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AKG

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## 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:

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.

**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.