Weill Medical College of Cornell University · Courses
MATH
133 courses with the subject MATH, each shown exactly as we captured it from the college's catalog, with every element we hold. Where the wording looks broken, that is our reading of the catalog, not the college's text.
MATH 1001Core Reasoning and Skills for STEM Courses2.5
A thorough review of pre-calculus: linear, power, exponential, logarithmic, and trigonometric functions; solving equations; working with graphical representations of common functions; foundational statistical skills. Emphasis on core numeracy skills used in the variety of quantitative courses students may find themselves in at Cornell. Each student will have opportunities to engage with the material at a depth that is appropriate to their math background.
MATH 1004Scaffolding Understanding Through Mathematical Systems2.5
This course helps students connect mathematics to real-world applications across multiple disciplines. Designed for those exploring majors or seeking deeper understanding, the course emphasizes problem-solving and reasoning rather than rote procedures. Students will work in small, collaborative groups using an inquiry-based approach. Each week introduces foundational concepts from different fields such as Sociology and Statistics, followed by large-scale problems that require applying these ideas in authentic scenarios. The course aims to develop a flexible mindset and the ability to synthesize concepts, while strengthening core math skills.
Reviews material presented in MATH 1106 lectures and provides further instruction for students who need reinforcement, including problem-solving techniques and tips as well as prelim review. Not a substitute for attending MATH 1106 lectures or discussions. Students should contact their college for the most up-to-date information regarding if and how credits for this course will count toward graduation, and/or be considered regarding academic standing.
Reviews material presented in MATH 1110 lectures and provides further instruction for students who need reinforcement, including problem-solving techniques and tips as well as prelim review. Not a substitute for attending MATH 1110 lectures or discussions. Students should contact their college for the most up-to-date information regarding if and how credits for this course will count toward graduation, and/or be considered regarding academic standing.
Reviews material presented in MATH 1120 lectures and provides further instruction for students who need reinforcement, including problem-solving techniques and tips as well as prelim review. Not a substitute for attending MATH 1120 lectures or discussions. Students should contact their college for the most up-to-date information regarding if and how credits for this course will count toward graduation, and/or be considered regarding academic standing.
Reviews material presented in MATH 2210 lectures and provides further instruction for students who need reinforcement, including problem-solving techniques and tips as well as prelim review. Not a substitute for attending MATH 2210 lectures or discussions. Students should contact their college for the most up-to-date information regarding if and how credits for this course will count toward graduation, and/or be considered regarding academic standing.
Introduces topics in calculus: limits, rates of change, definition of and techniques for finding derivatives, relative and absolute extrema, and applications. The calculus content of the course is similar to 1/3 of the content covered in MATH 1106 and MATH 1110. In addition, the course includes a variety of topics of algebra, with emphasis on the development of linear, power, exponential, logarithmic, and trigonometric functions. Because of the strong emphasis on graphing, students will have a better understanding of asymptotic behavior of these functions.
MATH 1105Finite Mathematics for the Life and Social Sciences3
Introduction to linear algebra, probability, and Markov chains that develops the parts of the theory most relevant for applications. Topics include equations of lines, the method of least squares, solutions of linear systems, matrices; basic concepts of probability, permutations, combinations, binomial distribution, mean and variance, and the normal approximation to the binomial distribution. Examples from biology and the social sciences are used.
MATH 1106Modeling with Calculus for the Life Sciences4
The goal of this course is to give students a strong basis in quantitative skills needed in the life and social sciences. We will focus 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, including predator-prey populations. We will discuss mathematical models describing the evolution of these populations, analyze quantitative and qualitative properties to make predictions about these populations, and discuss assumptions and limitations of these models. Derivatives and integrals will be covered with a more applied focus than in MATH 1110 or a typical high school calculus course. Students who plan to take more than one semester of calculus should take MATH 1110 rather than MATH 1106.
MATH 1110 can serve as a one-semester introduction to calculus or as part of a two-semester sequence in which it is followed by MATH 1120. 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.
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.
For students who wish to experience how mathematical ideas naturally evolve. The course emphasizes ideas and imagination rather than techniques and calculations. Homework involves students in actively investigating mathematical ideas. Topics vary depending on the instructor. Some assessment through writing assignments.
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.
