Mathematics
- COS 487/MAT 407: Theory of ComputationIntroduction to computability and complexity theory. Topics will include models of computation such as automata, and Turing machines; decidability and decidability; computational complexity; P, NP, and NP completeness; others.
- MAE 305/MAT 391/EGR 305/CBE 305: Mathematics in Engineering IA treatment of the theory and applications of ordinary differential equations with an introduction to partial differential equations. The objective is to provide the student with an ability to solve standard problems in this field.
- MAT 100: Calculus FoundationsIntroduction to limits and derivatives as preparation for further courses in calculus. Fundamental functions (polynomials, rational functions, exponential, logarithmic, trigonometric) and their graphs will be also reviewed. Other topics include tangent and normal lines, linearization, computing area and rates of change. The emphasis will be on learning to think independently and creatively in the mathematical setting.
- MAT 103: Calculus IFirst semester of calculus. Topics include limits, continuity, the derivative, basic differentiation formulas and applications (curve-sketching, optimization, related rates), definite and indefinite integrals, the fundamental theorem of calculus.
- MAT 104: Calculus IIContinuation of MAT103. Topics include techniques of integration, arclength, area, volume, convergence of series and improper integrals, L'Hopital's rule, power series and Taylor's theorem, introduction to differential equations and complex numbers.
- MAT 175: Mathematics for Economics/Life SciencesSurvey of topics from multivariable calculus as preparation for future course work in economics or life sciences. Topics include basic techniques of integration, average value, vectors, partial derivatives, gradient, optimization of multivariable functions, and constrained optimization with Lagrange multipliers.
- MAT 201: Multivariable CalculusVectors in the plane and in space, vector functions and motion, surfaces, coordinate systems, functions of two or three variables and their derivatives, maxima and minima and applications, double and triple integrals, vector fields and Stokes's theorem.
- MAT 202: Linear Algebra with ApplicationsCompanion course to MAT201. Matrices, linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvectors and their applications to quadratic forms and dynamical systems.
- MAT 203: Advanced Vector CalculusVector spaces, limits, derivatives of vector-valued functions, Taylor's formula, Lagrange multipliers, double and triple integrals, change of coordinates, surface and line integrals, generalizations of the fundamental theorem of calculus to higher dimensions. More abstract than 201 but more concrete than 218. Recommended for prospective physics majors and others with a strong interest in applied mathematics.
- MAT 214: Numbers, Equations, and ProofsAn introduction to classical number theory, to prepare for higher-level courses in the department. Topics include Pythagorean triples and sums of squares, unique factorization, Chinese remainder theorem, arithmetic of Gaussian integers, finite fields and cryptography, arithmetic functions and quadratic reciprocity. There will be a topic, chosen by the instructor, from more advanced or more applied number theory: possibilities include p-adic numbers, cryptography, and Fermat's Last Theorem. This course is suitable both for students preparing to enter the Mathematics Department and for non-majors interested in exposure to higher mathematics.
- MAT 215: Single Variable Analysis with an Introduction to ProofsAn introduction to the mathematical discipline of analysis, to prepare for higher-level course work in the department. Topics include rigorous epsilon-delta treatment of limits, convergence, and uniform convergence of sequences and series. Continuity, uniform continuity, and differentiability of functions. The Heine-Borel Theorem. The Riemann integral, conditions for integrability of functions and term by term differentiation and integration of series of functions, Taylor's Theorem.
- MAT 216: Multivariable Analysis and Linear Algebra IRigorous theoretical introduction to the foundations of analysis in one and several variables: basic set theory, vector spaces, metric and topological spaces, continuous and differential mapping between n-dimensional real vector spaces. Normally followed by MAT 218.
- MAT 300: Multivariable Analysis ITo familiarize the student with functions in many variables and higher dimensional generalization of curves and surfaces. Topics include: point set topology and metric spaces; continuous and differentiable maps in several variables; smooth manifolds and maps between them; Sard's theorem; vector fields and flows; differential forms and Stokes' theorem; differential equations; multiple integrals and surface integrals. An introduction to more advanced courses in analysis, differential equations, differential geometry, topology.
- MAT 320: Introduction to Real AnalysisIntroduction to real analysis, including the theory of Lebesgue measure and integration on the line and n-dimensional space, and the theory of Fourier series and Hilbert spaces.
- MAT 321/APC 321: Numerical MethodsIntroduction to numerical methods with emphasis on algorithms, applications and numerical analysis. Topics covered include solution of nonlinear equations; numerical differentiation, integration, and interpolation; direct and iterative methods for solving linear systems; computation of eigenvectors and eigenvalues; and approximation theory. Lectures include mathematical proofs where they provide insight and are supplemented with numerical demos using MATLAB.
