This course is designed to introduce a variety of mathematical topics stemming from the research of women mathematicians both past and present, from Hypatia to current professors. In discussing the work of these women, we will also discuss the gender issues that are associated with being a female mathematician. Course material will be covered in lecture, research, in-class visitors and activities. Course work will include research papers, a course project and problem sets related to the mathematician of discussion.

CORE REQUIREMENT MET: **MATH/SCI**

## Courses

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#### 104 - Women in Mathematics

#### 105 - Mathematics as a Liberal Art

Introduction to mathematical thinking. Investigation of mathematical patterns in counting, reasoning, motion and change, shape, symmetry, and position. *Not open to seniors. *

CORE REQUIREMENT MET:** MATH/SCI**

Calculus differs in some respects from the traditional courses offered at some secondary schools and most other colleges or universities. Occidental's program is based on scientific modeling, makes regular use of computers, and requires interpretation as well as computation. A variety of courses comprise this program, accommodating different levels of preparation. The core content is described below as Calculus 1 and 2. Actual courses suited to different levels of preparation are listed under each description.

**Calculus 1: Scientific Modeling and Differential Calculus**

Many mathematical models in the natural and social sciences take the form of systems of differential equations. This introduction to the calculus is organized around the construction and analysis of these models, focusing on the mathematical questions they raise. Models are drawn from biology, economics, and physics. The important elementary functions of analysis arise as solutions of these models in special cases.

The mathematical theme of the course is local linearity. Topics include the definition of the derivative, rules for computing derivatives, Euler's Method, Newton's Method, Taylor polynomials, error analysis, optimization, and an introduction to the differential calculus of functions of two variables.

**CALCULUS 2: S****cientific Modeling and Integral Calculus**

This course continues the study of the calculus through scientific modeling. While Calculus 1 is concerned with local changes in a function, Calculus 2 focuses on accumulated changes. Models solved by accumulation functions lead to the definition of the integral and the Fundamental Theorem of Calculus.

Additional topics include numerical and analytic techniques of integration, trigonometric functions and dynamical systems modeling periodic or quasiperiodic phenomena, local approximation of functions by Taylor polynomials and Taylor series, and approximation of periodic functions on an interval by trigonometric polynomials and Fourier series.

#### 108 - Unified Precalculus and Calculus 1-A

The first of a two course sequence enriching the material in Calculus 1 with additional study of elementary functions, algebra, trigonometry, graphing, and mathematical expression. *Weekly lab. Prerequisites: the Calculus Readiness Examination and less than four years of high school mathematics. *

#### 109 - Unified Precalculus and Calculus 1-B

Continuation of Mathematics 108. This course satisfies Calculus 1 prerequisites for subsequent courses. *Weekly lab. Prerequisite: Mathematics **108**. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 110 - Calculus 1

This course satisfies Calculus 1 prerequisites for subsequent courses. *Weekly lab. Prerequisites: the Calculus Readiness Examination and at least four years of high school mathematics. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 114 - Calculus 1 (Experienced)

This course satisfies Calculus 1 prerequisites for subsequent courses. *Weekly lab. Prerequisites: a year of prior calculus experience and either the Calculus Readiness Examination or an appropriate Advanced Placement Calculus score. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 120 - Calculus 2

This course satisfies Calculus 2 prerequisites for subsequent courses. We*ekly lab. Prerequisites: Mathematics **109** or **110** or **114**. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 128 - Calculus 2 (Advanced Placement)

A one-semester course focusing on infinite sequences and series, including power series and Taylor series. More advanced topics will be chosen by the professor, which may include techniques for solving differential equations, mathematical modeling, Fourier Series, the Laplace Transform, and the Fourier Transform. The course assumes mastery of the basic skills, particularly integration techniques and differentiation rules, from the successful completion of Advanced Placement (AB) calculus. This course satisfies Calculus 1 and Calculus 2 prerequisites for subsequent courses. Weekly lab. Prerequisite: permission of instructor or AP Calculus AB score of 4 or 5.

