| source Harvard (X) |
level |
department Mathematics (X) |
A tour of symplectic and contact topology. Hamiltonian and Reeb dynamics. Symplectomorphism groups. Symplectic fibrations and Lefschetz pencils. Open books. Surgery fillings. Holomorphic curves, Floer homology, and symplectic field theory.
Score: 6.8114195 Details | Listing | Web page
Continuation of Mathematics 231a. Spectral sequences and techniques of computation. Vector bundles and characteristic classes. Bott periodicity. K-theory, cobordism and stable cohomotopy as examples of cohomology theories.
Score: 6.8114195 Details | Listing | Web page
Fundamentals of Riemann surfaces. Topics may include sheaves and cohomology, potential theory, uniformization, and moduli.
Score: 6.8114195 Details | Listing | Web page
A continuation of Mathematics 230a. Topics in global Riemannian geometry: Ricci curvature and volume comparison; sectional curvature and distance comparison; Toponogov's theorem and applications; sphere theorems; Gromov's betti number bounds; Gromov-Hausdorff convergence; Cheeger's finiteness theorem, and convergence theorems.
Score: 6.8114195 Details | Listing | Web page
Continuation of Mathematics 212ar. The spectral theorem for self-adjoint operators in Hilbert space. Applications to partial differential equations.
Score: 6.8114195 Details | Listing | Web page
Groups and group actions, vector spaces and their linear transformations, bilinear forms and linear representations of finite groups.
Score: 6.8114195 Details | Listing | Web page
Rings and modules. Polynomial rings. Field extensions and the basic theorems of Galois theory. Structure theorems for modules.
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Introduction to linear algebra, including vectors, matrices, and applications. Calculus of functions of several variables, including partial derivatives, constrained and unconstrained optimization, and applications. Covers the topics from Mathematics 21a,b which are most important in applications to economics, the social sciences, and some other fields.
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Introduction to linear algebra, including vectors, matrices, and applications. Calculus of functions of several variables, including partial derivatives, constrained and unconstrained optimization, and applications. Covers the topics from Mathematics 21a,b which are most important in applications to economics, the social sciences, and some other fields.
Score: 6.8114195 Details | Listing | Web page
Affine and projective spaces, plane curves, Bezout's theorem, singularities and genus of a plane curve, Riemann-Roch theorem.
Score: 6.8114195 Details | Listing | Web page
Continuation of Mathematics 223a. Topics: adeles, global class field theory, duality, cyclotomic fields. Other topics may include: Tate's thesis or Euler systems.
Score: 6.8114195 Details | Listing | Web page
A graduate introduction to algebraic number theory. Topics: the structure of ideal class groups, groups of units, a study of zeta functions and L-functions, local fields, Galois cohomology, local class field theory, and local duality.
Score: 6.8114195 Details | Listing | Web page
Covering spaces and fibrations. Simplicial and CW complexes, Homology and cohomology, universal coefficients and Kunneth formulas. Hurewicz theorem. Manifolds and Poincare duality.
Score: 6.8114195 Details | Listing | Web page
A literature course in advanced topics in algebraic topology. Material will be taken from research papers, both classical and contemporary, and the presentation will involve significant participation from the students.
Score: 6.8114195 Details | Listing | Web page
Analytic functions of one complex variable: power series expansions, contour integrals, Cauchy's theorem, Laurent series and the residue theorem. Some applications to real analysis, including the evaluation of indefinite integrals. An introduction to some special functions.
Score: 6.8114195 Details | Listing | Web page
Lebesgue measure and integration; general topology; introduction to L p spaces, Banach and Hilbert spaces, and duality.
Score: 6.8114195 Details | Listing | Web page
An introduction to the p-adic Langlands correspondence. Precise details to be decided on nearer to the time.
Score: 6.8114195 Details | Listing | Web page
Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it.
Score: 6.8114195 Details | Listing | Web page
Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it.
Score: 6.8114195 Details | Listing | Web page
A discussion of the chromatic picture of stable homotopy theory, beginning with Quillen's work on cohomology theories and formal group laws and culminating with the resolution of the Ravenel conjectures via the work of Devinatz-Hopkins-Smith.
Score: 6.8114195 Details | Listing | Web page
Euclidean, spherical and hyperbolic geometry. No prior experience with proofs required.
Score: 6.8114195 Details | Listing | Web page
Topics include enumerative and algebraic combinatorics related to representations of the symmetric group, symmetric functions, and Young tableaux.
Score: 6.8114195 Details | Listing | Web page
A first course in commutative algebra: Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Noether normalization, the Nullstellensatz, localization, primary decomposition, discrete valuation rings, filtrations, completions and dimension theory.
Score: 6.8114195 Details | Listing | Web page
A second course in complex analysis: elliptic functions, canonical products, conformal mapping, extremal length, harmonic measure and capacity.
Score: 6.8114195 Details | Listing | Web page
Develops the theory of convex sets, normed infinite-dimensional vector spaces, and convex funtionals and applies it as a unifying principle to a variety of optimization problems such as resource allocation, production planning, and optimal control. Topics include Hilbert space, dual spaces, the Hahn-Banach theorem, the Riesz representation theorem, calculus of variations, and Fenchel duality. Students will be expected to understand and invent proofs of theorems in real and functional analysis.
Score: 6.8114195 Details | Listing | Web page