University of California Santa Cruz - Upper-Division (2024)

MATH100 Introduction to Proof and Problem Solving

Students learn the basic concepts and ideas necessary for upper-division mathematics and techniques of mathematical proof. Introduction to sets, relations, elementary mathematical logic, proof by contradiction, mathematical induction, and counting arguments.

MATH101 Mathematical Problem Solving

Students learn the strategies, tactics, skills and tools that mathematicians use when faced with a novel (new) problem. These include generalization, specialization, the optimization, invariance, symmetry, Dirichlet's box principle among others in the context of solving problems from number theory, geometry, calculus, combinatorics, probability, algebra, analysis, and graph theory. (Formerly, course 30.)

MATH103A Complex Analysis

Complex numbers, analytic and harmonic functions, complex integration, the Cauchy integral formula, Laurent series, singularities and residues, conformal mappings. (Formerly course 103.)

MATH103B Complex Analysis II

Conformal mappings, the Riemann mapping theorem, Mobius transformations, Fourier series, Fourier and Laplace transforms, applications, and other topics as time permits.

MATH105A Real Analysis

The basic concepts of one-variable calculus are treated rigorously. Set theory, the real number system, numerical sequences and series, continuity, differentiation.

MATH105B Real Analysis

Metric spaces, differentiation and integration of functions. The Riemann-Stieltjes integral. Sequences and series of functions.

MATH105C Real Analysis

The Stone-Weierstrass theorem, Fourier series, differentiation and integration of functions of several variables.

MATH106 Systems of Ordinary Differential Equations

Linear systems, exponentials of operators, existence and uniqueness, stability of equilibria, periodic attractors, and applications.

MATH107 Partial Differential Equations

Topics covered include first and second order linear partial differential equations, the heat equation, the wave equation, Laplace's equation, separation of variables, eigenvalue problems, Green's functions, Fourier series, special functions including Bessel and Legendre functions, distributions and transforms.

MATH110 Introduction to Number Theory

Prime numbers, unique factorization, congruences with applications (e.g., to magic squares). Rational and irrational numbers. Continued fractions. Introduction to Diophantine equations. An introduction to some of the ideas and outstanding problems of modern mathematics.

MATH111A Algebra

Group theory including the Sylow theorem, the structure of abelian groups, and permutation groups. Students cannot receive credit for this course and course 111T.

MATH111B Algebra

Introduction to rings and fields including polynomial rings, factorization, the classical geometric constructions, and Galois theory.

MATH111T Algebra

Introduction to groups, rings and fields; integers and polynomial rings; divisibility and factorization; hom*omorphisms and quotients; roots and permutation groups; and plane symmetry groups. Also includes an introduction to algebraic numbers, constructible numbers, and Galois theory. Focuses on topics most relevant to future K-12 teachers. Students cannot receive credit for this course and course 111A.

MATH114 Introduction to Financial Mathematics

Financial derivatives: contracts and options. Hedging and risk management. Arbitrage, interest rate, and discounted value. Geometric random walk and Brownian motion as models of risky assets. Ito's formula. Initial boundary value problems for the heat and related partial differential equations. Self-financing replicating portfolio; Black-Scholes pricing of European options. Dividends. Implied volatility. American options as free boundary problems.

MATH115 Graph Theory

Graph theory, trees, vertex and edge colorings, Hamilton cycles, Eulerian circuits, decompositions into isomorphic subgraphs, extremal problems, cages, Ramsey theory, Cayley's spanning tree formula, planar graphs, Euler's formula, crossing numbers, thickness, splitting numbers, magic graphs, graceful trees, rotations, and genus of graphs.

MATH116 Combinatorics

Based on induction and elementary counting techniques: counting subsets, partitions, and permutations; recurrence relations and generating functions; the principle of inclusion and exclusion; Polya enumeration; Ramsey theory or enumerative geometry.

MATH117 Advanced Linear Algebra

Review of abstract vector spaces. Dual spaces, bilinear forms, and the associated geometry. Normal forms of linear mappings. Introduction to tensor products and exterior algebras.

MATH118 Advanced Number Theory

Topics include divisibility and congruences, arithmetical functions, quadratic residues and quadratic reciprocity, quadratic forms and representations of numbers as sums of squares, Diophantine approximation and transcendence theory, quadratic fields. Additional topics as time permits.

MATH120 Coding Theory

An introduction to mathematical theory of coding. Construction and properties of various codes, such as cyclic, quadratic residue, linear, Hamming, and Golay codes; weight enumerators; connections with modern algebra and combinatorics.

MATH121A Differential Geometry

Topics include Euclidean space, tangent vectors, directional derivatives, curves and differential forms in space, mappings. Curves, the Frenet formulas, covariant derivatives, frame fields, the structural equations. The classification of space curves up to rigid motions. Vector fields and differentiable forms on surfaces; the shape operator. Gaussian and mean curvature. The theorem Egregium; global classification of surfaces in three space by curvature.

