# Mathematics (MATH)

### Courses

**MATH 1A. Pre-Calculus. 4 Workload Units. **

Basic equations and inequalities, linear and quadratic functions, and systems of simultaneous equations.

**MATH 1B. Pre-Calculus. 4 Units. **

Preparation for calculus and other mathematics courses. Exponentials, logarithms, trigonometry, polynomials, and rational functions. Satisfies no requirements other than contribution to the 180 units required for graduation.

Prerequisite: MATH 1A. Placement into MATH 1B via the Calculus Placement exam, or a score of 450 or higher on the Mathematics section of the SAT Reasoning Test.

Restriction: MATH 1B may not be taken for credit if taken after MATH 2A.

**MATH 2A. Single-Variable Calculus. 4 Units. **

Introduction to derivatives, calculation of derivatives of algebraic and trigonometric functions; applications including curve sketching, related rates, and optimization. Exponential and logarithm functions.

Prerequisite: MATH 1B or AP Calculus AB or SAT Mathematics or ACT Mathematics. MATH 1B with a grade of C or better. AP Calculus AB with a minimum score of 3. SAT Mathematics with a minimum score of 650. ACT Mathematics with a minimum score of 29. Placement via the Calculus Placement exam (fee required) is also accepted.

Overlaps with MATH 5A.

Restriction: School of Physical Sciences students have first consideration for enrollment. School of Engineering students have first consideration for enrollment. School of Info & Computer Sci students have first consideration for enrollment.

**(Vb)**

**MATH 2B. Single-Variable Calculus. 4 Units. **

Definite integrals; the fundamental theorem of calculus. Applications of integration including finding areas and volumes. Techniques of integration. Infinite sequences and series.

Prerequisite: MATH 2A or MATH 5A or AP Calculus AB or AP Calculus BC. AP Calculus AB with a minimum score of 4. AP Calculus BC with a minimum score of 3

Restriction: School of Physical Sciences students have first consideration for enrollment. School of Engineering students have first consideration for enrollment. School of Info & Computer Sci students have first consideration for enrollment.

**(Vb)**

**MATH 2D. Multivariable Calculus. 4 Units. **

Differential and integral calculus of real-valued functions of several real variables, including applications. Polar coordinates.

Prerequisite: MATH 2B or MATH 5B or AP Calculus BC. AP Calculus BC with a minimum score of 4

Restriction: School of Physical Sciences students have first consideration for enrollment. School of Engineering students have first consideration for enrollment. School of Info & Computer Sci students have first consideration for enrollment. Undeclared Majors have first consideration for enrollment.

**(Vb)**

**MATH 2E. Multivariable Calculus. 4 Units. **

The differential and integral calculus of vector-valued functions. Implicit and inverse function theorems. Line and surface integrals, divergence and curl, theorems of Greens, Gauss, and Stokes.

Prerequisite: MATH 2D or MATH H2D

Restriction: School of Physical Sciences students have first consideration for enrollment. School of Engineering students have first consideration for enrollment.

**MATH H2D. Honors Multivariable Calculus. 4 Units. **

Differential and integral calculus of real-valued functions of several real variables, including applications. Polar coordinates. Covers the same material as MATH 2D-E, but with a greater emphasis on the theoretical structure of the subject matter.

Prerequisite: MATH 2B or MATH 5B or (AP Calculus BC and (MATH H3A) or (MATH 3A and MATH 13)). MATH 2B with a grade of A or better. MATH 5B with a grade of A or better. AP Calculus BC with a minimum score of 5. MATH H3A with a grade of B- or better. MATH 3A with a grade of A or better. MATH 13 with a grade of A or better

Overlaps with MATH 2D.

**(Vb)**

**MATH H2E. Honors Multivariable Calculus. 4 Units. **

Differential and integral calculus of real-valued functions of several real variables, including applications. Polar coordinates. Covers the same material as MATH 2D-E, but with a greater emphasis on the theoretical structure of the subject matter.

Prerequisite: MATH H2D. MATH H2D with a grade of B- or better

Overlaps with MATH 2E.

**MATH 3A. Introduction to Linear Algebra. 4 Units. **

Systems of linear equations, matrix operations, determinants, eigenvalues and eigenvectors, vector spaces, subspaces, and dimension.

Prerequisite: MATH 2B or MATH 5B or AP Calculus BC. AP Calculus BC with a minimum score of 4

Overlaps with MATH 6G, I&C SCI 6N.

Restriction: School of Physical Sciences students have first consideration for enrollment. School of Engineering students have first consideration for enrollment. Undeclared Majors have first consideration for enrollment.

**(Vb)**

**MATH 3D. Elementary Differential Equations. 4 Units. **

Linear differential equations, variation of parameters, constant coefficient cookbook, systems of equations, Laplace tranforms, series solutions.

Prerequisite: (MATH 3A or MATH H3A) and (MATH 2D or MATH H2D) and (MATH 2B or AP Calculus BC). AP Calculus BC with a minimum score of 4

Restriction: School of Physical Sciences students have first consideration for enrollment. School of Engineering students have first consideration for enrollment.

**MATH H3A. Honors Introduction to Linear Algebra. 4 Units. **

Systems of linear equations, matrix operations, determinants, eigenvalues, eigenvectors, vector spaces, subspaces, and dimension.

Prerequisite: MATH 2B or MATH 5B or AP Calculus BC. MATH 2B with a grade of A or better. MATH 5B with a grade of A or better. AP Calculus BC with a minimum score of 5

Overlaps with MATH 6G, MATH 3A, I&C SCI 6N.

Restriction: School of Physical Sciences students only. School of Engineering students only. Mathematics Majors only. Undeclared Majors only.

**MATH 4. Mathematics for Economists. 4 Units. **

**MATH 5A. Calculus for Life Sciences. 4 Units. **

Differential calculus with applications to life sciences. Exponential, logarithmic, and trigonometric functions. Limits, differentiation techniques, optimization and difference equations.

