Mathematics (2022 - 2024)

Please note the following:

• Beginning with the fall 2022 semester, Algebra and Calculus (MATH-UA 9) is retitled as Algebra, Trigonometry, and Functions. There is no change to the content or level of the course. It is a course in precalculus and does not include any calculus.
• Beginning with the fall 2022 semester, the Mathematics for Economics I, II, III course sequence is renumbered as MATH-UA 131, 132, 133 (formerly MATH-UA 211, 212, 213). There is no change to the content or level of these courses, and they continue to satisfy the same major/minor requirements across the College.
• All SAT Subject Examinations (including those in mathematics, which are noted below as placing students into some MATH-UA courses) were discontinued as of January 2021 in the U.S. and after June 2021 internationally.

Algebra, Trigonometry, and Functions
MATH-UA 9 Formerly Algebra and Calculus. Prerequisite: high school mathematics or permission of the department. Offered every term. 4 points.
Serves as preparation for students who do not otherwise place into Discrete Mathematics (MATH-UA 120), Calculus I (MATH-UA 121), Mathematics for Economics I (MATH-UA 131; formerly MATH-UA 211), Linear Algebra (MATH-UA 140), Honors Linear Algebra (MATH-UA 148; an A- in MATH-UA 9 is required to enter MATH-UA 148), and some courses in other departments (such as chemistry and economics). Intermediate algebra and trigonometry; algebraic, exponential, logarithmic, and trigonometric functions and their graphs.

Discrete Mathematics
MATH-UA 120 Prerequisite: one of the following: a score of 670 or higher on the Mathematics portion of the SAT, 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, Trigonometry, and Functions (MATH-UA 9; formerly Algebra and Calculus) or equivalent, or a passing score on the departmental placement exam. Also acceptable as a prerequisite: for students entering NYU prior to fall 2021: an IB Mathematics SL score of 6 or higher, an IB Mathematical Studies SL score of 7, or an IB Mathematics HL score of 5; for students entering NYU in fall 2021 or later: an IB Analysis and Approaches SL score of 7, an IB Analysis and Approaches HL score of 5, or an IB Applications and Interpretations HL score of 5. 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: one of the following: a score of 670 or higher on the Mathematics portion of the SAT, 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, Trigonometry, and Functions (MATH-UA 9; formerly Algebra and Calculus) or equivalent, or a passing score on the departmental placement exam. Also acceptable as a prerequisite: for students entering NYU prior to fall 2021: an IB Mathematics SL score of 6 or higher, an IB Mathematical Studies SL score of 7, or an IB Mathematics HL score of 5; for students entering NYU in fall 2021 or later: an IB Analysis and Approaches SL score of 7, an IB Analysis and Approaches HL score of 5, or an IB Applications and Interpretations HL score of 5. 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: one of the following: Calculus I (MATH-UA 121) or equivalent with a grade of C or higher, a score of 4 or higher on the AP Calculus AB or BC exam, or a passing score on the departmental placement exam. Also acceptable as a prerequisite: an IB Analysis and Approaches HL score of 6 or an IB Applications and Interpretations HL score of 6. 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: one of the following: Calculus II (MATH-UA 122) or equivalent with a grade of C or higher, a score of 5 on the AP Calculus BC exam, or a passing score on the departmental placement exam. Also acceptable as a prerequisite: an IB Analysis and Approaches HL score of 7. 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.

Honors Calculus III
MATH-UA 129 Prerequisite: one of the following: Calculus II (MATH-UA 122) with a grade of A- or higher or an equivalent, a score of 5 on the Advanced Placement Calculus BC examination. Also acceptable as a prerequisite: an IB Analysis and Approaches HL score of 7.  Offered in the fall and spring. 4 points.
Covers more material, with more rigor and at a faster pace, than Calculus III (MATH-UA 123). Functions of several variables. Vectors in the plane and space. Partial derivatives with applications, especially Lagrange multipliers. Double and triple integrals. Spherical and cylindrical coordinates. Surface and line integrals. Divergence, gradient, and curl. Theorem of Gauss and Stokes.

