter 4, 5, 6;
[2] Chapter 5, 7, 8, 9;
[4] Chapter 2, 3, 5.



IV. Conformal Mapping, 

General properties of conformal mappings
, Analytic and mapping properties of linear fractional transformations
 References:* [1] Chapter 3, 8; [2] Chapter 3, 4; [4] Chapter 8.



References
[1]* L. Ahlfors, Complex Analysis, Third Edition, McGraw-Hill
[2]* E. Hille, Analytic Function Theory, Vols. 1, Ginn and Company.
[3]* W. Rudin, Principle of Mathematical Analysis, Third Edition.

[4] E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press.PAGE 1

 

[mathwonk]

Study Guide for Topology Exams

General Topology

topological spaces and continuous functions
product and quotient topology
connectedness and compactness
Urysohn lemma
complete metric spaces and function spaces

References: [2]

Algebraic Topology

fundamental group
van Kampen's theorem
classifications of surfaces
classifications of covering spaces
homology:
simplicial, singular and cellular: computations and applications
degree of maps
Euler characteristics
Lefschetz fixed point theorem

References: [1,3]

The weight of topics on the exam should be about 1/3 general topology and 2/3 algebraic topology.

References

[1] W. Massey, Algebraic Topology: An Introduction, Springer Verlag, 1977.
[2] J. Munkres, Topology, A First Course, Prentice-Hall, 1975.
[3] J. Munkres, Elements of Algebraic Topology, Addison-Wesley, 1984.

 

[mathwonk]

compare this modest syllabus with that at harvard, where one needs to know all 6 of the following areas, and the exam has one question on each. note their undergraduate algebra syllabus covers more than our graduate and undergradiate algebra syllabi combined.

Harvard:
The syllabus is divided into 6 areas. In each case we suggest (sections of) a book to more carefully define the syllabus. The examiners are asked to limit their questions to major topics covered in (these sections of) these books. We have tried to choose books we think are good. However there are many good books and others might better suit your needs. In each case we divide the syllabus into two sections. Section U is material which are usually covered in our undergraduate, not our graduate, courses. Section G is material usually taught at the graduate level. Where appropriate we list courses which will cover some of this material.

1) Algebra.
U: Dummit+Foote, Abstract Algebra, except chapters 16 and 17. (math 122, 123, 126)
G: Dummit+Foote, Abstract Algebra, chapter 17.

2) Algebraic Geometry
G: Harris, Algebraic geometry, a first course, lectures 1-7, 11, 13, 14, 18.

3) Complex Analysis
(Table of contents)
U: Ahlfors, Complex Analysis (2nd ed), chapters 1-4 and section 5.1. (math 113)
G: Ahlfors, Complex Analysis (2nd ed), section 5.4.

4) Algebraic Topology
U: Hatcher, Algebraic Topology, chapter 1 (but not the additional topics). (math 131)
G: Hatcher, Algebraic Topology, chapter 2 (including additional topics) and chapter 3 (without additional topics). (math 272a)

5) Differential Geometry
(Table of contents)
U: Boothby, An introduction to differentiable manifolds and Riemannian geometry, sections VII.1 , VIII.1 and VIII.2. (math 136)
G: Boothby, An introduction to differentiable manifolds and Riemannian geometry, chapters I - V and VII. (math 134, 135 and 230a)

6) Real Analysis
(Rudin: Table of contents)
(Birkhoff+Rota: Table of contents)
U: Rudin, Principles of mathematical analysis, chapters 1-8.
Birkhoff + Rota, Ordinary differential equations, chapters 1-4 and 6. (math 25, 55, 112)
G: Rudin, Principles of mathematical analysis, chapter 10.
Rudin, Functional analysis, chapters 1, 2, 3.1-3.14, 4, 6, 7.1-7.19 and 12.1-12.15. (math 212a)as usual it seems to be the real analysts who cannot bring themselves to shorten the syllabus.

 

[mathwonk]

here is the spring 2006 harvard prelim.