Prerequisite: MATH 1B or AP Calculus AB or SAT Mathematics or ACT Mathematics. MATH 1B with a grade of C or better. AP Calculus AB with a minimum score of 3. SAT Mathematics with a minimum score of 650. ACT Mathematics with a minimum score of 29. Placement via the Calculus Placement exam (fee required) is also accepted.

Overlaps with MATH 2A.

Restriction: School of Biological Sciences students have first consideration for enrollment.

**(Vb)**

**MATH 5B. Calculus for Life Sciences. 4 Units. **

Integral calculus and multivariable calculus with applications to life sciences. Integration techniques, applications of the integral, phase plane methods and basic modeling, basic multivariable methods.

Prerequisite: MATH 5A or MATH 2A or AP Calculus AB or AP Calculus BC. AP Calculus AB with a minimum score of 4. AP Calculus BC with a minimum score of 3

Restriction: School of Biological Sciences students have first consideration for enrollment. Cannot be taken for credit after MATH 2B.

**(Vb)**

**MATH 7A. Single-Variable Calculus I. 4 Units. **

Introduction to derivatives, calculation of derivatives of algebraic and trigonometric functions; applications including curve sketching, related rates, and optimization. Exponential and logarithm functions.

Prerequisite: MATH 1B or AP Calculus AB or SAT Mathematics or ACT Mathematics. MATH 1B with a grade of C or better. AP Calculus AB with a minimum score of 3. SAT Mathematics with a minimum score of 650. ACT Mathematics with a minimum score of 29. Placement via the Calculus Placement exam (fee required) is also accepted.

Overlaps with MATH 2A, MATH 5A.

Restriction: Mathematics Majors only.

**(Vb)**

**MATH 7B. Single-Variable Calculus II. 4 Units. **

Definite integrals; the fundamental theorem of calculus. Applications of integration including finding areas and volumes. Techniques of integration. Infinite sequences and series.

Prerequisite: MATH 2A or MATH 5A or AP Calculus AB or AP Calculus BC or MATH 7A. AP Calculus AB with a minimum score of 4. AP Calculus BC with a minimum score of 3

Overlaps with MATH 2B, MATH 5B.

Restriction: Mathematics Majors only.

**(Vb)**

**MATH 8. Explorations in Functions and Modeling. 4 Units. **

Explorations of applications and connections in topics in algebra, geometry, calculus, and statistics for future secondary math educators. Emphasis on nonstandard modeling problems.

Prerequisite or corequisite: MATH 2A or AP Calculus AB or AP Calculus BC. AP Calculus AB with a minimum score of 4. AP Calculus BC with a minimum score of 3

**MATH 9. Introduction to Programming for Numerical Analysis. 4 Units. **

Introduction to computers and programming using Matlab and Mathematica. Representation of numbers and precision, input/output, functions, custom data types, testing/debugging, reading exceptions, plotting data, numerical differentiation, basics of algorithms. Analysis of random processes using computer simulations.

Prerequisite: MATH 2A

Restriction: Mathematics Majors have first consideration for enrollment.

**(II and Vb ).**

**MATH 10. Introduction to Programming for Data Science. 4 Units. **

Intro to algorithms in data science using Python and R. Basic concepts of Python, store, access and manipulate data in lists, functions, methods and packages, NumPy, Numerical stability and accuracy. Gradient descent and Newton’s method. Basic of R Programming.

Prerequisite or corequisite: (MATH 2D or MATH H2D) and (MATH 3A or MATH H3A) and MATH 9

Restriction: Mathematics Majors have first consideration for enrollment.

Concurrent with MATH 3A.

**(II and VB ).**

**MATH 13. Introduction to Abstract Mathematics. 4 Units. **

Introduction to formal definition and rigorous proof writing in mathematics. Topics include basic logic, set theory, equivalence relations, and various proof techniques such as direct, induction, contradiction, contrapositive, and exhaustion.

Prerequisite: MATH 2A or I&C SCI 6D

Restriction: Mathematics Majors have first consideration for enrollment.

**MATH 105A. Numerical Analysis. 4 Units. **

Introduction to the theory and practice of numerical computation. Floating point arithmetic, roundoff; solving transcendental equations; quadrature; linear systems, eigenvalues, power method.

Corequisite: MATH 105LA

Prerequisite: MATH 3A or MATH H3A. Familiarity with computer programming is required.

Overlaps with ENGRMAE 185.

**MATH 105B. Numerical Analysis. 4 Units. **

Introduction to the theory and practice of numerical computation. Lagrange interpolation, finite differences, splines, Padé approximations; Gaussian quadrature; Fourier series and transforms.

Corequisite: MATH 105LB

Prerequisite: MATH 105A

**MATH 105LA. Numerical Analysis Laboratory. 1 Unit. **

Provides practical experience to complement the theory developed in Mathematics 105A.

Corequisite: MATH 105A

**MATH 105LB. Numerical Analysis Laboratory. 1 Unit. **

Provides practical experience to complement the theory developed in Mathematics 105B.

Corequisite: MATH 105B

**MATH 107. Numerical Differential Equations. 4 Units. **

**MATH 107L. Numerical Differential Equations Laboratory. 1 Unit. **

Provides practical experience to complement the theory developed in Mathematics 107.

Corequisite: MATH 107

**MATH 110A. Optimization I. 4 Units. **

**MATH 110B. Optimization II. 4 Units. **

**MATH 112A. Introduction to Partial Differential Equations and Applications. 4 Units. **

Introduction to ordinary and partial differential equations and their applications in engineering and science. Basic methods for classical PDEs (potential, heat, and wave equations). Classification of PDEs, separation of variables and series expansions, special functions, eigenvalue problems.

**MATH 112B. Introduction to Partial Differential Equations and Applications. 4 Units. **

Introduction to partial differential equations and their applications in engineering and science. Basic methods for classical PDEs (potential, heat, and wave equations). Green functions and integral representations, method of characteristics.

