College of Arts and Sciences

Mathematics and Statistics

Chairman Ralph D'Agostino

Associate Chairman Ashis Gangopadhyay

Director of Undergraduate Mathematical Instruction, Director of BA/MA Program Glen Hall

Director of Statistics Program Eric Kolaczyk

Professors Carpenter, D'Agostino, Devaney, Freedman, D. Fried, I. Fried, Grossberg, Hawkins, Kanamori, Kaper, Kon, Kopell, Kreimer, Previato, Rohrlich, Rosenberg, Stevens, Taqqu, Wayne

Associate Professors Blanchard, Gangopadhyay, Hall, Kimura, Kolaczyk, Lyasoff, Meuser, Sullivan, Weiner

Visiting Associate Professor Norbury

Assistant Professors Eden, Guasoni, Kardaras, Pollack, Ray, Szczesny

Professors Emeriti Esch, Granoff

Lecturers Acquista, Arazyan, Chibnik, Chung, Fernandez, Fitzgerald, Ginovyan, Gona, Heeren, Hoffman, Kezim, Kohl, Kristy, Levichev, Massaro, Moorthy, Panth, Pencina

The department's faculty and the courses they offer reflect a broad spectrum of areas in theory and application within the mathematical sciences. Students concentrating in mathematics may select programs emphasizing pure mathematics, classical applied mathematics, or statistics and probability. Aware of the importance of mathematics in a technological society and in keeping with the philosophy of liberal arts, the department encourages concentrators to pursue broad programs of study, building a sound foundation of core mathematical ability while learning about contemporary applications of their subject.

Students wishing to combine their interests in mathematics with those in computer science, economics, philosophy, or education may pursue joint concentrations in these areas. For those wishing to cap their undergraduate studies with advanced coursework at the master's degree level, combined BA/MA programs are available in mathematics and statistics. The department also offers minor concentration programs in mathematics, mathematical statistics, and statistical methods for students concentrating in other fields.

The Undergraduate Mathematics Association and a student chapter of the Mathematics Association of America are active groups enriching the department both socially and intellectually.

Classification of Elementary Courses For the purpose of stating prerequisites, elementary courses are grouped under the titles listed below. Any one course in a group satisfies a requirement, and only one course in each group may be taken for credit.

Elementary Statistics I: CAS MA 113, 115, 213

Elementary Statistics II: CAS MA 116, 214, 614

Calculus I: CAS MA 121, 123

Calculus II: CAS MA 122, 124, 127, 129 (MA 129 may be taken for credit by students with AP credit for CAS MA 124.)

Multivariate Calculus: CAS MA 225, 230

Differential Equations: CAS MA 226, 231

Linear Algebra: CAS MA 242, 442

Concentrations in Mathematics (1701)

Students concentrating in mathematics must satisfy one of the following sets of requirements. Courses may be credited toward the concentration only if a grade of C or higher is earned.

I. Requirements for a Specialty in Pure and Applied Mathematics

Lower Division Calculus: CAS MA 123 and 124, or 127, or 129, or equivalent; Multivariate Calculus: CAS MA 225 or 230; Elementary Differential Equations: CAS MA 226 or 231; Linear Algebra: CAS MA 242 or 442; CAS CS 111; and one other mathematics or computer science course selected in consultation with a mathematics faculty advisor.

Upper Division At least one of the two following sequences:

CAS MA 411, 412 Advanced Calculus and Complex Variables

CAS MA 511, 512 Introduction to Analysis

At least one of the following sequences:

CAS MA 531, 532 Computability and Logic and Foundations of Mathematics

CAS MA 539, 555 Scientific Computation and Numerical Analysis I

CAS MA 541, 542 Modern Algebra

CAS MA 555, 556 Numerical Analysis

CAS MA 561, 562 Methods of Applied Mathematics

CAS MA 563, 564 Differential Geometry/Topology

CAS MA 569, 570 Operations Research

CAS MA 573, 574 Differential Equations/Applied Dynamics

CAS MA 581, 582, 583 Probability and Statistics (any two)

Four additional CAS mathematics courses numbered 200 or above are required. Two of these four can be related courses. For specific information concerning related courses, contact the department.

II. Requirements for a Specialty in Statistics

Lower Division Calculus: CAS MA 123 and 124, or 127, or 129, or equivalent; Multivariate Calculus: CAS MA 225 or 230; Linear Algebra: CAS MA 242 or 442; Introduction to Probability and Statistics: CAS MA 213, 214; Introduction to Computer Science: CAS CS 111.

Upper Division

CAS MA 416 Analysis of Variance

CAS MA 575 Linear Models

Two of the following three courses:

CAS MA 581 Probability

CAS MA 582 Mathematical Statistics

CAS MA 583 Introduction to Stochastic Processes

Four additional CAS mathematics courses are required: two at the 200-level or above, and two chosen from among MA 500-599 and MA 684.

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Joint Concentration in Mathematics and Philosophy (1740)

This degree program is for students who wish to combine the fields of mathematics and philosophy, usually with an emphasis on logic.

Prerequisites CAS MA 123 and 124, or 127, or 129, or equivalent; MA 225 or 230; MA 293; and CAS PH 100 or 110.

Principal Courses: at least six courses in mathematics and six courses in philosophy at the 200 level or above, with a grade of C or higher.

