# Mathematics (2016 - 2018)

**Algebra and Calculus**

MATH-UA 9 *Prerequisite: high school mathematics or permission of the department. Offered every term. 4 points.*

Intensive study of intermediate algebra and trigonometry. Topics include algebraic, exponential, logarithmic, and trigonometric functions and their graphs.

**Discrete Mathematics**

MATH-UA 120 *Prerequisite: Calculus I (MATH-UA 121) or Mathematics for Economics I (MATH-UA 211) with a grade of C or better, or permission of the department. Offered every term. 4 points.*

Sets, algorithms, and induction. Combinatorics. Graphs and trees. Combinatorial circuits. Logic and Boolean algebra.

**Calculus I**

MATH-UA 121 *Prerequisite: a score of 650 or higher on the mathematics portion of the SAT or on either SAT Subject Test in Mathematics, an ACT mathematics score of 30 or higher, a score of 3 or higher on the AP Calculus AB exam or AB subscore, a score of 3 or higher on the AP Calculus BC exam, a grade of C or higher in Algebra and Calculus (MATH-UA 9) or equivalent, or a passing score on the departmental placement exam. Offered every term. 4 points.*

Derivatives, antiderivatives, and integrals of functions of one variable. Applications include graphing, maximizing, and minimizing functions. Definite integrals and the fundamental theorem of calculus. Areas and volumes.

**Calculus II**

MATH-UA 122 *Prerequisite: Calculus I (MATHUA 121) or equivalent with a grade of C or better, a score of 4 or higher on the AP Calculus AB or BC exam, or a passing score on the departmental placement exam. Offered every term. 4 points.*

Techniques of integration. Further applications. Plane analytic geometry. Polar coordinates and parametric equations. Infinite series, including power series.

**Calculus III**

MATH-UA 123 *Prerequisite: Calculus II (MATH-UA 122) or equivalent with a grade of C or better, a score of 5 on the AP Calculus BC exam, or a passing score on the departmental placement exam. Offered every term. 4 points.*

Functions of several variables. Vectors in the plane and space. Partial derivatives with applications. Double and triple integrals. Spherical and cylindrical coordinates. Surface and line integrals. Divergence, gradient, and curl. Theorem of Gauss and Stokes.

**Set Theory**

MATH-UA 130 *Identical to PHIL-UA 73. 4 points.*

Boolean operations on sets; set-theoretic representation of relations, functions, and orderings; the natural numbers; theory of transfinite cardinal and ordinal numbers; the axiom of choice and its equivalents; and the foundations of analysis. May also cover large cardinals or independence results.

**Linear Algebra**

MATH-UA 140 *Prerequisite: Calculus I (MATHUA 121) or Math for Economics I (MATH-UA 211) (for economics majors) with a grade of C or better, or equivalent. Offered every term. 4 points.*

Systems of linear equations. Gaussian elimination, matrices, determinants, and Cramer’s rule. Vectors, vector spaces, basis and dimension, linear transformations. Eigenvalues, eigenvectors, quadratic forms.

**Introduction to Computer Simulation**

MATH-UA 144 *Identical to CSCI-UA 330. Prerequisite: Calculus I (MATH-UA 121) or Math for Economics II (MATH-UA 212) (for economics majors) with a grade of C or better, and General Physics I (PHYS-UA 11). Offered in the spring. 4 points.*

Simulations of planetary orbits, epidemic and endemic disease, musical stringed instruments, and urban traffic flow. Simulations are based on mathematical models, numerical methods, and Matlab programming techniques taught in class. Emphasizes use of animation (and sound where appropriate).

**Mathematics for Economics I**

MATH-UA 211 *Prerequisite: same as for Calculus I (MATH-UA 121). Only for declared and prospective majors in economics. Economics majors pursuing a double or joint major in mathematics may substitute MATH-UA 211, 212, and 213 for the regular calculus sequence. Cannot apply both standard calculus courses and Mathematics for Economics courses toward the mathematics major. Offered every term. 4 points.*

Introduces elements of calculus and linear algebra and appropriate tools for the study of economics in the policy concentration. Examples and motivation are drawn from important topics in economics. Topics include derivatives of functions of one and several variables; interpretations of the derivatives; convexity; and constrained and unconstrained optimization.