- MAT 335: Analysis II: Complex AnalysisStudy of functions of a complex variable, with emphasis on interrelations with other parts of mathematics. Cauchy's theorems, singularities, contour integration, power series, infinite products. The gamma and zeta functions and the prime number theorem. Elliptic functions, theta functions, Jacobi's triple product and combinatorics. This course is the second semester of a four-semester sequence, but may be taken independently of the other semesters.
- MAT 340: Applied AlgebraAn applied algebra course that integrates the basics of theory and modern applications for students in MAT, APC, PHY, CBE, COS, ELE. This course is intended for students who have taken a semester of linear algebra and who have an interest in a course that treats the structures, properties and application of groups, rings, and fields. Applications and algorithmic aspects of algebra will be emphasized throughout.
- MAT 345: Algebra IThis course will cover the basics of symmetry and group theory, with applications. Topics include the fundamental theorem of finitely generated abelian groups, Sylow theorems, group actions, and the representation theory of finite groups, rings and modules.
- MAT 365: TopologyIntroduction to point-set topology, the fundamental group, covering spaces, methods of calculation and applications.
- MAT 377/APC 377: Combinatorial MathematicsThe course covers the basic combinatorial techniques as well as introduction to more advanced ones. The topics discussed include elementary counting, the pigeonhole principle, counting spanning trees, Inclusion-Exclusion, generating functions, Ramsey Theory, Extremal Combinatorics, Linear Algebra in Combinatorics, introduction to the probabilistic method, spectral graph theory, topological methods in combinatorics.
- MAT 419: Topics in Number Theory: Algebraic Number TheoryCourse on algebraic number theory. Topics covered include number fields and their integer rings, class groups, zeta and L-functions.
- MAT 425: Analysis III: Integration Theory and Hilbert SpacesThe theory of Lebesgue integration in n-dimensional space. Differentiation theory. Hilbert space theory and applications to Fourier Transforms, and partial differential equations. Introduction to fractals. This course is the third semester of a four-semester sequence, but may be taken independently of the other semesters.
- MAT 477: Advanced Graph TheoryAdvanced course in Graph Theory. Further study of graph coloring, graph minors, perfect graphs, graph matching theory. Topics covered include: stable matching theorem, list coloring, chi-boundedness, excluded minors and average degree, Hadwiger's conjecture, the weak perfect graph theorem, operations on perfect graphs, and other topics as time permits.
- MAT 500: Effective Mathematical CommunicationThis course is for second-year graduate students to help them develop their writing and speaking skills for communicating mathematics in a wide variety of settings, including teaching, grant applications, teaching statement, research statement, talks aimed at a general mathematical audience, and seminars, etc. In addition, responsible conduct in research (RCR) training is an integral part of this course.
- MAT 522/APC 522: Introduction to PDEThe course is an introduction to partial differential equations, problems associated to them and methods of their analysis. Topics may include: basic properties of elliptic equations, wave equation, heat equation, Schr\"{o}dinger equation, hyperbolic conservation laws, Fokker-Planck equation, basic function spaces and inequalities, regularity theory for linear PDE, De Giorgi method, basic harmonic analysis methods, existence results and long time behavior for classes of nonlinear PDE including the Navier-Stokes equations.
- MAT 527: Topics in Differential Equations: Elliptic partial differential equationsIn this course, we study various existence and regularity theories for elliptic partial differential equations and systems, though the emphasis is on regularity. I plan to treat the Leray-Schauder theory, the existence and regularity of critical points of elliptic integrands (scalar and vectorial), the solution of the XIX Hilbert problem (De Giorgi's proof and Moser's iteration), the theory of viscosity solutions of fully nonlinear elliptic equations.
- MAT 547: Topics in Algebraic Geometry: Algebraic K-stability TheoryWe discuss the algebraic K-stability theory of Fano varieties, specifically 1) the foundational theory of K-stability, including the original definition of K-stability, the valuative criterion of K-stability, etc., 2) the construction of general K-moduli stacks/spaces, including various necessary ingredients, and 3) verifying explicit examples of Fano varieties to be K-stable, including explicit K-moduli construction and estimate of stability of thresholds.
- MAT 549: Topics in Algebra: Introduction to \ell-adic etale cohomologyThe course explores connections between etale cohomology and both finite and Lie groups.
- MAT 550: Differential GeometryThis is an introductory graduate course covering questions and methods in differential geometry. As time permits, more specialized topics are covered as well, including minimal submanifolds, curvature and the topology of manifolds, the equations of geometric analysis and its main applications, and other topics of current interest.
