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MATH 121: Linear Algebra and Applications
4.00 Credits
Harvard University
Real and complex vector spaces, dual spaces, linear transformations and Jordan normal forms. Inner product spaces. Applications to differential equations, classical mechanics, and optimization theory. Emphasizes learning to understand and write proofs.
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MATH 121 - Linear Algebra and Applications
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MATH 122: Algebra I: Theory of Groups and Vector Spaces
4.00 Credits
Harvard University
Groups and group actions, vector spaces and their linear transformations, bilinear forms and linear representations of finite groups.
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MATH 122 - Algebra I: Theory of Groups and Vector Spaces
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MATH 123: Algebra II: Theory of Rings and Fields
4.00 Credits
Harvard University
Rings and modules. Polynomial rings. Field extensions and the basic theorems of Galois theory. Structure theorems for modules.
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MATH 123 - Algebra II: Theory of Rings and Fields
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MATH 124: Number Theory
4.00 Credits
Harvard University
Factorization and the primes; congruences; quadratic residues and reciprocity; continued fractions and approximations; Pell's equation; selected Diophantine equations; theory of integral quadratic forms.
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MATH 124 - Number Theory
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MATH 129: Number Fields
4.00 Credits
Harvard University
Algebraic number theory: number fields, unique factorization of ideals, finiteness of class group, structure of unit group, Frobenius elements, local fields, ramification, weak approximation, adeles, and ideles.
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MATH 129 - Number Fields
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MATH 130: Classical Geometry
4.00 Credits
Harvard University
Presents axioms for several geometries (affine, projective, Euclidean, spherical, hyperbolic). Develops models for these geometries using three-dimensional vector spaces over the reals, or over finite fields. Emphasis on reading and writing proofs.
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MATH 130 - Classical Geometry
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MATH 131: Topology I: Topological Spaces and the Fundamental Group
4.00 Credits
Harvard University
Abstract topological spaces; compactness, connectedness, continuity. Homeomorphism and homotopy, fundamental groups, covering spaces. Introduction to combinatorial topology.
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MATH 131 - Topology I: Topological Spaces and the Fundamental Group
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MATH 132: Topology II: Smooth Manifolds
4.00 Credits
Harvard University
Differential manifolds, smooth maps and transversality. Winding numbers, vector fields, index and degree. Differential forms, Stokes' theorem, introduction to cohomology.
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MATH 136: Differential Geometry
4.00 Credits
Harvard University
The exterior differential calculus and its application to curves and surfaces in 3-space and to various notions of curvature. Introduction to Riemannian geometry in higher dimensions and to symplectic geometry.
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MATH 136 - Differential Geometry
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MATH 137: Algebraic Geometry
4.00 Credits
Harvard University
Affine and projective spaces, plane curves, Bezout's theorem, singularities and genus of a plane curve, Riemann-Roch theorem.
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MATH 137 - Algebraic Geometry
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