Mathematics for Economics I
MATH-UA 131 Formerly 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 131, 132, and 133 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. 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 132 Formerly MATH-UA 212. Prerequisite: completion of Mathematics for Economics I (MATH-UA 131; formerly 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 131, 132, and 133 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 133 Formerly MATH-UA 213. Prerequisite: Mathematics for Economics II (MATH-UA 132; formerly MATH-UA 212). Economics majors pursuing a double or joint major in mathematics may substitute MATH-UA 131, 132, and 133 for the regular calculus sequence. Cannot apply both standard calculus courses and Mathematics for Economics courses toward the mathematics major. Offered in the fall and spring. 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 131, 132; formerly MATH-UA 211, 212) with prerequisite knowledge for higher-level mathematics courses.

Linear Algebra
MATH-UA 140 Prerequisite: one of the following: a score of 670 or higher on the Mathematics portion of the SAT, 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, Trigonometry, and Functions (MATH-UA 9; formerly Algebra and Calculus) or equivalent, or a passing score on the departmental placement exam. Also acceptable as a prerequisite: an IB Analysis and Approaches SL score of 7, an IB Analysis and Approaches HL score of 5, or an IB Applications and Interpretations HL score of 5. Offered in the fall and spring. 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 Mathematics for Economics II (MATH-UA 132; formerly MATH-UA 212) (for economics majors) with a grade of C or higher, 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).

Honors Linear Algebra
MATH-UA 148 Prerequisite: one of the following: a score of 670 or higher on the Mathematics portion of the SAT, 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 A- or higher in Algebra, Trigonometry, and Functions (MATH-UA 9; formerly Algebra and Calculus) or equivalent, or a passing score on the departmental placement exam. Also acceptable as a prerequisite: an IB Analysis and Approaches SL score of 7, an IB Analysis and Approaches HL score of 5, or an IB Applications and Interpretations HL score of 5. Offered in the fall and spring. 4 points.
Intended for well-prepared students who have already developed some mathematical maturity. Includes usual Linear Algebra (MATH-UA 140) topics, but is accelerated, covers additional topics, and delves more deeply. Vector spaces, linear dependence, basis and dimension, matrices, determinants, solving linear equations, eigenvalues and eigenvectors, quadratic forms, applications such as optimization or linear regression.

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 132; formerly MATH-UA 212) (for economics majors) or equivalent with a grade of B- or higher, 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 Honors Calculus III (MATH-UA 129) or Mathematics for Economics III (MATH-UA 133; formerly MATH-UA 213) (for economics majors) with a grade of C or higher. 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.

Set Theory
MATH-UA 232 Identical to PHIL-UA 73. Offered at the discretion of the Department of Philosophy. 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.

Theory of Probability
MATH-UA 233 Prerequisites: a grade of C or higher (B or higher strongly recommended) in Calculus III (MATH-UA 123) or Honors Calculus III (MATH-UA 129) or Mathematics for Economics III (MATH-UA 133; formerly MATH-UA 213) (for economics majors) or equivalent, and a grade of C or higher in Linear Algebra (MATH-UA 140), or equivalents. 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 higher 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 fall and 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 higher in Calculus II (MATH-UA 122) or Mathematics for Economics II (MATH-UA 132; formerly 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 an 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.

Honors Theory of Probability
MATH-UA 238 Prerequisites: Discrete Mathematics (MATH-UA 120), either Calculus III (MATH-UA 123) or Honors Calculus III (MATH-UA 129) or Mathematics for Economics III (MATH-UA 133; formerly MATH-UA 213) (for economics majors), and either Linear Algebra (MATH-UA 140) or Honors Linear Algebra (MATH-UA 148), all completed with a B+ or higher. Not open to students who have taken Theory of Probability (MATH-UA 233) or Probability and Statistics (MATH-UA 235). Offered in the spring. 4 points.
Covers more material, with more rigor and at a faster pace, than Theory of Probability (MATH-UA 233). Covers discrete and continuous probability and the most fundamental limit theorems (law of large numbers and central limit theorem).

Combinatorics
MATH-UA 240 Prerequisite: Calculus II (MATH-UA 122) or Mathematics for Economics II (MATH-UA 132; formerly MATH-UA 212) (for economics majors) with a grade of C or higher, 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 132; formerly MATH-UA 212) (for economics majors) with a grade of C or higher, 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: either Calculus III (MATH-UA 123) or Honors Calculus III (MATH-UA 129) or Mathematics for Economics III (MATH-UA 133; formerly 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+ or higher, and/or permission of the instructor. Offered in the fall and spring. 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 higher in either Calculus III (MATH-UA 123) or Honors Calculus III (MATH-UA 129) or Mathematics for Economics III (MATH-UA 133; formerly MATH-UA 213) (for economics majors) or permission of the instructor. Offered in the fall and spring. 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 Prerequisites: a grade of C or higher in either Calculus III (MATH-UA 123) or Honors Calculus III (MATH-UA 129) or Mathematics for Economics III (MATH-UA 133; formerly MATH-UA 213) (for economics majors), and either a grade of B- or higher in Linear Algebra (MATH-UA 140) or a grade of C or higher in Honors Linear Algebra (MATH-UA 148), or equivalents. Offered in the fall and 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.