Prerequisite: MATH 112A

**MATH 112C. Introduction to Partial Differential Equations and Applications. 4 Units. **

Nonhomogeneous problems and Green's functions, Sturm-Liouville theory, general Fourier expansions, applications of partial differential equations in different areas of science.

Prerequisite: MATH 112B

**MATH 113A. Mathematical Modeling in Biology. 4 Units. **

**MATH 113B. Mathematical Modeling in Biology. 4 Units. **

**MATH 115. Mathematical Modeling. 4 Units. **

**MATH 117. Dynamical Systems. 4 Units. **

**MATH 118. The Theory of Differential Equations. 4 Units. **

**MATH 120A. Introduction to Abstract Algebra: Groups. 4 Units. **

Axioms for group theory; permutation groups, matrix groups. Isomorphisms, homomorphisms, quotient groups. Advanced topics as time permits. Special emphasis on doing proofs.

Prerequisite: (MATH 3A or MATH H3A) and MATH 13. MATH 13 with a grade of C or better

Restriction: Mathematics Majors have first consideration for enrollment.

**MATH 120B. Introduction to Abstract Algebra: Rings and Fields. 4 Units. **

**MATH 120C. Introduction to Abstract Algebra: Galois Theory. 4 Units. **

Galois Theory: proof of the impossibility of certain ruler-and-compass constructions (squaring the circle, trisecting angles); nonexistence of analogues to the "quadratic formula" for polynomial equations of degree 5 or higher.

Prerequisite: MATH 120B

Restriction: Mathematics Majors have first consideration for enrollment.

**MATH H120A. Honors Introduction to Graduate Algebra I. 5 Units. **

Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, and symmetric operators. Introduction to groups, rings, and fields, including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials, and Galois groups.

Prerequisite: (MATH 3A or MATH H3A) and MATH 13 and (MATH 120A or MATH 121A). MATH 13 with a grade of A or better. MATH 120A with a grade of A or better. MATH 121A with a grade of A or better

Restriction: Mathematics Honors students only.

Concurrent with MATH 206A.

**MATH H120B. Honors Introduction to Graduate Algebra II. 5 Units. **

Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, and symmetric operators. Introduction to groups, rings, and fields, including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials, and Galois groups.

Prerequisite: MATH H120A

Restriction: Mathematics Honors students only.

Concurrent with MATH 206B.

**MATH H120C. Honors Introduction to Graduate Algebra III. 5 Units. **

Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, and symmetric operators. Introduction to groups, rings, and fields, including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials, and Galois groups.

Prerequisite: MATH H120B

Restriction: Mathematics Honors students only.

Concurrent with MATH 206C.

**MATH 121A. Linear Algebra. 4 Units. **

Introduction to modern abstract linear algebra. Special emphasis on students doing proofs. Vector spaces, linear independence, bases, dimension. Linear transformations and their matrix representations. Theory of determinants.

Prerequisite: (MATH 3A or MATH H3A) and MATH 13. MATH 13 with a grade of C or better

Restriction: Mathematics Majors have first consideration for enrollment.

**MATH 121B. Linear Algebra. 4 Units. **

Introduction to modern abstract linear algebra. Special emphasis on students doing proofs. Canonical forms; inner products; similarity of matrices.

Prerequisite: MATH 121A

Restriction: Mathematics Majors have first consideration for enrollment.

**MATH 130A. Probability and Stochastic Processes. 4 Units. **

Basic concepts of random variables, distributions, independence, correlations, moments, limit theorems, conditional probability, Markov chains, gambler's ruin, branching process, birth and death processes, numerical simulations in Matlab.

Prerequisite: (MATH 2A or AP Calculus BC or AP Calculus AB) and (MATH 2B or AP Calculus BC) and (MATH 3A or MATH H3A). AP Calculus BC with a minimum score of 3. AP Calculus AB with a minimum score of 4. AP Calculus BC with a minimum score of 4

Overlaps with MATH 131A, STATS 120A.

**MATH 130B. Probability and Stochastic Processes. 4 Units. **

Exponential distributions, Poisson processes, continuous time Markov chains, renewal theory, insurance ruin and claim problems, numerical simulations in Matlab.

Prerequisite: MATH 130A or MATH 131A or STATS 120A

**MATH 130C. Probability and Stochastic Processes. 4 Units. **

Martingales, Invariance Principle, Brownian motions and applications in option pricing, stationary processes and applications in Wiener filter, numerical simulations in Matlab.

Prerequisite: MATH 130B

**MATH 133A. Statistical Methods with Applications to Finance. 4 Units. **

Overview of probability, statistics, and financial concepts: distribution, point estimation, confidence interval, linear regression, hypothesis testing, principal component analysis, financial applications.

Prerequisite: MATH 130A or MATH 131A or STATS 120A

**MATH 133B. Statistical Methods with Applications to Finance. 4 Units. **

Overview of markets and options: asset modeling, Brownian motion, risk neutrality, option pricing, value at risk, MC simulations.

Prerequisite: MATH 133A

**MATH 133C. Statistical Methods with Applications to Finance. 4 Units. **

Overview of interest theory, time value of money, annuities/cash flows with payments that are not contingent, loans, sinking funds, bonds, general cash flow and portfolios, immunization, duration and convexity, swaps.

Prerequisite: MATH 133B

**MATH 140A. Elementary Analysis. 4 Units. **

Introduction to real analysis, including convergence of sequence, infinite series, differentiation and integration, and sequences of functions. Students are expected to do proofs.

Prerequisite: (MATH 2B or AP Calculus BC) and (MATH 2D or MATH H2D) and (MATH 3A or MATH H3A) and MATH 13. AP Calculus BC with a minimum score of 4. MATH 13 with a grade of C or better

Restriction: Mathematics Majors have first consideration for enrollment.

**MATH 140B. Elementary Analysis. 4 Units. **

Introduction to real analysis including convergence of sequences, infinite series, differentiation and integration, and sequences of functions. Students are expected to do proofs.

