MATH COURSES TAKEN 1. graduate Course Title Instructor Grade
advertisement
MATH COURSES TAKEN AJRASMUS@MATH.UCSB.EDU 1. graduate Course Title MA236A, B Homological Algebra Instructor Birge HuisgenZimmermann Martin Scharlemann John Douglas Moore Grade TBD Text and Topics Text: TBD Topics: TBD TBD MA246A Partial Differential Equations Gustavo Ponce A MA209 Set Theory Daryl Cooper A MA227A Topics in Algebraic and Geometric Topology Darren Long Text: TBD Topics: TBD Text: Introduction to Smooth Manifolds — Lee Topics: Smooth manifolds, tangent and cotangent bundles, immersions and imbeddings, Lie groups, Whitney’s imbedding theorem, vector fields, Lie bracket, the exterior derivative, integration of differential forms, Stokes’s Theorem, De Rham cohomology, the Poincare Lemma Text: Introduction to Partial Differential Equations — Folland Topics: Solutions of general first order equations, Cauchy-Kowalevski theorem, Holmgren uniqueness theorem, harmonic functions, fundamental solutions, Green’s functions, solutions of the wave equation Text: Notes on Logic and Set Theory — Johnstone Topics: Ordinal and cardinal numbers, ZFC set theory, theory of computation, recursive functions, universal algebra, propositional calculus, first order logic Text: None Topics: Knot theory, representation theory MA232A Algebraic Topology MA240A Differential Geometry A A 1 MA596 Directed Reading and Research MA260P Representations of Surface Groups MA260A Topological Quantum Field Theories MA221C Differential Topology MA221B Homotopy Theory MA221A Foundations of Topology MA220C Field and Galois Theory Ken Goodearl Not graded Text: Quantum Groups — Kassel Topics: Tensor products, Hopf algebras, modules over a Hopf algebra, quantum M (2), Hopf ∗-algebras, Lie algebras, enveloping algebras, quantum sl(2), connections with the Jones-Conway polynomial Sam Ballas A Text: None Topics: Residually finite groups, subgroup separability, models of hyperbolic space, hyperbolic structures on manifolds, developing maps, holonomy, Teichmüller space, convex projective geometry Zhenghan Not graded Text: Mathematical Foundations of Topological Quantum Computation Wang — Wang Topics: Basics of topological quantum field theories, category theory Martin A+ Text: Differential Topology — Guillemin and Pollack Scharlemann Topics: Immersions and submersions, transversality, homotopy and stability, Sard’s theorem, Whitney embedding theorem, manifolds with boundary, oriented intersection number, Lefschetz fixed point theory Stephen A Text: Algebraic Topology — Hatcher Bigelow Topics: Cell complexes, Brouwer fixed point theorem, Borsuk-Ulam theorem, induced homomorphisms, van Kampen’s theorem, classification of covering spaces, deck transformations, Cayley complexes Ken Millett A Text: General Topology — Willard Topics: Basic point-set topology, separation axioms, regularity, countability properties, local compactness and connectedness, paracompactness, compactification, Urysohn’s lemma, Tychonoff’s theorem Birge A+ Text: Algebra — Grove HuisgenTopics: Algebraic extensions, splitting fields, Galois correspondence, funZimmermann damental theorem of Galois theory, normal and separable extensions, finite fields, cyclotomic extensions, solving polynomials by radicals 2 MA220B Ring and Module Theory Ken Goodearl A+ MA220A Group Theory Ken Goodearl A+ MA201A, B, C Real Analysis Gustavo Ponce A Text: Algebra — Grove Topics: Polynomial rings, rings of fractions, factorization in commutative domains, UFDs, PIDs, unique factorization in polynomial rings, modules, group representations, direct sums of modules, free modules, semisimple modules and rings Text: Algebra — Grove Topics: Sylow theorems, solvable groups, normal and subnormal series, fundamental theorem of finitely generated abelian groups, free groups, generators and relations Texts: Real Analysis — Shakarchi and Stein Functional Analysis — Shakarchi and Stein Introduction to Nonlinear Dispersive Equations — Gustavo Ponce Topics: Measure and integration, differentiation of the integral, Hilbert spaces, Lp spaces and Banach spaces, tempered distributions, Baire category theorem, Fourier transforms, Hilbert transforms 2. Undergraduate Course Title MA484 Honors Research in Topology MA434 Topics in Abstract Algebra Instructor Scott Taylor Grade A+ Text and Topics Text: None Topic: Research on my honors thesis Leo Livshits A+ MA313 Differential Geometry Andreas Mal- A+ mendier Text: Algebras of Linear Transformations — Farenick Topics: Theory of algebras, invariant