How can you use writing to solve mathematical problems? Or even to discover problems that are worth solving? This course revisits high school mathematics with the goal of learning how mathematicians think about math. We will use writing to probe familiar math problems as well as discover new ones, and we will refine our ability to communicate their solutions. Readings will include excerpts from Lang’s Basic Mathematics and works by Gelfand. (These are high school math textbooks written by mathematicians with research careers.) The class will culminate in a research project: you will use writing to develop, investigate, and present your progress on a problem.
MATH 1900Introduction to Calculus for Engineers2.5
We will review material from differential calculus and introduce topics from integral calculus to prepare students for MATH 1910. The first half of the course reviews limits and continuity, differentiation, and linear approximation. The second half of the course introduces antiderivatives, definite and indefinite integrals, the Fundamental Theorem of Calculus, integration techniques (e.g., substitution and integration by parts), and some applications of integration.
Essentially a second course in calculus and the first in a sequence designed for engineers that assumes familiarity with differential calculus at the level of MATH 1110. 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.
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.
An introduction to linear algebra for students who plan to major or minor in mathematics or a related field. Topics include vector algebra, linear transformations, matrices, determinants, orthogonality, eigenvalues, and eigenvectors. Applications are made to linear differential or difference equations. Lectures will introduce students to formal proofs, and students will be required to produce some proofs in their homework and on exams. For a more applied version of this course, see MATH 2310.
An introduction to multivariable calculus for students who plan to major or minor in mathematics or a related field. Topics include differential and integral calculus of functions in several variables, line and surface integrals as well as the theorems of Green, Stokes and Gauss.
MATH 2230Theoretical Linear Algebra and Vector Calculus I5
Designed for students who have been extremely successful in their previous calculus courses and for whom the notion of solving very hard problems and writing careful proofs is highly appealing, MATH 2230-MATH 2240 provides an integrated treatment of linear algebra and multivariable calculus at a higher theoretical level than in MATH 2210-MATH 2220. Topics covered in MATH 2230 include vectors, matrices, and linear transformations; differential calculus of functions of several variables; inverse and implicit function theorems; quadratic forms, extrema, and manifolds; multiple and iterated integrals.
MATH 2240Theoretical Linear Algebra and Vector Calculus II5
Designed for students who have been extremely successful in their previous calculus courses and for whom the notion of solving very hard problems and writing careful proofs is highly appealing, MATH 2230-MATH 2240 provides an integrated treatment of linear algebra and multivariable calculus at a higher theoretical level than in MATH 2210-MATH 2220. Topics covered in MATH 2240 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.
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. Students who take MATH 2310 may need more foundational coursework before pursuing further study in mathematics.
An 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. MATH 2930 and MATH 2940 are independent and can be taken in either order; they should not be taken in the same semester.
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. MATH 2930 and MATH 2940 are independent and can be taken in either order; they should not be taken in the same semester.
A useful course for students who wish to improve their skills in mathematical proof and exposition, or who intend to study more advanced topics 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. We will study various methods of mathematical proof, starting with basic techniques in propositional and predicate calculus and in set theory and combinatorics, 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. Because cogent communication of mathematical ideas is important in the presentation of proofs, the course emphasizes clear, concise exposition.
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.
A manifold is a type of subset of Euclidean space that has a well-defined tangent space at every point. Such a set is amenable to the methods of multivariable calculus. After reviewing some relevant calculus, this course investigates manifolds and the structures they are endowed with, such as tangent vectors, boundaries, orientations, and differential forms. The notion of a differential form encompasses such ideas as area forms and volume forms, the work exerted by a force, the flow of a fluid, and the curvature of a surface, space or hyperspace. We re-examine the integral theorems of vector calculus (Green, Gauss, and Stokes) in the light of differential forms and apply them to problems in partial differential equations, topology, fluid mechanics, and electromagnetism.
MATH 3270Introduction to Ordinary Differential Equations3
A one-semester introduction to the theory and techniques of ordinary differential equations. Topics may include first-order and second-order differential equations, systems of linear differential equations, initial-value and two-point boundary-value problems, Sturm-Liouville theory, Sturm oscillation and comparison theory, the basic existence and uniqueness theorems, series solutions, special functions, and Laplace transforms. Applications from science and engineering may be included at the instructor’s discretion.
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.
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. Students considering graduate school in mathematics might consider taking MATH 4330 after MATH 3340.
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.