Prerequisite: MATH 140A. MATH 140A with a grade of C- or better

Restriction: Mathematics Majors have first consideration for enrollment.

**MATH 140C. Analysis in Several Variables . 4 Units. **

Rigorous treatment of multivariable differential calculus. Jacobians, Inverse and Implicit Function theorems.

Prerequisite: MATH 140B

**MATH H140A. Honors Introduction to Graduate Analysis I. 5 Units. **

Construction of the real number system, topology of the real line, concepts of continuity, differential and integral calculus, sequences and series of functions, equicontinuity, metric spaces, multivariable differential and integral calculus, implicit functions, curves and surfaces.

Prerequisite: (MATH 2E or MATH H2E) and (MATH 3A or MATH H3A) and MATH 13 and MATH 121A and MATH 140A and MATH 140B. MATH 2E with a grade of A or better. MATH H2E with a grade of A or better. MATH 13 with a grade of A or better. MATH 140A with a grade of A or better. MATH 140B with a grade of A or better

Restriction: Mathematics Honors students only.

Concurrent with MATH 205A.

**MATH H140B. Honors Introduction to Graduate Analysis II. 5 Units. **

Construction of the real number system, topology of the real line, concepts of continuity, differential and integral calculus, sequences and series of functions, equicontinuity, metric spaces, multivariable differential and integral calculus, implicit functions, curves and surfaces.

Prerequisite: MATH H140A

Restriction: Mathematics Honors students only.

Concurrent with MATH 205B.

**MATH H140C. Honors Introduction to Graduate Analysis III. 5 Units. **

Construction of the real number system; topology of the real line; concepts of continuity, differential, and integral calculus; sequences and series of functions, equicontinuity, metric spaces, multivariable differential, and integral calculus; implicit functions, curves and surfaces.

Prerequisite: MATH H140B

Restriction: Mathematics Honors students only.

Concurrent with MATH 205C.

**MATH 141. Introduction to Topology. 4 Units. **

The elements of naive set theory and the basic properties of metric spaces. Introduction to topological properties.

Prerequisite: MATH 140A

**MATH 147. Complex Analysis. 4 Units. **

Rigorous treatment of basic complex analysis: analytic functions, Cauchy integral theory and its consequences, power series, residue calculus, harmonic functions, conformal mapping. Students are expected to do proofs.

Prerequisite or corequisite: MATH 140A and MATH 140B

Restriction: MATH 114A may not be taken for credit after MATH 147.

**MATH 150. Introduction to Mathematical Logic. 4 Units. **

First order logic through the Completeness Theorem for predicate logic.

Prerequisite: MATH 13 or (I&C SCI 6B and I&C SCI 6D). MATH 13 with a grade of C- or better

Overlaps with LPS 105B, PHILOS 105B.

**MATH 161. Modern Geometry. 4 Units. **

Euclidean Geometry; Hilbert's Axioms; Absolute Geometry; Hyperbolic Geometry; the Poincare Models; and Geometric Transformations.

Prerequisite: MATH 13 or (I&C SCI 6B and I&C SCI 6D). MATH 13 with a grade of C- or better

Restriction: Mathematics Majors have first consideration for enrollment.

**MATH 162A. Introduction to Differential Geometry. 4 Units. **

**MATH 162B. Introduction to Differential Geometry. 4 Units. **

Applications of advanced calculus and linear algebra to the geometry of curves and surfaces in space.

Prerequisite: MATH 162A

**MATH 173A. Introduction to Cryptology. 4 Units. **

Introduction to some of the mathematics used in the making and breaking of codes, with applications to classical ciphers and public key systems. Includes topics from number theory, probability, and abstract algebra.

Prerequisite: (MATH 2B or AP Calculus BC) and (MATH 3A or MATH H3A) and MATH 13 or (I&C SCI 6B and I&C SCI 6D). AP Calculus BC with a minimum score of 4. MATH 13 with a grade of C or better

**MATH 173B. Introduction to Cryptology. 4 Units. **

Introduction to some of the mathematics used in the making and breaking of codes, with applications to classical ciphers and public key systems. The mathematics which is covered includes topics from number theory, probability, and abstract algebra.

Prerequisite: MATH 173A

**MATH 175. Combinatorics . 4 Units. **

Introduction to combinatorics including basic counting principles, permutations, combinations, binomial coefficients, inclusion-exclusion, derangements, ordinary and exponential generating functions, recurrence relations, Catalan numbers, Stirling numbers, and partition numbers.

Prerequisite: (MATH 2B or AP Calculus BC) and MATH 13. AP Calculus BC with a minimum score of 4. MATH 13 with a grade of C or better

**MATH 176. Mathematics of Finance. 4 Units. **

After reviewing tools from probability, statistics, and elementary differential and partial differential equations, concepts such as hedging, arbitrage, Puts, Calls, the design of portfolios, the derivation and solution of the Blac-Scholes, and other equations are discussed.

Prerequisite: MATH 3A or MATH H3A

Same as ECON 135.

Restriction: Business Economics Majors have first consideration for enrollment. Economics Majors have first consideration for enrollment. Quantitative Economics Majors have first consideration for enrollment. Mathematics Majors have first consideration for enrollment.

**MATH 180A. Number Theory. 4 Units. **

Introduction to number theory and applications. Divisibility, prime numbers, factorization. Arithmetic functions. Congruences. Quadratic residue. Diophantine equations. Introduction to cryptography.

Prerequisite: (MATH 3A or MATH H3A) and MATH 13. MATH 13 with a grade of C or better

Restriction: Mathematics Majors have first consideration for enrollment.

**MATH 180B. Number Theory. 4 Units. **

Introduction to number theory and applications. Analytic number theory, character sums, finite fields, discrete logarithm, computational complexity. Introduction to coding theory. Other topics as time permits.

Prerequisite: MATH 180A

Restriction: Mathematics Majors have first consideration for enrollment.

