Mathematics takes place in the imagination, in consciousness, unlimited either by finite measuring instruments, by the senses, or even by the feelings. At the same time, mathematics has strict criteria for right knowledge. The power of mathematics lies in bringing infinity out into the finite and making it useful in everyday life — from deciding which bank offers the best return on money, to medical imaging, to designing textiles, to creating a work of art, to putting a man on the moon.

In this course, students explore many different ways in which mathematics expresses, emerges from, and uses infinity and its self-interacting dynamics. They look at the foundation of mathematics in the infinitary processes of set theory, the universe of sets, different sizes of infinity, the continuum and its limit process, sequences and series, infinite replication, and applications of infinity in many areas of life. (2 units)

Arithmetic is the study of patterns, relations, and operations on numbers. Students study the arithmetic of integers, fractions, decimal fractions, ratios, and percents, with an emphasis on applications. (4 units)

The infinitely flexible language of algebra is used to quantify and model mathematical patterns and relationships. Topics include operations on algebraic expressions, linear equations, the coordinate plane, inequalities, factoring, and simple quadratic equations. (4 units)

This course extends Elementary Algebra to develop further algebraic models. Students study polynomials, rational expressions, quadratic equations, complex numbers, and graphing in the coordinate plane. (4 units) Prerequisite: MATH 152

A mathematical function quantifies the relationship between two related quantities and can be used to model change. Functions and their graphs are essential to all branches of mathematics and their applications. (4 units each)

Topics I: domain and range, average rate of change, graphs, functions (linear, exponential, logarithmic, and quadratic), and applications. (Prerequisite: MATH 153)

Topics II: trigonometry, algebra of functions, compositions and inverses of functions, functions (trigonometric, power, polynomial, and rational), and applications. (Prerequisite: MATH 161)

This course gives students a vision of the unified structure of modern mathematics grounded in the infinite, self-referral field of pure intelligence, the Unified Field of Natural Law. Students explore many different ways in which mathematics expresses, emerges from, and uses infinity and its self-interacting dynamics, and how the mathematical quantification of the infinite dynamism of the unified field leads to the great organizing power of modern mathematics. Topics include the development of set theory as a foundation of mathematics, the deductive structure of mathematics, algebraic symbolism and structure, elementary number theory, geometry, the continuum and its limit process, and applications of mathematics in many areas of our lives.

This course studies the mathematics of Veda, as explained by Maharishi. Topics include mathematical models of the self-referral structure of the Veda, mathematics as the intellectual expression of the structure of pure knowledge, mathematics in the Vedic Literature, and examination of the principles of modern mathematics in the light of Maharishi Vedic Science. (2–4 units)

Geometry, the study of shape and form, is an essential tool for the visual artist. Topics in this course include symmetry, Euclidean and non-Euclidean geometry, perspective and projective geometry, and fractals. Materials fee: $10 (4 units) No prerequisite

Geometry gives an understanding of shape, form, and structure that has many applications in mathematics, science, and technology. This course will study Euclidean and non-Euclidean geometries and their applications. (4 units) Prerequisite: Math 162

Discrete mathematics, the study of finite processes and discrete phenomena, is essential for computer science. Topics include logic and sets, relations and functions, vertex-edge graphs, recursion, and combinatorics. (4 units) Prerequisite: MATH 162

Linear algebra studies linearity, the simplest form of quantitative relationship and provides a basis for the study of many areas of pure and applied mathematics, as well as key applications in the physical, biological, and social sciences. Topics include systems of linear equations, vectors, vector equations, matrices, determinants, vector spaces, bases, and linear transformations. (4 units) Prerequisite: MATH 282

This course extends the calculus of a function of a single real variable to functions of several real variables. Topics include maxima and minima, curvilinear coordinates, line integrals, multiple integrals, change of variables, gradient fields, surface integrals, and the theorems of Green, Stokes, and Gauss. (4 units) Prerequisite: MATH 283

This course deepens and extends many of the topics covered in Linear Algebra I; additional topics include the Cayley-Hamilton theorem, Jordan canonical form, inner-product spaces, orthogonality, and spectral theory. (4 units) Prerequisite: MATH 286

The most concise mathematical expression that describes a continuously changing physical system is a differential equation, which uses derivatives to quantify all possible states of an evolving system in one equation. Topics include first-order differential equations, second-order linear differential equations, power-series solutions, Laplace transforms, numerical methods of solution, and systems of differential equations. (4 units) Prerequisite: MATH 283

Problem solving is a fundamental — and exciting — part of mathematics. In this course, students develop and practice many methods and techniques of mathematical problem solving. (4 units) Prerequisite: MATH 282

In this course students investigate a specialized area of mathematics in depth. Topics will vary. (4 units — may be repeated) Prerequisite: consent of the instructor

Complex analysis is one of the great achievements of modern mathematics, providing an extension of the real number line to a two-dimensional plane of numbers with surprising applications throughout most areas of mathematics. Topics include analytic functions, Cauchy-Riemann equations, contour integration, Cauchy’s Theorem and integral formulas, power series, residue theorem, and conformal mappings. (4 units) Prerequisite: MATH 304