Introduction to the theory and practice of mathematical modeling. We compare and contrast different types of mathematical models (discrete vs. continuous, deterministic vs. stochastic), focusing on advantages, disadvantages, and limits of applicability for each approach. Case-study format covers a variety of application areas including economics, physics, sociology, traffic engineering, urban planning, robotics, and resource management. Students learn how to implement mathematical models on the computer and how to interpret/describe the results of their computational experiments.
A mathematical study of the formal languages of standard first-order propositional and predicate logic, including their syntax, semantics, and deductive systems. The basic apparatus of model theory will be presented. Various formal results will be established, most importantly soundness and completeness.
This will be a course on standard set theory (first developed by Ernst Zermelo early in the 20th century): the basic concepts of sethood and membership, operations on sets, functions as sets, the set-theoretic construction of the Natural Numbers, the Integers, the Rational and Real numbers; time permitting, some discussion of cardinality.Course was formerly titled Foundations of Mathematics.
Modal logic is a general logical framework for systematizing reasoning about qualified and relativized truth. It has been used to study the logic of possibility, time, knowledge, obligation, provability, and much more. This course will explore both the theoretical foundations and the various philosophical applications of modal logic. On the theoretical side, we will cover basic metatheory, including Kripke semantics, soundness and completeness, correspondence theory, and expressive power. On the applied side, we will examine temporal logic, epistemic logic, deontic logic, counterfactuals, two-dimensional logics, and quantified modal logic.
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, and the transcendence of pi. Students will be expected to be comfortable writing proofs and will give presentations from original sources over 5000 years of mathematics.
Basic problem-solving techniques and strategies for contest problems. Contest problems differ from standard mathematical instruction in that they are "contextless": the problem does not tell you which field it is in and sometimes goes to great lengths to disguise it. Understanding how to solve contest problems has applications to research, where knowing how to work with unknown concepts, how to attack a problem fluidly from the perspective of multiple fields, and different tricks for simplifying problems is extremely useful. This course will cover topics that appear in standard competitions (e.g., the Putnam exam), including combinatorics, number theory, geometry, and calculus. It will also teach strategies for understanding and attacking problems, more advanced proof techniques, and effective proof writing.
Introduction to the rigorous theory underlying calculus, covering the real number system and functions of one variable. 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. Based entirely on proofs. The student is expected to know how to read and, to some extent, construct proofs before taking this course. More experience with proofs may be gained by first taking a 3000-level MATH course.
A 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, and ordinary differential equations.
A theoretical and rigorous introduction to complex variable theory recommended for students who plan to attend graduate school in mathematics. 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. Students will be expected to be comfortable writing proofs. For applications of complex analysis, consider MATH 4220 rather than MATH 4180.
MATH 4200Differential Equations and Dynamical Systems3
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. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
An introduction to nonlinear dynamics with applications to physics, engineering, biology, and chemistry. Emphasizes analytical methods, concrete examples, and geometric thinking. Topics include one-dimensional systems; bifurcations; phase plane; nonlinear oscillators; and Lorenz equations, chaos, strange attractors, fractals, iterated mappings, period doubling, renormalization. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
Covers complex variables, Fourier transforms, Laplace transforms and applications to partial differential equations. Additional topics may include an introduction to generalized functions. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course. Undergraduates who plan to attend graduate school in mathematics should take MATH 4180 rather than MATH 4220.
MATH 4250Numerical Analysis and Differential Equations4
Introduction to the fundamentals of numerical analysis: error analysis, approximation, interpolation, and numerical integration. In the second half of the course, we use these to build approximate solvers for ordinary and partial differential equations. Strong emphasis is placed on understanding the advantages, disadvantages, and limits of applicability for all the covered techniques. Computer programming is used to test the theoretical concepts throughout the course. Students will be expected to be comfortable writing proofs and have knowledge of programming. MATH 4250/CS 4210 and MATH 4260/CS 4220 can be taken independently from each other and in either order. Together they provide a comprehensive introduction to numerical analysis.
MATH 4260Numerical Analysis: Linear and Nonlinear Problems4
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.
MATH 4280Introduction to Partial Differential Equations4
Topics are selected from first-order quasilinear equations, classification of second-order equations, with emphasis on maximum principles, existence, uniqueness, stability, and Fourier series methods. Additional topics as time permits. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
Introduction to linear algebra, including the study of vector spaces, linear transformations, matrices, and systems of linear equations. Additional topics include quadratic forms and inner product spaces, canonical forms for various classes of matrices and linear transformations. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course. Undergraduates who plan to attend graduate school in mathematics should take MATH 4330 instead of MATH 4310.