Qualifying exam, Spring 2006, Day 1
(1) Let φ : A → B be a homomorphism of commutative rings, and let pB ⊂ B be
a maximal ideal. Set A
⊃ pA := φ− 1 (pB ).
(a) Show that pA is prime but in general non maximal.
(b) Assume that A, B are finitely generated algebras over a field k and φ is a
morphism of k-algebras. Show that in this case pA is maximal.

(2) Let V be a 4-dimensional vector space over k, and let Gr2 (V ) denote the set
of 2-dimensional vector subspaces of V . Set W = Λ2 (V ), and let P5 be the 5- dimensional pro jective space, thought of as the set of lines in W .
Define a map of sets Gr2 (V ) → P 5
that sends a 2-dimensional subspace U ⊂ V
to the line Λ2 (U ) ⊂ Λ 2 (V ) = W .
(a) Show that the above map is injective and identifies Gr2 (V ) with the set of points of a pro jective subvariety of P5 .
(b) Find the dimension of the above pro jective variety, and its degree.

(3) Are there any non-constant bounded holomorphic functions defined on the com-
plement C \ I of the unit interval
I = {a ∈ R | 0 ≤ a ≤ 1} ⊂ C in the complex plane C?
(4) Let X be the topological space obtained by removing one point from a Riemann surface of genus g ≥ 1. Compute the homotopy groups πn (X ).

(5) Let γ be a geodesic curve on a regular surface of revolution S
⊂ R3
. Let θ(p)
denote the angle the curve forms with the parallel at a point p
∈ γ and r(p) be the
distance to the axes of revolution. Prove Clairaut’s relation: r cos θ = const.

(6) Define the function f on the interval [0, 1] as follows. If x = 0.x1 x2 x3 ... is the
unique non-terminating decimal expansion of x
∈ (0, 1], define f (x) = maxn {xn }.
Prove that f is measurable.

 

[mathwonk]

here is another one from harvard. note they are testing group representations even though that is not on the syllabus. or maybe they have forgotten to say which edition of dummitt and foote they are using. at least group reps and character tables are in chapter 18 in both the latest two editions of DF. this sort of thing can really confuse students. no wait maybe they are calling that stuff undergrad material. i.e. their undergrad alg syllabus is everything in DF except homological algebra. wow. that's our undergrad syllabus plus our grad syllabus plus both our optional followup courses on commutaitve alg and group reps.

QUALIFYING EXAMINATION
Harvard University
Department of Mathematics
Tuesday 20 September 2005 (Day 1)
hmmm the browser will not accept this one for some reason so go to their website.

harvard math dept, info for grad students, quals syllabi.

 

[mathwonk]

here is the syllabus from Univ of Washington. More reasonable on algebra and reals, but more advanced on complex and quite advanced in manifolds and linear analysis. they do have a couple options to substitute a course for one exam or an oral for a written exam. Thats a lot of stuff to know.

Algebra

Topics: Linear algebra (canonical forms for matrices, bilinear forms, spectral theorems), commutative rings (PIDs, UFDs, modules over PIDs, prime and maximal ideals, noetherian rings, Hilbert basis theorem), groups (solvability and simplicity, composition series, Sylow theorems, group actions, permutation groups, and linear groups), fields (roots of polynomials, finite and algebraic extensions, algebraic closure, splitting fields and normal extensions, Galois groups and Galois correspondence, solvability of equations).
References: Dummit and Foote, Abstract Algebra, second edition; Lang, Algebra; MacLane and Birkhoff, Algebra; Herstein, Topics in Algebra; van der Waerden, Modern Algebra; Hungerford, Algebra.

Real Analysis
Topics: Elementary set theory, elementary general topology, connectedness, compactness, metric spaces, completeness. General measure theory, Lebesgue integral, convergence theorems, Lp spaces, absolute continuity.
References: Hewitt and Stromberg, Real and Abstract Analysis; Rudin, Real and Complex Analysis; Royden, Real Analysis; Folland, Real Analysis.

