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Course Criteria
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3.00 Credits
PQ: MATH 25400 or CMSC 27700, or consent of instructor. Students are encouraged to take both CMSC 21500 and 27700. Programming knowledge not required. R. Soare, W. Sterner. Winter.
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3.00 Credits
PQ: CMSC 15300, or MATH 25000 or 25500. This course is a basic introduction to computability theory and formal languages. Topics include automata theory, regular languages, context-free languages, and Turing machines. Autumn.
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3.00 Credits
PQ: CMSC 27100, or MATH 25000 or 25500; and experience with mathematical proofs. Computability topics are discussed (e.g., the s-m-n theorem and the recursion theorem, resource-bounded computation). This course introduces complexity theory. Relationships between space and time, determinism and non-determinism, NP-completeness, and the P versus NP question are investigated. Spring.
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3.00 Credits
PQ: MATH 25000 or 25400, or CMSC 27100, or consent of instructor. Experience with mathematical proofs. Methods of enumeration, construction, and proof of existence of discrete structures are discussed in conjunction with the basic concepts of probability theory over a finite sample space. Enumeration techniques are applied to the calculation of probabilities, and, conversely, probabilistic arguments are used in the analysis of combinatorial structures. Other topics include basic counting, linear recurrences, generating functions, Latin squares, finite projective planes, graph theory, Ramsey theory, coloring graphs and set systems, random variables, independence, expected value, standard deviation, and Chebyshev's and Chernoff' s inequalities . L. Babai. Spring.
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3.00 Credits
PQ: One year of calculus, two quarters of physics at any level, and PHYS 25000 or prior programming experience. For course description, see Physics. Winter. L.
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3.00 Credits
PQ: Completion of general education mathematics sequence. Consent of instructor and departmental counselor. Open only to students who are majoring in mathematics. Students are required to submit the College Reading and Research Course Form. Must be taken for a quality grade. Autumn, Winter, Spring.
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3.00 Credits
PQ: MATH 25500 or consent of instructor. CMSC 38000 is concerned with recursive (computable) functions and sets generated by an algorithm (recursively enumerable sets). Topics include various mathematical models for computations (e.g., Turing machines and Kleene schemata, enumeration and s-m-n theorems, the recursion theorem, classification of unsolvable problems, priority methods for the construction of recursively enumerable sets and degrees). CMSC 38100 treats classification of sets by the degree of information they encode, algebraic structure and degrees of recursively enumerable sets, advanced priority methods, and generalized recursion theory. This course is taught in alternate years. R. Soare. Winter, Spring.
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3.00 Credits
PQ: Consent of instructor. Part one consists of models for defining computable functions: primitive recursive functions, (general) recursive functions, and Turing machines; the Church-Turing Thesis; unsolvable problems; diagonalization; and properties of computably enumerable sets. Part two deals with Kolmogorov (resource bounded) complexity: the quantity of information in individual objects. Part three covers functions computable with time and space bounds of the Turing machine: polynomial time computability, the classes P and NP, NP-complete problems, polynomial time hierarchy, and P-space complete problems. A. Razborov. Winter.
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3.00 Credits
PQ: MATH 25500 or 25800. MATH 30900 covers completeness and compactness; elimination of quantifiers; omission of types; elementary chains and homogeneous models; two cardinal theorems by Vaught, Chang, and Keisler; categories and functors; inverse systems of compact Hausdorf spaces; and applications of model theory to algebra. In MATH 31000, we study saturated models; categoricity in power; the Cantor-Bendixson and Morley derivatives; the Morley theorem and the Baldwin-Lachlan theorem on categoricity; rank in model theory; uniqueness of prime models and existence of saturated models; indiscernibles; ultraproducts; and differential fields of characteristic zero. This course is offered in alternate years. Not offered 2009 C10; will be offered 201 0 -11.
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3.00 Credits
PQ: MATH 26200, 27000, 27200, and 27400; and consent of director or co-director of undergraduate studies. Topics include Lebesgue measure, abstract measure theory, and Riesz representation theorem; basic functional analysis ( Lp- spaces, elementary Hilbert space theory, Hahn-Banach, open mapping theorem, and uniform boundedness); Radon-Nikodym theorem, duality for Lp- spaces, Fubini's theorem, differentiation, Fourier transforms, locally convex spaces, weak topologies, and convexity; compact operators; spectral theorem and integral operators; Banach algebras and general spectral theory; Sobolev spaces and embedding theorems; Haar measure; and Peter-Weyl theorem, holomorphic functions, Cauchy' s theorem, harmonic functions, maximum modulus principle, meromorphic functions, conformal mapping, and analytic continuation . Autumn, Winter, Spring.
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