Honors version of a course in advanced linear algebra, which treats the subject from an abstract and axiomatic viewpoint. Topics include vector spaces, linear transformations, polynomials, determinants, tensor and wedge products, canonical forms, inner product spaces, and bilinear forms. Emphasis is on understanding the theory of linear algebra; homework and exams include at least as many proofs as computational problems. Strong proficiency in writing proofs is expected. More experience with proofs may be gained by first taking a 3000-level MATH course. MATH 4330-MATH 4340 is recommended for undergraduates who plan to attend graduate school in mathematics. For a less theoretical course that covers approximately the same subject matter as MATH 4330, see MATH 4310.
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. MATH 4330-MATH 4340 is recommended for undergraduates who plan to attend graduate school in mathematics. For a less theoretical course that covers subject matter similar to MATH 4340, see MATH 3340.
Introduction to algebraic geometry and computational algebra. Students will learn how to compute a Gröbner basis for polynomials in many variables. Covers the following applications: solving systems of polynomial equations in many variables, solving diophantine equations in many variables, 3-colorable graphs, and integer programming. Such applications arise, for example, in computer science, engineering, economics, and physics. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
Combinatorics studies discrete structures arising in mathematics, computer science, and many areas of application. Key topics include counting objects with specific properties (e.g., trees) and proving the existence of structures (e.g., matchings of all vertices in a graph). We cover basic questions in graph theory, including extremal graph theory (how large a graph must be to have a certain subgraph) and Ramsey theory (large objects are forced to have structure). An introduction to network flow theory and variations on matching theory, including theorems of Dilworth, Hall, König, and Birkhoff, are discussed. Methods of enumeration (inclusion/exclusion, Möbius inversion, and generating functions) are applied to problems of counting permutations, partitions, and triangulations. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
Continuation of MATH 4410, although formally independent of the material covered there. The emphasis here is the study of certain combinatorial structures, such as Latin squares and combinatorial designs (which are of use in statistical experimental design), classical finite geometries and combinatorial geometries (also known as matroids, which arise in many areas from algebra and geometry through discrete optimization theory). We introduce partially ordered sets and lattices, including general Möbius inversion and its application, as well as the Polya theory of counting in the presence of symmetries. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
Algebraic combinatorics studies combinatorial problems with algebraic tools and algebraic problems with combinatorial tools. Topics covered include the study of the ring of symmetric functions, the symmetric group, and tableaux combinatorics.
Matrix groups are central to mathematics and important in physics and engineering. The objects of study are classes of matrices (e.g., orthogonal, unitary, or symplectic) with both algebraic (groups) and geometric/topological (manifolds) structure. Thus the course is a mixture of algebra, geometry/topology, and a little analysis. Topics include Lie algebras (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 different ways groups can be represented by matrices). Concrete examples will be emphasized. Background not included in the prerequisites will be developed as needed. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
MATH 4520Classical Geometries and Modern Applications4
An introduction to projective, hyperbolic, and spherical geometry and their modern applications. The course will be divided into short modules with an emphasis on participation, discovery, and student projects and presentations. In addition to proving theorems, students will have the opportunity to make, build, 3D print, or program something related to the course material as a project component. We will cover classical theorems and techniques (e.g., stereographic projection and conics) and see how classical geometry is used in and relates to other areas of mathematics (e.g., topology, via Euler characteristic) and applications such as computer vision, networks, or architectural drawing. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
Topology may be described briefly as qualitative geometry. This course begins with basic point-set topology, including connectedness, compactness, and metric spaces. Later topics may include the classification of surfaces (such as the Klein bottle and Möbius band), elementary knot theory, or the fundamental group and covering spaces. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
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. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
An introduction to probability theory that 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 central limit theorem are stated and their implications for statistics are discussed.
Introduction to classical theory of parametric statistical inference that builds on the material covered in STSCI 3080. Topics include: sampling distributions, principles of data reduction, likelihood, parameter estimation, hypothesis testing, interval estimation, and basic asymptotic theory.