Probability provides precise descriptions of the laws underlying random events, with applications in quantum physics, statistics, computer science, and control theory. Topics include permutations and combinations, conditional probability, random variables, discrete and continuous distributions, expectation, and the central limit theorem. (4 units) Prerequisite: MATH 282

Probability provides precise mathematical descriptions of the laws underlying random events, and statistics uses this mathematical theory to make inferences from empirical data and assess their reliability. Topics include probability, random variables, probability distributions, mean and standard deviation, central limit theorem, tests of hypotheses, linear regression, and correlation. (4 units) Prerequisite: MATH 161

In this course, the topics of Probability and Statistics are studied more deeply, with emphasis on their mathematical foundations. (4 units) Prerequisites: MATH 353 and MATH 283

Mathematical logic is the mathematical description of the structure and function of the symbolic language of mathematics. This course develops a rigorous symbolic language, suitable for expressing all mathematical concepts, demonstrates the soundness and completeness of the language, and shows the inherent limitations of such formal systems indicated by Gödel’s Incompleteness Theorems. (4 units) Prerequisite: consent of the instructor

Under the direction of a senior faculty member, students prepare and give lectures, lead tutorial sessions, and write and grade quizzes and exams for a college-level mathematics course. (4 units) Prerequisite: consent of the instructor

This course provides an opportunity for students to do original research under the supervision of a faculty member. (1 unit) Prerequisite: consent of the instructor

In these courses, students apply the theoretical knowledge they have gained in previous mathematics courses to an applied problem taken from a real-life situation in business or industry. Problems differ from year to year. (4 units each — may be repeated) Prerequisite: consent of the instructor

Scientific and engineering applications of computers require advanced numerical techniques of manipulating and solving complex systems of equations with great efficiency and minimum error. Topics include numerical solutions of systems of linear equations, curve fitting, interpolation, numerical integration, solution of algebraic equations, and error analysis. (4 units) Prerequisite: MATH 282

Analysis is the mathematically rigorous development of calculus based on the theory of infinite sets. The analysis sequence begins with the application of the infinitary methods of set theory to construct the uncountable continuum of real numbers, and then shows how the basic principles of calculus can be logically unfolded from a set-theoretic understanding of the continuum. (4 units each)

Topics I: infinite sets, completeness, open sets, closed sets, compact sets, connected sets, and continuous functions. Prerequisite: MATH 283

Topics II: properties of continuous functions, differentiation, mean value theorem, Riemann integral, numerical sequences and series. Prerequisite: MATH 423

Algebra is the study of sets of elements together with operations or relations as well as the structure-preserving transformations between these sets. (4 units each)

Topics Algebra I: groups and subgroups, quotient groups, group homomorphisms, direct sum, kernel, image, Noether isomorphism theorems, and the structure of finitely generated abelian groups. Prerequisite: MATH 286

Topics Algebra II: rings, integral domains, fields, principal ideal domains, unique factorization domains, modules and submodules, tensor products, and exact sequences. Prerequisite: MATH 431

Set theory provides a unified foundation for the diverse theories of modern mathematics based upon the single concept of a set. Topics include axioms of set theory, ordinals, transfinite induction, the universe of sets, cardinal arithmetic, large cardinals, and independence results. (4 units) Prerequisite: MATH 370

This course introduces recent developments in foundational areas that have provided important new insights into the structure of the foundations of mathematics. Topics covered in the course vary from year to year. (4 units) Prerequisite: MATH 370

Topics vary from year to year and may include large cardinals and elementary embeddings; applications of set theory in topology and analysis; applications of set theory in algebra; introduction to the theory of forcing; Gödel’s constructible universe; descriptive set theory. (4 units) Prerequisite: consent of instructor

Topology shows how all mathematical aspects of shape, structure, and form can be expressed in terms of set theory. Students study topologies and their properties of separation, connectedness and compactness, topological mappings, and the fundamental group of a topological space. (4 units) Prerequisites: MATH 423 and 431

Students focus on formal abstract models of computation and capabilities of abstract machines in relation to their increasing ability to recognize more general classes of formal languages. Topics include formal grammars, finite-state machines, equivalence of finite-state machines, right-linear and left-linear grammars, pushdown automata, context-free languages, Turing machines, unsolvable problems, and recursive functions. (4 units) Prerequisite: MATH 272

Students write a substantial paper unifying the knowledge gained from the courses taken during their major and relating this knowledge to deep principles from Maharishi’s Vedic Science. This paper may take the form of (1) An integrated summary of main ideas from the courses taken during their major, addressing themes and questions to be provided by the Department of Mathematics, or (2) A paper written in accord with the guidelines for submissions for the Raja Raam Award and submitted for that award (3) A report of research conducted by the student on a mathematical topic or problem chosen in conjunction with the Department of Mathematics.

(variable units) Prerequisite: consent of the Department faculty