**Mathematics for Economics II**

MATH-UA 212 *Prerequisite: completion of Mathematics for Economics I (MATH-UA 211) with a C or higher, or placement by departmental exam. Only for declared and prospective majors in economics. Economics majors pursuing a double or joint major in mathematics may substitute MATH-UA 211, 212, and 213 for the regular calculus sequence. Cannot apply both standard calculus courses and Mathematics for Economics courses toward the mathematics major. Offered every term. 4 points.*

A continuation of Mathematics for Economics I. Matrix algebra; eigenvalues; ordinary differential equations and stability analysis; multivariable integration; and (time permitting) dynamic optimization.

**Mathematics for Economics III**

MATH-UA 213 *Prerequisite: Mathematics for Economics II (MATH-UA 212). Offered every term. 4 points.*

Further topics in vector calculus. Vector spaces, matrix analysis, and linear and nonlinear programming with applications to game theory. Provides economics majors who have taken Mathematics for Economics I, II (MATH-UA 211, 212) with prerequisite knowledge for higher-level mathematics courses.

**Vector Analysis**

MATH-UA 224 *Prerequisite: a grade of C or better in Analysis (MATH-UA 325). Offered in the spring. 4 points.*

Brief review of multivariate calculus: partial derivatives, chain rule, Riemann integral, change of variables, line integrals. Lagrange multipliers. Inverse and implicit function theorems and their applications. Introduction to calculus on manifolds: definition and examples of manifolds, tangent vectors and vector fields, differential forms, exterior derivative, line integrals and integration of forms. Gauss’s and Stokes’s theorems on manifolds.

**Earth’s Atmosphere and Ocean: Fluid Dynamics and Climate**

MATH-UA 228 *Identical to ENVST-UA 360. Prerequisite: Calculus I (MATH-UA 121) or Mathematics for Economics II (MATH-UA 212) (for economics majors) or equivalent with a grade of B-minus or better, and familiarity with introductory physics (at least at the advanced high school level). Recommended: Calculus III (MATH-UA 123). Offered in the spring. 4 points.*

The unifying principles of planetary fluid dynamics. Topics: global energy balance, convection and radiation (greenhouse effect), effects of planetary rotation (Coriolis force), structure of atmospheric circulation (Hadley cell and wind patterns), structure of oceanic circulation (wind-driven currents and thermohaline circulation), and climate and climate variability (including anthropogenic warming).

**Introduction to Fluid Dynamics**

MATH-UA 230 *Identical to PHYS-UA 180. Prerequisite: Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213) (for economics majors) with a grade of C or better. Recommended: Mathematical Physics (PHYS-UA 106). Offered in the spring. 4 points.*

Key concepts: formalism of continuum mechanics; conservation of mass, energy, and momentum in a fluid; Euler and Navier-Stokes equations; viscosity and vorticity. Concepts applied to potential flow around a cylinder, propagation of sound and gravity waves, and onset of instability in shear flow.

**Theory of Probability**

MATH-UA 233 *Prerequisite: a grade of C or better (B or better strongly recommended) in Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213) (for economics majors) or equivalent, and a grade of C or better in Linear Algebra (MATH-UA 140) or equivalent. The course is intended for mathematics majors and other students with a strong interest in mathematics and requires fluency in multi-variable integration. Not open to students who have taken Probability and Statistics (MATH-UA 235). Offered every term. 4 points.*

Mathematical techniques of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, Markov chains, and applications.

**Mathematical Statistics**

MATH-UA 234 *Prerequisite: a grade of C or better in Theory of Probability (MATH-UA 233) or equivalent. Not open to students who have taken Probability and Statistics (MATH-UA 235). Offered in the spring. 4 points.*

Mathematical foundations and techniques of statistical analysis used in the interpretation of data in quantitative sciences. Mathematical theory of sampling; normal populations and distributions; chi-square, t, and F distributions; hypothesis testing; estimation; confidence intervals; sequential analysis; correlation, regression, and analysis of variance. Applications to the sciences.