subspaces for algebras of linear transformations, nilpotent algebras, structure of simple and semisimple algebras, free algebras, tensor products, representation theory Text: Elementary Differential Geometry — Pressley Topics: Curves, surfaces, first fundamental form, Gauss and Weingarten maps, parallel transport, the covariant derivative, Gaussian, mean, and principle curvatures, theorema egregium 3 MA332 Numerical Analysis Jan Holly A+ MA439 Topics in Real Analysis Fernando Gouvêa A+ MA320 Introduction to Topology (Budapest Semesters in Math) MA241 Introduction to Combinatorics (BSM) Ágnes Szilárd A+ MA220 Real Analysis (BSM) Szilárd Szabó A MA230 Number Theory (BSM) Csaba Szabó A+ MA311 Ordinary Differential Equations Jan Holly A+ Dezső Miklos A+ Text: Numerical Analysis — Burden and Faires Topics: Order of convergence, polynomial interpolation, numerical differentiation, Richardson’s extrapolation, numerical integration, solutions of linear systems, solutions of differential equations, least squares fitting Text: Naı̈ve Lie Theory — Stillwell Topics: Quaternions, isometries of Rn , classical Lie groups, maximal tori and centers of classical Lie groups, connectedness, discrete normal subgroups, matrix exponential and logarithm, tangent spaces and Lie algebras, simple Lie algebras, representations Text: Topology — Munkres Topics: Basic point-set topology, connectedness, compactness, the fundamental group, covering spaces, retractions and deformation retractions, classification of surfaces, connected sums Text: A Walk through Combinatorics — Bóna Topics: Basic counting rules, occupancy problems, generating functions, recurrence relations, inclusion and exclusion, graph theory, pigeonhole principle, Ramsey theory, extremal graph theory, combinatorial design Text: Elements of Real Analysis — Bartle Topics: Topology of metric spaces, sequences, uniform convergence, convergence tests for series, power series, differentiation in Rn , inverse and implicit function theorems, the Riemann-Stieltjes integral, Fourier series Text: Elementary Methods in Number Theory — Nathanson Topics: Euclidean algorithm, Fermat numbers, Wilson’s theorem, EulerFermat theorem, Chinese remainder theorem, order and primitive roots, quadratic residues, arithmetic functions, Diophantine equations Text: Differential Equations — Blanchard, Devaney, Hall Topics: Existence and uniqueness of solutions, separation of variables, integrating factors, bifurcation diagrams, linear systems, equilibrium point analysis for nonlinear systems, Laplace transforms, numerical methods 4 MA352 Complex Variables Andreas Mal- A mendier MA333 Abstract Algebra Otto Bretscher MA274 Introduction to Abstract Mathematical Thought Scott bert MA376 History of Mathematics Fernando Gouvêa A MA253 Linear Algebra Fernando Gouvêa A+ MA302 Vector Calculus Otto Bretscher A+ MA122 Series and Multivariable Calculus Scott Taylor A+ A+ Lam- A+ Text: Fundamentals of Complex Analysis — Saff and Snider Topics: Standard complex functions, stereographic projection, limits and continuity, holomorphic functions, branch cuts, complex integration, Cauchy’s integral theorem, Laurent series, residue theory Text: Abstract Algebra — Herstein Topics: Groups, rings, and fields, Lagrange’s theorem, homomorphism theorems, normal subgroups, quotient groups, symmetric groups, maximal ideals, polynomial rings, field extensions, compass and straightedge problems, Galois theory, solutions of equations by radicals Text: Chapter Zero: Fundamental Notions in Abstract Mathematics — Schumacher Topics: Logic, set theory, induction, equivalence relations, functions, binary operations, cardinality, constructions of the integers, rational numbers, and real numbers Text: Mathematics Emerging — Stedall Topics: Early geometry and algebra, development of calculus, solutions of equations, group theory, combinatorics, and probability Text: Linear Algebra with Applications — Bretscher Topics: Solutions of systems of linear equations, linear transformations, matrix representations, orthogonality, determinants, eigenvalues and eigenvectors Text: Vector Calculus — Colley Topics: Parametrized curves and surfaces, arc length, vector fields, line integrals, Green’s theorem, surface integrals, Stokes’s theorem, Gauss’s theorem Text: Calculus: Single and Multivariable — Hughes-Hallett, McCallum, et al. Topics: Taylor series, convergence of sequences and series, partial and directional derivatives, gradients, extrema of functions of several variables, multiple integration 5
