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): Zeev Dvir 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): James D. Olsen 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): Otis Chodosh 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 An intensive and rigorous treatment of algebra and trigonometry as preparation for further courses in calculus or statistics. Topics include functions and their graphs, equations involving polynomial and rational functions, exponentials, logarithms and trigonometry. 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): Jonathan Michael Fickenscher 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 Hanselman 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): Jennifer Michelle Johnson 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 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): Oanh Thi Hoang Nguyen 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): Tetiana Shcherbyna 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. 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): Charles Louis Fefferman, 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 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. 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 be devoted to Galois theory, developing the necessary group theory, including the Sylow theorems, and algebra from scratch. 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): Paul Chien-Ping Yang 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: Arithmetic of Elliptic Curves The study of the arithmetic of elliptic curves is an extremely rich area of mathematics, incorporating ideas from complex analysis, algebraic geometry, group and Galois theory, and of course number theory. Many fundamental problems in number theory, such as Fermat's Last Theorem and the congruent number problem, lead naturally to the study of elliptic curves. The purpose of this course is to explore some of the basic ideas in this area, including the group law on elliptic curves, the structure of this group over various fields, and the Birch and Swinnerton-Dyer conjeture. 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): Mihalis Dafermos MAT 449 Topics in Algebra: Representation Theory An introduction to representation theory of Lie groups and Lie algebras. The goal is to cover roughly the first half of Knapp's book. Instructor(s): Sophie Marguerite Morel 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 490/APC 390 Mathematical Introduction to Machine Learning This course gives a mathematical introduction to machine learning. It is not about proving theorems in machine learning, but rather a unified understanding of the models and algorithms used in machine learning. It begins with a simple introduction to supervised and unsupervised learning, including regression, classification, density estimation, clustering, and dimension reduction. Simple models such as linear regression, support vector machines, and k-means will be introduced, followed by focus on deep learning. 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