Complex Analysis
Topics: Cauchy theory and applications. Series and product expansions of holomorphic and meromorphic functions. Classification of isolated singularities. Theory and applications of normal families. Riemann mapping theorem; mappings defined by elementary functions; construction of explicit conformal maps. Runge's theorem and applications. Picard's theorems and applications. Harmonic functions; the Poisson integral; the Dirichlet problem. Analytic continuation and the monodromy theorem. The reflection principle.
References: Ahlfors, Complex Analysis; Conway, Functions of One Complex Variable, vol. 1; Rudin, Real and Complex Analysis (the chapters devoted to complex analysis).

Manifolds
Topics: Elementary manifold theory; the fundamental group and covering spaces; submanifolds, the inverse and implicit function theorems, immersions and submersions; the tangent bundle, vector fields and flows, Lie brackets and Lie derivatives, the Frobenius theorem, tensors, Riemannian metrics, differential forms, Stokes's theorem, the Poincaré lemma, deRham cohomology; elementary properties of Lie groups and Lie algebras, group actions on manifolds, the exponential map.
References: Lee, Introduction to Topological Manifolds (Chapters 1-12) and Introduction to Smooth Manifolds (all but Chapter 16); Massey, Algebraic Topology: An Introduction or A Basic Course in Algebraic Topology (Chapters 1-5); Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry (Chapters 1-6); and Warner, Foundations of Differentiable Manifolds and Lie Groups (Chapters 1-4).

Linear Analysis
Topics: Linear algebra (spectral theory and resolvents, canonical forms and factorization theorems for matrices), ordinary differential equations (existence and uniqueness theory, linear systems, numerical approximations), Fourier analysis (Fourier series and transforms, convolutions, applications to PDE), functional analysis (theory and examples of Banach and Hilbert spaces and linear operators, spectral theory of compact operators, distribution theory).
References: Kato, A Short Introduction to Perturbation Theory for Linear Operators; Horn and Johnson, Matrix Analysis; Birkhoff and Rota, Ordinary Differential Equations; Coddington and Levinson, Theory of Ordinary Differential Equations; Lambert, Numerical Methods for Ordinary Differential Systems; Dym and McKean, Fourier Series and Integrals; Folland, Fourier Analysis and its Applications; Jones, Lebesgue Integration on Euclidean Space; Riesz and Nagy, Functional Analysis; Retherford, Hilbert Space: Compact Operators and the Trace Theorem; Schechter, Principles of Functional Analysis; Friedlander, Introduction to the Theory of Distributions.

 

[mathwonk]

these requirements are really quite different from place to place. i recommend you look at the website of the UPenn math dept e.g. They are more focused on making sure the incoming students know basic undergrad math than cramming a lot of new topics down their throats. so they have a preliminary masters test covering genuinely undergrad stuff, and they list a syllabus for that. then later they give 2 oral exams on grad material the student has studied there. then they are on their way to thesis work. that seems an enlightened approach to creating research mathematicians without requiring them to learn the whole history of mathematics first.at princeton, the process is advanced but informal. almost no guidance is provided from the dept but the students help each other out with old questions etc...

Princeton:

General examination

The student must stand for an oral exam administered by a committee of three professors, including the advisor who serves as chair of the committee. A typical exam can last 2 to 3 hours. Areas covered are algebra, and real and complex variables. The student must also choose two (2) special or advanced topics. These two additional topics are expected to come from distinct major areas of mathematics, and the student's choice is subject to the approval of the Department. Usually in the second year, and sometimes even in the first, students begin investigations of their own that lead to the doctoral dissertation. For the student interested in mathematical physics, the general examination is adjusted to include mathematical physics as one of the two special topics. There are three general examination periods each academic year--October, January, and May. It has been a tradition of the students to post their exams as a resource and study guide for other students, see Graduate Students' Guide to Generals.

 

[courtrigrad]

I came across this link http://wuphys.wustl.edu/~katz/scientist.html and many others from PhDs.org. What are your opinions about his experiences/advice? Also, is it very competitive to become a professor? Are there major differences in becoming a professor at an undergraduate only institution versus becoming a professor at both a undergraduate and graduate institution? Which is more competitive?

