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Course Criteria
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3.00 Credits
Decision theory, estimation, confidence intervals, hypothesis testing. Introduces large sample theory. Asymptotic efficiency of estimates. Exponential families. Sequential analysis.
Prerequisite:
Prereq: Permission of instructor
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3.00 Credits
Students present and discuss the subject matter taken from current journals or books. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.
Prerequisite:
Prereq: 18.100; 18.06, 18.510, 18.511, 18.700, or 18.701
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3.00 Credits
Propositional and predicate logic. Zermelo-Frenkel set theory. Ordinals and cardinals. Axiom of choice and transfinite induction. Elementary model theory: completeness, compactness, and Lowenheim-Skolem theorems. Godel's incompleteness theorem.
Prerequisite:
Prereq: None
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3.00 Credits
Church's thesis and models of computation. Elementary computability theory: enumeration and recursion theorems, the halting problem, relative computability, Turing degrees, and basic priority constructions. Post's problem. Truth vs. provability, Godel's incompleteness theorem. Decidable and undecidable problems in number theory and other areas of mathematics.
Prerequisite:
Prereq: None
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3.00 Credits
More rigorous treatment of basic mathematical logic, Godel's theorems, and Zermelo-Fraenkel set theory. First-order logic. Models and satisfaction. Deduction and proof. Soundness and completeness. Compactness and its consequences. Quantifier elimination. Recursive sets and functions. Incompleteness and undecidability. Ordinals and cardinals. Set-theoretic formalization of mathematics.
Prerequisite:
Prereq: Permission of instructor
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3.00 Credits
Vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. More emphasis on theory and proofs than in 18.06.
Prerequisite:
Prereq: Calculus II (GIR)
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3.00 Credits
No course description available.
Prerequisite:
Prereq: 18.700, 18.100, or permission of instructor
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3.00 Credits
More extensive and theoretical than the 18.700-18.703 sequence. Experience with proofs necessary. First term: group theory, geometry, and linear algebra. Second term: group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, Galois theory.
Prerequisite:
Prereq: 18.701
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3.00 Credits
Focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics: group theory, emphasizing finite groups; ring theory, including ideals and unique factorization in polynomial and Euclidean rings; field theory, including properties and applications of finite fields. 18.700 and 18.703 together form a standard algebra sequence.
Prerequisite:
Prereq: Calculus II (GIR)
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3.00 Credits
Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Some experience with proofs required. Enrollment limited.
Prerequisite:
Prereq: 18.702
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