MATH121B Differential Geometry and Topology

Examples of surfaces of constant curvature, surfaces of revolutions, minimal surfaces. Abstract manifolds; integration theory; Riemannian manifolds. Total curvature and geodesics; the Euler characteristic, the Gauss-Bonnet theorem. Length-minimizing properties of geodesics, complete surfaces, curvature and conjugate points covering surfaces. Surfaces of constant curvature; the theorems of Bonnet and Hadamard.

MATH124 Introduction to Topology

Topics include introduction to point set topology (topological spaces, continuous maps, connectedness, compactness), hom*otopy relation, definition and calculation of fundamental groups and hom*ology groups, Euler characteristic, classification of orientable and nonorientable surfaces, degree of maps, and Lefschetz fixed-point theorem.

MATH128A Classical Geometry: Euclidean and Non-Euclidean

Euclidean, projective, spherical, and hyperbolic (non-Euclidean) geometries. Begins with the thirteen books of Euclid. Surveys the other geometries. Attention paid to constructions and visual intuition as well as logical foundations. Rigid motions and projective transformations covered.

MATH128B Classical Geometry: Projective

Theorems of Desargue, Pascal, and Pappus; projectivities; hom*ogeneous and affine coordinates; conics; relation to perspective drawing and some history.

MATH129 Algebraic Geometry

Algebraic geometry of affine and projective curves, including conics and elliptic curves; Bezout's theorem; coordinate rings and Hillbert's Nullstellensatz; affine and projective varieties; and regular and singular varieties. Other topics, such as blow-ups and algebraic surfaces as time permits.

MATH130 Celestial Mechanics

Solves the two-body (or Kepler) problem, then moves onto the N-body problem where there are many open problems. Includes central force laws; orbital elements; conservation of linear momentum, energy, and angular momentum; the Lagrange-Jacobi formula; Sundman's theorem for total collision; virial theorem; the three-body problem; Jacobi coordinates; solutions of Euler and of Lagrange; and restricted three-body problem.

MATH134 Cryptography

Introduces different methods in cryptography (shift cipher, affine cipher, Vigenere cipher, Hill cipher, RSA cipher, ElGamal cipher, knapsack cipher). The necessary material from number theory and probability theory is developed in the course. Common methods to attack ciphers discussed.

MATH140 Industrial Mathematics

Introduction to mathematical modeling of industrial problems. Problems in air quality remediation, image capture and reproduction, and crystallization are modeled as ordinary and partial differential equations then analyzed using a combination of qualitative and quantitative methods.

MATH145 Introductory Chaos Theory

The Lorenz and Rossler attractors, measures of chaos, attractor reconstruction, and applications from the sciences. Students cannot receive credit for this course and AM 114.

MATH145L Introductory Chaos Laboratory

Laboratory sequence illustrating topics covered in course 145. One three-hour session per week in microcomputer laboratory.

MATH148 Numerical Analysis

A survey of the basic numerical methods which are used to solve scientific problems, including mathematical analysis and computing assignments. Some prior experience with Matlab (or similar) is helpful but not required. Some typical topics are: computer arithmetic; Newton's method for non-linear equations; linear algebra; interpolation and approximation; numerical differentiation and integration; numerical solutions of systems of ordinary differential equations and some partial differential equations; convergence and error bounds.

MATH148L Numerical Analysis Laboratory

Laboratory sequence illustrating topics covered in course 148. One three-hour session per week in the computer laboratory.

MATH152 Programming for Mathematics

Introduces programming in Python with applications to advanced mathematics. Students apply data structures and algorithms to topics such as numerical approximation, number theory, linear algebra, and combinatorics. No programming experience is necessary, but a strong mathematics background is required.

MATH160 Mathematical Logic I

Propositional and predicate calculus. Resolution, completeness, compactness, and Lowenheim-Skolem theorem. Recursive functions, Godel incompleteness theorem. Undecidable theories. Hilbert's 10th problem.

MATH161 Mathematical Logic II

Naive set theory and its limitations (Russell's paradox); construction of numbers as sets; cardinal and ordinal numbers; cardinal and ordinal arithmetic; transfinite induction; axiom systems for set theory, with particular emphasis on the axiom of choice and the regularity axiom and their consequences (such as, the Banach-Tarski paradox); continuum hypothesis.

MATH181 History of Mathematics

A survey from a historical point of view of various developments in mathematics. Specific topics and periods to vary yearly.

MATH188 Supervised Teaching

Supervised tutoring in self-paced courses. May not be repeated for credit. Students submit petition to sponsoring agency.

MATH192 Dir Stu Teach

MATH194 Senior Seminar

Designed to expose the student to topics not normally covered in the standard courses. The format varies from year to year. In recent years each student has written a paper and presented a lecture on it to the class.

MATH195 Senior Thesis

Students research a mathematical topic under the guidance of a faculty sponsor and write a senior thesis demonstrating knowledge of the material. Prerequisite(s): satisfaction of the Entry Level Writing and Composition requirements. Students submit petition to sponsoring agency.

MATH199 Tutorial

Students submit petition to sponsoring agency.

MATH199F Tutorial

Tutorial