**MATH 184. History of Mathematics. 4 Units. **

Topics vary from year to year. Some possible topics: mathematics in ancient times; the development of modern analysis; the evolution of geometric ideas. Students will be assigned individual topics for term papers.

Corequisite: MATH 184L

Prerequisite: MATH 120A and MATH 140A

Restriction: Mathematics Majors have first consideration for enrollment.

**MATH 184L. History of Mathematics Lesson Lab. 1 Unit. **

**MATH 189. Special Topics in Mathematics. 4 Units. **

Offered from time to time, but not on a regular basis. Content and prerequisites vary with the instructor.

Prerequisite: Prerequisites vary.

Repeatability: Unlimited as topics vary.

**MATH 192. Studies in the Learning and Teaching of Secondary Mathematics. 2 Units. **

Focus is on historic and current mathematical concepts related to student learning and effective math pedagogy, with fieldwork in grades 6-14.

Grading Option: Pass/no pass only.

Repeatability: May be taken for credit 2 times.

**MATH 194. Problem Solving Seminar. 2 Units. **

Develops ability in analytical thinking and problem solving, using problems of the type found in the Mathematics Olympiad and the Putnam Mathematical Competition. Students taking the course in fall will prepare for and take the Putnam examination in December.

Grading Option: Pass/no pass only.

Repeatability: May be taken for credit 2 times.

**MATH 199A. Special Studies in Mathematics. 2-4 Units. **

Supervised reading. For outstanding undergraduate Mathematics majors in supervised but independent reading or research of mathematical topics.

Repeatability: Unlimited as topics vary.

**MATH 199B. Special Studies in Mathematics. 2-4 Units. **

Supervised reading. For outstanding undergraduate Mathematics majors in supervised but independent reading or research of mathematical topics.

Repeatability: Unlimited as topics vary.

**MATH 199C. Special Studies in Mathematics. 2-4 Units. **

Supervised reading. For outstanding undergraduate Mathematics majors in supervised but independent reading or research of mathematical topics.

Repeatability: Unlimited as topics vary.

**MATH 205A. Introduction to Graduate Analysis. 5 Units. **

Construction of the real number system, topology of the real line, concepts of continuity, differential and integral calculus, sequences and series of functions, equicontinuity, metric spaces, multivariable differential and integral calculus, implicit functions, curves and surfaces.

Prerequisite: (MATH 2E or MATH H2E) and (MATH 3A MATH H3A) and MATH 13. MATH 2E with a grade of A or better. MATH H2E with a grade of A or better. MATH 13 with a grade of C or better

Concurrent with MATH H140A.

**MATH 205B. Introduction to Graduate Analysis. 5 Units. **

Prerequisite: MATH 205A

Concurrent with MATH H140B.

**MATH 205C. Introduction to Graduate Analysis. 5 Units. **

Prerequisite: MATH 205B

Concurrent with MATH H140C.

**MATH 206A. Introduction to Graduate Algebra. 5 Units. **

Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, symmetric operators. Introduction to groups, rings, and fields including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials, and Galois groups.

Prerequisite: MATH 3A or MATH H3A

Concurrent with MATH H120A.

**MATH 206B. Introduction to Graduate Algebra. 5 Units. **

Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, symmetric operators. Introduction to groups, rings, and fields including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials, and Galois groups.

Prerequisite: MATH 206A

Concurrent with MATH H120B.

**MATH 206C. Introduction to Graduate Algebra. 5 Units. **

Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, symmetric operators. Introduction to groups, rings, and fields including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials, and Galois groups.

Prerequisite: MATH 206B

Concurrent with MATH H120C.

**MATH 210A. Real Analysis. 4 Units. **

Measure theory, Lebesgue integral, signed measures, Radon-Nikodym theorem, functions of bounded variation and absolutely continuous functions, classical Banach spaces, Lp spaces, integration on locally compact spaces and the Riesz-Markov theorem, measure and outer measure, product measure spaces.

Prerequisite: MATH 140C

**MATH 210B. Real Analysis. 4 Units. **

Measure theory, Lebesgue integral, signed measures, Radon-Nikodym theorem, functions of bounded variation and absolutely continuous functions, classical Banach spaces, Lp spaces, integration on locally compact spaces and the Riesz-Markov theorem, measure and outer measure, product measure spaces.

Prerequisite: MATH 210A

**MATH 210C. Real Analysis. 4 Units. **

Measure theory, Lebesgue integral, signed measures, Radon-Nikodym theorem, functions of bounded variation and absolutely continuous functions, classical Banach spaces, Lp spaces, integration on locally compact spaces and the Riesz-Markov theorem, measure and outer measure, product measure spaces.

Prerequisite: MATH 210B

**MATH 211C. Topics in Analysis . 4 Units. **

**MATH 218A. Introduction to Manifolds and Geometry. 4 Units. **

General topology and fundamental groups, covering space; Stokes theorem on manifolds, selected topics on abstract manifold theory.

Prerequisite: MATH 205C

**MATH 218B. Introduction to Manifolds and Geometry. 4 Units. **

General topology and fundamental groups, covering space; Stokes theorem on manifolds, selected topics on abstract manifold theory.

Prerequisite: MATH 218A

**MATH 218C. Introduction to Manifolds and Geometry. 4 Units. **

General topology and fundamental groups, covering space; Stokes theorem on manifolds, selected topics on abstract manifold theory.

Prerequisite: MATH 218B

**MATH 220A. Analytic Function Theory. 4 Units. **

Standard theorems about analytic functions. Harmonic functions. Normal families. Conformal mapping.

Prerequisite: MATH 140C

**MATH 220B. Analytic Function Theory. 4 Units. **

Standard theorems about analytic functions. Harmonic functions. Normal families. Conformal mapping.

Prerequisite: MATH 220A

**MATH 220C. Analytic Function Theory. 4 Units. **

Standard theorems about analytic functions. Harmonic functions. Normal families. Conformal mapping.

