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Course Criteria
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3.00 Credits
Introduction to the analytic theory of automorphic forms. Constructing L-functions with analytic continuation and functional equation, spectral theory of automorphic forms, Eisenstein and Poincare series, character sums, central values of L-functions and applications. Background in elementary number theory, e.g., 18.781, strongly recommended.
Prerequisite:
Prereq: 18.112
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3.00 Credits
An introduction to algebraic number theory. Dedekind domains, unique factorization of prime ideals. Number fields, splitting of primes, class group. Lattice methods, finiteness of the class number, Dirichlet's units theorem. Local fields, ramification, discriminants. Background in elementary number theory (e.g., 18.781) strongly recommended.
Prerequisite:
Prereq: 18.100; 18.702
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3.00 Credits
Topics vary from year to year.
Prerequisite:
Prereq: Permission of instructor
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3.00 Credits
Guided research in mathematics, employing the scientific method. Students confront puzzling and complex mathematical situations, through the acquisition of data by computer, pencil and paper, or physical experimentation, and attempt to explain them mathematically. Students choose three projects from a large collection of options. Each project results in a laboratory report subject to revision; oral presentation on one or two projects. Projects drawn from many areas, including dynamical systems, number theory, algebra, fluid mechanics, asymptotic analysis, knot theory, and probability. Enrollment limited.
Prerequisite:
Prereq: Two mathematics subjects numbered 18.100 or above
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3.00 Credits
Introduces topology, covering topics fundamental to modern analysis and geometry. Topological spaces and continuous functions, connectedness, compactness, separation axioms, and selected further topics such as function spaces, embedding theorems, dimension theory, or covering spaces and the fundamental group.
Prerequisite:
Prereq: 18.100 or permission of instructor
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3.00 Credits
Topics vary from year to year and include the fundamental group and covering spaces. Time permitting, also covers the relationship between these objects and the theory of knots. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.
Prerequisite:
Prereq: 18.901
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3.00 Credits
Singular homology, CW complexes, universal coefficient and Kunneth theorems, cohomology, cup products, Poincare duality.
Prerequisite:
Prereq: 18.701 or 18.703; 18.901
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3.00 Credits
Continues the introduction to Algebraic Topology from 18.905. Topics include basic homotopy theory, spectral sequences, characteristic classes, and cohomology operations.
Prerequisite:
Prereq: 18.905
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3.00 Credits
Study and discussion of important original papers in the various parts of algebraic topology. Open to all students who have taken 18.906 or the equivalent, not only prospective topologists.
Prerequisite:
Prereq: 18.906
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3.00 Credits
Content varies from year to year. Introduces new and significant developments in algebraic topology with the focus on homotopy theory and related areas.
Prerequisite:
Prereq: 18.906
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