CORE REQUIREMENT MET: **MATH/SCI**

#### 146 - Statistics

Comprehensive study of measures of central tendency, variation, probability, the normal distribution, sampling, estimation, confidence intervals and hypothesis testing. Introduction to use of technology in statistics. Real-life problems are used to illustrate methods. *Weekly lab. Not open to students who have completed or are currently enrolled in Psychology **201**, Biology **368**, Mathematics **150**, or any Mathematics course above 200. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 150 - Statistical Data Analysis

An introductory course in statistics emphasizing modern techniques of data analysis. Exploratory data analysis and graphical methods; random variables, statistical distributions, and linear models; classical, robust, and nonparametric methods for estimation and hypothesis testing; introduction to modern multivariate methods. Students will make significant use of a computer application specifically designed for data exploration. The course is strongly recommended for students who are going to use graphical techniques and statistics for research in their fields. *Weekly lab. Prerequisite: a Calculus 1 course or permission of instructor. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 160 - Creative Problem-Solving

Formal and informal techniques for problem-solving, developed by working on an intriguing collection of puzzles and problems which go beyond those encountered in the usual curriculum. These include problems which can be posed in elementary mathematical or logical terms but which require strategy and ingenuity to solve. *Prerequisite: a genuine desire to solve problems! *

2 units

#### 186 - Network Models

This course treats network and graphical models arising especially in biological and cognitive sciences. Methods include networks, graphs, and matrices; probability, conditional probability, and Markov chains; discrete-time dynamics and recurrent neural networks; Bayesian statistical inference on graphical models; and optimization on graphs, including dynamic programming. In the computing laboratory component (a separately-scheduled 1.5 hour session), students will learn to use MATLAB to build and analyze models. Students will complete projects in each major area of the course. Calculus is not a pre-requisite. While open to all students, this course is intended as an alternative to calculus as a first course in college-level mathematics.

CORE REQUIREMENT MET: **MATH/SCI**

#### 195 - Directed Research in Mathematics

Intensive study in an area of mathematics or computer science of the student's choosing under the direct supervision of a member of the faculty. *Prerequisite: permission of the supervising instructor. May be repeated once for credit. *

1 unit

#### 201 - Mathematics, Education, and Access to Power

This seminar course is a writing-intensive Community Based Learning based course designed to expose students to the complicated ways that mathematics affects the community. The CBL component of this course involves tutoring and mathematics assistance at Franklin High School in nearby Highland Park. The seminar component involves meeting weekly with processing discussions and discussion of readings. Topics will include the teaching and learning of mathematics as well as the role of mathematics in individuals' lives and their community.* May be taken twice for credit. *

2 units

#### 210 - Discrete Mathematics

The language of sets and logic, including propositional and predicate calculus. Formal and informal proofs using truth tables, formal rules of inference and mathematical induction. Congruences and modular arithmetic. Elementary counting techniques. Discrete probability. Abstract relations including equivalence relations and orders. *Prerequisite: a Calculus 2 course. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 212 - Multivariable Calculus

Calculus of functions of several variables, parametric curves and surfaces, and vector fields in 2- and 3-space, with applications. Vectors, graphs, contour plots. Differentiation, with application to optimization. Lagrange multipliers. Multiple and iterated integrals, change of variable and the Jacobian. Line and surface integrals. Vector analysis, Green's, Gauss', and Stokes' Theorems. Applications to physics, economics, chemistry, and mathematics. *Prerequisite: a Calculus 2 course. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 214 - Linear Algebra

Introduction to basic concepts of linear algebra through the study of linear algebraic systems and matrix algebra. Topics include an introduction to planes and lines, vector operations, subspaces, Gaussian elimination, linear independence, invertibility of matrices, orthogonality, matrix operations, eigenvalues and eigenspaces. The course stresses definitions and mathematical proof.. *Prerequisite: a Calculus 2 course. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 295 - Topics in Mathematics

Topics in mathematics, selected largely by student interest and faculty agreement. *Prerequisite: a Calculus 2 course or permission of instructor. *

2 or 4 units

#### 300 - Junior Colloquium

Preparation for the comprehensive examination and senior project. Completion of Third Year Writing Requirement. Emphases on problem-solving, clear written expression and verbal presentation. *Open to junior mathematics majors. *

2 units

#### 310 - Real Analysis

A beginning course in advanced calculus and real analysis. Properties of the real number system, sequences and series of real numbers, the Heine-Borel and Bolzano-Weierstrass Theorems, continuity and uniform continuity, sequences and series of functions. *Prerequisite: Mathematics **210**. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 312 - Complex Analysis

The differential and integral calculus of complex-valued functions of a complex variable, emphasizing geometry and applications. The complex number system, analytic functions and the Cauchy-Riemann equations, elementary functions and conformal mappings, contour integration, Taylor and Laurent series, function theory. Applications to physics, engineering and real analysis. *Prerequisite: Mathematics **212**.*

CORE REQUIREMENT MET: **MATH/SCI**

#### 320 - Algebra

A first course in group theory: basic axioms and theorems, subgroups, cosets, normal subgroups, homomorphisms, and extension of the theory to rings and fields. *Prerequisites: Mathematics **210** and **214**. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 322 - Number Theory