Prerequisite: MATH 220B

**MATH 222A. Several Complex Variables and Complex Geometry. 4 Units. **

**MATH 225A. Introduction to Numerical Analysis and Scientific Computing. 4 Units. **

Introduction to fundamentals of numerical analysis from an advanced viewpoint. Error analysis, approximation of functions, nonlinear equations.

Prerequisite: MATH 3D and (MATH 105A and MATH 105B) or (MATH 140A and MATH 140B) and MATH 121A and (MATH 112A or ENGRMAE 140)

Restriction: Graduate students only.

**MATH 225B. Introduction to Numerical Analysis and Scientific Computing. 4 Units. **

Introduction to fundamentals of numerical analysis from an advanced viewpoint. Numerical linear algebra, numerical solutions of differential equations; stability.

Prerequisite: MATH 225A

Restriction: Graduate students only.

**MATH 225C. Introduction to Numerical Analysis and Scientific Computing. 4 Units. **

Introduction to fundamentals of numerical analysis from an advanced viewpoint. Numerical linear algebra, numerical solutions of differential equations; stability.

Prerequisite: MATH 225B

Restriction: Graduate students only.

**MATH 226A. Computational Differential Equations. 4 Units. **

Finite difference and finite element methods. Quick treatment of functional and nonlinear analysis background: weak solution, Lp spaces, Sobolev spaces. Approximation theory. Fourier and Petrov-Galerkin methods; mesh generation. Elliptic, parabolic, hyperbolic cases in 226A-B-C, respectively.

Prerequisite: MATH 3D and (MATH 112A or ENGRMAE 140) and (MATH 140B or MATH 105B)

**MATH 226B. Computational Differential Equations. 4 Units. **

Finite difference and finite element methods. Quick treatment of functional and nonlinear analysis background: weak solution, Lp spaces, Sobolev spaces. Approximation theory. Fourier and Petrov-Galerkin methods; mesh generation. Elliptic, parabolic, hyperbolic cases in 226A-B-C, respectively.

Prerequisite: MATH 226A

**MATH 226C. Computational Differential Equations. 4 Units. **

Finite difference and finite element methods. Quick treatment of functional and nonlinear analysis background: weak solution, Lp spaces, Sobolev spaces. Approximation theory. Fourier and Petrov-Galerkin methods; mesh generation. Elliptic, parabolic, hyperbolic cases in 226A-B-C, respectively.

Prerequisite: MATH 226B

**MATH 227A. Mathematical and Computational Biology. 4 Units. **

Analytical and numerical methods for dynamical systems, temporal-spatial dynamics, steady state, stability, stochasticity. Application to life sciences: genetics, tissue growth and patterning, cancers, ion channels gating, signaling networks, morphogen gradients. Analytical methods.

Prerequisite: (MATH 2A or AP Calculus BC or AP Calculus AB) and (MATH 2B or AP Calculus BC or MATH 5B) and (MATH 3A or MATH H3A). AP Calculus BC with a minimum score of 3. AP Calculus AB with a minimum score of 4. AP Calculus BC with a minimum score of 4

**MATH 227B. Mathematical and Computational Biology. 4 Units. **

Analytical and numerical methods for dynamical systems, temporal-spatial dynamics, steady state, stability, stochasticity. Application to life sciences: genetics, tissue growth and patterning, cancers, ion channels gating, signaling networks, morphogen gradients. Numerical simulations.

Prerequisite: MATH 227A

**MATH 227C. Mathematical and Computational Biology . 4 Units. **

Analytical and numerical methods for dynamical systems, temporal-spatial dynamics, steady state, stability, stochasticity. Application to life sciences: genetics, tissue growth and patterning, cancers, ion channels gating, signaling networks, morphogen gradients. Probabilistic methods.

Prerequisite: MATH 227A

Same as COMPSCI 285.

**MATH 230A. Algebra. 4 Units. **

**MATH 230B. Algebra. 4 Units. **

Elements of the theories of groups, rings, fields, modules. Galois theory. Modules over principal ideal domains. Artinian, Noetherian, and semisimple rings and modules.

Prerequisite: MATH 230A

**MATH 230C. Algebra. 4 Units. **

Elements of the theories of groups, rings, fields, modules. Galois theory. Modules over principal ideal domains. Artinian, Noetherian, and semisimple rings and modules.

Prerequisite: MATH 230B

**MATH 232A. Algebraic Number Theory. 4 Units. **

Algebraic integers, prime ideals, class groups, Dirichlet unit theorem, localization, completion, Cebotarev density theorem, L-functions, Gauss sums, diophantine equations, zeta functions over finite fields. Introduction to class field theory.

Prerequisite: MATH 230C

**MATH 232B. Algebraic Number Theory. 4 Units. **

Algebraic integers, prime ideals, class groups, Dirichlet unit theorem, localization, completion, Cebotarev density theorem, L-functions, Gauss sums, diophantine equations, zeta functions over finite fields. Introduction to class field theory.

Prerequisite: MATH 232A

**MATH 232C. Algebraic Number Theory. 4 Units. **

Algebraic integers, prime ideals, class groups, Dirichlet unit theorem, localization, completion, Cebotarev density theorem, L-functions, Gauss sums, diophantine equations, zeta functions over finite fields. Introduction to class field theory.

Prerequisite: MATH 232B

**MATH 233A. Algebraic Geometry. 4 Units. **

Basic commutative algebra and classical algebraic geometry. Algebraic varieties, morphisms, rational maps, blow ups. Theory of schemes, sheaves, divisors, cohomology. Algebraic curves and surfaces, Riemann-Roch theorem, Jacobians, classification of curves and surfaces.

Prerequisite: MATH 230C

**MATH 233B. Algebraic Geometry. 4 Units. **

Basic commutative algebra and classical algebraic geometry. Algebraic varieties, morphisms, rational maps, blow ups. Theory of schemes, sheaves, divisors, cohomology. Algebraic curves and surfaces, Riemann-Roch theorem, Jacobians, classification of curves and surfaces.