Thanks

 

[mathwonk]

i think there is a lot of truth in what he says, but he still seems rather cynical.

there are always people out there to discourage you. be aware of the problems they expose. if those are too much for you, ok, but if you want to do science anyway, try to figure out how. the competition is very tough.

I read an inspirational story by glenn clark once about his disappointment at not being the writer he wanted to be, but only an english teacher. he decided to make the best of it, maybe hoping to help a student become a writer. then one day he wrote down some inspirational ideas he had, and published his first book. that was his beginning as a writer.

it is true many of us will get a phd only to struggle to find an academic job. still there must be many openings in prep schools and junior colleges and high schools for good people. our high schools have very few well qualified math and science teachers in my opinion. if more trained people took such jobs we could begin to reverse the pitiful quality of our high chool teaching in US.

i seem to recall einstein only had a job as a patent clerk; of course we mostly do not expect to repeat his achievements. still it shows what even talented people must do to survive. andre weil had no job in the 1940's and had to go to south america to find employment. look him up if you do not know who he is. alexander grothendieck was denied a position in france in his last days in academe even though people are still working out the ramifications of the few letters and hints he wrote down at the end of his career (esquisse d'un programme - dessins des enfants, [children's drawings?])we are not guranteed a good salary and respect for pursuing our own dream. the dream has to be enough.

hang in there. do not give up too easily, but do not be naive either.

 

[mathwonk]

perhaps we should ask professor katz to think back and recall his own struggles to realize the career he now enjoys. he is worried abut the plight of others, but might not even take his own advice if he were young again.

 

[mathwonk]

Becoming a professor is not as hard as getting a job, many places. e.g. at harvard, you don't even get the job unless you are already a famous professor. curt mcmullen went there after getting the fields medal e.g.

some places granting of tenure is very strict and stingy, and other places essentially everyone who gets hired and does not self destruct eventually gets tenure and promotion. even those places however it does not feel that way to the candidate, it feels like a very difficult struggle. academic life is hard, standards are high, competition from smart young energetic people is strong, concrete rewards are minimal, almost everyone i know feels strapped for money to carry out their work and have even some of the good life. outstanding people with world class reputations are earning less than an average person in some other fields. although they often work essentially almost all the time, many people think of academicians as lazy people who sit around in an ivory tower doing nothing and living off the public dole. this can be frustrating.

but people who want to spend their time talking to smart people about science and math, and interacting with students, still seek this environment.

It may be true however that in some cases working in an industry like the internet, where you can use your skills and learn more everyday on the job, and get paid well, may be more exciting and rewarding.

Although academics like academia, i have only known one person, a lawyer, who wished to return to academia after getting a taste of what it is like to earn real money in the private sector.

the others who left were students or colleagues who did not get tenure, and who revisited later earning several times our salaries and apparently very happy about it.

but the training they got in math or physics apparently helped them succeed at what they did later in the "real" world.

 

[mathwonk]

perhaps the previous post acknowledges what professor katz was advising.

 

[mathwonk]

I want to thank you for your questions here as the advice I have been giving about reading original papers and working, has motivated me to return to work and reading and I am just now trying to read an expository paper on cohomology of moduli spaces of curves by frances kirwan.

 

[courtrigrad]

Would it be advantageous to obtain two bachelors degrees in mathematics (one in pure math and the other in applied math). Would this combination open up more career options (i.e. a quant/financial engineer, operations research, etc..) than just majoring in pure or applied math alone. I am leaning towards becoming a quant/financial engineer in some company in the future. Also, the two bachelor degrees in pure and applied math would be obtained in 5 years versus 8 years. So I figure that this is a good deal. What are your opinions?

 

[Dragonfall]

http://wuphys.wustl.edu/~katz/scientist.html

This has got to be the most discouraging thing I read since the deletion of the 'Frivolous Theorem of Arithmetic' on Wikipedia.

 

[mathwonk]

more recommended reading: The Calculus Affair, by Herge'.

