Homework Helper

## Main Question or Discussion Point

I'm going into my fourth and final year of my undergraduate program in mathematics, and I'm looking for advice as to what courses to choose. Of the following, how would you rank them from being most essential to a good undergraduate education to least essential:

1. Differential Topology
Smooth manifolds, Sard’s theorem and transversality. Morse theory. Immersion and embedding theorems. Intersection theory. Borsuk-Ulam theorem. Vector fields and Euler characteristic. Hopf degree theorem. Additional topics may vary.

2. Algebraic Topology
Introduction to homology theory: singular and simplicial homology; homotopy invariance, long exact sequence, excision, Mayer-Vietoris sequence; applications. Homology of CW complexes; Euler characteristic; examples. Singular cohomology; products; cohomology ring. Topological manifolds; orientation; Poincare duality.

3. Introduction to Commutative Algebra and Algebraic Geometry
Basic notions of algebraic geometry, with emphasis on commutative algebra or geometry according to the interests of the instructor. Algebraic topics: localization, integral dependence and Hilbert’s Nullstellensatz, valuation theory, power series rings and completion, dimension theory. Geometric topics: affine and projective varieties, dimension and intersection theory, curves and surfaces, varieties over the complex numbers.

4. Differential Geometry
Riemannian metrics and connections. Geodesics. Exponential map. Complete manifolds. Hopf-Rinow theorem. Riemannian curvature. Ricci and scalar curvature. Tensors. Spaces of constant curvature. Isometric immersions. Second fundamental form. Topics from: Cut and conjugate loci. Variation energy. Cartan-Hadamard theorem. Vector bundles.

5. Ordinary Differential Equations II
Sturm-Liouville problem and oscillation theorems for second-order linear equations. Qualitative theory; integral invariants, limit cycles. Dynamical systems; invariant measures; bifurcations, chaos. Elements of the calculus of variations. Hamiltonian systems. Analytic theory; singular points and series solution. Laplace transform.

Note, to this point I've already taken:

Analysis I and II
Real Analysis I
Complex Analysis I
Intro to Topology (Point-Set topology only)
ODE I
Algebra I (Linear Algebra) and Algebra II (More lin. alg. with an intro to group theory)
Groups, Rings, and Fields
Intro to Differential Geometry (Geometry of curves and surfaces in 3-spaces. Curvature and geodesics. Minimal surfaces. Gauss-Bonnet theorem for surfaces. Surfaces of constant curvature.)
Mathematical Logic
Combinatorics
Intro to Number Theory

In my fourth year, I will be taking Real Analysis II, Complex Analysis II, Classical Geometries, and Set Theory. In addition, depending on how the scheduling and everything works out, I will take as many of the five courses in bold above as I can. This is why I'm asking which are most essential, so if I only have room to take 2 or 3 (or even just 1) then I will take the 2 or 3 most essential ones.

Thanks.

MathematicalPhysicist
Gold Member
after three years at uni, haven't you gained a like/dislike to particular courses?
i see youv'e taken intros to differential geometry and to topology, if you liked them perhaps you could take courses which are relevant to them, which are a continuation to them.

If you're unsure which courses you would like, ask yourself which professors are teaching which courses.

If there is a professor you particularly liked, or a particularly famous professor, you would want to consider those as you may appreciate the opportunity later. Plus, these are likely the professors who will write references and letters of recommendation for you...

matt grime
Homework Helper
Depends what you think is important in mathematics. I'd rate them most important to least important as

1. Alg Geom
2. Alg Top
3. Diff Geom
3. Diff Top
5. DEs

(note joints 3rd for the diff courses).

If you were an applied mathematician you'd probably have them in exactly the reverse order.

J77
matt grime said:
If you were an applied mathematician you'd probably have them in exactly the reverse order.
Yep Homework Helper
Thanks for the responses everyone.

loop quantum gravity, if I were to choose based on what I liked, I wouldn't take Complex Analysis II given that I hated the first installment, but I've been told by the people working in the math department that, especially since I want to go to graduate school, it's essential that I take it. That said, I'm taking set theory and classical geometries for interest, but I'm trying to balance interest with what's important.

jbusc, I can't choose based on professors since they haven't decided who will be teaching what yet.

matt grime, thanks a lot, that was what I needed and I was hoping to get your opinion. I'll probably be taking Alg Geom and Alg Top.