A one-semester introduction to stochastic processes which develops the theory together with applications. Covers Markov chains in discrete and continuous time and Poisson processes. Other topics may include queuing theory, martingales, Brownian motion, and option pricing. This course may be useful to graduate students in the biological sciences or other disciplines who encounter stochastic models in their work but who do not have the background for more advanced courses such as ORIE 6500. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
First course in mathematical logic providing precise definitions of the language of mathematics and the notion of proof (propositional and predicate logic). The completeness theorem says that we have all the rules of proof we could ever have. The Gödel incompleteness theorem says that they are not enough to decide all statements even about arithmetic. The compactness theorem exploits the finiteness of proofs to show that theories have unintended (nonstandard) models. Possible additional topics: the mathematical definition of an algorithm and the existence of noncomputable functions; the basics of set theory to cardinality and the uncountability of the real numbers. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking CS 2800 or a 3000-level MATH course.
Topics chosen from propositional logic, first-order logic, and higher-order logic, both classical and intuitionistic versions, including completeness, incompleteness, and compactness results. Natural deduction and tableaux style logics and connection to the lambda calculus and programming languages and logics, and program verification. Other topics chosen from equational logic, Herbrand universes and unification, rewrite rules and Knuth-Bendix method, and the congruence-closure algorithm and lambda-calculus reduction strategies. Modal logics, intuitionistic logic, computational logics and programming languages (e.g., LISP, ML, or Nuprl). Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
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.
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.
MATH 4980Special Study for Mathematics Teaching1-3
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 e.math.cornell.edu/classes/math5080) 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. Does not count toward the math major or math minor and will not count as degree credits for A&S students.
This independent study course offers math majors (i.e., undergraduates whose applications to affiliate with the math major have been approved) an opportunity to reflect on concepts from mathematics as they were encountered and applied in a recent internship. Students write a short paper describing their work experience and how it connects to the educational objectives of the mathematics major.
Examines principles underlying the content of the secondary school mathematics curriculum, including connections with the history of mathematics, technology, and mathematics education research.
MATH 5200Differential Equations and Dynamical Systems3
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. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
Covers complex variables, Fourier transforms, Laplace transforms and applications to partial differential equations. Additional topics may include an introduction to generalized functions. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
MATH 5250Numerical Analysis and Differential Equations4
Introduction to the fundamentals of numerical analysis: error analysis, approximation, interpolation, and numerical integration. In the second half of the course, we use these to build approximate solvers for ordinary and partial differential equations. 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. Students will be expected to be comfortable writing proofs and have knowledge of programming.
Combinatorics studies discrete structures arising in mathematics, computer science, and many areas of application. Key topics include counting objects with specific properties (e.g., trees) and proving the existence of structures (e.g., matchings of all vertices in a graph). We cover basic questions in graph theory, including extremal graph theory (how large a graph must be to have a certain subgraph) and Ramsey theory (large objects are forced to have structure). An introduction to network flow theory and variations on matching theory, including theorems of Dilworth, Hall, König, and Birkhoff, are discussed. Methods of enumeration (inclusion/exclusion, Möbius inversion, and generating functions) are applied to problems of counting permutations, partitions, and triangulations. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
Continuation of MATH 5410, although formally independent of the material covered there. The emphasis here is the study of certain combinatorial structures, such as Latin squares and combinatorial designs (which are of use in statistical experimental design), classical finite geometries and combinatorial geometries (also known as matroids, which arise in many areas from algebra and geometry through discrete optimization theory). We introduce partially ordered sets and lattices, including general Möbius inversion and its application, as well as the Polya theory of counting in the presence of symmetries. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.
Algebraic combinatorics studies combinatorial problems with algebraic tools and algebraic problems with combinatorial tools. Topics covered include the study of the ring of symmetric functions, the symmetric group, and tableaux combinatorics.
Provides graduate students with an introduction to core concepts in teaching and learning mathematics. Participants read and discuss articles and videos, reflect on their own teaching and learning experiences, and engage in collaborative activities that help them become more effective teachers, learners, and communicators. Students will observe peers and faculty and reflect on their observations. Especially valuable for those considering teaching mathematics at some point in their careers, or who want to improve their own mathematics learning skills. Students are expected to have taken the Math TA training prior to enrolling in the course.
MATH 6110-MATH 6120 are the core analysis courses in the mathematics graduate program. MATH 6110 covers abstract measure and integration theory, and related topics such as the Lebesgue differentiation theorem, the Radon-Nikodym theorem, the Hardy-Littlewood maximal function, the Brunn-Minkowski inequality, rectifiable curves and the isoperimetric inequality, Hausdorff dimension and Cantor sets, and an introduction to ergodic theory.
MATH 6110-6120 are the core analysis courses in the mathematics graduate program. MATH 6120 covers complex analysis, Fourier analysis, and distribution theory.