 

[Dragonfall]

http://wuphys.wustl.edu/~katz/scientist.html

This was written 7 years ago, before the tech bubble burst.

 

[Dragonfall]

With that no one can disagree, I think. I wonder how much of it applies to mathematics?

 

[courtrigrad]

Anyone heard of Gottingen? How are the PhD programs from math outside the US? Is Oxford the best for getting a math/applied math PhD? In general, are international graduate schools more competitive than US graduate schools? How is the University of Waterloo/Mcgill (and other Canadian universities) for math/applied math?

Thanks

 

[mathwonk]

Andrew Granville, a terrific young number theorist, got his PhD at Queens and is now at Montreal.

I think the famous (recently deceased) Raoul Bott got his degree at McGill?

 

[mathwonk]

no it says Bott studied engineering at mcgill then got phd at carnegie tech.

 

[mathwonk]

well i only recognize famous people who have been around, but even i notice the names of tom dieck, sandy patterson, and yuri tschinkel at gottingen.

 

[mathwonk]

one of my most brilliant friends, Fabrizio Catanese, is chaired professor at Bayreuth, and like many Italian mathematicians of his generation, note he does not have a phd.

he also formerly held the Gauss chair at Gottingen.

 

[George Jones]

one of my most brilliant friends, Fabrizio Catanese, is chaired professor at Bayreuth, and like many Italian mathematicians of his generation, note he does not even have a phd at all.

he also formerly held the Gauss chair at Gottingen.

Physicist/mathematician/author Freeman Dyson never bothered to get a phd.

 

[mathwonk]

Barry Mazur supposedly does not even have a BA. [The story goes that at the time MIT had an ROTC requirement which he declined to complete.]

 

[interested_learner]

There is a Nobel Prize winner in Physics (Kirby) who only has a master's degree and you are right Dyson has only a BA. But these guys are exceptional. Mazur has a Phd.

 

[mathwonk]

here another exceptional one - Andrew Gleason, who I believe contributed to the solution of hilberts 5th problem, and is a professor emeritus at harvard, reportedly has no phd.

 

[mathwonk]

here are my day one algebra notes, what think you?

8000 fall 2006 day one.

Introduction:
We will begin with the study of commutative groups, i.e. modules over the integers Z. We will prove that all fin gen abelian groups are products of cyclic groups. In particular we will classify all finite abelian groups. Then we will observe that the same proof works for modules with an action by any Euclidean or principal ideal domain, and generalize these results to classify f.g. modules over such rings, especially over k[X] where k isa field. This will allow us to deduce the usual classification theorems for linear operators on a finite dimensional vector space, since a pair (V,T) where T is a linear operator on the k space V, is merely a k[X] module structure on V.

For completeness sake we recall some familiar definitions.

A group is a set G with a bjnary operation GxG--->G which satisfies:
(i) associativity, a(bc) = (ab)c, for all a,b,c, in G;
(ii) existence of identity: there is an element e: ea = ae = a for all a in G.
(iii) existence of inverses: for every a in G, there is a b : ab = ba = e.

A subgroup of G is a subset H ⊂ G which is also a group with the same operation. H has the same identity as G, and the inverse for any element in H is its inverse in G.

A group G is commutative, or abelian, if also
(iv) ab = ba for every a,b, in G.

Remarks:
We will study mostly commutative groups in the first part of the course, and we will usually write them additively instead of multiplicatively, thus we write the identity as 0 . Two advantages of commutative groups are the following, which will make more sense shortly: if G (commutative) has elements a,b, such that na = 0 = mb, where n,m are positive integers, and if p = lcm(n,m), then G has an element c such that pc = 0. Also the subset of elements a of G such that na = 0 for some integer n>0, the set of elments of "finite order", is a subgroup of G. Thus it is easier to understand the "orders" of the elements of a commutative group. Also it is easier to construct the "coproduct", sometimes called the "direct sum", of a family of commutative groups.

Blanket assumption, all groups are assumed commutative until we say otherwise.