Linear and Nonlinear Optimization
MATH-UA 253 Prerequisites: a grade of C or higher in Calculus III (MATH-UA 123) or Honors Calculus III (MATH-UA 129) or Mathematics for Economics III (MATH-UA 133; formerly MATH-UA 213) (for economics majors) or the equivalent, and a grade of C or higher in either Linear Algebra (MATH-UA 140) or Honors Linear Algebra (MATH-UA 148) or the equivalent. 4 points.
An application-oriented introduction to linear programming and convex optimization. Numerical implementations often require programming (no experience necessary). Algorithmic topics: the simplex method for linear programming, techniques for smooth multidimensional optimization, solution techniques for L1-type optimizations, and stochastic gradient descent. Applications in economics, data science, and operations research.

Mathematics in Medicine and Biology
MATH-UA 255 Identical to BIOL-UA 255. Prerequisites: either Calculus I (MATH-UA 121) or Mathematics for Economics II (MATH-UA 132; formerly 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 higher, 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 higher in either Calculus III (MATH-UA 123) or Honors Calculus III (MATH-UA 129) or Mathematics for Economics III (MATH-UA 133; formerly MATH-UA 213) (for economics majors), and a grade of C or higher in either Linear Algebra (MATH-UA 140) or Honors Linear Algebra (MATH-UA 148), or equivalents. Offered in the fall and spring. 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 higher in Ordinary Differential Equations (MATH-UA 262) or Honors Ordinary Differential Equations (MATH-UA 268) or equivalent. Offered in the fall and spring. 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 Prerequisites: a grade of C or higher in either Calculus II (MATH-UA 122) or Mathematics for Economics II (MATH-UA 132; formerly MATH-UA 212) (for economics majors), and a grade of C or higher in either Linear Algebra (MATH-UA 140) or Honors Linear Algebra (MATH-UA 148), or equivalents. Offered in the spring. 4 points.
Dynamics of maps and of first-order and second-order 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.

Honors Ordinary Differential Equations
MATH-UA 268 Prerequisite: An A- or higher in Analysis (MATH-UA 325) or a B+ or higher in Honors Analysis I (MATH-UA 328). Offered in the fall. 4 points.
Examines the existence and uniqueness of solutions to ODEs for both linear and nonlinear problems, set on the real line or the complex plane. Behavior of the solutions, with possible connections to dynamical systems theory. Applications in physics and biology.

Functions of a Complex Variable
MATH-UA 282 Prerequisites: a grade of C or higher in either Calculus III (MATH-UA 123) or Honors Calculus III (MATH-UA 129) or Mathematics for Economics III (MATH-UA 133; formerly MATH-UA 213) (for economics majors), and a grade of C or higher in either Linear Algebra (MATH-UA 140) or Honors Linear Algebra (MATH-UA 148), or equivalents. 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 higher in either Calculus III (MATH-UA 123) or Honors Calculus III (MATH-UA 129) or Mathematics for Economics III (MATH-UA 133; formerly MATH-UA 213) (for economics majors), and a grade of C or higher in either Linear Algebra (MATH-UA 140) or Honors Linear Algebra (MATH-UA 148), or equivalents. Offered in the fall and spring. 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 A- or higher in either Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 133; formerly MATH-UA 213) (for economics majors), and a grade of A- or higher in Linear Algebra (MATH-UA 140), or equivalents. Recommended: a B+ or higher in both Honors Calculus III (MATH-UA 129) and Honors Linear Algebra (MATH-UA 148). Offered in the fall and spring. 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: either a grade of C or higher 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 higher, or the equivalents. 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 higher in either Calculus III (MATH-UA 123) or Honors Calculus III (MATH-UA 129) or Mathematics for Economics III (MATH-UA 133; formerly MATH-UA 213) (for economics majors), and a grade of C or higher in either Linear Algebra (MATH-UA 140) or Honors Linear Algebra (MATH-UA 148), or equivalents. Strongly recommended: Analysis (MATH-UA 325). Offered in the fall and spring. 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 A- or higher in either Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 133; formerly MATH-UA 213) (for economics majors), and a grade of A- in Linear Algebra (MATH-UA 140), or equivalents. Recommended: a B+ or higher in both Honors Calculus III (MATH-UA 129) and Honors Linear Algebra (MATH-UA 148). 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 Prerequisite: a grade of C or higher 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 higher in Analysis (MATH-UA 325) or Honors Analysis I (MATH-UA 328) 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 higher in either Calculus III (MATH-UA 123) or Honors Calculus III (MATH-UA 129) or Mathematics for Economics III (MATH-UA 133; formerly MATH-UA 213) (for economics majors), and a grade of C or higher in either Linear Algebra (MATH-UA 140) or Honors Linear Algebra (MATH-UA 148), or equivalents. Recommended: Analysis (MATH-UA 325) or Honors Analysis I (MATH-UA 328). Offered in the spring. 4 points.
The geometry of curves and surfaces in Euclidean space. Frenet formulas, the isoperimetric inequality, local theory of surfaces in Euclidean space, first and second fundamental forms. Gaussian and mean curvature, isometries, geodesics, parallelism, the Gauss-Bonnet Theorem.