- MAT 555: Topics in Differential Geometry: Minimal SurfacesOver the course of the semester, we discuss the theory of minimal surfaces, including some recent developments and their applications.
- MAT 558: Topics in Conformal and Cauchy-Rieman (CR) Geometry: Recent Developments in Conformal GeometryOver the course of the semester, we develop the analysis of CR structure along the lines for conformal geometry. The main analytic tools are several conformally covariant operators, which give rise to curvature invariants. The course covers the embedding problem, the CR version of Yamabe problem and Q-prime curvature equations.
- MAT 559: Topics in Geometry: Discrete Geometry: Incidence Theorems and their ApplicationsThe course focuses on algebraic methods in geometric incidence problems. Most topics fall under one of these three themes: (1) Kakeya type problems: packing lines in different directions (2) Szemeredi-Trotter type problems: Counting incidences between lines/points/curves etc. (3) Sylvester-Gallai type problems: finding structure in point sets with many local dependencies (e.g., many collinear triples). Applications in theoretical computer science are also discussed.
- MAT 566: Topics in Differential Topology: Symplectic Methods in Low-dimensional TopologyHeegaard Floer homology is an invariant for low-dimensional manifolds constructed using methods in symplectic geometry (Lagrangian Floer homology). A related invariant for knots can also be constructed, whose Euler characteristic, in a suitable sense, is the Alexander polynomial of that knot. This course gives the construction of Heegaard Floer homology and the knot invariant, and with a view towards topological applications, and a special emphasis on modern computational tools.
- MAT 567: Topics in Low Dimensional Topology: Smooth Surfaces in 4-manifoldsThe first half of the course is a detailed proof of the h-cobordism theorem. We then address topics that may include: the topological Schoenflies theorem, stable homeomorphisms of R^n, topological isotopy extension, engulfing and uniqueness of smooth structures on R^n for n>4.
- MAT 569: Topics in Topology: Contact and Symplectic TopologyThis course explores stable homotopical underpinnings of the enumerate theory of pseudo-holomorphic curves in symplectic manifolds.
- MAT 579: Topics in Discrete Mathematics: Hadwiger's ConjectureA graph is a "minor" of another if it can be obtained from a subgraph of the second by contracting edges. Perhaps the most well-known open question about minors is Hadwiger's conjecture, extending the four-color theorem, that for all t, every graph with no K_{t+1} minor can be t-colored. After an introduction to minors of graphs, the course focuses on aspects of Hadwiger's conjecture, particularly on recent results giving upper bounds on the chromatic number of graphs with no K_{t+1} minor.
- MAT 589: Topics in Probability, Statistics and Dynamics: Modern Discrete Probability TheoryThe aim of this course is to survey some of the fundamentals of modern discrete probability, particularly random processes on networks and discrete structures. The course introduces topics including concentration of measure, random graphs, percolation, convergence of Markov chains and other topics depending on student interest.
- MAT NFO1: Princeton Calculus OrientationMAT NFO1/MAT NFO2 are designed to serve as both an aid to placement decisions and a review of material covered in earlier classes. Students who expect to take a 100-level math course (MAT100, 103, 104 or 175) should sign up for MAT NFO1. Students who expect to take MAT 201 or 203 should sign up for MAT NFO2.Likely to take MAT 100? Students should select a section of NFO1 that meets at 8:30 or at 11. Likely to take MAT 175? Select a section of NFO1 that meets at 8:30, 11 or 1:30. Likely to take MAT 203? Select a section of NFO2 that meets at 11 or 1:30.More information at: https://www.math.princeton.edu/undergraduate/placement/MAT-INFO
- MAT NFO2: Princeton Calculus Orientation IIMAT NFO1/MAT NFO2 are designed to serve as both an aid to placement decisions and a review of material covered in earlier classes. Students who expect to take a 100-level math course (MAT100, 103, 104 or 175) should sign up for MAT NFO1. Students who expect to take MAT 201 or 203 should sign up for MAT NFO2.Likely to take MAT 100? Students should select a section of NFO1 that meets at 8:30 or at 11. Likely to take MAT 175? Select a section of NFO1 that meets at 8:30, 11 or 1:30. Likely to take MAT 203? Select a section of NFO2 that meets at 11 or 1:30.More information at: https://www.math.princeton.edu/undergraduate/placement/MAT-INFO
- ORF 309/EGR 309/MAT 380: Probability and Stochastic SystemsAn introduction to probability and its applications. Topics include: basic principles of probability; Lifetimes and reliability, Poisson processes; random walks; Brownian motion; branching processes; Markov chains