Course Schedule

COS 487/MAT 407 Theory of Computation Introduction 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. Instructor(s): Gillat Kol EGR 191/MAT 191/PHY 191 An Integrated Introduction to Engineering, Mathematics, Physics Taken concurrently with EGR/MAT/PHY 192, this course offers an integrated presentation of the material from PHY 103 (General Physics: Mechanics and Thermodynamics) and MAT 201 (Multivariable Calculus) with an emphasis on applications to engineering. Physics topics include: mechanics with applications to fluid mechanics; wave phenomena; and thermodynamics. Instructor(s): M. Zahid Hasan, Peter Daniel Meyers EGR 192/MAT 192/PHY 192/APC 192 An Integrated Introduction to Engineering, Mathematics, Physics Taken concurrently with EGR/MAT/PHY 191, this course offers an integrated presentation of the material from PHY 103 (General Physics: Mechanics and Thermodynamics) and MAT 201 (Multivariable Calculus) with an emphasis on applications to engineering. Math topics include: vector calculus; partial derivatives and matrices; line integrals; simple differential equations; surface and volume integrals; and Green's, Stokes', and divergence theorems. Instructor(s): Casey Lynn Kelleher MAE 305/MAT 391/EGR 305/CBE 305 Mathematics in Engineering I A 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. Instructor(s): Howard A. Stone MAT 100 Calculus Foundations Introduction 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. Instructor(s): Tatiana Katarzyna Howard, Jennifer Michelle Johnson MAT 103 Calculus I First 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. Instructor(s): Daniel Alvarez-Gavela, Chiara Damiolini, Charles Louis Fefferman, Chao Li, Yakov Mordechai Shlapentokh-Rothman, Sophie Theresa Spirkl, Jingwei Xiao MAT 104 Calculus II Continuation 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. Instructor(s): Jonathan Michael Fickenscher, Henry Theodore Horton, Tetiana Shcherbyna MAT 175 Mathematics for Economics/Life Sciences Survey 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. Instructor(s): Boyu Zhang MAT 201 Multivariable Calculus Vectors 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. Instructor(s): Gabriele Di Cerbo, Jiequn Han, Sameer Subramanian Iyer, Y. Baris Kartal, János Kollár, Joaquin Moraga, Maxime C.R Van De Moortel, Jonathan Julian Zhu MAT 202 Linear Algebra with Applications Companion 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. Instructor(s): Kenneth Brian Ascher, Yaim Cooper, Duncan Dauvergne, Michele Fornea, Ary Shaviv MAT 203 Advanced Vector Calculus Vector 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. Instructor(s): Adam Wade Marcus MAT 214 Numbers, Equations, and Proofs An 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. Instructor(s): Mark Weaver McConnell MAT 215 Honors Analysis (Single Variable) An 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. Instructor(s): Sun-Yung Alice Chang, Javier Gómez-Serrano MAT 216 Accelerated Honors Analysis I Rigorous 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 MAT218. Instructor(s): Robert Clifford Gunning, John Vincent Pardon MAT 320 Introduction to Real Analysis Introduction 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. Instructor(s): Paul M.N. Feehan MAT 321/APC 321 Numerical Methods Introduction 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. Instructor(s): Nicolas Boumal MAT 335 Analysis II: Complex Analysis Study 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. Instructor(s): Assaf Naor MAT 340 Applied Algebra An 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. Instructor(s): Mark Weaver McConnell MAT 345 Algebra I This 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. Instructor(s): Yunqing Tang MAT 350 Differential Manifolds Fundamentals of multivariable analysis and calculus on manifolds. Topics to be covered include: differentiation and integration in multiple dimensions; smooth manifolds; vector fields and differential forms; stokes' theorem; de Rham cohomology; Frobenius' theorem on integrability of plane fields. This course will serve as a preparation for advanced courses in differential geometry and topology. Instructor(s): Ian Michael Zemke MAT 365 Topology Introduction to point-set topology, the fundamental group, covering spaces, methods of calculation and applications. Instructor(s): Zoltán Szabó MAT 377/APC 377 Combinatorial Mathematics Introduction to combinatorics, a fundamental mathematical discipline as well as an essential component of many mathematical areas. While in the past many of the basic combinatorial results were at first obtained by ingenuity and detailed reasoning, modern theory has grown out of this early stage and often relies on deep, well-developed tools. Topics include Ramsey Theory, Turan Theorem and Extremal Graph Theory, Probabilistic Argument, Algebraic Methods and Spectral Techniques. Showcases the gems of modern combinatorics. Instructor(s): Noga Mordechai Alon MAT 419 Topics in Number Theory: The Arithmetic of Quadratic Forms An introduction to the arithmetic and geometry of quadratic forms. Topics will include lattices, class number, local-global principle, Gauss composition, and other connections to algebraic number theory. Applications to the representation of integers by quadratic forms will be discussed, with the eventual goal of discussing a proof (joint work with Jonathan Hanke) of Conway's 290-Conjecture: if a positive-definite integral quadratic form takes the values 1, 2, ... , 290, then it takes all (positive) integer values. Instructor(s): Manjul Bhargava MAT 425 Analysis III: Integration Theory and Hilbert Spaces The 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. Instructor(s): Aleksandr Logunov MAT 457 Algebraic Geometry Introduction to affine and projective algebraic varieties over fields. Instructor(s): Joe Allen Waldron MAT 477 Advanced Graph Theory Advanced 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. Instructor(s): Maria Chudnovsky MAT 486 Random Processes Wiener measure. Stochastic differential equations. Markov diffusion processes. Linear theory of stationary processes. Ergodicity, mixing, central limit theorem for stationary processes. If time permits, the theory of products of random matrices and PDE with random coefficients will be discussed. Instructor(s): Allan M. Sly MAT 490/APC 490 Mathematical Introduction to Machine Learning This course gives a mathematical introduction to machine learning. There are three major components in this course. (1) Machine learning models (kernel methods, shallow and deep neural network models) for both supervised and unsupervised learning problems. (2) Optimization algorithms (gradient descent, stochastic gradient descent, EM). (3) Mathematical analysis of these models and algorithms. Instructor(s): Weinan E MAT INFO Princeton Calculus Orientation A successful self-placement in University-level calculus has two components. Beyond the straightforward list of calculus topics covered, there is an additional component of independent problem-solving skills at the University level. These problem-solving workshops that highlight the latter component are designed to assist students in their calculus placement decisions. These workshops will meet the Wednesday and Friday of the first week of classes. By Friday afternoon of the first week, students will register for MAT100, MAT103, MAT104, or MAT175. MAT INFO Princeton Calculus Orientation A successful self-placement in University-level calculus has two components. Beyond the straightforward list of calculus topics covered, there is an additional component of independent problem-solving skills at the University level. These problem-solving workshops that highlight the latter component are designed to assist students in their calculus placement decisions. These workshops will meet the Wednesday and Friday of the first week of classes. By Friday afternoon of the first week, students will register for MAT100, MAT103, MAT104, or MAT175. ORF 309/EGR 309/MAT 380 Probability and Stochastic Systems An 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 Instructor(s): Mykhaylo Shkolnikov