Required Mathematics Courses: CAS MA 242 or 442; MA 531 (or PH 461 or PH 661) and MA 532 (or PH 462 or PH 662); MA 411–412 or 511–512; any one of the following: MA 294, 341, or any course numbered 400 or higher.

Required Philosophy Courses: CAS PH 468; three courses in philosophy above the 100 level; any two of the following: CAS PH 422, 443, 460, 474.

Joint Concentration in Mathematics and Computer Science (1721)

Principal Courses

Lower Division CAS MA 123 and 124, or 127, or 129, or equivalent, 225 or 230, 242 or 442, 293, and 294. CAS CS 111 and 112 or equivalent, and CS 210.

Upper Division CAS CS 330, 332, and 350; two additional computer science courses numbered above 400; one two-course sequence selected from: CAS MA 531–532, MA 541–542, MA 555–556, MA 569–570; two additional MA courses numbered 200 or above.

Joint Concentration in Economics and Mathematics (1704)

Prerequisites CAS MA 123 and 124, or 127, or 129, or equivalent and CAS EC 101 or 111 and EC 102 or 112.

Principal Courses A minimum of eight economics courses (distributed according to the description given in the "Economics" section of this site). A minimum of six mathematics courses (in addition to Calculus I and II as above) distributed as follows: CAS CS 111 or 113, and CAS MA 225 or 230, MA 242 or 442, 569; plus two others with at least one being CAS MA 400 or above, selected from among CAS MA 213, 214, 226 or 231, 416, 581, 582, CAS CS 112.

Finally, students must satisfy a statistics requirement, either by taking CAS MA 213, 214 or by taking courses with comparable statistics content offered in the Department of Economics.

BA in Economics and Mathematics (1704) BA/MA in Economics (2204)

This BA/MA program is identical to that of the BA/MA in economics described in the Economics section of this site, except that the candidate must satisfy all requirements for the joint concentration in economics and mathematics as well as the MA in economics. The departmental BA/MA statement and further information are available from the Graduate Studies Coordinator in the Department of Economics. Please refer to Special Courses and Programs under College of Arts and Sciences on this site for general information about BA/MA programs.

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Mathematics and Mathematics Education (1706)

This program combines a general bachelor's degree mathematics program with the mathematics education courses required for high school teaching certification. Advisors in both the CAS Mathematics Department and the SED Mathematics Education Department should be consulted, as the scheduling of course programs requires careful planning.

Required CAS Courses:

CAS MA 123 Calculus I

CAS MA 124 Calculus II

CAS MA 213 Basic Statistics and Probability

CAS MA 225 Multivariate Calculus

CAS MA 226 Differential Equations

CAS MA 242 Linear Algebra

CAS MA 293 Discrete Mathematics

CAS MA 341 Introduction to Number Theory

CAS MA 505 History of Mathematics

CAS MA 528 Introduction to Modern Geometry

or

SED ME 559 Mathematics for Teachers: Geometry

CAS MA 541 Modern Algebra I

One additional course must be chosen from the following list: CAS CS 111, PY 211, or any CAS mathematics course numbered 200 or above.

Note: The following substitutions are permitted: MA 123 -124 may be replaced by either MA 127 or MA 129, and MA 225, 226, and 242 may be replaced by MA 230, 231, and 442 respectively.

Required SED Courses:

SED CT 575 General Methods of Instruction, 5–12

SED ED 100 Introduction to Education (normally taken in the first year; can be taken in the second year)

SED ED 410/412 Social Context of Education/Civic Context of Education (taken in the junior year)

SED ME 508 Student-Teaching Practicum (second semester of the senior year; on-site practice in teaching).

SED ME 547 Methods of Teaching Mathematics: High School

SED ME 563 Problem-Solving Lab

Note: The following 2-credit courses are also required: SED DS 502 (Adolescent Psychology) and SED SE 251 (Special Education).

BA/MA in Mathematics (1701)

Students entering the BA/MA program in mathematics will elect one of two options: mathematics or statistics. Both options emphasize breadth of coverage of basic topics of current general importance, as opposed to narrow specialization. The objective of the program in statistics is to train general practitioners capable of making immediate contributions in industry, the medical sciences, government, or private educational and research institutions. Potential applicants must first complete at least one upper-level course (400-level or higher statistics course for the statistics option, and 500-level or higher mathematics course for the mathematics option). As part of the application, one of the letters of recommendation should be provided by the instructor of such a course. Students in the BA/MA program must elect and complete the BA concentration in mathematics (1701). In addition, students must complete the master's degree requirements, which consist of a coherent program of eight graduate-level (500 or above) mathematics courses and comprehensive exams. No overlap is permitted between mathematics courses taken to satisfy the BA requirements and those taken to satisfy the MA requirements. The total number of courses taken for both degrees, including distribution courses and electives, can be reduced to as few as 36, provided that all requirements are met.

For further specific information concerning the BA/MA program in mathematics, please consult an advisor in the department. Please refer to Special Courses and Programs under College of Arts and Sciences on this site for general information about BA/MA programs.

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BA/MA in Mathematics and Cognitive and Neural Systems

The BA/MA in mathematics and cognitive and neural systems is an interdepartmental program in the College of Arts and Sciences and the Graduate School of Arts and Sciences. The program allows undergraduate concentrators in mathematics to begin working toward an MA in cognitive and neural systems while still completing Mathematics Department BA requirements.