Classical theory of numbers, from ancient to modern. Prime numbers and factorization. Divisors, numerical functions, linear and quadratic congruences. Diophantine problems, including the Fermat conjecture. Factoring methods. *Prerequisite: Mathematics **210**.*

CORE REQUIREMENT MET: **MATH/SCI **

#### 330 - Probability

Standard methods of calculus are used to study probability: sample spaces, random variables, distribution theory, estimating unknown parameters of distributions. Various applications to real life problems will be discussed. Moment-generating functions and other techniques to calculate moments and characterize distributions. Probabilistic inequalities and the central limit theorem. Point estimators and unbiasedness. *Prerequisites: Mathematics **212** and **214**. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 332 - Mathematical Statistics

Theory and applications of statistical inference. Both Bayesian and classical parametric methods are considered. Point and interval estimation, hypothesis testing. Limit theorems and their use in approximation, maximum likelihood estimation and the generalized likelihood ratio test. Introduction to linear models, nonparametric methods, and decision theory. *Prerequisite: Mathematics **330**. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 340 - Ordinary Differential Equations

This course will focus on theoretical, qualitative, and quantitative analyses of ordinary differential equations from a modern perspective using analytical, graphical, and numerical points of view. Topics can include first-order linear and nonlinear differential equations, first order linear systems, second-order linear differential equations, autonomous differential equations and their bifurcations, boundary value problems, solutions using infinite series, Laplace transforms and various numerical approximation techniques. *Prerequisite: Mathematics 212 and 214 or permission of instructor.*

CORE REQUIREMENT MET: **MATH/SCI**

#### 342 - Partial Differential Equations

An introduction to the study of partial differential equations. This course will include the study of Fourier series, the separation of variables methods, and specifically the wave, heat and Laplace's equations as well as other elementary topics is PDEs. Numerical approximation techniques and applications to specific topics such as traffic flow, dispersive waves or other areas may be included. *Given in alternate years.Prerequisite: Mathematics *

*212*

*, 214 and 341*

CORE REQUIREMENT MET:

**MATH/SCI**

#### 350 - Mathematical Logic

A metamathematical investigation of the main formal language used to symbolize ordinary mathematics: first order logic. The focus is on the two fundamental theorems of logic: completeness and compactness. Gödel's completeness theorem says that every intuitively valid consequence is formally provable from the hypotheses, while compactness says that every intuitively valid consequence of an infinite premise set really depends on only finitely many premises. *Prerequisite: Mathematics **210** or permission of instructor. Given in alternate years. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 352 - Computability and Complexity

The logical foundation of the notion of a computable function underlying the workings of modern computers. Representation of the informal mathematical idea of calculability by canonical proxies: "general recursive functions," "Turing computable functions." Discussion of Church's Thesis, which asserts the adequacy of these representations. Survey of decidable and undecidable problems. *Prerequisites: Mathematics **210 **or permission of instructor. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 354 - Set Theory and Foundations of Mathematics

Cantor's naïve theory of sets and equinumerosity. Paradoxes and axiomatic set theory. Finite and infinite cardinal numbers, fixed point theory, applications to computer science. Well orderings, transfinite induction and recursion, the Axiom of Choice and its consequences, ordinal numbers and the cumulative hierarchy of sets. Discussion of the Continuum Hypothesis and its relation to models of set theory.* Prerequisites: Mathematics **210 **or permission of instructor. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 360 - Axiomatic Geometry

Axiomatic development of Euclidian and non-Euclidian geometries, including neutral and hyperbolic geometries, and, possibly, brief introductions to elliptic and projective geometries. The course will emphasize a rigorous and axiomatic approach to geometry and consequences of Euclid's Parallel Postulate and its negations. *Prerequisite: Mathematics 210 or permission of instructor. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 362 - Topology

General topology studies those properties (such as connectedness and compactness) which are preserved by continuous mappings. A disk and a solid square are topologically equivalent; so are a doughnut and a coffee cup; but a disk is different from a doughnut. This course enables you to construct your own proofs and counterexamples while getting to know the basic concepts behind modern mathematics. *Prerequisites: Mathematics **210,** and Mathematics 212 or 214, or permission of instructor. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 370 - Numerical Analysis