Important Examples: i) the set Z of integers is a group for addition; ii) if n is an integer, the multiples of n, form a subgroup nZ ⊂ Z; iii) the rationals form a subgroup of the reals for addition Q ⊂ R; the positive rationals form a multiplicative subgroup of the positive reals Q+ ⊂ R+; S1 = the multiplicative group of complex numbers of length one, is also called the circle group.

It is efficient to use only a few elements of a group to represent all others, and the number of elements so needed also helps measure the size of the group.
A subset S ⊂G generates G if there is no subgroup containing S except G, equivalently if every non zero element of F can be written as a finite linear combination n1a1 + ...+nkak, where all ai are in S and the ni are integers. If G is written multiplicatively it means all elements except e can be written as a finite product ∏ aini, where where all ai are in S and the ni are integers.

Examples: {1} generates Z, as does {-1}. The empty set generates the trivial group {0}. The interval (0,d) generates (R,+) if d>0. The set of positive primes generates Q+.

We say G is finitely generated, or fin gen, or f.g., if there is a finite set of generators for G.
We proceed now to the classification of all fin gen abelian groups. The relevant concepts we will use are products, quotients, isomorphisms, and other linear maps.

Fundamental constructions (on abelian groups):
I) Products: Given an indexed family of (always abelian) groups {Gi}I, form the cartesian product set ∏IGi of all functions from the index set I to the union of the groups Gi and where the vaue of a function at i lies in Gi. We define the operation pointwise on functions, i,.e. we multiply or add the values of the functions. If the set I is finite of cardinality n, this means the elements are ordered n tuples if elements, one from each Gi, and we add them componentwise, like vectors. The identity is the function whose value at every i is the identity of Gi.

II) Coproducts: This is the same construction as above, except that the function must have the value 0 except possibly at a finite number of indices. Hence it is exactly the same in case the index set I is finite. It is denoted by an upside down product or by a summation sign, ∑ Gi. Obviously the coproduct of a family of groups is a subgroup of their product.

If all the groups Gi are equal to the integers Z, we call their coproduct a "free abelian group" on the set I, i.e. a group of form ∑I Zi.
We also write Zn for the product (or sum) of n copies of Z. The standard basis vectors
{ei = (0,...,0,1,0,...0) where the 1 is in the ith place, for i =1,...,n}, generate Zn.
Other commonly encountered products groups are Rn, S1xR, and S1x S1= the torus group.

III) Quotients: If H ⊂ G is a subgroup, define the quotient group G/H as the set of equivalence classes of elements of G for the relation x ≡ y iff x-y belongs to H. Write [x] for the equivalence s of x and add by setting [x]+[y] =[x+y], after checking this is independent of choice of representative elements of the classes.

The basic quotient group is Z/nZ, the additive group of integers "mod n".

When we define isomorphism, we will see that the circle group is isomorphic to a quotient group S1 ≅ R/Z. and also S1xS1 ≅ (R/Z)x(R/Z) ≅ (RxR)/(ZxZ). The interchange of quotients and products is more subtle than it may appear here, and will play a crucial role in the proof of the fundamental theorem we are seeking. The fact that renders the interchange easy here is that each factor in the denominator is a subgroup of a factor group in the numerator. When this is not the case the problem is more difficult.

One way to construct a finite abelian group is to form a product (Z/n1)x(Z/n2)x...x(Z/nk). Our goal is to show that these examples give essentially all finite abelian groups. To make the phrase "essentially all" precise, we must define how we will compare two groups, and when we will say two groups are essentially the same.

A map of groups f:G-->H (abelian or not) is a homomorphism, sometimes called simply a map, if for all a,b, in G, we have f(ab) = f(a)f(b), or additively, if f(a+b) = f(a) + f(b).
It follows that f(0) = 0 , and f(-x) = -f(x).

The set of homomorphisms from G to H is denoted Hom(G,H). When G,H are abelian it is also an abelian group under pointwise addition, [but it is not even a group if H is not abelian].

