Applied Mathematics < University of California, Berkeley (2024)

Terms offered: Fall 2024, Fall 2023, Fall 2022
Metric spaces and general topological spaces. Compactness and connectedness. Characterization of compact metric spaces. Theorems of Tychonoff, Urysohn, Tietze. Complete spaces and the Baire category theorem. Function spaces; Arzela-Ascoli and Stone-Weierstrass theorems. Partitions of unity. Locally compact spaces; one-point compactification. Introduction to measure and integration. Sigma algebras of sets. Measures and outer measures. Lebesgue measure on the line and Rn. Construction of the integral. Dominated convergence theorem.
Introduction to Topology and Analysis: Read More [+]

Terms offered: Spring 2024, Spring 2023, Spring 2022
Measure and integration. Product measures and Fubini-type theorems. Signed measures; Hahn and Jordan decompositions. Radon-Nikodym theorem. Integration on the line and in Rn. Differentiation of the integral. Hausdorff measures. Fourier transform. Introduction to linear topological spaces, Banach spaces and Hilbert spaces. Banach-Steinhaus theorem; closed graph theorem. Hahn-Banach theorem. Duality; the dual of LP. Measures on locally compact spaces; the dual of C(X). Weak and weak-* topologies; Banach-Alaoglu theorem. Convexity and the Krein-Milman theorem. Additional topics chosen may include compact operators, spectral theory of compact operators, and applications to integral equations.
Introduction to Topology and Analysis: Read More [+]

Terms offered: Fall 2022, Fall 2016, Spring 2016
Rigorous theory of ordinary differential equations. Fundamental existence theorems for initial and boundary value problems, variational equilibria, periodic coefficients and Floquet Theory, Green's functions, eigenvalue problems, Sturm-Liouville theory, phase plane analysis, Poincare-Bendixon Theorem, bifurcation, chaos.
Ordinary Differential Equations: Read More [+]

Terms offered: Spring 2024, Spring 2023, Spring 2022
Normal families. Riemann Mapping Theorem. Picard's theorem and related theorems. Multiple-valued analytic functions and Riemann surfaces. Further topics selected by the instructor may include: harmonic functions, elliptic and algebraic functions, boundary behavior of analytic functions and HP spaces, the Riemann zeta functions, prime number theorem.
Theory of Functions of a Complex Variable: Read More [+]

Terms offered: Fall 2024, Fall 2023, Fall 2022
Spectrum of an operator. Analytic functional calculus. Compact operators. Hilbert-Schmidt operators. Spectral theorem for bounded self-adjoint and normal operators. Unbounded self-adjoint operators. Banach algebras. Commutative Gelfand-Naimark theorem. Selected additional topics such as Fredholm operators and Fredholm index, Calkin algebra, Toeplitz operators, semigroups of operators, interpolation spaces, group algebras.
Functional Analysis: Read More [+]

Terms offered: Spring 2023, Spring 2022, Spring 2021
Basic theory of C*-algebras. Positivity, spectrum, GNS construction. Group C*-algebras and connection with group representations. Additional topics, for example, C*-dynamical systems, K-theory.
C*-algebras: Read More [+]

Terms offered: Spring 2024, Spring 2017, Spring 2014
Basic theory of von Neumann algebras. Density theorems, topologies and normal maps, traces, comparison of projections, type classification, examples of factors. Additional topics, for example, Tomita Takasaki theory, subfactors, group actions, and noncommutative probability.
Von Neumann Algebras: Read More [+]

Terms offered: Fall 2023, Fall 2021, Fall 2019
Power series developments, domains of holomorphy, Hartogs' phenomenon, pseudo convexity and plurisubharmonicity. The remainder of the course may treat either sheaf cohom*ology and Stein manifolds, or the theory of analytic subvarieties and spaces.
Several Complex Variables: Read More [+]

Terms offered: Fall 2024, Spring 2024, Fall 2022
This is an introduction to abstract differential topology based on rigorous mathematical proofs. The topics include Smooth manifolds and maps, tangent and normal bundles. Sard's theorem and transversality, Whitney embedding theorem. differential forms, Stokes' theorem, Frobenius theorem. Basic degree theory. Flows, Lie derivative, Lie groups and algebras. Additional topics selected by instructor.
Differential Topology: Read More [+]