Topics Courses

Special Topics
MATH-UA 395, 396 Topics determine prerequisites. Offered in the fall and spring respectively. 4 points.
Topics vary by semester.

Honors Topics Courses

Honors I
MATH-UA 393 Prerequisite: permission of the department. May be repeated once for credit as topics change. Offered in the fall of even years. 4 points.
Advanced topics, which vary yearly.

Honors II
MATH-UA 394 Prerequisite: permission of the department. May be repeated once for credit as topics change. Offered in the spring of odd years. 4 points.
Advanced topics, which vary yearly.

Honors III
MATH-UA 397 Prerequisite: permission of the department. May be repeated once for credit as topics change. Offered in the fall of odd years. 4 points.
Advanced topics, which vary yearly.

Honors IV
MATH-UA 398 Prerequisite: permission of the department. May be repeated once for credit as topics change. Offered in the spring of even years. 4 points.
Advanced topics, which vary yearly.

Internship and Independent Study

Internship
MATH-UA 897, 898 Prerequisite: permission of the department. Student must be a declared mathematics major, have a mathematics GPA of 3.5 and an overall GPA of 3.0, and have at least half of the mathematics major courses completed. Offered every term. 2 or 4 points per term.
Students must complete an online enrollment request form (contact the department for details) and have the approval of the director of undergraduate studies.

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 Center for Data Science (CDS) and 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 websites of the CDS and 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 graduate-level credit is allowed for them in CNS or GSAS.

CENTER FOR DATA SCIENCE

Introduction to Data Science
DS-GA 1001

Statistical and Mathematical Methods
DS-GA 1002

DEPARTMENT OF MATHEMATICS

Computing in Finance
MATH-GA 2041

Scientific Computing
MATH-GA 2043

Computational Methods for Finance
MATH-GA 2045

Advanced Econometric Modeling and Big Data
MATH-GA 2046

Scientific Computing in Finance
MATH-GA 2048

Topology I
MATH-GA 2310

Differential Geometry I
MATH-GA 2350

Ordinary Differential Equations
MATH-GA 2470

Partial Differential Equations I
MATH-GA 2490

Fluid Dynamics
MATH-GA 2702

Applied Stochastic Analysis
MATH-GA 2704

Time Series Analysis and Statistical Arbitrage
MATH-GA 2707

Algorithmic Trading and Quantitative Strategies
MATH-GA 2708

Mechanics
MATH-GA 2710

Risk and Portfolio Management with Econometrics
MATH-GA 2751

Active Portfolio Management
MATH-GA 2752

Advanced Risk Management
MATH-GA 2753

Regulation and Regulatory Risk Models
MATH-GA 2757

Derivative Securities
MATH-GA 2791

Continuous Time Finance
MATH-GA 2792

Securitized Products and Energy Derivatives
MATH-GA 2796

Credit Markets and Models
MATH-GA 2797

Interest Rates & Fx Models
MATH-GA 2798

Basic Probability
MATH-GA 2901

Stochastic Calculus
MATH-GA 2902