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It will emphasize precise thinking and the presentation of mathematical results, both orally and in written form. The course is intended for students who are considering majoring in mathematics but wish additional training. Terms offered: Spring , Fall , Spring Topics to be covered and the method of instruction to be used will be announced at the beginning of each semester that such courses are offered.

See department bulletins. Designed for students who do not meet the prerequisites for Offered through the Student Learning Center. Summer: 6 weeks - 10 hours of workshop per week 8 weeks - 10 hours of workshop per week. Summer: 3 weeks - hours of directed group study per week 8 weeks - 1.

Final exam not required. Terms offered: Spring , Fall , Spring Berkeley Connect is a mentoring program, offered through various academic departments, that helps students build intellectual community. Over the course of a semester, enrolled students participate in regular small-group discussions facilitated by a graduate student mentor following a faculty-directed curriculum , meet with their graduate student mentor for one-on-one academic advising, attend lectures and panel discussions featuring department faculty and alumni, and go on field trips to campus resources.

Students are not required to be declared majors in order to participate. Terms offered: Spring , Spring , Fall Supervised independent study by academically superior, lower division students. A written proposal must be submitted to the department chair for pre-approval.

Prerequisites: Restricted to freshmen and sophomores only. Consent of instructor. Credit Restrictions: Enrollment is restricted; see the Introduction to Courses and Curricula section of this catalog. Terms offered: Fall , Fall , Spring Selected topics illustrating the application of mathematics to economic theory. This course is intended for upper-division students in Mathematics, Statistics, the Physical Sciences, and Engineering, and for economics majors with adequate mathematical preparation.

No economic background is required. Introduction to Mathematical Economics: Read Less [-]. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral. Terms offered: Fall , Fall , Fall Honors section corresponding to Recommended for students who enjoy mathematics and are good at it.

Greater emphasis on theory and challenging problems. Terms offered: Spring , Spring , Spring Differential calculus in Rn: the derivative as a linear map; the chain rule; inverse and implicit function theorems.

Lebesgue integration on the line; comparison of Lebesgue and Riemann integrals. Convergence theorems. Fourier series, L2 theory. Fubini's theorem, change of variable. Terms offered: Not yet offered A rigorous development of the basics of modern probability theory based on a self-contained treatment of measure theory.

The topics covered include: probability spaces; random variables; expectation; convergence of random variables and expectations; laws of large numbers; zero-one laws; convergence in distribution and the central limit theorem; Markov chains; random walks; the Poisson process; and discrete-parameter martingales. Terms offered: Spring , Fall , Summer 8 Week Session Matrices, vector spaces, linear transformations, inner products, determinants. QR factorization. Quadratic forms and Rayleigh's principle.

Jordan canonical form, applications. Linear functionals. Prerequisites: 54 or a course with equivalent linear algebra content. Terms offered: Fall , Fall , Fall Honors section corresponding to course for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems. The integers, congruences, and the Fundamental Theorem of Arithmetic.

Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions. Introduction to Abstract Algebra: Read Less [-]. Terms offered: Spring , Spring , Spring Honors section corresponding to Recommended for students who enjoy mathematics and are willing to work hard in order to understand the beauty of mathematics and its hidden patterns and structures.

Terms offered: Spring , Spring , Spring Further topics on groups, rings, and fields not covered in Math Possible topics include the Sylow Theorems and their applications to group theory; classical groups; abelian groups and modules over a principal ideal domain; algebraic field extensions; splitting fields and Galois theory; construction and classification of finite fields.

Terms offered: Spring , Fall , Summer 8 Week Session Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions, additive problems. Terms offered: Fall , Fall , Fall Construction and analysis of simple cryptosystems, public key cryptography, RSA, signature schemes, key distribution, hash functions, elliptic curves, and applications.

Terms offered: Spring , Spring , Spring Introduction to signal processing including Fourier analysis and wavelets. Theory, algorithms, and applications to one-dimensional signals and multidimensional images. Terms offered: Fall , Fall , Fall Intended for students in the physical sciences who are not planning to take more advanced mathematics courses.

Rapid review of series and partial differentiation, complex variables and analytic functions, integral transforms, calculus of variations. Terms offered: Spring , Spring , Spring Intended for students in the physical sciences who are not planning to take more advanced mathematics courses.