Admission to the BA/MA Program in Mathematics and Cognitive and Neural Systems College of Arts and Sciences students currently in or entering the junior year are eligible to apply for admission. Students admitted to the BA/MA program will typically have completed at least one CNS course, and be making satisfactory progress toward a BA in mathematics.

The application should include a letter from the student's Department of Mathematics advisor. Application forms for admission to the BA/MA program may be obtained from the Graduate School of Arts and Sciences Office, College of Arts and Sciences, Room 112.

Requirements Students are required to complete all requirements for the BA in Mathematics as specified in this bulletin, plus all requirements for the MA in cognitive and neural systems, as specified in the Graduate School of Arts and Sciences Bulletin. In particular, 32 courses (128 credits) are required for the BA and eight courses (32 credits) are required for the MA degree. Please refer to Special Courses and Programs under College of Arts and Sciences on this site for general information about BA/MA programs, and contact Professor Gail Carpenter in the Department of Cognitive and Neural Systems for details.

Minor Concentration Programs (1701, 1702)

Students must earn a minimum grade of C in courses taken toward a minor concentration. All minor concentrators develop their programs in consultation with an advisor.

Mathematics (1701) The prerequisite for the minor concentration is CAS MA 123 and 124, or 127 or 129 or equivalent. The principal courses required are CAS MA 225 or 230, 226 or 231, 242 or 442, and at least two CAS MA courses numbered 400 or above. In addition, it is strongly recommended that students take CAS CS 111 Introduction to Computer Science.

Mathematical Statistics (1702) The prerequisite for the minor concentration is CAS MA 123 and 124, or 127 or 129 or equivalent. Principal courses required are CAS MA 213, 214, 225 or 230, and two chosen from 581, 582, and 583. In addition, it is strongly recommended that students take CAS CS 111 Introduction to Computer Science.

Statistical Methods (1702) Principal courses required are CAS MA 115 or 213, 116 or 214, 416, 684; one course at the CAS MA 400-600 or CS 400-600 level, and one additional MA or CS course. The two latter courses are to be selected with the written approval of a statistics advisor.

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Courses

Courses marked with a (†) satisfy divisional studies requirements.

Mathematics Courses Primarily for Nonconcentrators See also the statistics listings below. For computer science courses, refer to the Computer Science Department.

CAS MA 118 College Algebra and Trigonometry

CAS MA 118 may not be taken for credit by any student who has completed any MA course numbered 121 or higher. Functions and graphs. Linear and quadratic equations. Exponents; logarithms. Right and oblique triangles; trigonometric functions. Optimization. Specifically intended to prepare students with insufficient background in mathematics for the study of calculus. This course may not be used in fulfillment of the distribution requirement. Satisfies the mathematics requirement in the College program. 4 cr, either sem.

†CAS MA 120 Applied Mathematics for Social and Management Sciences

CAS MA 120 may not be taken for credit by any student who has completed any MA course numbered 124 or higher. Linear equations, systems of linear equations, matrix algebra, exponential functions and logarithms, elements of differential calculus, optimization, probability. Applications in economics, finance, and management. Satisfies both mathematics requirement and distribution requirement. 4 cr, either sem. (MCS)

Calculus Courses

†CAS MA 121 Calculus for the Life and Social Sciences I

Students may receive credit for either CAS MA 121 or 123, but not both. Differentiation and integration of functions of one variable. Same topics as CAS MA 123, but with less emphasis on mathematical generality and more on application. Especially suitable for students concentrating in the biological and social sciences. 4 cr, either sem. (MCS)

†CAS MA 122 Calculus for the Life and Social Sciences II

Students may receive credit for not more than one of the following courses: CAS MA 122, MA 124, MA 127, or MA 129. Prereq: CAS MA 121 or MA 123. Continuation of CAS MA 121. Review of univariate calculus, calculus of the elementary transcendental functions, elementary differential equations, elementary multivariate calculus. Applications to exponential growth, optimization, equilibrium, and dynamic modeling problems. 4 cr, 2nd sem. (MCS)

†CAS MA 123 Calculus I

Students may receive credit for either CAS MA 121 or MA 123, but not both. Limits; derivatives; differentiation of algebraic functions. Applications to maxima, minima, and convexity of functions. The definite integral; the fundamental theorem of integral calculus; applications of integration. 4 cr, either sem. (MCS)

†CAS MA 124 Calculus II

Prereq: CAS MA 121 or MA 123. Students may receive credit for not more than one of the following courses: CAS MA 122, MA 124, MA 127, or MA 129. Logarithmic, exponential, and trigonometric functions. Sequences and series; Taylor's series with the remainder. Methods of integration. Calculus I and II together constitute an introduction to calculus of a function of a single real variable. 4 cr, either sem. (MCS)

†CAS MA 127 Enriched Calculus

Prereq: CAS MA 123 or equivalent. Students may receive credit for not more than one of the following courses: CAS MA 122, MA 124, MA 127, or MA 129. For students with prior experience in calculus. Emphasizes material from the end of Calculus 2 (sequences, Taylor series), reviews more challenging problems from Calculus 1. 4 cr, 1st sem. (MCS)

†CAS MA 129 Honors Calculus

Prereq: consent of instructor. Students may receive credit for not more than one of the following courses: CAS MA 122, MA 124, MA 127, or MA 129. Alternative to CAS MA 127 for advanced students with a deep interest in mathematics. Continuity, degrees of differentiability, integrability. Transcendental functions. Techniques of differentiation and integration. Sequences, series, summability, uniform convergence. Complex numbers, Taylor series. Application to mechanics, probability. 4 cr, 1st sem. (MCS)

CAS MA 142 Introduction to Linear Algebra

Coreq: CAS MA 122 or 124 or 127 or 129. Systems of linear equations; matrices. Vector spaces and linear transformations. Determinants. Eigenvalues and eigenvectors. Applications. (Cannot be taken for credit in addition to CAS MA 242, MA 442, or ENG EK 102.) 2 cr, either sem.

