# Upper-Division

## MATH 100 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.

### Credits

5

#### Requirements

Prerequisite(s): satisfaction of the Entry Level Writing and Composition requirements; MATH 11A or MATH 19A or MATH 20A; and MATH 21 or AM 10 or AMS 10A.

MF

#### Quarter offered

Fall, Winter, Spring, Summer

## MATH 101 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 21 and MATH 100.

PR-E

Fall

## MATH 103A Complex Analysis

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

### Credits

5

#### Requirements

Prerequisite(s): MATH 23B; and either MATH 100 or CSE 101.

#### Quarter offered

Winter, Spring, Summer

## MATH 103B Complex Analysis II

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

### Credits

2

#### Requirements

Prerequisite(s): MATH 103A.

## MATH 105A Real Analysis

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

### Credits

5

#### Requirements

Prerequisite(s):MATH 22 or MATH 23B and either MATH 100 or CSE 101.

## MATH 105B Real Analysis

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

### Credits

5

#### Requirements

Prerequisite(s): MATH 105A.

Spring

## MATH 105C Real Analysis

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

### Credits

5

#### Requirements

Prerequisite(s): MATH 105B.

## MATH 106 Systems of Ordinary Differential Equations

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

### Credits

5

#### Requirements

Prerequisite(s): MATH 21 or AM 10; and either MATH 24 or AM 20; and either MATH 100 or CSE 101.

Winter, Summer

## MATH 107 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 21 or AM 10; and MATH 24 or AM 20; and either MATH 100 or CSE 101; MATH 106 is recommended as preparation.

Spring

## MATH 110 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 100 or CSE 101.

#### Quarter offered

Fall, Winter, Summer

## MATH 111A Algebra

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

### Credits

5

#### Requirements

Prerequisite(s): MATH 21 or AM 10 and either MATH 100 or CSE 101.

Fall, Winter

## MATH 111B Algebra

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

### Credits

5

#### Requirements

Prerequisite(s): MATH 111A.

Spring

## MATH 111T Algebra

Introduction to groups, rings and fields; integers and polynomial rings; divisibility and factorization; homomorphisms 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 MATH 111A.

### Credits

5

#### Requirements

Prerequisite(s): MATH 100.

Spring, Summer

## MATH 114 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.

### Credits

5

#### Requirements

Corequisite(s): STAT 131 or CSE 107.

## MATH 115 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 21 or AM 10 and either MATH 100 or CSE 101.

Fall

## MATH 116 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 100 or CSE 101. Enrollment is restricted to sophomores juniors, and seniors. Familiarity with basic group theory is recommended.

Spring

## MATH 117 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 21 or AM 10 and either MATH 100 or CSE 101.

#### Quarter offered

Fall, Spring, Summer

## MATH 118 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 110 or MATH 111A

## MATH 120 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 21.

## MATH 121A 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 21 and MATH 23B and either MATH 100 or CSE 101. MATH 105A strongly recommended.

Winter

## MATH 121B 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 121A.

## MATH 124 Introduction to Topology

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

### Credits

5

#### Requirements

Prerequisite(s): MATH 100; MATH 111A recommended.

Fall

## MATH 128A 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.

### Credits

5

#### Requirements

Prerequisite(s): either MATH 100 or CSE 101.

Spring

## MATH 128B Classical Geometry: Projective

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

### Credits

5

#### Requirements

Prerequisite(s): MATH 21.

## MATH 129 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 21 and MATH 100.

Winter

## MATH 130 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 19A and 19B; and MATH 23A or PHYS 5A or PHYS 6A; MATH 21 and MATH 24 strongly recommended.

## MATH 134 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 100 or CSE 101; MATH 110 is recommended as preparation.

Winter

## MATH 140 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.

### Credits

5

The Staff

#### Requirements

Prerequisite(s): MATH 24; and either MATH 100 or CSE 101; and MATH 105A.

## MATH 145 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 22 or MATH 23A; MATH 21; MATH 100 or CSE 101. Concurrent enrollment in MATH 145L is required.

## MATH 145L Introductory Chaos Laboratory

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

### Credits

1

#### Requirements

Concurrent enrollment in MATH 145 is required.

## MATH 148 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. Students cannot receive credit for this course and AM 147.

### Credits

5

#### Requirements

Prerequisite(s): MATH 22 or MATH 23A; and MATH 21 or AM 10; and MATH 24 or AM 20; and MATH 100 or CSE 101. Concurrent enrollment in MATH 148L is required.

Spring

## MATH 148L Numerical Analysis Laboratory

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

### Credits

1

#### Requirements

Concurrent enrollment in MATH 148 is required.

Spring

## MATH 152 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 100.

MF

Winter

## MATH 160 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 100 or CSE 101.

Spring

## MATH 161 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.

### Credits

5

#### Requirements

Prerequisite(s): MATH 100 or equivalent, or by permission of instructor.

## MATH 181 History of Mathematics

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

### Credits

5

#### Requirements

Prerequisite(s): MATH 19B or MATH 20B. MATH 100 is strongly recommended for preparation.

TA

Winter, Summer

5

## MATH 194 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.

### Credits

5

#### Requirements

Prerequisite(s): satisfaction of the Entry Level Writing and Composition requirements; MATH 103A or MATH 105A or MATH 110 or MATH 111A or MATH 111T or MATH 117. Enrollment priority is given to seniors; juniors may request permission from the undergraduate vice chair.

Winter, Spring

## MATH 195 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.

### Credits

5

Yes

#### Quarter offered

Fall, Winter, Spring

## MATH 199 Tutorial

Students submit petition to sponsoring agency.

### Credits

5

Yes

#### Quarter offered

Fall, Winter, Spring

Tutorial

2

Yes