Examples of homomorphisms: The inclusion of a subgroup H ⊂ G is a homomorphism; The map
G--->G/H taking an element x to the class [x] is a homomorphism; The ith projection ∏IGi--->Gi taking a function to its value at i, that is taking a vector to its ith component, is a homomorphism. The injection Gi--->∑I Zi taking an element x of Gi to the function having value x at i and value 0 elsewhere, is a homomorphism. [This puts x in the ith component of a vector and 0's elsewhere.] The map R--->S1 taking t to e^(2πit) is a homomorphism.

Important invariants of a homomorphism:
To understand homomorphisms we focus on what goes to 0, and what things get "hit" by it.
If f:G-->H is a homomorphism of groups (abelian or not), the subset kerf = {x in G : f(x) = 0} is called the kernel of f; it is a subgroup of G. The subset Im(f) = {y in H: y = f(x) for some x in G} = the image of f, is a subgroup of H.

The quotient H/Im(f), defined for abelian groups only, is the cokernel of f.

An isomorphism is a homomorphism with an inverse homomorphism. I.e. a homomorphism f:G-->H , is an isomorphism if there is a homomorphism g--->G such that fog = id(H), and gof=id(G).

How to recognize an isomorphism:
A homomorphism f:G-->H is an isomorphism if and only if it is bijective,
if and only if kerf = {0} and Im(f)= H.

How to define homomorphisms:
1) To define a homomorphism into a product G--->∏IGi is equivalent to defining one homomorphism G--->Gi into each Gi. I.e. Hom(G, ∏IGi) ≅ ∏IHom(G,Gi), via the map taking the homomorphism
f:G---> ∏IGi to the family of compositions πiof where πi is the projection ∏IGi --->Gi .

2) To define a homomorphism out of a coproduct ∑ Gi--->H, is equivalent to defining one map out of each summand Gi--->H, i.e. Hom(∑Gi,H) ≅ ∏IHom(Gi,H) via the map taking f:∑ Gi--->H to the family of compositions foßi where ßi is the injection Gi--->∑ Gi.
3) To define a map from Zn--->Zm by 1) and 2), it suffices to define mn maps Z--->Z, i.e,. to give mn integers, in the form of an mxn matrix, where the ith column is the image under the map of the ith standard basis vector ei = (0,...,0,1,0,...0).

4) To define a map out of a quotient G/H--->K, is equivalent to defining a homomorphism f:G---K such that f(H) = {0}, i.e. Hom(G/H,K) ≅ Hom((G,H), (K,{0})) (maps of pairs), via the map taking
f:G/H--->K, to the composition foπ:G--->K, where π:G--->G/H is the projection.

Examples: The map R--->S1 sending t to e^(2πit) induces an isomorphism R/Z--->S1, via the correspondence in 3) above.
The maps Z--->(Z/riZ), induce a map Zn--->(Z/r1Z)x...x(Z/rnZ) which induces an isomorphism
Zn/[(r1Z)x...x(rnZ)]--->(Z/r1Z)x...x(Z/rnZ).
If f:G--->H is a surjective homomorphism, it induces an isomorphism G/kerf--->H.

Our first main theorem is the following.
Theorem: If G is any finitely generated abelian group, then there exist integers n,m≥ 0, and a sequence of integers r1,...,rm ≥ 2, such that G ≅ Zn x (Z/r1Z)x...x(Z/rmZ). Moreover these r's can always be chosen so that r1| r2|...,|rm-1|rm, i.e. each one divides the next one, and if this is done, all the integers are uniquely determined by the isomorphism class of )G. We call n the rank of G and the integers r1,...,rm the torsion coefficients, or invariant factors. Thus G is completely determined by the sequence (n, r1,...,rm). If n=m=0, G ={0}.

Exercises: if Tor(G) = { x in G such that there is an integer n>0 with nx = 0} then Tor(G) is a subgroup of G, called the torsiion subgroup. [This is false if G is not abelian.]