CAS MA 193 Discrete Mathematics for Engineering

Propositional logic, set theory, relations, combinatorics, graphs. Topics are treated with particular emphasis on applications to the engineering discipline. (Cannot be taken for credit in addition to ENG EC 333.) 2 cr, either sem.

CAS MA 225 Multivariate Calculus

Prereq: CAS MA 124 or 127 or 129. Vectors, lines, planes. Multiple integration, cylindrical and spherical coordinates. Partial derivatives, directional derivatives, scalar and vector fields, the gradient, potentials, approximation, multivariate minimization, Stokes's and related theorems. (Cannot be taken for credit in addition to CAS MA 230.) 4 cr, either sem.

CAS MA 226 Differential Equations

Prereq: CAS MA 225 or 230. First-order linear and separable equations. Second-order equations and first-order systems. Linear equations and linearization. Numerical and qualitative analysis. Laplace transforms. Applications and modeling of real phenomena throughout. (Cannot be taken for credit in addition to CAS MA 231.) 4 cr, either sem.

CAS MA 230 Honors-Level Vector Calculus

Prereq: CAS MA 124 or 127 or 129. Linear algebra: linear transformations and matrices, inner products, cross products. Differentiation: tangent spaces, directional derivatives, gradients, vector fields and flow, divergence, curl. Integration: multiple integrals, line integrals, Green's theorem, surface integrals, Stokes's theorem, the divergence theorem. (Cannot be taken for credit in addition to CAS MA 225.) 4 cr, 2nd sem.

CAS MA 231 Honors-Level Differential Equations

Prereq: CAS MA 225 or 230. Ordinary differential equations with applications to mechanics, biology, and physics. Linear and non-linear systems. Numerical and qualitative analysis. Laplace transforms. Modeling. Special topics. (Cannot be taken for credit in addition to CAS MA 226.) 4 cr, 1st sem.

Principal Mathematics Courses Statistics courses are listed in a separate section.

CAS MA 242 Linear Algebra

Coreq: CAS MA 122 or 124 or 127 or 129. Matrix algebra, solution of linear systems, determinants, Gaussian elimination, fundamental theory, row-echelon form. Vector spaces, bases, norms. Computer methods. Eigenvalues and eigenvectors, canonical decomposition. Applications. (Cannot be taken for credit in addition to CAS MA 142, MA 442, or ENG EK 102.) 4 cr, either sem.

CAS MA 293 Discrete Mathematics

Prereq: CAS MA 123. Propositional logic, set theory. Elementary probability theory. Number theory. Combinatorics with applications. 4 cr, either sem.

CAS MA 294 Applied Abstract Algebra

Prereq: CAS MA 293 or consent of instructor. Abstract algebra and its applications to combinatorics. A first exposure to groups, rings, and fields via significant combinatorial applications. Students who have already received credit for MA 541 or MA 542 may not subsequently receive credit for MA 294. 4 cr, either sem.

CAS MA 341 Introduction to Number Theory

Prereq: CAS MA 242 or consent of instructor. Study of integers and basic results of number theory. Topics include Linear Diophantine equations, prime numbers and factorization, congruences, and quadratic reciprocity. 4 cr, 2nd sem.

CAS MA 381 Elementary Probability

Prereq: CAS MA 225 or 230. A calculus-based introductory course in probability. Language of probability, combinatorics, conditional probability, discrete and continuous random variables, the Poisson process, joint probability distribution, moment generating function, law of large numbers and central limit theorem, sample estimators for mean and variance. (Cannot be taken for credit in addition to MA 581 or MA 590.) 4 cr, either sem.

CAS MA 401, 402 Senior Independent Work

Prereq: approval of the Honors Committee. 4 cr each, 1st & 2nd sem.

CAS MA 411 Advanced Calculus

Prereq: CAS MA 225 or 230 and MA 242 or 442. Extends concepts and techniques of calculus and develops further applications. Topics include higher dimensional calculus, applications of vector analysis, uniform convergence of series, complex series, improper integrals, gamma and beta functions, Stirling's formula, Fourier series and transform. 4 cr, 1st sem.

CAS MA 412 Complex Variables

Prereq: CAS MA 225 or 230. Basic concepts, results, and applications of complex analysis. Emphasis on computation and applications. Complex plane and functions, differentiability, Cauchy-Riemann conditions, contour integrals, Cauchy formulas, complex series, residue calculus, applications. Extends the concepts of the calculus to the complex setting. 4 cr, 2nd sem.

CAS MA 442 Honors-Level Linear Algebra

Prereq: CAS MA 225 or 230. A thorough development of the fundamentals of linear algebra. Systems of linear equations; matrices, linear transformations, duality; determinants, characteristic and minimal polynomials; diagonalization and normal forms of linear transformations; inner products, unitary and self-adjoint operators, and spectral theory. Applications to physics, probability, and statistics. (Cannot be taken for credit in addition to CAS MA 142, MA 242, or ENG EK 102.) 4 cr, either sem.