Terms offered: Fall 2024, Fall 2023, Fall 2022
Fundamental group and covering spaces, simplicial and singular hom*ology theory with applications, cohom*ology theory, duality theorem. hom*otopy theory, fibrations, relations between hom*otopy and hom*ology, obstruction theory, and topics from spectral sequences, cohom*ology operations, and characteristic classes. Sequence begins fall.
Algebraic Topology: Read More [+]

Terms offered: Spring 2024, Spring 2023, Spring 2022
Fundamental group and covering spaces, simplicial and singular hom*ology theory with applications, cohom*ology theory, duality theorem. hom*otopy theory, fibrations, relations between hom*otopy and hom*ology, obstruction theory, and topics from spectral sequences, cohom*ology operations, and characteristic classes. Sequence begins fall.
Algebraic Topology: Read More [+]

Terms offered: Fall 2024, Fall 2023, Fall 2022
The course is designed as a sequence with Statistics C205B/Mathematics C218B with the following combined syllabus. Measure theory concepts needed for probability. Expection, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations, martingales and martingale convergence theorems. Markov chains. Stationary processes. Brownian motion.
Probability Theory: Read More [+]

Terms offered: Spring 2024, Spring 2023, Spring 2022
The course is designed as a sequence with with Statistics C205A/Mathematics C218A with the following combined syllabus. Measure theory concepts needed for probability. Expection, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations, martingales and martingale convergence theorems. Markov chains. Stationary processes. Brownian motion.
Probability Theory: Read More [+]

Terms offered: Fall 2024, Fall 2023, Spring 2022
Diffeomorphisms and flows on manifolds. Ergodic theory. Stable manifolds, generic properties, structural stability. Additional topics selected by the instructor.
Dynamical Systems: Read More [+]

Terms offered: Spring 2012, Spring 2011, Spring 2010
Brownian motion, Langevin and Fokker-Planck equations, path integrals and Feynman diagrams, time series, an introduction to statistical mechanics, Monte Carlo methods, selected applications.
Introduction to Probabilistic Methods in Mathematics and the Sciences: Read More [+]

Terms offered: Fall 2024, Fall 2023, Spring 2022
Direct solution of linear systems, including large sparse systems: error bounds, iteration methods, least square approximation, eigenvalues and eigenvectors of matrices, nonlinear equations, and minimization of functions.
Advanced Matrix Computations: Read More [+]

Terms offered: Fall 2024, Fall 2023, Fall 2022
The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Laplace's equation, heat equation, wave equation, nonlinear first-order equations, conservation laws, Hamilton-Jacobi equations, Fourier transform, Sobolev spaces.
Partial Differential Equations: Read More [+]

Terms offered: Spring 2024, Spring 2023, Spring 2022
The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Second-order elliptic equations, parabolic and hyperbolic equations, calculus of variations methods, additional topics selected by instructor.
Partial Differential Equations: Read More [+]

Terms offered: Fall 2020, Fall 2016, Fall 2015, Fall 2014
The topics of this course change each semester, and multiple sections may be offered. Advanced topics in probability offered according to students demand and faculty availability.
Advanced Topics in Probability and Stochastic Process: Read More [+]

Terms offered: Spring 2024, Spring 2023, Spring 2022
The topics of this course change each semester, and multiple sections may be offered. Advanced topics in probability offered according to students demand and faculty availability.
Advanced Topics in Probability and Stochastic Processes: Read More [+]

Terms offered: Fall 2024, Fall 2023, Fall 2022
Introduction to the theory of distributions. Fourier and Laplace transforms. Partial differential equations. Green's function. Operator theory, with applications to eigenfunction expansions, perturbation theory and linear and non-linear waves. Sequence begins fall.
Mathematical Methods for the Physical Sciences: Read More [+]

Terms offered: Spring 2015, Spring 2014, Spring 2013
Introduction to the theory of distributions. Fourier and Laplace transforms. Partial differential equations. Green's function. Operator theory, with applications to eigenfunction expansions, perturbation theory and linear and non-linear waves. Sequence begins fall.
Mathematical Methods for the Physical Sciences: Read More [+]

