Course Schedule

COS 433/MAT 473 Cryptography An introduction to the theory of modern cryptography. Topics covered include private key and public key encryption schemes, digital signatures, pseudorandom generators and functions, zero-knowledge proofs, and some advanced topics. Instructor(s): Alex Lombardi 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 MAT 101 History of Mathematics Throughout the course, we will be studying some of the most beautiful and timeless mathematical problems and solutions (theorems and proofs), and their discoverers, as well as the historical developments that led to each breakthrough. Rather than going deeply into a single complete theory as we understand it today, in this course the material is drawn from a broad variety of sources and topics and arranged roughly chronologically. One should leave this course with a bird's-eye view of many developments in mathematics from antiquity up to the 21st century. This makes the course both fun and interesting, and also challenging. Instructor(s): Alex Kontorovich 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, Tongmu He, Kimoi Kemboi, Stan Palasek, Sahana Vasudevan, Mingjia Zhang MAT 104 Calculus II Continuation of MAT 103. 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): Hongyi Liu, Mark Weaver McConnell, Anubhav Mukherjee, Bogdan Zavyalov 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): Tatyana Chmutova 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): Louis Esser, Casey Lynn Kelleher, János Kollár, Tristan Jean-Bernard Leger, Andrew O'Desky, Sung Gi Park, John Thomas Sheridan, David Villalobos MAT 202 Linear Algebra with Applications Companion course to MAT 201. 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): Fraser Malcolm Watt Binns, Matija Bucic, Tangli Ge, Sepehr Hajebi, Jennifer Michelle Johnson, Justin Lacini, Jennifer Li 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): David Gabai, Stan Palasek, Liyang Yang MAT 210 One Variable Calculus with Proofs MAT 210 will survey the main ideas of calculus in a single variable incorporating an introduction to formal mathematical proofs. The course will place equal emphasis on theory (how to construct formal mathematical definitions and rigorous, logical proofs) and on practice (concrete computational examples involving integration and infinite sequences and series). This course provides a more theoretical foundation in single variable calculus than MAT104, intended to prepare students better for a first course in real analysis (MAT215), but it covers all the computational tools needed to continue to multivariable calculus (MAT201 or MAT203). Instructor(s): Jonathan Hanselman 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): Wei Ho MAT 215 Single Variable Analysis with an Introduction to Proofs 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, Jacob Shapiro MAT 216 Multivariable Analysis and Linear Algebra 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 MAT 218. Instructor(s): Joshua Xu Wang MAT 300 Multivariable Analysis I To 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. Instructor(s): Sun-Yung Alice Chang 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): Semon Rezchikov MAT 321/APC 321 Numerical Analysis and Scientific Computing 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 or Python. Instructor(s): Marc Aurèle Tiberius Gilles 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): Lue Pan 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): Jakub Witaszek 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 The 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. Instructor(s): Noga Mordechai Alon MAT 385 Probability Theory An introduction to probability theory. The course begins with the measure theoretic foundations of probability theory, expectation, distributions and limit theorems. Further topics include concentration of measure, Markov chains and martingales. Instructor(s): Dmitry Krachun MAT 418 Topics in Algebraic Number Theory: Algebraic Number Theory Topics introducing various aspects of algebraic number theory, including number fields and their rings of integers, cyclotomic fields, and class groups. Instructor(s): Christopher McLean Skinner 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): Chenyang Xu MAT 449 Topics in Algebra: Representation Theory An introduction to representation theory of Lie groups and semisimple Lie algebras. Instructor(s): Shou-Wu Zhang 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 90 Schemes and Cohomology No description available Instructor(s): Chenyang Xu MAT 91 Cyclotomic Fields and Iwasawa Conjecture No description available Instructor(s): Shou-Wu Zhang MAT 92 Diffeomorphisms of Manifolds No description available Instructor(s): David Gabai MAT 93 Random Matrix Theory No description available Instructor(s): Jacob Shapiro MAT 94 Euler Systems No description available Instructor(s): Christopher McLean Skinner MAT 95 J-Holomorphic Curves in Symplectic Topology No description available Instructor(s): Semon Rezchikov MAT 981 Junior Independent Work No Description Available 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): Mark Cerenzia