CAS MA 471 Chaotic Dynamical Systems

Prereq: CAS MA 225 or 230. Iteration of functions in one and higher dimensions. Periodic points; hyperbolicity; recurrence; topological conjugacy and structural stability; chaotic dynamics. Some numerical experimentation involving personal computers. 4 cr, 1st sem.

CAS MA 491, 492 Directed Study

Prereq: consent of instructor and application made through the Mathematics Department. Variable cr, 1st & 2nd sem.

CAS MF 502 Fundamentals of Finance

Prereq: CAS MA 123, 124, and 242. Financial systems, financial statements, time value of money, interest rates, return sales, spot rates, forward rates, inflation rates, cost of capital rates, tax rates, bonds, cash-flow models, risk management. This course is part of the Mathematical Finance Program but is open to all students. 4 cr, Summer II.

CAS MA 505 History of Mathematics

Prereq: one year of college-level calculus. Patterns of mathematical thought from antiquity to the seventeenth century. Emphasis throughout on the background and origins of the mathematical revolution of the seventeenth century in which Descartes, Newton, and Leibniz played key roles. 4 cr, either sem.

CAS MA 511 Introduction to Analysis I

Prereq: CAS MA 225 or 230. Fundamental concepts of mathematical reasoning. Properties of the real-number system, elementary point-set theory, metric spaces. Limits, sequences, series, convergence, uniform convergence, continuity. Differentiability for functions of a single variable, Riemann-Stieltjes integration. 4 cr, 1st sem.

CAS MA 512 Introduction to Analysis II

Prereq: CAS MA 511. Background of MA 511 used to develop further topics of calculus. Exponential and logarithmic functions, Taylor series, power series, real analytic functions. Differential and integral calculus for functions of several variables. Line and surface integrals, divergence theorem, Stokes's theorem, inverse and implicit function theorems, change of variables. Fourier analysis. 4 cr, 2nd sem.

CAS MA 528 Introduction to Modern Geometry

Prereq: CAS MA 225 or 230 or consent of instructor. The foundations of Euclidean geometry. Transformation and symmetries in the plane, inversive and projective planes, coordinates, conics and quadrics, the Golden Section, intermediary and Dedekind's axiom, models for non-Euclidean geometries. 4 cr, 2nd sem.

CAS MA 531 Mathematical Logic

Prereq: CAS MA 293 or consent of instructor. The syntax and semantics of sentential and quantificational logic, culminating in the Gödel Completeness Theorem. The Gödel Incompleteness Theorem and its ramifications for computability and philosophy. Students may receive credit for only one of the following: CAS MA 531, PH 461, or PH 661. 4 cr, 1st sem.

CAS MA 532 Foundations of Mathematics

Prereq: CAS MA 531. Axiomatic set theory as a foundation for mathematics and as a field of mathematics: Axiom of Choice, the Continuum Hypothesis, and consistency results. Students may receive credit for only one of the following courses: CAS MA 532, PH 462, or PH 662. 4 cr, 2nd sem.

CAS MA 539 Methods of Scientific Computing

(Meets with CAS CS 539.) Prereq: CAS MA 225 or 230, MA 226 or 231, MA 242 or 442, and CAS CS 330, or consent of instructor. Topics include computational linear algebra, fast Fourier transform, wavelets, data compression, numerical integration, simulated annealing, random number generation, Monte Carlo methods. 4 cr, either sem.

CAS MA 541 Modern Algebra I

Basic properties of groups, Sylow theorems, basic properties of rings and ideals, Euclidean rings, polynomial rings. 4 cr, 1st sem.

CAS MA 542 Modern Algebra II

Prereq: CAS MA 541. Vector spaces and modules, Galois theory, linear transformations and matrices, canonical forms, bilinear and quadratic forms. 4 cr, 2nd sem.

CAS MA 549 Geometry and Symmetry

Prereq: CAS MA 547 and 548 or consent of instructor. Problem-oriented seminar in modern geometry focusing on invariants of transformation groups. Specific topics may include Euclidean and plane geometry, Hilbert's Axioms, conics, tilings, finite, projective, spherical and/or hyperbolic geometry, tessellations, applications to number theory, Platonic Solids. 4 cr, Summer II.

CAS MA 555 Numerical Analysis I

Prereq: CAS MA 225 or 230. Numerical solutions of equations, iterative methods, analysis of sequences. Theory of interpolation and functional approximation, divided differences. Numerical differentiation and integration. Polynomial theory. Ordinary differential equations. 4 cr, 2nd sem.

CAS MA 556 Numerical Analysis II

Prereq: CAS MA 242 or 442 and MA 555, or consent of instructor. Numerical linear algebra; norms, elimination methods, error analysis, conditioning, eigenvalues, iterative methods, least squares and nonlinear functional minimization. Partial differentiation equation boundary value and initial value problems. Finite element methods. Legendre and Chebyshev polynomials. Treatment in greater depth of selected topics from CAS MA 555. 4 cr, 1st sem.

CAS MA 557 Mathematical Structures in Physics I

Prereq: CAS MA 226 or 231, and CAS MA 242 or 442, or CAS PY 313 or 354. Relativistic wave equations, quantum equations of motion, Feynman graphs, combinatorics of perturbative expansions and Hopf algebras, renormalization and elimination of divergences, locality of fields, scaling transformations and renormalization group, basic applications to particle physics and condensed matter theory. 4 cr, either sem.

