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  • Ma 110 abc: Analysis, I

    9.00 Credits
    California Institute of Technology

    First term: integration theory and basic real analysis: topological spaces, Hilbert space basics, Fejer’s theorem, measure theory, measures as functionals, product measures, L -spaces, Baire category, Hahn- Banach theorem, Alaoglu’s theorem, Krein-Millman theorem, countably normed spaces, tempered distributions and the Fourier transform. Second term: basic complex analysis: analytic functions, conformal maps and fractional linear transformations, idea of Riemann surfaces, elementary and some special functions, infinite sums and products, entire and meromorphic functions, elliptic functions. Third term: harmonic analysis; operator theory. Harmonic analysis: maximal functions and the Hardy-Littlewood maximal theorem, the maximal and Birkoff ergodic theorems, harmonic and subharmonic functions, theory of H -spaces and boundary values of analytic functions. Operator theory: compact operators, trace and determinant on a Hilbert space, orthogonal polynomials, the spectral theorem for bounded operators. If time allows, the theory of commutative Banach algebras. Instructors: Simon, Rains.
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  • Ma 111 a: Analysis, II

    9.00 Credits
    California Institute of Technology

    This course will discuss advanced topics in analysis, which vary from year to year. Topics from previous years include potential theory, bounded analytic functions in the unit disk, probabilistic and combinatorial methods in analysis, operator theory, C*-algebras, functional analysis. The third term will cover special functions: gamma functions, hypergeometric functions, beta/Selberg integrals and $q$-analogues. Time permitting: orthogonal polynomials, Painlev\’e transcendents and/or elliptic analogues Instructor: Chipeniuk. Prerequisite:    Ma 110 or instructor’s permission.
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  • Ma 112 ab: Statistics

    9.00 Credits
    California Institute of Technology

    The first term covers general methods of testing hypotheses and constructing confidence sets, including regression analysis, analysis of variance, and nonparametric methods. The second term covers permutation methods and the bootstrap, point estimation, Bayes methods, and multistage sampling. Instructor: Alberts.
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  • Ma 116 abc: Mathematical Logic and Axiomatic Set Theory

    9.00 Credits
    California Institute of Technology

    Propositional logic, predicate logic, formal proofs, Gödel completeness theorem, the method of resolution, elements of model theory. Computability, undecidability, Gödel incompleteness theorems. Axiomatic set theory, ordinals, transfinite induction and recursion, iterations and fixed points, cardinals, axiom of choice. Not offered 2012–13.
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  • Ma 117 abc: Computability Theory

    9.00 Credits
    California Institute of Technology

    Various approaches to computability theory, e.g., Turing machines, recursive functions, Markov algorithms; proof of their equivalence. Church’s thesis. Theory of computable functions and effectively enumerable sets. Decision problems. Undecidable problems: word problems for groups, solvability of Diophantine equations (Hilbert’s 10th problem). Relations with mathematical logic and the Gödel incompleteness theorems. Decidable problems, from number theory, algebra, combinatorics, and logic. Complexity of decision procedures. Inherently complex problems of exponential and superexponential difficulty. Feasible (polynomial time) computations. Polynomial deterministic vs. nondeterministic algorithms, NP-complete problems and the P = NP question. Instructors: Kechris, Sokic.
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  • Ma 118: Topics in Mathematical Logic: Geometrical Paradoxes

    9.00 Credits
    California Institute of Technology

    This course will provide an introduction to the striking paradoxes that challenge our geometrical intuition. Topics to be discussed include geometrical transformations, especially rigid motions; free groups; amenable groups; group actions; equidecomposability and invariant measures; Tarski’s theorem; the role of the axiom of choice; old and new paradoxes, including the Banach-Tarski paradox, the Laczkovich paradox (solving the Tarski circle-squaring problem), and the Dougherty-Foreman paradox (the solution of the Marczewski problem). Not offered 2012–13.
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  • Ma 120 abc: Abstract Algebra

    9.00 Credits
    California Institute of Technology

    Basic theory of groups, rings, modules, and fields, including free groups; Sylow’s theorem; solvable and nilpotent groups; factorization in commutative rings; integral extensions; Wedderburn theorems; Jacobson radical; semisimple, projective, and injective modules; tensor products; chain conditions; Galois theory; cyclotomic extensions; separability; transcendental extensions. Instructors: Tian, Hedayatazadeh, Flach.
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  • Ma 121 abc: Combinatorial Analysis

    9.00 Credits
    California Institute of Technology

    A survey of modern combinatorial mathematics, starting with an introduction to graph theory and extremal problems. Flows in networks with combinatorial applications. Counting, recursion, and generating functions. Theory of partitions. (0, 1)-matrices. Partially ordered sets. Latin squares, finite geometries, combinatorial designs, and codes. Algebraic graph theory, graph embedding, and coloring. Instructors: Wilson, Rains, Omar.
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  • Ma 122 abc: Topics in Group Theory

    9.00 Credits
    California Institute of Technology

    Topics to be decided by instructor. Not offered 2012–13.
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  • Ma 123: Classification of Simple Lie Algebras

    9.00 Credits
    California Institute of Technology

    This course is an introduction to Lie algebras and the classification of the simple Lie algebras over the complex numbers. This will include Lie’s theorem, Engel’s theorem, the solvable radical, and the Cartan Killing trace form. The classification of simple Lie algebras proceeds in terms of the associated reflection groups and a classification of them in terms of their Dynkin diagrams. Not offered 2012–13.
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