Terms offered: Fall 2024, Fall 2023, Fall 2022
Metamathematics of predicate logic. Completeness and compactness theorems. Interpolation theorem, definability, theory of models. Metamathematics of number theory, recursive functions, applications to truth and provability. Undecidable theories. Sequence begins fall.
Metamathematics: Read More [+]

Terms offered: Spring 2024, Spring 2023, Spring 2022
Metamathematics of predicate logic. Completeness and compactness theorems. Interpolation theorem, definability, theory of models. Metamathematics of number theory, recursive functions, applications to truth and provability. Undecidable theories. Sequence begins fall.
Metamathematics: Read More [+]

Terms offered: Spring 2021, Fall 2015, Fall 2013
Recursive and recursively enumerable sets of natural numbers; characterizations, significance, and classification. Relativization, degrees of unsolvability. The recursion theorem. Constructive ordinals, the hyperarithmetical and analytical hierarchies. Recursive objects of higher type. Sequence begins fall.
Theory of Recursive Functions: Read More [+]

Terms offered: Fall 2024, Fall 2023, Fall 2022
Ordinary differential equations: Runge-Kutta and predictor-corrector methods; stability theory, Richardson extrapolation, stiff equations, boundary value problems. Partial differential equations: stability, accuracy and convergence, Von Neumann and CFL conditions, finite difference solutions of hyperbolic and parabolic equations. Finite differences and finite element solution of elliptic equations.
Numerical Solution of Differential Equations: Read More [+]

Terms offered: Spring 2024, Spring 2023, Spring 2022
Ordinary differential equations: Runge-Kutta and predictor-corrector methods; stability theory, Richardson extrapolation, stiff equations, boundary value problems. Partial differential equations: stability, accuracy and convergence, Von Neumann and CFL conditions, finite difference solutions of hyperbolic and parabolic equations. Finite differences and finite element solution of elliptic equations.
Numerical Solution of Differential Equations: Read More [+]

Terms offered: Spring 2019, Spring 2015, Spring 2013
Syntactical characterization of classes closed under algebraic operations. Ultraproducts and ultralimits, saturated models. Methods for establishing decidability and completeness. Model theory of various languages richer than first-order.
Theory of Models: Read More [+]

Terms offered: Fall 2024, Spring 2024, Fall 2018
Axiomatic foundations. Operations on sets and relations. Images and set functions. Ordering, well-ordering, and well-founded relations; general principles of induction and recursion. Ranks of sets, ordinals and their arithmetic. Set-theoretical equivalence, similarity of relations; definitions by abstraction. Arithmetic of cardinals. Axiom of choice, equivalent forms, and consequences. Sequence begins fall.
Theory of Sets: Read More [+]

Terms offered: Fall 2021, Fall 2014, Fall 2010
Various set theories: comparison of strength, transitive, and natural models, finite axiomatizability. Independence and consistency of axiom of choice, continuum hypothesis, etc. The measure problem and axioms of strong infinity.
Metamathematics of Set Theory: Read More [+]

Terms offered: Spring 2011, Fall 2008, Spring 2008
Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry.
Discrete Mathematics for the Life Sciences: Read More [+]

Terms offered: Spring 2013
Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry.
Discrete Mathematics for the Life Sciences: Read More [+]

Terms offered: Fall 2022, Fall 2021, Fall 2019
Riemannian metric and Levi-Civita connection, geodesics and completeness, curvature, first and second variations of arc length. Additional topics such as the theorems of Myers, Synge, and Cartan-Hadamard, the second fundamental form, convexity and rigidity of hypersurfaces in Euclidean space, hom*ogeneous manifolds, the Gauss-Bonnet theorem, and characteristic classes.
Riemannian Geometry: Read More [+]

Terms offered: Spring 2024, Spring 2023, Spring 2021
Riemann surfaces, divisors and line bundles on Riemann surfaces, sheaves and the Dolbeault theorem on Riemann surfaces, the classical Riemann-Roch theorem, theorem of Abel-Jacobi. Complex manifolds, Kahler metrics. Summary of Hodge theory, groups of line bundles, additional topics such as Kodaira's vanishing theorem, Lefschetz hyperplane theorem.
Complex Manifolds: Read More [+]