Special functions, series solutions of ordinary differential equations, partial differential equations arising in mathematical physics, probability theory.

Terms offered: Fall , Fall , Fall Existence and uniqueness of solutions, linear systems, regular singular points. Other topics selected from analytic systems, autonomous systems, Sturm-Liouville Theory. Terms offered: Spring , Spring , Spring An introduction to computer programming with a focus on the solution of mathematical and scientific problems. Basic programming concepts such as variables, statements, loops, branches, functions, data types, and object orientation.

Examples from a wide range of mathematical applications such as evaluation of complex algebraic expressions, number theory, combinatorics, statistical analysis, efficient algorithms, computational geometry, Fourier analysis, and optimization.

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Terms offered: Fall , Fall , Spring Introduction to mathematical and computational problems arising in the context of molecular biology. Theory and applications of combinatorics, probability, statistics, geometry, and topology to problems ranging from sequence determination to structure analysis. Terms offered: Spring , Fall , Spring Programming for numerical calculations, round-off error, approximation and interpolation, numerical quadrature, and solution of ordinary differential equations.

Practice on the computer. Terms offered: Spring , Spring , Spring Iterative solution of systems of nonlinear equations, evaluation of eigenvalues and eigenvectors of matrices, applications to simple partial differential equations.

Summer: 8 weeks - 4 hours of web-based lecture and 4 hours of web-based discussion per week. Final exam required, with common exam group. The Platonic solids and their symmetries.

Crystallographic groups. Projective geometry. Hyperbolic geometry. Terms offered: Fall , Fall , Fall Set-theoretical paradoxes and means of avoiding them. Sets, relations, functions, order and well-order. Proof by transfinite induction and definitions by transfinite recursion.

Cardinal and ordinal numbers and their arithmetic. Construction of the real numbers. Axiom of choice and its consequences. Introduction to the Theory of Sets: Read Less [-]. Terms offered: Spring , Spring , Spring Functions computable by algorithm, Turing machines, Church's thesis. Unsolvability of the halting problem, Rice's theorem. Recursively enumerable sets, creative sets, many-one reductions. Self-referential programs. Godel's incompleteness theorems, undecidability of validity, decidable and undecidable theories.

Incompleteness and Undecidability: Read Less [-]. Terms offered: Spring , Spring , Spring Frenet formulas, isoperimetric inequality, local theory of surfaces in Euclidean space, first and second fundamental forms.

Terms offered: Fall , Fall , Fall Manifolds in n-dimensional Euclidean space and smooth maps, Sard's Theorem, classification of compact one-manifolds, transversality and intersection modulo 2. Elementary Differential Topology: Read Less [-]. Terms offered: Fall , Fall , Fall The topology of one and two dimensional spaces: manifolds and triangulation, classification of surfaces, Euler characteristic, fundamental groups, plus further topics at the discretion of the instructor.

Terms offered: Spring , Spring , Spring Introduction to basic commutative algebra, algebraic geometry, and computational techniques. Main focus on curves, surfaces and Grassmannian varieties. Simplification and Factorisation. Arithmetic of Algebraic Fractions. Formulae and Transposition. Basic Concepts of Functions.

Graphs of Functions and Parametric Form. One-to-one and Inverse Functions. Characterising Functions. The Straight Line. The Circle. Some Common Functions. Solving Linear Equations. Solving Quadratic Equations. Solving Polynomial Equations. Solving Simultaneous Linear Equations. Solving Inequalities. Partial Fractions. Right-angled Triangles. Trigonometric Functions. Trigonometric Identities. Applications of Trigonometry to Triangles.

Applications of Trigonometry to Waves. The Modelling Cycle and Functions. Quadratic Functions and Modelling. Oscillating Functions and Modelling.

Inverse Square Law Modelling. The Exponential Function. The Hyperbolic Functions. The Logarithmic Function. Modelling Exercises. Log-linear Graphs. Introduction to Matrices. Matrix Multiplication. The Inverse of a Matrix. Solution by Cramer's Rule.

Solution by Inverse Matrix Method. Solution by Gauss Elimination. Basic Concepts of Vectors. Cartesian Components of Vectors. The Scalar Product. The Vector Product. Lines and Planes. Complex Arithmetic. Argand Diagrams and the Polar Form.

The Exponential Form of a Complex Number. De Moivre's Theorem. Introducing Differentiation. Using a Table of Derivatives.