Cor: If G ≅ Zn x (Z/r1Z)x...x(Z/rmZ), then Tor(G) ≅ (Z/r1Z)x...x(Z/rmZ), and G/Tor(G) ≅ Zn . Thus the torsion part of G is a uniquely defined subgroup of G, and the free part of G is a uniquely defined quotient group.

Proposition: If G is a fin gen abelian group, there is a homomorphism f:Zn--->Zm , such that G ≅ coker(f) = Zm/f( Zn).

Proposition: If A is the matrix of f:Zn--->Zm , and if B is a matrix obtained by elementary row and column operations from A, then the cokernels Zn/A(Zm) ≅ Zn/B(Zm), are isomorophic.

Proposition: Every matrix A of integers can be reduced by elementary row and column operations, to a diagonal matrix B.

Corollary: The theorem is true.

 

[mathwonk]

do you think i am going too fast, too slow? [for some reason, all isomoprhism and equivalence relation symbols came out as subsets. when i saw the first one, which really was a subset i was encouraged briefly.]

 

[mathwonk]

this is for a first year grad course by the way, but many schools teach this in undergrad even sophomore year at places like Brandeis in the 1960's.

 

[mathwonk]

apparently i was going way too fast. i need to do the case of products of 2 groups before doing the case of arbitrary indexed families. and do some detailed examples. ill do that friday.

heres some homework problems, show that defining a homomorphism from AxB to G (abelian groups) is the same as defining one A-->G and another oen B--->G. then show that defining a homomorphism G--->AxB is the same as defining one G---<>A and another one G--->B.

then show that the multiplicative group C* of non zero compelx numbers is isomorphic to the product group RxS^1, where R is the aditive group of reals and S^1 is the multiplicative circle group of complex numbers of length one.

then show that C* is aLSO ISOMORPHIC TO THE PRODUCT R*xS^1 where R* is the multiplicative group of positive real numbers.

then find a homomorphism from RxR --->C* that has kernel equal to {0}xZ. [is that right?]

 

[mathwonk]

hey guys and gals, i want to brag on and advertise my colleague here at uga, Valery Alexeev, who will speak next week at the ICM, in madrid, international congress of matehmaticians. here is the schedule of talks in algebraic geometry for the first day of sectional talks.


Jaroslaw Wlodarczyk, Purdue University, West Lafayette, USA
Algebraic Morse theory and factorization of birational maps Abstract
Chair: Phillip Griffiths

Lawrence Ein, University of Illinois at Chicago, Chicago, USA and University of California at Irvine, Irvine, USA
Invariants of singularities of pairs Abstract
Chair: Phillip Griffiths

Valery Alexeev, University of Georgia, Athens, USA
Higher-dimensional analogues of stable curves Abstract
Chair: Phillip Griffiths

Tom Graber, University of California, Berkeley, USA
Rational curves and rational points Abstract
Chair: Óscar García-Prada

Tomohide Terasoma, University of Tokyo, Tokyo, Japan
Geometry of multiple zeta values Abstract
Chair: Óscar García-Prada

Jun-Muk Hwang, Korea Institute for Advanced Study, Seoul, Korea
Geometric structures arising from varieties of minimal rational tangents Abstract
Chair: Luc Illusie

Tom Bridgeland, University of Sheffield, Sheffield, UK
Derived categories of coherent sheaves Abstract
Chair: Luc Illusie

Yuri Tschinkel, Georg-August Universität Göttingen, Göttingen, Germany and Courant Institute, New York University, USA.
Geometry over nonclosed fields Abstract
Chair: Ignacio Sols

Jean-Benoît Bost, Université Paris-Sud, Orsay, France
Evaluation maps, slopes, and algebraization Abstract
Chair: Ignacio Sols

 

[ircdan]

Hey mathwonk,

I just read all the way up to the part where it says "Fundamental constructions (on abelian groups):". Anyways my point is, I have not formally studied algebra at school(I start a course in it in a few days), and I was able to understand it, so I think that's a good indication maybe that up to that point the notes are very good. I take it the prerequisites for this course are for students who have had 1-2 semesters of undergraduate algebra.