Reviews material presented in MATH 1106 lectures, provides problem-solving techniques and tips as well as prelim review. Provides further instruction for students who need reinforcement. Not a substitute for attending MATH 1106 lectures or discussions.

Full details for MATH 1006 - Academic Support for MATH 1106

Reviews material presented in MATH 1110 lectures, provides problem-solving techniques and tips as well as prelim review. Provides further instruction for students who need reinforcement. Not a substitute for attending MATH 1110 lectures or discussions.

Full details for MATH 1011 - Academic Support for MATH 1110

Reviews material presented in MATH 1120 lectures, provides problem-solving techniques and tips as well as prelim review. Provides further instruction for students who need reinforcement. Not a substitute for attending MATH 1120 lectures or discussions.

Full details for MATH 1012 - Academic Support for MATH 1120

Reviews material presented in MATH 2210 lectures, provides problem-solving techniques and tips as well as prelim review. Provides further instruction for students who need reinforcement. Not a substitute for attending MATH 2210 lectures or discussions.

Full details for MATH 1021 - Academic Support for MATH 2210

The goal of this course is to give students a strong basis in some quantitative skills needed in the life and social sciences. There will be an emphasis on modeling, using fundamental concepts from calculus developed in the course, including: derivatives, integrals, and introductory differential equations. Examples from the life sciences are used throughout the course. To give a concrete example, we will study predator-prey populations. We will write down mathematical models that describe the evolution of these populations, analyze both quantitative and qualitative properties to make predictions about the future of these populations, and discuss the assumptions and limitations of the models. Note that while we will cover the topics of derivatives and integrals, this course has a different, much more applied, focus from courses such as MATH 1110 or a typical high school calculus course.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 1106 - Modeling with Calculus for the Life Sciences

Topics include functions and graphs, limits and continuity, differentiation and integration of algebraic, trigonometric, inverse trig, logarithmic, and exponential functions; applications of differentiation, including graphing, max-min problems, tangent line approximation, implicit differentiation, and applications to the sciences; the mean value theorem; and antiderivatives, definite and indefinite integrals, the fundamental theorem of calculus, and the area under a curve.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 1110 - Calculus I

Focuses on integration: applications, including volumes and arc length; techniques of integration, approximate integration with error estimates, improper integrals, differential equations and their applications. Also covers infinite sequences and series: definition and tests for convergence, power series, Taylor series with remainder, and parametric equations.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 1120 - Calculus II

We apply mathematical reasoning to problems arising in the social sciences. We discuss game theory and its applications to questions of governing and the analysis of political conflicts. The problem of finding fair election procedures to choose among three or more alternatives is analyzed.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 1340 - Strategy, Cooperation, and Conflict

Essentially a second course in calculus. Topics include techniques of integration, finding areas and volumes by integration, exponential growth, partial fractions, infinite sequences and series, tests of convergence, and power series.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 1910 - Calculus for Engineers

Introduction to multivariable calculus. Topics include partial derivatives, double and triple integrals, line and surface integrals, vector fields, Green's theorem, Stokes' theorem, and the divergence theorem.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 1920 - Multivariable Calculus for Engineers

Topics include vector algebra, linear transformations, matrices, determinants, orthogonality, eigenvalues, and eigenvectors. Applications are made to linear differential or difference equations. The lectures introduce students to formal proofs. Students are required to produce some proofs in their homework and on exams.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 2210 - Linear Algebra

Differential and integral calculus of functions in several variables, line and surface integrals as well as the theorems of Green, Stokes, and Gauss.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 2220 - Multivariable Calculus

Topics include vector fields; line integrals; differential forms and exterior derivative; work, flux, and density forms; integration of forms over parametrized domains; and Green's, Stokes', and divergence theorems.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 2240 - Theoretical Linear Algebra and Calculus

An introduction to linear algebra for students interested in applications to data science. The course diverges from traditional linear algebra courses by emphasizing data science applications while teaching similar concepts. Key topics include matrices as data tables, high-dimensional datasets, singular value decomposition for data compression, and linear transformations in computer graphics.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 2310 - Linear Algebra for Data Science