CAS MA 561 Methods of Applied Mathematics I

Prereq: CAS MA 226 or 231. Derivation and analysis of the classical equations of mathematical physics; heat equation, wave equation, and potential equation. Initial boundary value problems, method of separation of variables, eigenvalue problems, eigenfunction expansions. Fourier analysis. Existence and uniqueness of solution. 4 cr, 1st sem.

CAS MA 562 Methods of Applied Mathematics II

Prereq: CAS MA 561. Calculus of variations, first-order non-linear partial differential equations, Hamilton-Jacobi theory, Rayleigh-Ritz procedure, perturbation methods. 4 cr, 2nd sem.

CAS MA 563 Introduction to Differential Geometry

Prereq: CAS MA 226 or 231 and MA 411. Study of local properties of curves and surfaces in the three-dimensional Euclidean space; curvature, torsion, Frenet equations, tangent and normal planes; first and second fundamental form; developable surfaces, principal, mean and Gaussian curvature; vector fields, covariant differentiation, geodesics, surfaces of constant curvature. 4 cr, 2nd sem.

CAS MA 564 Introduction to Topology

Prereq: CAS MA 411. Introduction to point set and algebraic topology. Topological spaces and continuity. Compactness and connectedness. Metrizable topological spaces. Product topology and Tychonoff's theorem. The fundamental group and van Kampen's theorem. Covering spaces and the universal cover. 4 cr, 1st sem.

CAS MA 565 Mathematical Models in the Life Sciences

Prereq: CAS MA 226 or 231. An introduction to mathematical modeling, using applications in the biological sciences. Mathematics includes linear difference and differential equations, and an introduction to nonlinear phenomena and qualitative methods. An elementary knowledge of differential equations and linear algebra is assumed. 4 cr, either sem.

CAS MA 566 Geometric Methods in Mechanics

Prereq: CAS MA 226 or 231, CAS MA 242 or 442, CAS MA 411, or consent of instructor. Modern geometric theories applied to motion of physical objects. Differential forms. Symplectic manifolds. Lie groups and their Lie algebras. Hamiltonian and Lagrangian systems; Liouville's theorem. Poincaré's return theorem, Noether's theorem. Additional topics according to instructor. 4 cr, 1st sem.

CAS MA 569 Optimization Methods of Operations Research

Prereq: CAS MA 225 or 230, MA 242 or 442. Optimization of linear functions: linear programming, simplex method; transportation, assignment, and network problems. Optimization of non-linear functions: unconstrained optima, constrained optima and Lagrange multipliers, Kuhn-Tucker conditions, calculus of variations, and Euler's equation. 4 cr, 1st sem.

CAS MA 570 Stochastic Methods of Operations Research

Prereq: CAS MA 225 or 230 and MA 242 or 442. Poisson processes, Markov chains, queuing theory. Matrix differential equations, differential-difference equations, probability-generating functions, single- and multiple-channel queues, steady-state and transient distributions. 4 cr, 2nd sem.

CAS MF 572 Introduction to Mathematical Finance

Prereq: CAS MA 226 or 231 and MA 381 or 581. A rigorous mathematical introduction to developments in the field of finance. Mathematics of modern portfolio theory, capital asset pricing model, and arbitrage pricing theory. Derivation of pricing models for options, futures, and swaps based on concepts from Itô calculus. 4 cr, either sem.

CAS MA 573 Qualitative Theory of Ordinary Differential Equations

Prereq: CAS MA 226 or 231, MA 242 or 442, and MA 411. Eigenvalues, eigen-vectors, Jordan normal forms. Linear systems of differential equations, Phase portrait, Hamiltonian systems, stability theory. Applications to systems arising in mechanics, economics, ecology, electrical circuit theory, etc. 4 cr, 1st sem.

CAS MA 574 Applied Nonlinear Dynamics

Prereq: CAS MA 573. Attractors and invariant measures for nonlinear dynamical systems. Measures of chaos such as Lyapunov exponents. Time series analysis. Multiple time scales and singular perturbation theory. Synchronization in coupled oscillators. Strong emphasis on applications to realistic biological and mechanical systems. 4 cr, 2nd sem.

CAS MA 590 Introduction to Probability Theory

Prereq: CAS MA 225 or 230. Combinatorics. Conditional Probability. Independence. Discrete and continuous random variables. Sigma algebras. Joint, marginal, and conditional distributions. Conditional and unconditional expectations and variance. Derived distributions. Characteristic functions. Convergence of random variables. Limit theorems. Unbiased estimates of mean and variance. Cochran's Theorem. Cannot be taken for credit in addition to CAS MA 381 or MA 581. 4 cr, Summer II.

CAS MF 593 Statistical Analysis of Financial Data

Prereq: CAS MA 381 or 581 or 590. Statistical/probabilistic techniques commonly used in Financial Engineering and Risk Management. Topics include estimation of diffusion processes, elements of linear (ARMA) and nonlineary ((G)ARCH) time series-analysis, multivariate data anlysis, extreme value theory, and Monte Carlo methods. 4 cr, 1st sem.

CAS MF 594 Stochastic Optimal Control and Investment

Prereq: CAS MA 225 or 230, MA 226 or 231, and MA 242 or 442. Sample topics: Brownian motion, Itô's lemma, computer simulation, dynamic programming, Bellmann's equation, contingent claims analysis, investment opportunities/timing, project/investment valuation, dynamic equilibrium, sequential investment. 4 cr, either sem.