Terms offered: Fall 2024, Fall 2023, Fall 2021
Basic topics: symplectic linear algebra, symplectic manifolds, Darboux theorem, cotangent bundles, variational problems and Legendre transform, hamiltonian systems, Lagrangian submanifolds, Poisson brackets, symmetry groups and momentum mappings, coadjoint orbits, Kahler manifolds.
Symplectic Geometry: Read More [+]

Terms offered: Spring 2015, Spring 2014
A graduate seminar class in which a group of students will closely examine recent computational methods in high-throughput sequencing followed by directly examining interesting biological applications thereof.
Seq: Methods and Applications: Read More [+]

Terms offered: Fall 2017, Fall 2015, Spring 2014
Structures defined by operations and/or relations, and their hom*omorphisms. Classes of structures determined by identities. Constructions such as free objects, objects presented by generators and relations, ultraproducts, direct limits. Applications of general results to groups, rings, lattices, etc. Course may emphasize study of congruence- and subalgebra-lattices, or category-theory and adjoint functors, or other aspects.
General Theory of Algebraic Structures: Read More [+]

Terms offered: Fall 2024, Spring 2024, Spring 2023
(I) Enumeration, generating functions and exponential structures, (II) Posets and lattices, (III) Geometric combinatorics, (IV) Symmetric functions, Young tableaux, and connections with representation theory. Further study of applications of the core material and/or additional topics, chosen by instructor.
Algebraic Combinatorics: Read More [+]

Terms offered: Fall 2024, Fall 2023, Fall 2022
Group theory, including the Jordan-Holder theorem and the Sylow theorems. Basic theory of rings and their ideals. Unique factorization domains and principal ideal domains. Modules. Chain conditions. Fields, including fundamental theorem of Galois theory, theory of finite fields, and transcendence degree.
Groups, Rings, and Fields: Read More [+]

Terms offered: Spring 2024, Spring 2023, Spring 2022
Development of the main tools of commutative and hom*ological algebra applicable to algebraic geometry, number theory and combinatorics.

Commutative Algebra: Read More [+]

Terms offered: Fall 2021, Fall 2016, Spring 2013
Topics such as: Noetherian rings, rings with descending chain condition, theory of the radical, hom*ological methods.
Ring Theory: Read More [+]

Terms offered: Fall 2021, Fall 2020, Fall 2015
Structure of finite dimensional algebras, applications to representations of finite groups, the classical linear groups.
Representation Theory: Read More [+]

Terms offered: Spring 2023, Fall 2016, Fall 2014
Modules over a ring, hom*omorphisms and tensor products of modules, functors and derived functors, hom*ological dimension of rings and modules.
hom*ological Algebra: Read More [+]

Terms offered: Fall 2024, Fall 2023, Fall 2022
Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, P-adic analysis, and transcendental numbers. Sequence begins fall.
Number Theory: Read More [+]

Terms offered: Spring 2024, Spring 2023, Spring 2022
Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, P-adic analysis, and transcendental numbers. Sequence begins fall.
Number Theory: Read More [+]

Terms offered: Fall 2022, Spring 2019, Fall 2014
Elliptic curves. Algebraic curves, Riemann surfaces, and function fields. Singularities. Riemann-Roch theorem, Hurwitz's theorem, projective embeddings and the canonical curve. Zeta functions of curves over finite fields. Additional topics such as Jacobians or the Riemann hypothesis.
Algebraic Curves: Read More [+]

Terms offered: Fall 2024, Fall 2023, Fall 2022
Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohom*ology. Riemann-Roch theorem and selected applications. Sequence begins fall.
Algebraic Geometry: Read More [+]

Terms offered: Spring 2024, Spring 2023, Spring 2022
Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohom*ology. Riemann-Roch theorem and selected applications. Sequence begins fall.
Algebraic Geometry: Read More [+]

Terms offered: Spring 2021, Spring 2018, Spring 2014
Topics such as: generators and relations, infinite discrete groups, groups of Lie type, permutation groups, character theory, solvable groups, simple groups, transfer and cohom*ological methods.
Group Theory: Read More [+]