Introduction to ordinary and partial differential equations. Topics include: first-order equations (separable, linear, homogeneous, exact); mathematical modeling (e.g., population growth, terminal velocity); qualitative methods (slope fields, phase plots, equilibria, and stability); numerical methods; second-order equations (method of undetermined coefficients, application to oscillations and resonance, boundary-value problems and eigenvalues); and Fourier series. A substantial part of this course involves partial differential equations, such as the heat equation, the wave equation, and Laplace's equation. (This part must be present in any outside course being considered for transfer credit to Cornell as a substitute for MATH 2930.)

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 2930 - Differential Equations for Engineers

Linear algebra and its applications. Topics include: matrices, determinants, vector spaces, eigenvalues and eigenvectors, orthogonality and inner product spaces; applications include brief introductions to difference equations, Markov chains, and systems of linear ordinary differential equations. May include computer use in solving problems.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 2940 - Linear Algebra for Engineers

In mathematics, the methodology of proof provides a central tool for confirming the validity of mathematical assertions, functioning much as the experimental method does in the physical sciences. In this course, students learn various methods of mathematical proof, starting with basic techniques in propositional and predicate calculus and in set theory and combinatorics, and then moving to applications and illustrations of these via topics in one or more of the three main pillars of mathematics: algebra, analysis, and geometry. Since cogent communication of mathematical ideas is important in the presentation of proofs, the course emphasizes clear, concise exposition.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 3040 - Prove It!

Provides a transition from calculus to real analysis. Topics include rigorous treatment of fundamental concepts in calculus: including limits and convergence of sequences and series, compact sets; continuity, uniform continuity and differentiability of functions. Emphasis is placed upon understanding and constructing mathematical proofs.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 3110 - Introduction to Analysis

An introductory course on number theory, the branch of algebra that studies the deeper properties of integers and their generalizations. Usually includes most of the following topics: the Euclidean algorithm, continued fractions, Pythagorean triples, Diophantine equations such as Pell's equation, congruences, quadratic reciprocity, binary quadratic forms, Gaussian integers, and factorization in quadratic number fields. May include a brief introduction to Fermat's Last Theorem.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 3320 - Introduction to Number Theory

An introduction to structures of abstract algebra, including groups, rings, fields, factorization of polynomials and integers, congruences, and the structure of finite abelian groups. Additional topics include modules over Euclidean domain and Sylow theorems.

Catalog Distribution: (SMR-AS)

Full details for MATH 3340 - Abstract Algebra

Introduction to the concepts and methods of abstract algebra that are of interest in applications. Covers the basic theory of groups, rings and fields and their applications to such areas as public-key cryptography and error-correcting codes. Applications include the RSA cryptosystem and use of finite fields to construct error-correcting codes. Topics include elementary number theory, Euclidean algorithm, prime factorization, congruences, theorems of Fermat and Euler, elementary group theory, Chinese remainder theorem, factorization in the ring of polynomials, and classification of finite fields.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 3360 - Applicable Algebra

Development of mathematics from Babylon and Egypt and the Golden Age of Greece through its nineteenth century renaissance in the Paris of Cauchy and Lagrange and the Berlin of Weierstrass and Riemann. Covers basic algorithms underlying algebra, analysis, number theory, and geometry in historical order. Theorems and exercises cover the impossibility of duplicating cubes and trisecting angles, which regular polygons can be constructed by ruler and compass, the impossibility of solving the general fifth degree algebraic equation by radicals, the transcendence of pi. Students give presentations from original sources over 5000 years of mathematics.