**Probability and Statistics**

MATH-UA 235 *Prerequisite: a grade of C or better in Calculus II (MATH-UA 122) or Mathematics for Economics II (MATH-UA 212) (for economics majors) or equivalent. Not open to students who have taken Theory of Probability (MATH-UA 233) or Mathematical Statistics (MATH-UA 234). Offered in the spring. 4 points.*

Combination of MATH-UA 233 and 234 at a more elementary level to acquaint students with both probability and statistics in a single term. In probability: mathematical treatment of chance; combinatorics; binomial, Poisson, and Gaussian distributions; law of large numbers and the normal distribution; application to coin-tossing; radioactive decay. In statistics: sampling; normal and other useful distributions; testing of hypotheses; confidence intervals; correlation and regression; applications to scientific, industrial, and financial data.

**Combinatorics**

MATH-UA 240 *Prerequisite: Calculus II (MATH-UA 122) or Mathematics for Economics II (MATH-UA 212) (for economics majors) with a grade of C or better, or equivalent. Offered every spring. 4 points.*

Techniques for counting and enumeration, including generating functions, the principle of inclusion and exclusion, and Polya counting. Graph theory. Modern algorithms and data structures for graph theoretic problems.

**Theory of Numbers**

MATH-UA 248 *Prerequisite: Calculus II (MATH-UA 122) or Mathematics for Economics II (MATH-UA 212) (for economics majors) with a grade of C or better, or equivalent. Offered in the fall. 4 points.*

Divisibility and prime numbers. Linear and quadratic congruences. The classical number-theoretic functions. Continued fractions. Diophantine equations.

**Mathematics of Finance**

MATH-UA 250 *Prerequisites: Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213) (for economics majors) and one of the following: Theory of Probability (MATH-UA 233), Probability and Statistics (MATH-UA 234), Statistics (ECON-UA 18), or Analytical Statistics (ECON-UA 20) with a grade of C-plus or better, and/or permission of the instructor. Offered every term. 4 points.*

Linear programming with application to pricing. Interest rates and present value. Basic probability, random walks, central limit theorem, Brownian motion, log-normal model of stock prices. Black- Scholes theory of options. Dynamic programming with application to portfolio optimization.

**Introduction to Mathematical Modeling**

MATH-UA 251 *Prerequisite: a grade of C or better in Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213) (for economics majors) or permission of the instructor. Offered every term. 4 points.*

Dimensional analysis, optimization, simulation, probability, and elementary differential equations are applied to natural and social sciences. Necessary mathematical and scientific background is developed as needed.

**Numerical Analysis**

MATH-UA 252 *Prerequisite: a grade of C or better in both Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213) (for economics majors) and Linear Algebra (MATH-UA 140), or equivalent. Offered in the spring. 4 points.*

Computer analysis and solutions of mathematical problems. Theory and practical examples using Matlab are combined to explore topics ranging from simple root-finding procedures to differential equations and the finite element method.

**Mathematics in Medicine and Biology**

MATH-UA 255 *Identical to BIOL-UA 255. Prerequisites: Calculus I (MATH-UA 121) or Mathematics for Economics II (MATH-UA 212) (for economics majors) and Principles of Biology I (BIOL-UA 11), or permission of the instructor. Offered in the fall. 4 points.*

Primarily for prehealth students. Topics include control of the heart, optimal principles in the lung, cell membranes, electrophysiology, countercurrent exchange in the kidney, acid-base balance, cardiac catheterization, and computer diagnosis. Material from the physical sciences is introduced and developed as needed.

**Computers in Medicine and Biology**

MATH-UA 256 *Identical to BIOL-UA 256. Prerequisite: Mathematics in Medicine and Biology (MATH-UA 255) with a grade of C or better, or permission of the instructor. Familiarity with a programming language such as Pascal, Fortran, or BASIC is recommended. Offered in the spring. 4 points.*

Introduces the student of biology or mathematics to the use of computers as tools for modeling physiological phenomena. Construction of computer models (circulation, gas exchange in the lung, control of cell volume, renal countercurrent mechanism). Simulated physiological experiments.