Homework Helper
Well my department just made some changes, and most of the courses I had to choose from were cancelled, whereas a couple which had previously been cancelled were resurrected. So I have a new dilemma:

1. Differential Topology
Smooth manifolds, Sard’s theorem and transversality. Morse theory. Immersion and embedding theorems. Intersection theory. Borsuk-Ulam theorem. Vector fields and Euler characteristic. Hopf degree theorem. Additional topics may vary.

OR

2. Algebraic Curves
Projective geometry. Curves and Riemann surfaces. Algebraic methods. Intersection of curves; linear systems; Bezout’s theorem. Cubics and elliptic curves. Riemann-Roch theorem. Newton polygon and Puiseux expansion; resolution of singularities.

Recommendations?

EDIT: The Differential Topology course (number 1 in this post as well as in the first post) is the only one of the original 5 that survived. But who knows, maybe they'll change their minds again and bring back the other courses. So ideally, I'd like to see how you guys would rank all 6 courses - the original 5 together with Algebraic Curves. Thanks again.

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mathwonk
Homework Helper
those all look fundamental to me. i myself know essentially all of those courses, and have taught all of them except certain topics on the ode course.

but sturm liouville problems were taught in my advanced calc course as a college junior, from dieudonne, and I taught them myself in a 2nd course on ode.

just to give you an idea of what people know who are mathematicians.

the ones i know best are alg geom, diff top, and alg top.

i mean how can you not know those things and survive in mathematics, think about it: how to apply algebra to topology, how to apply calculus to topology, how to apply algebra to geometry, how to apply calculus to geometry, and how to understand functions by knowing things about their derivatives. how can you not want and need to know all those things?

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mathwonk
Homework Helper
so take the one with the best professor.

mathwonk
Homework Helper
if guillemin and pollack is the text for diff top, i strongly recommend reading also milnors topology from the differentiable viewpoint.

mathwonk
Homework Helper
more detail: there are many ways to say which is most valuble. e.g. which subject possesses the most powerful far reaching tools and methods that touch the most other subjects, including number theory, physics, topology, etc.. that would probably be algebraic geometry, which matt listed first.

but one can also wonder how easy it is to understand a subject that big and powerful in isolation. i.e. many of the most fundamental tools of algebraic geometry evolved from more intuitive concepts in other subjecdts, like complex analysis and algebraic topology.

for instance the most basicinvariant of a smooth algebraic variety is the canonical line bundle, i.e. the determinant oif the dual bundle of its algebraic tangent bundle. obviously that came from differential geometry. and riemann's riemann rioch theorem for curves is based on integrals of differential forms over homolofy clases of loops, which combines complex analysis, integral calculus, and algebraic topology. the genus of a curve, now defiend as the dimension of some cohomology group, was originally defined by riemann by topological dissections of the surface.

it is not a accident that riemnn wrote papers on topology, complex analysis, diff geom, number theory, and physics.

so look at another criterion, namely which ideas are the most basic? there are many kinds of geometry and certainly the most fundamental, or simplest if you like, or crudest, is topology. thus alg top and diff top might be considered as more fundamental than alg geom and diff geom.

diff equations on the other hand is a bit technical, although unbelievably important and powerful, as spivak calls his section on pde in his diff geom book " a word from our sponsor"

so i might list the topics in order of fundamental nature

1) alg top
2) diff top
3) alg geom (which grows out of and strengthens both the previous two)
4) diff geom
5) diff eq.

but both diff top and diff geom need theroems from ode and pde, as does alg geom! one of the most fundamental breakthroughs in geometry of complex jacobian varieties due to andreotti and mayer in the 1960's was from applying the heat equation to the deformation theory of the theta divisor of a jacobian of a curve! thn welters gave apurely deformation theoretic interpretation of the heat equation in pure algebraic geometry. you have to know tht stuff before you can see how to use it a new way, needless to say.

mathwonk