Course Schedule

APC 199/MAT 199 Math Alive Mathematics has profoundly changed our world, from the way we communicate with each other and listen to music, to banking and computers. This course is designed for those without college mathematics who want to understand the mathematical concepts behind important modern applications. The course consists of individual modules, each focusing on a particular application (e.g., digital music, sending secure emails, and using statistics to explain, or hide, facts). The emphasis is on ideas, not on sophisticated mathematical techniques, but there will be substantial problem-set requirements. Students will learn by doing simple examples. Instructor(s): Adam Wade Marcus APC 350/CEE 350/MAT 322 Introduction to Differential Equations This course will introduce the basic theory, models and techniques for ordinary and partial differential equations. Emphasis will be placed on the connection with other disciplines of science and engineering. We will try to strike a balance between the theoretical (e.g. existence and uniqueness issues, qualitative properties) and the more practical issues such as analytical and numerical approximations. Instructor(s): Jiequn Han COS 433/MAT 473 Cryptography An introduction to the theory and practice of modern cryptography, with an emphasis on the fundamental ideas. Topics covered include private key and public key encryption schemes, digital signatures, pseudorandom generators and functions, chosen ciphertext security, and some advanced topics. Instructor(s): Mark Landry Zhandry COS 488/MAT 474 Introduction to Analytic Combinatorics Analytic Combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the scientific analysis of algorithms in computer science and for the study of scientific models in many other disciplines. This course combines motivation for the study of the field with an introduction to underlying techniques, by covering as applications the analysis of numerous fundamental algorithms from computer science. The second half of the course introduces Analytic Combinatorics, starting from basic principles. Instructor(s): Robert Sedgewick 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 problems in this field. Instructor(s): Stanislav Yefimovic Shvartsman MAE 306/MAT 392 Mathematics in Engineering II This course covers a range of fundamental mathematical techniques and methods that can be employed to solve problems in contemporary engineering and the applied sciences. Topics include algebraic equations, vectors and tensors, numerical integration, analytical and numerical solution of ordinary and partial differential equations, time-series data and the Fourier transform, and calculus of variations. The course synthesizes descriptive observations, mathematical theories, numerical methods, and engineering consequences. Instructor(s): Mikko Petteri Haataja 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): Tatiana Katarzyna Howard, Ana Menezes 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, Tetiana Shcherbyna, Boyu Zhang 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): Chiara Damiolini, 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): Theodore Dimitrios Drivas, Henry Theodore Horton, Chao Li 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): Clark Wilmer Butler, Mark Weaver McConnell, Rafael Montezuma Pinheiro Cabral, Remy van Dobben de Bruyn MAT 204 Advanced Linear Algebra with Applications Companion course to MAT203. Linear systems of equations, linear independence and dimension, linear transforms, determinants, (real and complex) eigenvectors and eigenvalues, orthogonality, spectral theorem, singular value decomposition, Jordan forms, other topics as time permits. More abstract than MAT202 but more concrete than MAT217. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or MAT215 or equivalent. Instructor(s): Andrew V Yarmola MAT 215 Honors Analysis (Single Variable) An introduction to the mathematicsal 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 Rieman integral, conditions for integrability of functions and term by term differentiation and integration of series of functions, Taylor's Theorem. Instructor(s): Stephen Edward McKeown MAT 217 Honors Linear Algebra A rigorous course in linear algebra with an emphasis on proof rather than applications. Topics include vector spaces, linear transformations, inner product spaces, determinants, eigenvalues, the Cayley-Hamilton theorem, Jordan form, the spectral theorem for normal transformations, bilinear and quadratic forms. Instructor(s): Yunqing Tang, Ian Michael Zemke MAT 218 Accelerated Honors Analysis II Continuation of the rigorous introduction to analysis in MAT216 Instructor(s): Robert Clifford Gunning MAT 325 Analysis I: Fourier Series and Partial Differential Equations Basic facts about Fourier Series, Fourier Transformations, and applications to the classical partial differential equations will be covered. Also Finite Fourier Series, Dirichlet Characters, and applications to properties of primes. Instructor(s): Yakov Mordechai Shlapentokh-Rothman MAT 330 Complex Analysis with Applications The theory of functions of one complex variable, covering analyticity, contour integration, residues, power series expansions, and conformal mapping. The goal in the course is to give adequate treatment of the basic theory and also demonstrate the use of complex analysis as a tool for solving problems. Instructor(s): Javier Gómez-Serrano MAT 346 Algebra II Local Fields and the Galois theory of Local Fields. Instructor(s): Shou-Wu Zhang MAT 355 Introduction to Differential Geometry Introduction to geometry of surfaces. Surfaces in Euclidean space, second fundamental form, minimal surfaces, geodescis, Gauss curvature, Gauss-Bonnet formula. Then differential forms and the higher-dimensional Gauss-Bonnet, as time permits. Instructor(s): Otis Chodosh MAT 375/COS 342 Introduction to Graph Theory The fundamental theorems and algorithms of graph theory. Topics include: connectivity, matchings, graph coloring, planarity, the four-color theorem, extremal problems, network flows, and related algorithms. Instructor(s): Paul Douglas Seymour MAT 378 Theory of Games Games in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations, solution concepts; related topics and applications. Instructor(s): Jonathan Michael Fickenscher MAT 419 Topics in Number Theory: Algebraic Number Theory Course on algebraic number theory. Topics covered include number fields and their integer rings, class groups, zeta and L-functions. Instructor(s): Francesc Castella MAT 429 Topics in Analysis: Distribution Theory, PDE & Basic Inequalities of Analysis Introduction to Geometric Partial Differential Equations. The course will review some basic topics in Elliptic theory and give a comprehensive introduction to linear and nonlinear wave equations with applications to relativistic filed theories including General Relativity. Instructor(s): Sergiu Klainerman MAT 447 Commutative Algebra This course will cover the standard material in a first course on commutative algebra. Topics include: ideals in and modules over commutative rings, localization, primary decomposition, integral dependence, Noetherian rings and chain conditions, discrete valuation rings and Dedekind domains, completion; and dimension theory. Instructor(s): Hansheng Diao MAT 468 Topics in Knot Theory, Modern Knot Invariantiants & Applications Knot theory involves the study of smoothly embedded circles in three-dimensional space. There ar lots of different techniques to study knots: combinatorial invariants, algebraic topology, hyperbolic geometry, Khovanov homology and mathematical gauge theory. This course will cover some of the modern techniques and recent developments in the field. Instructor(s): Zoltán Szabó MAT 478 Topics In Combinatorics, Extremal Combinatorics This course will cover topics in Extremal Combinatorics including ones motivated by questions in other areas like Computer Science, Information Theory, Number Theory and Geometry. The subjects that will be covered include Graph powers, the Shannon capacity and the Witsenhausen rate of graphs, Szemeredi's Regularity Lemma and its applications in graph property testing and in the study of sets with no 3 term arithmetic progressions, the Combinatorial Nullstellensatz and its applications, the capset problem, Containers and list coloring, and related topics as time permits. Instructor(s): Noga Mordechai Alon 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): Ramon van Handel PHY 403/MAT 493 Mathematical Methods of Physics Mathematical methods and terminology which are essential for modern theoretical physics. These include some of the traditional techniques of mathematical analysis, but also more modern tools such as group theory, functional analysis, calculus of variations, non-linear operator theory and differential geometry. Mathematical theories are not treated as ends in themselves; the goal is to show how mathematical tools are developed to solve physical problems. Instructor(s): Bogdan Andrei Bernevig