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18 745: Introduction to Lie Algebras
3.00 Credits
Massachusetts Institute of Technology
Structure of finite-dimensional Lie algebras. Theorems of Engel and Lie. Cartan subalgebras and regular elements. Trace form and Cartan's criterion. Chevalley's conjugacy theorem. Classification and construction of semisimple Lie algebras. Weyl group. Universal enveloping algebra and the Casimir operator. Weyl's complete reducibility theorem, Levi and Maltsev theorems. Verma modules. Classification of irreducible finite-dimensional representations of semisimple Lie algebras. Weyl's character and dimension formulas.
Prerequisite:
Prereq: 18.701 or 18.703
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18 745 - Introduction to Lie Algebras
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18 747: Infinite-dimensional Lie Algebras
3.00 Credits
Massachusetts Institute of Technology
Topics vary from year to year.
Prerequisite:
Prereq: 18.745
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18 747 - Infinite-dimensional Lie Algebras
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18 755: Introduction to Lie Groups
3.00 Credits
Massachusetts Institute of Technology
A general introduction to manifolds and Lie groups. The role of Lie groups in mathematics and physics. Exponential mapping. Correspondence with Lie algebras. Homogeneous spaces and transformation groups. Adjoint representation. Covering groups. Automorphism groups. Invariant differential forms and cohomology of Lie groups and homogeneous spaces. 18.101 recommended but not required.
Prerequisite:
Prereq: 18.100; 18.700 or 18.701
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18 755 - Introduction to Lie Groups
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18 757: Representations of Lie Groups
3.00 Credits
Massachusetts Institute of Technology
Covers representations of locally compact groups, with emphasis on compact groups and abelian groups. Includes Peter-Weyl theorem and Cartan-Weyl highest weight theory for compact Lie groups.
Prerequisite:
Prereq: 18.745 or 18.755
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18 757 - Representations of Lie Groups
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18 758: Representations of Lie Groups
3.00 Credits
Massachusetts Institute of Technology
Introduction to unitary representations of semi-simple Lie groups: compact groups and the Borel-Weil theorem; parabolic induction; Zuckerman construction; unipotent representations.
Prerequisite:
Prereq: 18.757
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18 758 - Representations of Lie Groups
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18 769: Topics in Lie Theory
3.00 Credits
Massachusetts Institute of Technology
Topics vary from year to year.
Prerequisite:
Prereq: Permission of instructor
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18 769 - Topics in Lie Theory
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18 781: Theory of Numbers
3.00 Credits
Massachusetts Institute of Technology
An elementary introduction to number theory with no algebraic prerequisites. Primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, partitions.
Prerequisite:
Prereq: None
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18 781 - Theory of Numbers
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18 782: Introduction to Arithmetic Geometry
3.00 Credits
Massachusetts Institute of Technology
Uses ideas from algebra to study rational points on plane conics and elliptic curves. Includes an introduction to p-adic numbers and a brief introduction to algebraic geometry.
Prerequisite:
Prereq: 18.702
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18 782 - Introduction to Arithmetic Geometry
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18 783: Elliptic Curves
4.00 Credits
Massachusetts Institute of Technology
Computationally focused introduction to elliptic curves, with applications to number theory and cryptography. Topics include point-counting, isogenies, pairings, and the theory of complex multiplication. Explores how the rich theory of elliptic curves facilitates the design of efficient algorithms that play a key role in practical applications, including integer factorization, primality proving, and, in particular, elliptic curve cryptography. Students should have a working knowledge of standard algebraic structures (groups, rings, and fields) and computational complexity (big-O notation).
Prerequisite:
Prereq: None. Coreq: 18.702, 18.703, or permission of instructor
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18 783 - Elliptic Curves
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18 784: Seminar in Number Theory
3.00 Credits
Massachusetts Institute of Technology
Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.
Prerequisite:
Prereq: 18.06, 18.100
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18 784 - Seminar in Number Theory
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