Prerequisite: MATH 233A

**MATH 233C. Algebraic Geometry. 4 Units. **

Basic commutative algebra and classical algebraic geometry. Algebraic varieties, morphisms, rational maps, blow ups. Theory of schemes, sheaves, divisors, cohomology. Algebraic curves and surfaces, Riemann-Roch theorem, Jacobians, classification of curves and surfaces.

Prerequisite: MATH 233B

**MATH 235A. Mathematics of Cryptography. 4 Units. **

Mathematics of public key cryptography: encryption and signature schemes; RSA; factoring; primality testing; discrete log based cryptosystems, elliptic and hyperelliptic curve cryptography and additional topics as determined by the instructor.

Prerequisite: MATH 230C

**MATH 240A. Differential Geometry. 4 Units. **

Riemannian manifolds, connections, curvature and torsion. Submanifolds, mean curvature, Gauss curvature equation. Geodesics, minimal submanifolds, first and second fundamental forms, variational formulas. Comparison theorems and their geometric applications. Hodge theory applications to geometry and topology.

**MATH 240B. Differential Geometry. 4 Units. **

Riemannian manifolds, connections, curvature and torsion. Submanifolds, mean curvature, Gauss curvature equation. Geodesics, minimal submanifolds, first and second fundamental forms, variational formulas. Comparison theorems and their geometric applications. Hodge theory applications to geometry and topology.

Prerequisite: MATH 240A

**MATH 240C. Differential Geometry. 4 Units. **

Riemannian manifolds, connections, curvature and torsion. Submanifolds, mean curvature, Gauss curvature equation. Geodesics, minimal submanifolds, first and second fundamental forms, variational formulas. Comparison theorems and their geometric applications. Hodge theory applications to geometry and topology.

Prerequisite: MATH 240B

**MATH 245A. Topics in Differential Geometry. 4 Units. **

**MATH 245B. Topics in Differential Geometry. 4 Units. **

**MATH 245C. Topics in Differential Geometry. 4 Units. **

**MATH 250A. Algebraic Topology. 4 Units. **

Provides fundamental materials in algebraic topology: fundamental group and covering space, homology and cohomology theory, and homotopy group.

Prerequisite: MATH 230A

**MATH 250B. Algebraic Topology. 4 Units. **

Provides fundamental materials in algebraic topology: fundamental group and covering space, homology and cohomology theory, and homotopy group.

Prerequisite: MATH 250A

**MATH 250C. Algebraic Topology. 4 Units. **

Provides fundamental materials in algebraic topology: fundamental group and covering space, homology and cohomology theory, and homotopy group.

Prerequisite: MATH 250B

**MATH 260A. Functional Analysis. 4 Units. **

Normed linear spaces, Hilbert spaces, Banach spaces, Stone-Weierstrass Theorem, locally convex spaces, bounded operators on Banach and Hilbert spaces, the Gelfand-Neumark Theorem for commutative C*-algebras, the spectral theorem for bounded self-adjoint operators, unbounded operators on Hilbert spaces.

**MATH 260B. Functional Analysis. 4 Units. **

Normed linear spaces, Hilbert spaces, Banach spaces, Stone-Weierstrass Theorem, locally convex spaces, bounded operators on Banach and Hilbert spaces, the Gelfand-Neumark Theorem for commutative C*-algebras, the spectral theorem for bounded self-adjoint operators, unbounded operators on Hilbert spaces.

Prerequisite: MATH 260A

**MATH 260C. Functional Analysis. 4 Units. **

Normed linear spaces, Hilbert spaces, Banach spaces, Stone-Weierstrass Theorem, locally convex spaces, bounded operators on Banach and Hilbert spaces, the Gelfand-Neumark Theorem for commutative C*-algebras, the spectral theorem for bounded self-adjoint operators, unbounded operators on Hilbert spaces.

Prerequisite: MATH 260B

**MATH 270A. Probability. 4 Units. **

**MATH 270B. Probability. 4 Units. **

Probability spaces, distribution and characteristic functions. Strong limit theorems. Limit distributions for sums of independent random variables. Conditional expectation and martingale theory. Stochastic processes.

Prerequisite: MATH 270A

**MATH 270C. Probability. 4 Units. **

Probability spaces, distribution and characteristic functions. Strong limit theorems. Limit distributions for sums of independent random variables. Conditional expectation and martingale theory. Stochastic processes.

Prerequisite: MATH 270B

**MATH 271A. Stochastic Processes. 4 Units. **

**MATH 271B. Stochastic Processes. 4 Units. **

**MATH 271C. Stochastic Processes. 4 Units. **

**MATH 274. Topics in Probability. 4 Units. **

Selected topics, such as theory of stochastic processes, martingale theory, stochastic integrals, stochastic differential equations.

Prerequisite: MATH 270C

Repeatability: Unlimited as topics vary.

**MATH 277A. Topics in Mathematical Physics . 4 Units. **

Studies in selected areas of mathematical physics. Topics addressed vary each quarter.

Repeatability: May be repeated for credit unlimited times.

**MATH 277B. Topics in Mathematical Physics . 4 Units. **

Studies in selected areas of mathematical physics. Topics addressed vary each quarter.

Prerequisite: MATH 277A

Repeatability: May be repeated for credit unlimited times.

**MATH 277C. Topics in Mathematical Physics . 4 Units. **

Studies in selected areas of mathematical physics. Topics addressed vary each quarter.

Prerequisite: MATH 277B

Repeatability: May be repeated for credit unlimited times.

**MATH 280A. Mathematical Logic. 4 Units. **

Basic set theory; models, compactness, and completeness; basic model theory; Incompleteness and Gödel's Theorems; basic recursion theory; constructible sets.

**MATH 280B. Mathematical Logic. 4 Units. **

Basic set theory; models, compactness, and completeness; basic model theory; Incompleteness and Gödel's Theorems; basic recursion theory; constructible sets.