**Ordinary Differential Equations**

MATH-UA 262 *Prerequisites: a grade of C or better in both Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213) (for economics majors) and Linear Algebra (MATH-UA 140), or equivalent. Offered every term. 4 points.*

First- and second-order equations. Series solutions. Laplace transforms. Introduction to partial differential equations and Fourier series.

**Partial Differential Equations**

MATH-UA 263 *Prerequisite: a grade of C or better in Ordinary Differential Equations (MATH-UA 262) or equivalent. Offered every term. 4 points.*

The wave equation, the diffusion equation, and Laplace’s equation. Nonlinear conservation laws and the theory of shock waves. Applications to physics, chemistry, biology, and population dynamics.

**Chaos and Dynamical Systems**

MATH-UA 264 *Prerequisite: a grade of C or better in both Calculus II (MATH-UA 122) or Mathematics for Economics II (MATH-UA 212) (for economics majors) and Linear Algebra (MATH-UA 140), or equivalent. Offered in the spring. 4 points.*

Dynamics of maps and of first-order and secondorder differential equations: stability, bifurcations, limit cycles, and dissection of systems with fast and slow timescales. Geometric viewpoint, including phase planes. Chaotic behavior introduced in the context of one-variable maps (the logistic), fractal sets, etc. Applications from physics and biology. Computer lab sessions.

**Transformations and Geometries**

MATH-UA 270 *Prerequisite: a grade of C or better in Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213) (for economics majors) or equivalent. Strongly recommended: Linear Algebra (MATH-UA 140). This course is only open to mathematics education majors and prospective majors; does not count toward the CAS mathematics major. Offered every fall. 4 points.*

Axiomatic and algebraic study of Euclidean, non- Euclidean, affine, and projective geometries. Special attention is given to group-theoretic methods.

**Functions of a Complex Variable**

MATH-UA 282 *Prerequisites: a grade of C or better in both Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213) (for economics majors) and Linear Algebra (MATH-UA 140), or equivalent. Offered in the spring. 4 points.*

Complex numbers and complex functions. Differentiation and the Cauchy-Riemann equations. Cauchy’s theorem and the Cauchy integral formula. Singularities, residues, Taylor and Laurent series. Fractional linear transformations and conformal mapping. Analytic continuation. Applications to fluid flow.

**Analysis**

MATH-UA 325 *Prerequisites: a grade of C or better in Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213) (for economics majors) and Linear Algebra (MATH-UA 140), or equivalent. Offered every term. 4 points.*

Rigorous analysis on the real line. Topics: the real number system, sequences and series of numbers, functions of a real variable (continuity and differentiability), the Riemann integral, basic topological notions in a metric space, and sequences and series of functions (including Taylor and Fourier series).

**Honors Analysis I**

MATH-UA 328 *Prerequisites: a grade of C or better in Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213) (for economics majors) and Linear Algebra (MATH-UA 140), or equivalent. Recommended: the honors section of Calculus III and Honors Linear Algebra (MATH-UA 148). Offered in the fall. 4 points.*

Rigorous treatment of the foundations of real analysis in one variable, based entirely on proofs. Topics: properties of the real number system, sequences, continuous functions, topology of the real line, compactness, derivatives, the Riemann integral, sequences of functions, uniform convergence, infinite series, and Fourier series. Additional topics may include: Lebesgue measure and integral on the real line, metric spaces, and analysis on metric spaces.

**Honors Analysis II**

MATH-UA 329 *Prerequisites: a grade of C or better in Honors Analysis I (MATH-UA 328) or a grade of A in Analysis (MATH-UA 325) with permission of instructor, and Honors Linear Algebra (MATH-UA 148) with a grade of C or better or the equivalent. Offered in the spring. 4 points.*

Metric spaces, differentiation of functions of several real variables, the implicit and inverse function theorems, Riemann integral on R^n, Lebesgue measure on R^n, the Lebesgue integral.