Terms offered: Fall 2024, Fall 2023, Fall 2021
Basic properties of Fourier series, convergence and summability, conjugate functions, Hardy spaces, boundary behavior of analytic and harmonic functions. Additional topics at the discretion of the instructor.
Harmonic Analysis: Read More [+]

Terms offered: Fall 2024, Fall 2023, Fall 2022
Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall.
Lie Groups: Read More [+]

Terms offered: Spring 2024, Spring 2023, Spring 2022
Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall.
Lie Groups: Read More [+]

Terms offered: Spring 2024, Fall 2023, Spring 2023
This course will give introductions to research-related topics in mathematics. The topics will vary from semester to semester.
Advanced Topics Course in Mathematics: Read More [+]

Terms offered: Fall 2024, Fall 2023, Spring 2019
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
Interdisciplinary Topics in Mathematics: Read More [+]

Terms offered: Spring 2022, Spring 2016, Spring 2014
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
Topics in Numerical Analysis: Read More [+]

Terms offered: Fall 2023, Spring 2023, Fall 2022
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
Topics in Algebra: Read More [+]

Terms offered: Spring 2024, Spring 2023, Fall 2021
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
Topics in Applied Mathematics: Read More [+]

Terms offered: Spring 2021, Fall 2017, Spring 2016
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
Topics in Topology: Read More [+]

Terms offered: Spring 2023, Fall 2022, Fall 2021
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
Topics in Differential Geometry: Read More [+]

Terms offered: Fall 2024, Spring 2024, Fall 2021
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
Topics in Analysis: Read More [+]

Terms offered: Fall 2024, Fall 2023, Spring 2023
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
Topics in Partial Differential Equations: Read More [+]

Terms offered: Spring 2017, Spring 2015, Fall 2014
Topics in foundations of mathematics, theory of numbers, numerical calculations, analysis, geometry, topology, algebra, and their applications, by means of lectures and informal conferences; work based largely on original memoirs.
Seminars: Read More [+]

Terms offered: Summer 2016 10 Week Session, Spring 2016, Fall 2015
Intended for candidates for the Ph.D. degree.
Individual Research: Read More [+]

Terms offered: Summer 2022 8 Week Session, Summer 2021 8 Week Session, Summer 2006 10 Week Session
Intended for candidates for the Ph.D. degree.
Individual Research: Read More [+]

Terms offered: Prior to 2007
This is an independent study course designed to provide structure for graduate students engaging in summer internship opportunities. Requires a paper exploring how the theoretical constructs learned in academic courses were applied during the internship.
General Academic Internship: Read More [+]

Terms offered: Fall 2018, Fall 2017, Fall 2016
Investigation of special problems under the direction of members of the department.
Reading Course for Graduate Students: Read More [+]

Terms offered: Fall 2018, Spring 2018, Fall 2017
May be taken for one unit by special permission of instructor. Tutoring at the Student Learning Center or for the Professional Development Program.
Undergraduate Mathematics Instruction: Read More [+]

Terms offered: Summer 2002 10 Week Session, Summer 2001 10 Week Session
Mandatory for all graduate student instructors teaching summer course for the first time in the Department. The course consists of practice teaching, alternatives to standard classroom methods, guided group and self-analysis, classroom visitations by senior faculty member.
Teaching Workshop: Read More [+]

Terms offered: Spring 2017, Spring 2016, Fall 2015
Meeting with supervising faculty and with discussion sections. Experience in teaching under the supervision of Mathematics faculty.
Professional Preparation: Supervised Teaching of Mathematics: Read More [+]

Terms offered: Summer 2006 10 Week Session, Fall 2005, Spring 2005
Individual study for the comprehensive or language requirements in consultation with the field adviser.
Individual Study for Master's Students: Read More [+]

Terms offered: Fall 2019, Fall 2018, Fall 2016
Individual study in consultation with the major field adviser intended to provide an opportunity for qualified students to prepare themselves for the various examinations required for candidates for the Ph.D. Course does not satisfy unit or residence requirements for doctoral degree.
Individual Study for Doctoral Students: Read More [+]