Prerequisite: MATH 280A

**MATH 280C. Mathematical Logic. 4 Units. **

Basic set theory; models, compactness, and completeness; basic model theory; Incompleteness and Gödel's Theorems; basic recursion theory; constructible sets.

Prerequisite: MATH 280B

**MATH 281A. Set Theory. 4 Units. **

Ordinals, cardinals, cardinal arithmetic, combinatorial set theory, models of set theory, Gödel's constructible universe, forcing, large cardinals, iterate forcing, inner model theory, fine structure.

Prerequisite: MATH 280C

**MATH 281B. Set Theory. 4 Units. **

Ordinals, cardinals, cardinal arithmetic, combinatorial set theory, models of set theory, Gödel's constructible universe, forcing, large cardinals, iterate forcing, inner model theory, fine structure.

Prerequisite: MATH 281A

**MATH 281C. Set Theory. 4 Units. **

Ordinals, cardinals, cardinal arithmetic, combinatorial set theory, models of set theory, Gödel's constructible universe, forcing, large cardinals, iterate forcing, inner model theory, fine structure.

Prerequisite: MATH 281B

**MATH 282A. Model Theory. 4 Units. **

Languages, structures, compactness and completeness. Model-theoretic constructions. Omitting types theorems. Morley's theorem. Ranks, forking. Model completeness. O-minimality. Applications to algebra.

Prerequisite: MATH 280C

**MATH 282B. Model Theory. 4 Units. **

Languages, structures, compactness and completeness. Model-theoretic constructions. Omitting types theorems. Morley's theorem. Ranks, forking. Model completeness. O-minimality. Applications to algebra.

Prerequisite: MATH 282A

**MATH 282C. Model Theory. 4 Units. **

Languages, structures, compactness and completeness. Model-theoretic constructions. Omitting types theorems. Morley's theorem. Ranks, forking. Model completeness. O-minimality. Applications to algebra.

Prerequisite: MATH 282B

**MATH 285A. Topics in Mathematical Logic. 4 Units. **

**MATH 285B. Topics in Mathematical Logic. 4 Units. **

**MATH 285C. Topics in Mathematical Logic. 4 Units. **

**MATH 290A. Methods in Applied Mathematics. 4 Units. **

Introduction to ODEs and dynamical systems: existence and uniqueness. Equilibria and periodic solutions. Bifurcation theory. Perturbation methods: approximate solution of differential equations. Multiple scales and WKB. Matched asymptotic. Calculus of variations: direct methods, Euler-Lagrange equation. Second variation and Legendre condition.

**MATH 290B. Methods in Applied Mathematics. 4 Units. **

Introduction to ODEs and dynamical systems: existence and uniqueness. Equilibria and periodic solutions. Bifurcation theory. Perturbation methods: approximate solution of differential equations. Multiple scales and WKB. Matched asymptotic. Calculus of variations: direct methods, Euler-Lagrange equation. Second variation and Legendre condition.

Prerequisite: MATH 290A

**MATH 290C. Methods in Applied Mathematics. 4 Units. **

Introduction to ODEs and dynamical systems: existence and uniqueness. Equilibria and periodic solutions. Bifurcation theory. Perturbation methods: approximate solution of differential equations. Multiple scales and WKB. Matched asymptotic. Calculus of variations: direct methods, Euler-Lagrange equation. Second variation and Legendre condition.

Prerequisite: MATH 290B

**MATH 295A. Partial Differential Equations. 4 Units. **

**MATH 295B. Partial Differential Equations. 4 Units. **

Theory and techniques for linear and nonlinear partial differential equations. Local and global theory of partial differential equations: analytic, geometric, and functional analytic methods.

Prerequisite: MATH 295A

**MATH 295C. Partial Differential Equations. 4 Units. **

Theory and techniques for linear and nonlinear partial differential equations. Local and global theory of partial differential equations: analytic, geometric, and functional analytic methods.

Prerequisite: MATH 295B

**MATH 296. Topics in Partial Differential Equations. 4 Units. **

**MATH 297. Mathematics Colloquium. 1 Unit. **

Weekly colloquia on topics of current interest in mathematics.

Grading Option: Satisfactory/unsatisfactory only.

Repeatability: May be repeated for credit unlimited times.

**MATH 298A. Seminar . 2 Units. **

Seminars organized for detailed discussion of research problems of current interest in the Department. The format, content, frequency, and course value are variable.

Grading Option: Satisfactory/unsatisfactory only.

Repeatability: Unlimited as topics vary.

**MATH 298B. Seminar . 2 Units. **

Seminars organized for detailed discussion of research problems of current interest in the Department. The format, content, frequency, and course value are variable.

Prerequisite: MATH 298A

Grading Option: Satisfactory/unsatisfactory only.

Repeatability: Unlimited as topics vary.

**MATH 298C. Seminar . 2 Units. **

Seminars organized for detailed discussion of research problems of current interest in the Department. The format, content, frequency, and course value are variable.

Prerequisite: MATH 298B

Grading Option: Satisfactory/unsatisfactory only.

Repeatability: Unlimited as topics vary.

**MATH 299A. Supervised Reading and Research. 1-12 Units. **

Supervised reading and research with Mathematics faculty.

Repeatability: May be repeated for credit unlimited times.

**MATH 299B. Supervised Reading and Research. 1-12 Units. **

Supervised reading and research with Mathematics faculty.

Prerequisite: MATH 299A

Repeatability: May be repeated for credit unlimited times.

**MATH 299C. Supervised Reading and Research. 1-12 Units. **

Supervised reading and research with Mathematics faculty.

Prerequisite: MATH 299B

Repeatability: May be repeated for credit unlimited times.

**MATH 399. University Teaching. 1-4 Units. **

Limited to Teaching Assistants.

Grading Option: Satisfactory/unsatisfactory only.

Repeatability: May be repeated for credit unlimited times.