CAS MA 671 Chaotic Dynamical Systems

Prereq: CAS MA 225 or 230 or equivalent. This course is not open to CAS students; it is intended for graduate students in disciplines outside mathematics. Iterations of functions of one or several variables. Periodicity, stability, chaos, fractals, bifurcations. Julia sets and the Mandelbrot set. Students are required to perform several experiments on personal computers. 4 cr, 1st sem.

Statistics Courses

†CAS MA 113 Elementary Statistics

CAS MA 113 may not be taken for credit by any student who has completed any MA course numbered 300 or higher. Students may receive credit for not more than one of the following courses: CAS MA 113, MA 115, or MA 213. Basic concepts of estimation and tests of hypotheses, ideas from probability; one-, two-, and multiple-sample problems. Applications in social sciences. Primarily for students in the social sciences who require a one-semester introduction to statistics; others should consider CAS MA 115 or MA 213. 4 cr, either sem. (MCS)

†CAS MA 115 Statistics I

CAS MA 115 may not be taken for credit by any student who has completed any MA course numbered 300 or higher. Students may receive credit for not more than one of the following courses: CAS MA 113, MA 115, or MA 213. Numerical and graphical summaries of univariate and bivariate data. Basic probability, random variables, binomial distribution, normal distribution. One-sample statistical inference for normal means and binomial probabilities. Primarily for students in the social sciences with limited mathematics preparation. 4 cr, either sem. (MCS)

†CAS MA 116 Statistics II

CAS MA 116 may not be taken for credit by any student who has completed any MA course numbered 300 or higher. Prereq: CAS MA 115 or equivalent. Students may receive credit for not more than one of the following courses: CAS MA 116, MA 214, or MA 614. One- or two-sample inference for normal means and binomial probabilities, analysis of variance, simple linear regression, multiple regression, analysis of categorical data. Introduction to survey design and design of experiments. Primarily for students in the social sciences with limited mathematics preparation. 4 cr, either sem. (MCS)

†CAS MA 213 Basic Statistics and Probability

Prereq: good background in high school algebra. Students may receive credit for not more than one of the following courses: CAS MA 113, MA 115, or MA 213. Elementary treatment of probability densities, means, variances, correlation, independence, the binomial distribution, the central limit theorem. Stresses understanding and theoretical manipulation of statistical concepts. 4 cr, either sem. (MCS)

†CAS MA 214 Applied Statistics

Prereq: CAS MA 213 or consent of instructor. Students may receive credit for not more than one of the following courses: CAS MA 116, MA 214, or MA 614. Inference about proportions, goodness of fit, student's t-distribution, tests for normality; two-sample comparisons, regression and correlation, tests for linearity and outliers, residual analysis, contingency tables, analysis of variance. 4 cr, either sem. (MCS)

CAS MA 416 Analysis of Variance

Prereq: CAS MA 116 or 214 or equivalent. Fundamental concepts and analytical skills in analysis of variance, including crossed and nested designs, as well as fixed- and random-effect models. Trend analysis for repeated measures, expected mean squares, and nonparametric techniques. SAS is used throughout the course. 4 cr, 1st sem.

CAS MA 421 Modern Statistical Modeling and Data Analysis with Splus

Prereq: CAS MA 115 or 213 and MA 116 or 214. Provides a non-technical descriptive introduction to modern techniques in data modeling via software Splus. Topics include linear and nonlinear, nonparametric and semiparametric regression, Bayesian models and computations, introduction to data mining, association rules, decision trees, and neural network algorithms. 4 cr, Summer I.

CAS MA 568 Statistical Analysis of Point Process Data

Prereq: CAS MA 213 and CAS MA 214 or consent of instructor. Introduces the theory of point processes and develops practical problem-solving skills to construct models, assess goodness-of-fit, and perform estimation from point process data. Applications to neural data, earthquake analysis, financial modeling, and queuing theory. 4 cr, 1st sem.

CAS MA 570 Stochastic Methods of Operations Research

Prereq: CAS MA 225 or 230 and MA 242 or 442. Poisson processes, Markov chains, queuing theory. Matrix differential equations, differential-difference equations, probability- and moment-generating functions, single- and multiple-channel queues, steady-state and transient distributions. 4 cr, 2nd sem.

CAS MA 572 Introduction to Mathematical Finance

Prereq: CAS MA 226 or 231 and MA 381 or 581. A rigorous mathematical introduction to developments in the field of finance. Mathematics of modern portfolio theory, capital asset pricing model, and arbitrage pricing theory. Derivation of pricing models for options, futures, and swaps based on concepts from Itô calculus. 4 cr, either sem.

CAS MA 575 Linear Models

Prereq: CAS MA 214 and CAS MA 242 and CAS MA 581 or consent of instructor. Post-introductory course in linear models, with focus on both principles and practice. Simple and multiple linear regression, weighted and generalized least squares, polynomials and factors, transformations, regression diagnostics, variable selection, and a selection from topics on extensions of linear models. 4 cr, 1st sem.

CAS MA 576 Generalized Linear Models

Prereq: CAS MA 575 or consent of instructor. Covers topics in linear models beyond MA 575: generalized linear models, analysis of binary and polytomous data, log-linear models, multivariate response models, non-linear models, graphical models, and relevant model selection techniques. Additional topics in modern regression as time allows. 4 cr, 2nd sem.