Catalog Distribution: (SMR-AS)

Full details for MATH 4030 - History of Mathematics

Introduction to the rigorous theory underlying calculus, covering the real number system and functions of one variable. Based entirely on proofs. The student is expected to know how to read and, to some extent, construct proofs before taking this course. Topics typically include construction of the real number system, properties of the real number system, continuous functions, differential and integral calculus of functions of one variable, sequences and series of functions.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 4130 - Honors Introduction to Analysis I

Proof-based introduction to further topics in analysis. Topics may include the Lebesgue measure and integration, functions of several variables, differential calculus, implicit function theorem, infinite dimensional normed and metric spaces, Fourier series, ordinary differential equations.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 4140 - Honors Introduction to Analysis II

Theoretical and rigorous introduction to complex variable theory. Topics include complex numbers, differential and integral calculus for functions of a complex variable including Cauchy's theorem and the calculus of residues, elements of conformal mapping.

Catalog Distribution: (SMR-AS)

Full details for MATH 4180 - Complex Analysis

Covers ordinary differential equations in one and higher dimensions: qualitative, analytic, and numerical methods. Emphasis is on differential equations as models and the implications of the theory for the behavior of the system being modeled and includes an introduction to bifurcations.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 4200 - Differential Equations and Dynamical Systems

Introduction to the fundamentals of numerical linear algebra: direct and iterative methods for linear systems, eigenvalue problems, singular value decomposition. In the second half of the course, the above are used to build iterative methods for nonlinear systems and for multivariate optimization. Strong emphasis is placed on understanding the advantages, disadvantages, and limits of applicability for all the covered techniques. Computer programming is required to test the theoretical concepts throughout the course.

Catalog Distribution: (SMR-AS)

Full details for MATH 4260 - Numerical Analysis: Linear and Nonlinear Problems

Topics are selected from first-order quasilinear equations, classification of second-order equations, with emphasis on maximum principles, existence, uniqueness, stability, Fourier series methods. Additional topics as time permits.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 4280 - Introduction to Partial Differential Equations

Introduction to linear algebra, including the study of vector spaces, linear transformations, matrices, and systems of linear equations. Additional topics are quadratic forms and inner product spaces, canonical forms for various classes of matrices and linear transformations.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 4310 - Linear Algebra

Honors version of a course in abstract algebra, which treats the subject from an abstract and axiomatic viewpoint, including universal mapping properties. Topics include groups, groups acting on sets, Sylow theorems; rings, Euclidean domains, factorization, structure theorem of finitely generated modules over a principal ideal domain; fields, root adjunction, finite fields, introduction to Galois theory. The course emphasizes understanding the theory with proofs in both homework and exams.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 4340 - Honors Introduction to Algebra

An introduction to a topic that is central to mathematics and important in physics and engineering. The objects of study are certain classes of matrices, such as orthogonal, unitary, or symplectic matrices. These classes have both algebraic structure (groups) and geometric/topological structure (manifolds). Thus the course will be a mixture of algebra and geometry/topology, with a little analysis as well. The topics will include Lie algebras (which are an extension of the notion of vector multiplication in three-dimensional space), the exponential mapping (a generalization of the exponential function of calculus), and representation theory (which studies the different ways in which groups can be represented by matrices). Concrete examples will be emphasized. Background not included in the prerequisites will be developed as needed.

Catalog Distribution: (SMR-AS)

Full details for MATH 4500 - Matrix Groups

Differential geometry involves using calculus to study geometric concepts such as curvature and geodesics. This introductory course focuses on the differential geometry of curves and surfaces. It may also touch upon the higher-dimensional generalizations, Riemannian manifolds, which underlie the study of general relativity.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 4540 - Introduction to Differential Geometry

Introduction to probability theory, which prepares the student to take MATH 4720. The course begins with basics: combinatorial probability, mean and variance, independence, conditional probability, and Bayes formula. Density and distribution functions and their properties are introduced. The law of large numbers and the central limit theorem are stated and their implications for statistics are discussed.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 4710 - Basic Probability

Statistics have proved to be an important research tool in nearly all of the physical, biological, and social sciences. This course serves as an introduction to statistics for students who already have some background in calculus, linear algebra, and probability theory. Topics include parameter estimation, hypothesis testing, and linear regression. The course emphasizes both the mathematical theory of statistics and techniques for data analysis that are useful in solving scientific problems.

