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18 785: Analytic Number Theory
3.00 Credits
Massachusetts Institute of Technology
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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18 785 - Analytic Number Theory
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18 786: Algebraic Number Theory
3.00 Credits
Massachusetts Institute of Technology
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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18 786 - Algebraic Number Theory
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18 787: Topics in Number Theory
3.00 Credits
Massachusetts Institute of Technology
Topics vary from year to year.
Prerequisite:
Prereq: Permission of instructor
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18 787 - Topics in Number Theory
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18 821: Project Laboratory in Mathematics
3.00 Credits
Massachusetts Institute of Technology
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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18 821 - Project Laboratory in Mathematics
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18 901: Introduction to Topology
3.00 Credits
Massachusetts Institute of Technology
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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18 901 - Introduction to Topology
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18 904: Seminar in Topology
3.00 Credits
Massachusetts Institute of Technology
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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18 904 - Seminar in Topology
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18 905: Algebraic Topology I
3.00 Credits
Massachusetts Institute of Technology
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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18 905 - Algebraic Topology I
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18 906: Algebraic Topology II
3.00 Credits
Massachusetts Institute of Technology
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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18 906 - Algebraic Topology II
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18 915: Graduate Topology Seminar
3.00 Credits
Massachusetts Institute of Technology
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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18 915 - Graduate Topology Seminar
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18 917: Topics in Algebraic Topology
3.00 Credits
Massachusetts Institute of Technology
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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18 917 - Topics in Algebraic Topology
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