Linear Algebra. Vector spaces, linear transformations, duality, diagonalization, primary and cyclic decomposition, Jordan canonical form, inner product spaces, orthogonal reduction of symmetric matrices, spectral theorem, bilinear forms, multilinear functions. A much more abstract course than MATH or Algebraic Structures. Permutation groups, matrix groups, groups of linear transformations, symmetry groups; finite abelian groups. Residue class rings, algebra of matrices, linear maps, and polynomials.
Real and complex numbers, rational functions, quadratic fields, finite fields. Topics in Mathematics. Topics may focus on matrix theory, analysis, algebra, geometry, or applied and computational mathematics.
Repeat rules: May be repeated for credit; may be repeated in the same term for different topics; 12 total credits. Nonlinear Dynamics. Interdisciplinary introduction to nonlinear dynamics and chaos. Fixed points, bifurcations, strange attractors, with applications to physics, biology, chemistry, finance.
Probability II. Foundations of probability. Basic classical theorems. Modes of probabilistic convergence. Central limit problem.
Generating functions, characteristic functions. Conditional probability and expectation. Enumerative Combinatorics.
Basic counting; partitions; recursions and generating functions; signed enumeration; counting with respect to symmetry, plane partitions, and tableaux. Combinatorial Structures. Graph theory, matchings, Ramsey theory, extremal set theory, network flows, lattices, Moebius inversion, q-analogs, combinatorial and projective geometries, codes, and designs. Introductory Analysis. Requires knowledge of advanced calculus. Elementary metric space topology, continuous functions, differentiation of vector-valued functions, implicit and inverse function theorems.
Topics from Weierstrass theorem, existence and uniqueness theorems for differential equations, series of functions. Complex Analysis. A rigorous treatment of complex integration, including the Cauchy theory. Elementary special functions, power series, local behavior of analytic functions. Qualitative Theory of Differential Equations. Requires knowledge of linear algebra. Existence and uniqueness theorems, linear and nonlinear systems, differential equations in the plane and on surfaces, Poincare-Bendixson theory, Lyapunov stability and structural stability, critical point analysis.
Scientific Computation I. Requires some programming experience and basic numerical analysis. Error in computation, solutions of nonlinear equations, interpolation, approximation of functions, Fourier methods, numerical integration and differentiation, introduction to numerical solution of ODEs, Gaussian elimination.
Scientific Computation II. Theory and practical issues arising in linear algebra problems derived from physical applications, e.
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Linear systems, linear least squares, eigenvalue problems, singular value decomposition. Methods of Applied Mathematics I. Requires an undergraduate course in differential equations. Methods of Applied Mathematics II. Modules, Linear Algebra, and Groups. Requires knowledge of linear algebra and algebraic structures. Modules over rings, canonical forms for linear operators and bilinear forms, multilinear algebra, groups and group actions.
Groups, Representations, and Fields. Internal structure of groups, Sylow theorems, generators and relations, group representations, fields, Galois theory, category theory. Geometry of Curves and Surfaces.
Topics include curves Frenet formulas, isoperimetric inequality, theorems of Crofton, Fenchel, Fary-Milnor; surfaces fundamental forms, Gaussian and mean curvature, special surfaces, geodesics, Gauss-Bonnet theorem. Requisites: Prerequisite, advanced calculus. Introductory Topology. Topological spaces, connectedness, separation axioms, product spaces, extension theorems.
Classification of surfaces, fundamental group, covering spaces. Topics In Mathematics. Permission of the department. Directed study of an advanced topic in mathematics.
Statistical Reasoning. Burkhart 2nd ed. Must have concurrent teaching assistant appointment in mathematics. Differentiation and integration of exponential, logarithmic, and inverse trigonometric functions; techniques of integration and applications; indeterminate forms; improper integrals; sequences and series of numbers; power series. More information about this seller Contact this seller 7.
Topics will vary. Honors Research in Mathematics. Readings in mathematics and the beginning of directed research on an honors thesis.
Honors Thesis in Mathematics. Completion of an honors thesis under the direction of a member of the faculty.
Required of all candidates for graduation with honors in mathematics. Introduction to Partial Differential Equations. Basic methods in partial differential equations. Topics may include: Cauchy-Kowalewski Theorem, Holmgren's Uniqueness Theorem, Laplace's equation, Maximum Principle, Dirichlet problem, harmonic functions, wave equation, heat equation.
Measure and Integration. Lebesgue and abstract measure and integration, convergence theorems, differentiation, Radon-Nikodym theorem, product measures, Fubini theorem, Lebesgue spaces, invariance under transformations, Haar measure and convolution. Introductory Functional Analysis. Hahn-Banach and separation theorems.
Normed and locally convex spaces, duals of spaces and maps, weak topologies; closed graph and open mapping theorems, uniform boundedness theorem, linear operators. Advanced Complex Analysis. Laurent series; Mittag-Leffler and Weierstrass Theorems; Riemann mapping theorem; Runge's theorem; additional topics chosen from: harmonic, elliptic, univalent, entire, meromorphic functions; Dirichlet problem; Riemann surfaces.
Several Complex Variables. Elementary theory, the Cousin problems, domains of holomorphy, Runge domains and polynomial approximation, local theory, complex analytic structures, coherent analytic sheaves and Stein manifolds, Cartan's theorems. Single, multistep methods for ODEs: stability regions, the root condition; stiff systems, backward difference formulas; two-point BVPs; stability theory; finite difference methods for linear advection diffusion equations.
Mathematical Modeling I. Nondimensionalization and identification of leading order physical effects with respect to relevant scales and phenomena; derivation of classical models of fluid mechanics lubrication, slender filament, thin films, Stokes flow ; derivation of weakly nonlinear envelope equations.