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2 Topology, Measure, Probability. 6 3 Topology and Measure in Infinite- Dimensional Spaces. 13 Measure, Topology, Probability in Cosmology .. invariant measure on any linear space is often referred to as a Lebesgue.
Table of contents
Vector spaces and degrees of extensions. Adjoining roots of polynomials. Finite fields.
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Straight edge and compass constructions. Specific content determined by instructor. MATH Linear Optimization 3 NW Maximization and minimization of linear functions subject to constraints consisting of linear equations and inequalities; linear programming and mathematical modeling. Simplex method, elementary games and duality. MATH Nonlinear Optimization 3 NW Maximization and minimization of nonlinear functions, constrained and unconstrained; nonlinear programming problems and methods.
Lagrange multipliers; Kuhn-Tucker conditions, convexity. Quadratic programming. MATH Discrete Optimization 3 NW Maximization and minimization problems in graphs and networks shortest paths, minimum spanning trees, maximum flows, minimum cost flows ; transportation and trans-shipment problems, NP-completeness. MATH Introduction to Modern Algebra for Teachers 3 NW Basic concepts of abstract algebra with an emphasis on problem solving, constructing proofs, and communication of mathematical ideas. Cannot be used as elective credit for either BS program in mathematics.
Offered: WS. MATH History of Mathematics 3 NW Survey of the development of mathematics from its earliest beginnings through the first half of the twentieth century. Offered: S. Metric spaces: Euclidean space. Bolzano-Weierstrass property.
Sequences and limits of sequences. Cauchy sequences and completeness. The Heine-Borel Theorem. Uniform continuity. Connected sets and the intermediate value theorem. MATH Fundamental Concepts of Analysis 3 NW One-variable differential calculus: chain rule, inverse function theorem, Rolle's theorem, intermediate value theorem, Taylor's theorem, and intermediate value theorem for derivatives. Multivariable differential calculus: mean value theorem, inverse and implicit function theorems, and Lagrange multipliers. Construction of the Lebesgue integral and its basic properties.
Integration of series. Continuity and differentiability theorems for functions defined by integrals. Introduction to general measures and integration. MATH Complex Analysis 3 NW Complex numbers; analytic functions; sequences and series; complex integration; Cauchy integral formula; Taylor and Laurent series; uniform convergence; residue theory; conformal mapping. Topics chosen from: Fourier series and integrals, Laplace transforms, infinite products, complex dynamics; additional topics chose by instructor. MATH Differential Geometry 3 NW Examines curves in the plane and 3-spaces, surfaces in 3-space, tangent planes, first and second fundamental forms, curvature, the Gauss-Bonnet Theorem, and possible other selected topics.
ARMS, E. DEVINATZ Further examines curves in the plane and 3-spaces, surfaces in 3-space, tangent planes, first and second fundamental forms, curvature, the Gauss-Bonnet Theorem, and possible other selected topics. MATH Geometry for Teachers 3 NW Concepts of geometry from multiple approaches; discovery, formal and informal reasoning, transformations, coordinates, exploration using computers and models.
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Topics selected from Euclidean plane and space geometry, spherical geometry, non-Euclidean geometries, fractal geometry. Designed for teaching majors.
Offered: SpS. MATH Combinatorial Theory 3 NW Selected topics from among: block designs and finite geometries, coding theory, generating functions and other enumeration methods, graph theory, matroid theory, combinatorial algorithms, applications of combinatorics. MATH Numerical Analysis I 3 NW Basic principles of numerical analysis, classical interpolation and approximation formulas, finite differences and difference equations. Numerical methods in algebra, systems of linear equations, matrix inversion, successive approximations, iterative and relaxation methods. Numerical differentiation and integration.
Solution of differential equations and systems of such equations. Basic stochastic analysis tools, including stochastic integrals, stochastic differential equations, Ito's formula, theorems of Girsanov and Feynman-Kac, Black-Scholes option pricing, American and exotic options, bond options. Offered: jointly with STAT Cannot be repeated for credit. Designed for the improvement of teachers of mathematics. MATH Modern Algebra 5 First quarter of a three-quarter sequence covering group theory; field theory and Galois theory; commutative rings and modules, linear algebra, theory of forms; representation theory, associative rings and modules; commutative algebra and elementary algebraic geometry.
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Prerequisite: MATH or equivalent. Prerequisite: MATH MATH Algebraic Structures 3 First quarter of a three-quarter sequence covering homological algebra, advanced commutative algebra, and Lie algebras and representation theory. Specific topics include chain complexes, resolutions and derived functors, dimension theory, Cohen-Macaulay modules, Gorenstein rings, local cohomology, local duality, triangulated and derived categories, group cohomology, and structure and representation. MATH Algebraic Structures 3 Second quarter of a three-quarter sequence covering homological algebra, advanced commutative algebra, and Lie algebras and representation theory.
MATH Algebraic Structures 3 Third quarter of a three-quarter sequence covering homological algebra, advanced commutative algebra, and Lie algebras and representation theory. MATH Networks and Combinatorial Optimization 3 Mathematical foundations of combinatorial and network optimization with an emphasis on structure and algorithms with proofs. Topics include combinatorial and geometric methods for optimization of network flows, matching, traveling salesmen problem, cuts, and stable sets on graphs. Special emphasis on connections to linear and integer programming, duality theory, total unimodularity, and matroids.
MATH Optimization: Fundamentals and Applications 5 Maximization and minimization of functions of finitely many variables subject to constraints. Basic problem types and examples of applications; linear, convex, smooth, and nonsmooth programming. Optimality conditions. Saddlepoints and dual problems. Penalties, decomposition. Overview of computational approaches. Desirable: optimization, e. Math , and scientific programming experience in Matlab, Julia or Python.
MATH Numerical Optimization 3 Methods of solving optimization problems in finitely many variables, with or without constraints.
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Steepest descent, quasi-Newton methods. Quadratic programming and complementarity. Exact penalty methods, multiplier methods. Sequential quadratic programming. Cutting planes and nonsmooth optimization. Controllability, optimality, maximum principle. Relaxation and existence of solutions. Techniques of nonsmooth analysis. Fourier analysis of distributions, central limit problem and infinitely divisible laws, conditional expectations, martingales.
MATH Real Analysis 5 First quarter of a three-quarter sequence covering the theory of measure and integration, point set topology, Banach spaces, Lp spaces, applications to the theory of functions of one and several real variables. Additional topics to be chosen by instructor. Working knowledge of real variables, general topology, complex variables.
MATH Complex Analysis 5 First quarter of a three-quarter sequence covering complex numbers, analytic functions, contour integration, power series, analytic continuation, sequences of analytic functions, conformal mapping of simply connected regions, and related topics. MATH Topology and Geometry of Manifolds 5 First quarter of a three-quarter sequence covering general topology, the fundamental group, covering spaces, topological and differentiable manifolds, vector fields, flows, the Frobenius theorem, Lie groups, homogeneous spaces, tensor fields, differential forms, Stokes's theorem, deRham cohomology.
Additional topics to be chosen by the instructor, such as connections in vector bundles and principal bundles, symplectic geometry, Riemannian comparison theorems, symmetric spaces, complex manifolds, Hodge theory.