**Algebra**

MATH-UA 343 *Prerequisites: a grade of C or better in both Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213) (for economics majors) and Linear Algebra (MATH-UA 140), or equivalent. Strongly recommended: Analysis (MATH-UA 325). Offered every term. 4 points.*

Introduction to abstract algebraic structures, including groups, rings, and fields. Sets and relations. Congruences and unique factorization of integers. Groups, permutation groups, homomorphisms, and quotient groups. Rings and quotient rings, Euclidean rings, polynomial rings. Fields, finite extensions.

**Honors Algebra I**

MATH-UA 348 *Prerequisites: a grade of C or better in both Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213) (for economics majors) and Linear Algebra (MATH-UA 140), or equivalent. Offered in the fall. 4 points.*

Introduction to abstract algebraic structures, including groups, rings, and fields. Sets and relations. Congruences and unique factorization of integers. Groups, permutation groups, group actions, homomorphisms and quotient groups, direct products, classification of finitely generated abelian groups, Sylow theorems. Rings, ideals and quotient rings, Euclidean rings, polynomial rings, unique factorization.

**Honors Algebra II**

MATH-UA 349 *Prerequisites: a grade of C or better in Honors Algebra I (MATH-UA 348), or a grade of A in Algebra (MATH-UA 343) and permission of instructor. Offered in the spring. 4 points.*

Principal ideal domains, polynomial rings in several variables, unique factorization domains. Fields, finite extensions, constructions with ruler and compass, Galois theory, solvability by radicals.

**Topology**

MATH-UA 375 *Prerequisite: a grade of C or better in Analysis (MATH-UA 325) or permission of the department. Offered in the spring. 4 points.*

Metric spaces, topological spaces, compactness, connectedness. Covering spaces and homotopy groups.

**Differential Geometry**

MATH-UA 377 *Prerequisite: a grade of C or better in Honors Analysis II (MATH-UA 329) or permission of the department. Offered in the spring. 4 points.*

The differential properties of curves and surfaces. Introduction to manifolds and Riemannian geometry.

**Honors I**

MATH-UA 393 *Prerequisite: permission of the department. Offered in the fall of even years. 4 points.*

Advanced topics, which vary yearly.

**Honors II**

MATH-UA 394 *Prerequisite: permission of the department. Offered in the spring of odd years. 4 points.*

Advanced topics, which vary yearly.

**Honors III**

MATH-UA 397 *Prerequisite: permission of the department. Offered in the fall of odd years. 4 points.*

Advanced topics, which vary yearly.

**Honors IV**

MATH-UA 398 *Prerequisite: permission of the department. Offered in the spring of even years. 4 points.*

Advanced topics, which vary yearly.

**Independent Study**

MATH-UA 997, 998 *Prerequisite: permission of the department. 2 or 4 points per term.*

Students must have a faculty sponsor and submit a research proposal to the director of undergraduate studies.

### Graduate Courses Open to Undergraduates

Qualified students may take certain mathematics courses in the Graduate School of Arts and Science (GSAS), provided they obtain permission from the director of undergraduate studies or vice chair for undergraduate affairs. A few such courses are listed below. Students should consult the GSAS Bulletin and the website of the Department of Mathematics for prerequisites, points per course, and descriptions. If these courses are used toward fulfillment of the requirements for the baccalaureate degree, no advanced credit is allowed for them in GSAS.

**Numerical Methods**

MATH-GA 2010, MATH-GA 2020

**Scientific Computing**

MATH-GA 2043

**Linear Algebra**

MATH-GA 2111

**Algebra**

MATH-GA 2130, MATH-GA 2140

**Number Theory**

MATH-GA 2210

**Topology**

MATH-GA 2320

**Differential Geometry I, II**

MATH-GA 2350, MATH-GA 2360

**Real Variables**

MATH-GA 2430

**Ordinary Differential Equations**

MATH-GA 2470

**Methods of Applied Mathematics**

MATH-GA 2701

**Fluid Dynamics**

MATH-GA 2702

**Advanced Topics in Math Biology (Mathematical Models of Primitive Organisms)**

MATH-GA 2851

**Probability: Limit Theorems I, II**

MATH-GA 2911, 2912