CAS MA 577 Mathematics of Financial Derivatives

Prereq: CAS MA 581 or consent of instructor. Develops the probabilistic tools used in finance and presents the methodologies that are used in the pricing of financial derivatives. No previous knowledge of finance is required. 4 cr, either sem.

CAS MA 578 Bayesian Statistics

Prereq: CAS MA 581 and 582. The principles and methods of Bayesian statistics. Subjective probability, Bayes rule, posterior distributions, predictive distributions. Computationally based inference using Monte Carlo integration and Markov chain simulation. Hierarchical models, mixture models, model checking, and methods for Bayesian model selection. 4 cr, 2nd sem.

CAS MA 581 Probability

Prereq: CAS MA 225 or 230 or consent of instructor. Basic probability, conditional probability, independence. Discrete and continuous random variables, mean and variance, functions of random variables, moment generating function. Jointly distributed random variables, conditional distributions, independent random variables. Methods of transformations, law of large numbers, central limit theorem. (Cannot be taken for credit in addition to CAS MA 381 or MA 590.) 4 cr, 1st sem.

CAS MA 582 Mathematical Statistics

Prereq: CAS MA 381 or 581. Point estimation including unbiasedness, efficiency, consistency, sufficiency, minimum variance unbiased estimator, Rao-Blackwell theorem, and Rao-Cramer inequality. Maximum likelihood and method of moment estimations; interval estimation; tests of hypothesis, uniformly most powerful tests, uniformly most powerful unbiased tests, likelihood ratio test, and chi-square test. 4 cr, 2nd sem.

CAS MA 583 Introduction to Stochastic Processes

Prereq: CAS MA 381 or 581 or consent of instructor. Basic concepts and techniques of stochastic process as they are most often used to construct models for a variety of problems of practical interest. Topics include Markov chains, Poisson process, birth and death processes, queuing theory, renewal processes, and reliability. 4 cr, 2nd sem.

CAS MA 584 Multivariate Statistical Analysis

Prereq: CAS MA 213 and CAS MA 242 and CAS MA 581 or consent of instructor. Presents statistical concepts and methods, and their application for the exploration, regression, testing, visualization, and clustering of multivariate data. Both classical and modern techniques are developed, including methods for analysis of high dimensional and non-euclidean data. 4 cr, 2nd sem.

CAS MA 585 Time Series and Forecasting

Prereq: CAS MA 581 or consent of instructor. Autocorrelation and partial autocorrelation functions; stationary and nonstationary processes; ARIMA and Seasonal ARIMA model identification, estimation, diagnostics, and forecasting. Modeling financial data via ARCH and GARCH models. Volatility estimation; additional topics, including long-range dependence and state-space models. 4 cr, either sem.

CAS MA 586 The Design of Experiments

Prereq: CAS MA 582 or equivalent or consent of instructor. Randomized blocks, Latin and Graeco-Latin squares, factorial arrangements with confounding and fractional replication, split-plot, crossover, and response surface designs. Treatment of missing data, group sizes, relative efficiency, relationship between design and analysis. 4 cr, 2nd sem.

CAS MA 587 Sampling Design: Theory and Methods

Prereq: CAS MA 582 or equivalent or consent of instructor. Stratified, cluster, systematic, multistage, double, and inverse sampling; optimum sample size, relative efficiency, sampling with unequal probabilities, types of estimators (ratio and regression) and their properties. Measurement error nonresponse and randomized response models. 4 cr, 1st sem.

CAS MA 588 Nonparametric Statistics

Prereq: CAS MA 582 or equivalent, or consent of instructor. The theory and logic in the development of nonparametric techniques including order statistics, tests based on runs, goodness of fit, rank-order (for location and scale), measures of association, analysis of variance, asymptotic relative efficiency. 4 cr, 2nd sem.

CAS MA 614 Statistical Methods

For graduate students in education and the social sciences. Not open to CAS students. Students may receive credit for not more than one of the following courses: CAS MA 116, MA 214, or MA 614. Second course in statistics, embodying basic statistical methods used in educational and social science research. Reviews all basic concepts covered in a first statistics course and presents, in detail, more advanced topics such as analysis of variance, covariance, experimental design, correlation, regression, and selected nonparametric techniques. A problem-solving course; students carry out analysis of data taken from educational and other social science sources. 4 cr, 1st sem.

CAS MA 684 Applied Multiple Regression and Multivariable Methods

Prereq: one year of statistics. Application of multivariate data analytic techniques. Multiple regression and correlation, confounding and interaction, variable selection, categorical predictors and outcomes, logistic regression, factor analysis, MANOVA, discriminant analysis, regression with longitudinal data, repeated measures, ANOVA. 4 cr, 2nd sem.

CAS MA 685 Advanced Topics in Applied Statistical Analysis

Prereq: CAS MA 684 or consent of instructor. Continues topics of CAS MA 684 at a more advanced level. Canonical correlation, multivariate analysis of variance, multivariate regressions. Categorical dependent variables techniques; discriminant analysis, logistic regression, log-linear analysis. Factor analysis; principal-axes, rotations, factor scores. Cluster analysis. Power analysis. Extensive use of statistical software. 4 cr, 1st sem.

Program in Cognitive and Neural Systems

See the Cognitive and Neural Systems section of this site.

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