This course emphasizes the "classical" aspects of partial differential equations (PDEs) — analytic methods for linear second-order PDEs and first-order nonlinear PDEs – without relying on more modern tools of functional analysis. The usual topics include fundamental solutions for the Laplace/Poisson, heat and wave equations in n dimensions, mean-value properties, maximum principles, energy methods, Duhamel's principle, and an introduction to nonlinear first-order equations, including shocks and weak solutions. Additional topics may include Hamilton-Jacobi equations, Euler-Lagrange equations, similarity solutions, transform methods, asymptotics, power series methods, homogenization, distribution theory, and the Fourier transform.
This course highlights applications of functional analysis to the theory of partial differential equations (PDEs). It covers parts of the basic theory of linear (elliptic and evolutionary) PDEs, including Sobolev spaces, existence and uniqueness of solutions, interior and boundary regularity, maximum principles, and eigenvalue problems. Additional topics may include: an introduction to variational problems, Hamilton-Jacobi equations, and other modern techniques for non-linear PDEs.
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.
This course is a self-contained introduction to the modern theory of optimal control and differential games. Dynamic programming uses Hamilton-Jacobi partial differential equations (PDEs) to encode the optimal behavior in cooperative and adversarial sequential decision-making problems. The same PDEs have an alternative interpretation in the context of front propagation problems. We show how both interpretations are useful in constructing efficient numerical methods. We also consider a wide range of applications, including robotics, computational geometry, path-planning, computer vision, photolithography, economics, seismic imaging, ecology, financial engineering, crowd dynamics, and aircraft collision avoidance. Assumes no prior knowledge of non-linear PDEs or numerical analysis.
A high-level survey course on ten influential algorithms that shaped modern computational science and optimization. Each week focuses on a single algorithm - such as the Metropolis algorithm, simplex method, conjugate gradient, QR factorization, fast Fourier transform, fast multipole method - covering the historical context, key mathematical ideas, and modern applications. Students must have a background in linear algebra, calculus, and basic analysis.
Topics include existence and uniqueness theorems for ODEs; Poincaré-Bendixon theorem and global properties of two-dimensional flows; limit sets, nonwandering sets, chain recurrence, pseudo-orbits, and structural stability; linearization at equilibrium points: stable manifold theorem and the Hartman-Grobman theorem; and generic properties: transversality theorem and the Kupka-Smale theorem. Examples include expanding maps and Anosov diffeomorphisms; hyperbolicity: the horseshoe and the Birkhoff-Smale theorem on transversal homoclinic orbits; rotation numbers; Herman's theorem; and characterization of structurally stable systems. Prior exposure to topology (e.g., MATH 4530) will be helpful.
Topics include review of planar (single-degree-of-freedom) systems; local and global analysis; structural stability and bifurcations in planar systems; center manifolds and normal forms; the averaging theorem and perturbation methods; Melnikov's method; discrete dynamical systems, maps and difference equations, homoclinic and heteroclinic motions, the Smale Horseshoe and other complex invariant sets; global bifurcations, strange attractors, and chaos in free and forced oscillator equations; and applications to problems in solid and fluid mechanics.
This course studies the iteration of holomorphic maps in one complex variable, with emphasis on the dynamics of polynomials and rational maps on the Riemann sphere. We will cover Julia and Fatou sets, normal families, local and global behavior of periodic points, polynomial dynamics, puzzle techniques and local connectivity problems, quasiconformal methods, and renormalization. Central results include Sullivan’s no wandering domains theorem, the Douady– Hubbard straightening theorem, Yoccoz’s proof of the Siegel Brjuno theorem, and Thurston’s topological characterization of rational functions. The course will also give a glimpse of the connection between complex dynamics with Teichmuller theory, hyperbolic geometry, and geometric group theory.
MATH 6302Lattices: Geometry, Cryptography, and Algorithms3
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.
MATH 6310-MATH 6320 are the core algebra courses in the mathematics graduate program. MATH 6310 covers group theory, especially finite groups; rings and modules; ideal theory in commutative rings; arithmetic and factorization in principal ideal domains and unique factorization domains; introduction to field theory; tensor products and multilinear algebra. (Optional topic: introduction to affine algebraic geometry.)
MATH 6310-MATH 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.
An introduction to the theory of noncommutative rings and modules. Topics vary by semester and include semisimple modules and rings, the Jacobson radical and Artinian rings, group representations and group algebras, characters of finite groups, representations of the symmetric group, central simple algebras and the Brauer group, representation theory of finite-dimensional algebras, and Morita theory.