This is a first course in studying numerical techniques in solving a variety of problems in mathematics using computer algorithms. Topics can include methods for approximating solutions to algebraic equations, iterative methods for linear and nonlinear systems, interpolation and approximation theory, numerical integration, explicit and implicit methods for solving initial value problems and boundary value problems. Error estimation, stability and performance are themes throughout. Students will be expected to be able to execute standard algorithms using a computer programming language by the end of the course. Prerequisite: Mathematics 212 and 214 or permission of instructor. *Prerequisite: Mathematics **212** and **214** or permission of instructor. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 372 - Operations Research

Optimal decision-making and modeling of deterministic and stochastic systems. Different systems of constraints lead to different methods. Linear, integer, dynamic programming, and combinatorial algorithms. Practical problems from economics and game theory. Inventory strategies and stochastic models are analyzed by queuing theory. *Prerequisites: Mathematics **210** and **214**. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 380 - Combinatorics

Investigation of the existence and classification of arrangements. Topics to include principles of enumeration, inclusion-exclusion, the pigeon-hole principle, Ramsey theory, generating functions, special counting sequences, and introductory graph theory. *Prerequisite: Mathematics **210**. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 382 - Graph Theory

Graph Theory is a beautiful area of mathematics with many applications. It is used in computer science, biology, urban planning, and many other contexts. Like other areas of discrete mathematics, Graph Theory has the property that the problems are often quite approachable and understandable. Sometimes the solutions to Graph Theory problems can be complex and often require clever arguments, thus the subject is quite pleasing to study. This class will build a solid foundation in Graph Theory for the students. Possible topics are graph isomorphisms, coverings, and colorings; independence number, clique number, connectivity, network flows, and matching theory. *Prerequisite: Mathematics **210**. Suggested co-requisite: Mathematics **380**. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 392 - Mathematical Models in Biology

This course is intended to introduce students to common models used in biology. A variety of models in terms of both biology and mathematics will be covered. Biological topics include action potential generation, genetic spread, cell motion and pattern formation, and circulation. These topics span a range of mathematical models as well, including finite difference equations and differential equations, both linear and non-linear. The focus will be on model analysis and the translation between the mathematical language and the biological meaning. Such analysis will be done both quantitatively and qualitatively. Towards this end, topics seen in previous mathematical courses, such as eigenvalues, phase portraits, and stability, will be revisited. Relevant biology will be presented with each model. The course will be project based. *Prerequisite: Mathematics **212** and **214**, or permission of instructor. *

CORE REQUIREMENT MET: **MATH/SCI**

#### 395 - Special Topics in Advanced Mathematics

Special topics in advanced mathematics, selected largely by student interest and faculty agreement. *May be repeated for credit.*

**Advanced Differential Equations.** The course will consists of advanced topics in differential equations not usually seen in either ordinary differential equations or partial differential equations such as delay differential equations, stochastic differential equations. boundary value problems, numerical methods, and infinite series solutions. *Prerequisite: Math 340 or Math 342 or permission of instructor.*

**Knot Theory.** An introduction to Knot Theory: how knots are described mathematically, how one can distinguish different knots, create new knots, classify knots. Topics include: Reidemeister moves, links, knot colorings, alternating knots, braids, knots and graphs, knot invariants, mirror images, unknotting number, crossing number, applications to biology and chemistry. *Prerequisite: Math 210, Math 212 or Math 214 or permission of instructor.*

**Computational Differential Geometry. **Differential geometry combines calculus, algebra and geometry. It is an important tool in the simulation and analysis of nonlinear objects. The focus of this course is extrinsic and intrinsic analysis of curves and surfaces. Topics include differential forms, frame fields, the shape operator, Gaussian and mean curvature, the Gauss-Bonnet Theorem, and geodesics. Applications to computer graphics, image processing, statistics, differential equations and physics. The computational component uses Mathematica, but no prior knowledge of Mathematica is assumed.* Prerequisite: Math 212 and Math 214, or permission of the instructor.*

CORE REQUIREMENT MET: **MATH/SCI**

#### 396 - Mathematical Modeling

A project-oriented introduction to mathematical modeling. Techniques from calculus, linear algebra and other areas of mathematics will be used to solve problems from the life, physical and social sciences. Familiarity with a programming language is desirable but not required. *This course may be taken up to two times for credit. Prerequisites: Mathematics 212 and 214.*

2 units

#### 397 - Independent Study in Mathematics

Directed individual study of advanced topics. *Prerequisite: permission of instructor.*

2 or 4 units

#### 400 - Senior Colloquium in Mathematics

Senior comprehensive projects. *Required of senior mathematics majors.*

2 units

#### 499 - Honors in Mathematics

Students who have been accepted by the Department to do honors should register for Math 499 in lieu of Math 400.*Prerequisite: permission of department.*