Catalog Distribution: (SDS-AS, SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 4720 - Statistics

A one-semester introduction to stochastic processes which develops the theory together with applications. The course will always cover Markov chains in discrete and continuous time and Poisson processes. Depending upon the interests of the instructor and the students, other topics may include queuing theory, martingales, Brownian motion, and option pricing.

Catalog Distribution: (SMR-AS) (MQL-AG, OPHLS-AG)

Full details for MATH 4740 - Stochastic Processes

An independent research course by arrangement with an individual professor.  The goal is for the student to perform an independent investigation into a specific mathematical question.  The student and professor will set expectations and grading policies at the beginning of the term.

Full details for MATH 4900 - Supervised Research

An independent reading course by arrangement with an individual professor. The goal is for the student to master a body of mathematics outside the normal curriculum. The student and professor will set expectations and grading policies at the beginning of the term.

Full details for MATH 4901 - Supervised Reading

Examines principles underlying the content of the secondary school mathematics curriculum, including connections with the history of mathematics, technology, and mathematics education research. One credit is awarded for attending two Saturday workshops (see math.cornell.edu/math-5080) and writing a paper. Other credit options are available for students completing additional work, such as tutoring at a local middle school or completing a research paper or project.

Full details for MATH 4980 - Special Study for Mathematics Teaching

Examines principles underlying the content of the secondary school mathematics curriculum, including connections with the history of mathematics, technology, and mathematics education research.

Full details for MATH 5080 - Special Study for Teachers

Covers ordinary differential equations in one and higher dimensions: qualitative, analytic, and numerical methods. Emphasis is on differential equations as models and the implications of the theory for the behavior of the system being modeled and includes an introduction to bifurcations.

Full details for MATH 5200 - Differential Equations and Dynamical Systems

MATH 6110-6120 are the core analysis courses in the mathematics graduate program. MATH 6120 covers complex analysis, Fourier analysis, and distribution theory.

Full details for MATH 6120 - Complex Analysis

Functional analysis is a branch of mathematical analysis that mainly focuses on the study of infinite-dimensional vector spaces and the operators acting upon them. It builds upon results and ideas from linear algebra and real and complex analysis to develop general frameworks that can be used to study analytical problems. Functional analysis plays a pivotal role in several areas of mathematics, physics, engineering, and even in some areas of computer science and economics. This course will cover the basic theory of Banach, Hilbert, and Sobolev spaces, as well as explore several notable applications, from analyzing partial differential equations (PDEs), numerical analysis, inverse problems, control theory, optimal transportation, and machine learning.

Full details for MATH 6220 - Applied Functional Analysis

A mathematically rigorous course on lattices. Lattices are periodic sets of vectors in high-dimensional space. They play a central role in modern cryptography, and they arise naturally in the study of high-dimensional geometry (e.g., sphere packings). We will study lattices as both geometric and computational objects. Topics include Minkowski's celebrated theorem, the famous LLL algorithm for finding relatively short lattice vectors, Fourier-analytic methods, basic cryptographic constructions, and modern algorithms for finding shortest lattice vectors. We may also see connections to algebraic number theory.

Full details for MATH 6302 - Lattices: Geometry, Cryptography, and Algorithms

MATH 6310-6320 are the core algebra courses in the mathematics graduate program. MATH 6320 covers Galois theory, representation theory of finite groups, and introduction to homological algebra.

Full details for MATH 6320 - Algebra

An introduction to number theory focusing on the algebraic theory. Topics include, but are not limited to, number fields, Dedekind domains, class groups, Dirichlet's unit theorem, local fields, ramification, decomposition and inertia groups, and the distribution of primes.

Full details for MATH 6370 - Algebraic Number Theory

MATH 6510-MATH 6520 are the core topology courses in the mathematics graduate program. MATH 6510 is an introductory study of certain geometric processes for associating algebraic objects such as groups to topological spaces. The most important of these are homology groups and homotopy groups, especially the first homotopy group or fundamental group, with the related notions of covering spaces and group actions. The development of homology theory focuses on verification of the Eilenberg-Steenrod axioms and on effective methods of calculation such as simplicial and cellular homology and Mayer-Vietoris sequences. If time permits, the cohomology ring of a space may be introduced.