A first course on homological algebra. Topics will include a brief introduction to categories and functors, chain and cochain complexes, operations on complexes, (co)homology, standard resolutions (injective, projective, flat), classical derived functors, Tor and Ext, Yoneda's interpretation of Ext, homological dimension, rings of small dimensions, introduction to group cohomology.
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.
Lie groups, Lie algebras, and their representations play an important role in much of mathematics, particularly in number theory, mathematical physics, and topology. This is an introductory course, meant to be useful for more advanced topics and applications. The relationship between Lie groups and Lie algebras will be highlighted throughout the course. A different viewpoint is that of algebraic groups. We will endeavor to discuss this along with the smooth viewpoint. Some knowledge of differential and algebraic geometry are helpful.
An introduction to enumerative combinatorics from an algebraic, geometric and topological point of view. Topics include, but are not limited to, permutation statistics, partitions, generating functions and combinatorial species, various types of posets and lattices (distributive, geometric, and Eulerian), Mobius inversion, face numbers, shellability, and relations to the Stanley-Reisner ring.
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.
MATH 6510-MATH 6520 are the core topology courses in the mathematics graduate program. This course is an introduction to geometry and topology from a differentiable viewpoint, suitable for beginning graduate students. The objects of study are manifolds and differentiable maps. The collection of all tangent vectors to a manifold forms the tangent bundle, and a section of the tangent bundle is a vector field. Alternatively, vector fields can be viewed as first-order differential operators. We will study flows of vector fields and prove the Frobenius integrability theorem. We will examine the tensor calculus and the exterior differential calculus and prove Stokes' theorem. If time permits, de Rham cohomology, Morse theory, or other optional topics will be covered.
An introduction to topological K-theory and characteristic classes. Topological K-theory is a generalized cohomology theory which is surprisingly simple and useful for computation while still containing enough structure for proving interesting results. The class will begin with the definition of K-theory, Chern classes, and the Chern character. Additional topics may include the Hopf invariant 1 problem, the J-homomorphism, Stiefel-Whitney classes and Pontrjagin classes, cobordism groups and the construction of exotic spheres, and the Atiyah-Singer Index Theorem.
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.
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.
Symplectic geometry is a branch of differential geometry which studies manifolds endowed with a nondegenerate closed 2-form. The field originated as the mathematics of classical (Hamiltonian) mechanics, and it has connections to (at least!) complex geometry, algebraic geometry, representation theory, and mathematical physics. In this introduction to symplectic geometry, the class will begin with linear symplectic geometry, discuss canonical local forms (Darboux-type theorems), and examine related geometric structures including almost complex structures and Kähler metrics. Further topics may include symplectic and Hamiltonian group actions, the orbit method, the topology and geometry of momentum maps, toric symplectic manifolds, Hamiltonian dynamics, symplectomorphism groups, and symplectic embedding problems.
An introduction to the topology and geometry of hyperbolic manifolds. The class will begin with the geometry of hyperbolic n-space, including the upper half-space, Poincare disc, Klein, and Lorentzian models. We will cover both synthetic and computational approaches. We will then discuss hyperbolic structures on surfaces and 3-manifolds, and the corresponding groups of isometries (i.e. Fuchsian and Kleinian groups). Additional topics may include: Geodesic and horocycle flows and their properties, counting closed geodesics and simple closed geodesics, Mostow rigidity, infinite area surfaces.
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.
Measure theory, independence, distribution of sums of iid random variables, laws of large numbers, and central limit theorem. Other topics as time permits.
The second course in a graduate probability series. Topics include conditional expectation, martingales, Markov chains, Brownian motion, and (time permitting) elements of stochastic integration.
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.
Covers theory of effectively computable functions; classification of recursively enumerable sets; degrees of recursive unsolvability; applications to logic; hierarchies; recursive functions of ordinals and higher type objects; generalized recursion theory.
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.
MATH 7290Seminar on Scientific Computing and Numerics1
Talks on various methods in scientific computing, the analysis of their convergence properties and computational efficiency, and their adaptation to specific applications.
Topics will vary depending on the instructor and the level of the audience. They range from representation theory of Lie algebras and of real and p-adic Lie groups, geometric representation theory, quantum groups and their representations, invariant theory to applications of Lie theory to other parts of mathematics.
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.
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.