Full details for MATH 6510 - Algebraic Topology

This course is an introduction to some of the fundamentals of homotopy theory. Homotopy theory studies spaces up to homotopy equivalence, not just up to homeomorphism. This allows for a variety of algebraic techniques which are not available when working up to homeomorphism. This class studies the fundamentals and tools of homotopy theory past homology and cohomology. Topics may include computations of higher homotopy groups, simplicial sets, model categories, spectral sequences, and rational homotopy theory.

Full details for MATH 6540 - Homotopy Theory

Topics include linear connections, Riemannian metrics and parallel translation; covariant differentiation and curvature tensors; the exponential map, the Gauss Lemma and completeness of the metric; isometries and space forms, Jacobi fields and the theorem of Cartan-Hadamard; the first and second variation formulas; the index form of Morse and the theorem of Bonnet-Myers; the Rauch, Hessian, and Laplacian comparison theorems; the Morse index theorem; the conjugate and cut loci; and submanifolds and the Second Fundamental form.

Full details for MATH 6620 - Riemannian Geometry

A first course in algebraic geometry. Affine and projective varieties. The Nullstellensatz. Schemes and morphisms between schemes. Dimension theory. Potential topics include normalization, Hilbert schemes, curves and surfaces, and other choices of the instructor.

Full details for MATH 6670 - Algebraic Geometry

The second course in a graduate probability series. Topics include conditional expectation, martingales, Markov chains, Brownian motion, and (time permitting) elements of stochastic integration.

Full details for MATH 6720 - Probability Theory II

This course will focus on the finite sample theory of statistical inference, emphasizing estimation, hypothesis testing, and confidence intervals.  Specific topics include: uniformly minimum variance unbiased estimators, minimum risk equivariant estimators, Bayes estimators, minimax estimators, the Neyman-Pearson theory of hypothesis testing, and the construction of optimal invariant tests.

Full details for MATH 6730 - Mathematical Statistics I

Covers basic topics in mathematical logic, including propositional and predicate calculus; formal number theory and recursive functions; completeness and incompleteness theorems, compactness and Skolem-Loewenheim theorems. Other topics as time permits.

Full details for MATH 6810 - Logic

An introduction to (mostly Euclidean) harmonic analysis. Topics usually include convergence of Fourier series, harmonic functions and their conjugates, Hilbert transform, Calderon-Zygmund theory, Littlewood-Paley theory, pseudo-differential operators, restriction theory of the Fourier transform, connections to PDE. Applications to number theory and/or probability theory may also be discussed, as well as Fourier analysis on groups.

Full details for MATH 7150 - Fourier Analysis

Selection of advanced topics from partial differential equations. Content varies.

Full details for MATH 7160 - Topics in Partial Differential Equations

Talks on various methods in scientific computing, the analysis of their convergence properties and computational efficiency, and their adaptation to specific applications.

Full details for MATH 7290 - Seminar on Scientific Computing and Numerics

Selection of advanced topics from algebra. Course content varies.

Full details for MATH 7310 - Topics in Algebra

Selection of advanced topics from number theory. Course content varies.

Full details for MATH 7370 - Topics in Number Theory

Selection of advanced topics in combinatorics. Course content varies.

Full details for MATH 7410 - Topics in Combinatorics

Selection of advanced topics from modern geometry. Content varies.

Full details for MATH 7620 - Topics in Geometry

A twice weekly seminar in logic. Typically, a topic is selected for each semester, and at least half of the meetings of the course are devoted to this topic with presentations primarily by students. Opportunities are also provided for students and others to present their own work and other topics of interest.

Full details for MATH 7820 - Seminar in Logic

Supervised research for the doctoral dissertation.

Full details for MATH 7900 - Supervised Reading and Research