## Mathematics

- COS 433/MAT 473: CryptographyAn 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.
- 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 101: History of MathematicsThroughout 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.
- 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 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.
- 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 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.
- 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 210: One Variable Calculus with ProofsMAT 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).
- 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 Analysis and Scientific ComputingIntroduction 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.
- 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 385: Probability TheoryAn 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.
- MAT 418: Topics in Algebraic Number Theory: Algebraic Number TheoryTopics introducing various aspects of algebraic number theory, including number fields and their rings of integers, cyclotomic fields, and class groups.
- MAT 447: Commutative AlgebraThis 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.
- MAT 449: Topics in Algebra: Representation TheoryAn introduction to representation theory of Lie groups and semisimple Lie algebras.
- 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 520: Functional AnalysisBasic introductory course to modern methods of analysis. The possible topics may include Lp spaces, Banach spaces, uniform boundedness principle, closed graph theorem, locally convex spaces, distributions, Fourier transform, Riesz interpolation theorem, Hardy-Littlewood maximal function, Calderon-Zygmund theory, oscillatory integrals, almost orthogonality, Sobolev spaces, restriction theorems, spectral theory of compact operators, applications to partial differential equations.
- 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 526: Topics in Geometric Analysis: The PDEs of incompressible fluid dynamicsThis course first covers some classical topics about the Euler and Navier-Stokes equations, such as classical existence theories of smooth solutions and weak solutions, global well-posedness in two dimensions and local well-posedness in three dimensions, the Ladyzenskaya-Prodi-Serrin conditional regularity theorem, Caffarelli-Kohn-Nirenberg local regularity theory for Navier-Stokes and various a-priori estimates. We then get to more recent results, such as the Escauriaza-Seregin-Sverak theorem, more recent conditional regularity results, Liouville theorems and results under special symmetry assumptions.
- MAT 547: Topics in Algebraic Geometry: Arithmetic Algebraic GeometryEtale cohomology can be applied to concrete problems in number theory and beyond. For these applications, it is often important to bound the dimension of cohomology groups. I will discuss existing approaches to this problem as well as some new ideas.
- MAT 549: Topics in Algebra: Monodromy and Some of its ApplicationsThe course will be devoted to monodromy and some of its applications.
- 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 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: Diffeomorphisms of ManifoldsThis course focuses on spaces of smooth embeddings from one manifold to another. Topics may include embedding spaces as infinite dimensional manifolds, spaces of diffeomorphisms of surfaces, diffeomorphisms of the 3-sphere, pseudo-isotopy theory, Cerf theory, and diffeomorphisms of 4-manifolds.
- MAT 579: Topics in Discrete Mathematics: Induced SubgraphsThis course begins with a general introduction to the standard theorems about induced subgraphs, then go on to survey some recent work. Two central conjectures are (i) the Erdos-Hajnal conjecture, that every graph H, there exists c>0 such that every graph G not containing H as an induced subgraph has a clique or stable set of size at least |G|^c, and (ii) the Gyarfas-Sumner conjecture, that for every tree H and every integer k, there exists c such that every graph not containing H as an induced subgraph and with clique number at most k, has chromatic number at most c. Attempts to prove these two conjectures are themes throughout.
- 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
- PHY 521/MAT 597: Introduction to Mathematical PhysicsAn introduction to the statistical mechanic of classical and quantum spin systems. Among the topics to be discussed are phase transitions, emergent structures, critical phenomena, and scaling limits. The goal is to present the physics embodied in the subject along with mathematical methods, from probability and analysis, for rigorous results concerning the phenomena displayed by, and within, the subject's essential models. The lectures start with a brisk review of what was covered in the course in Spring 2021, and then continue beyond that with a more extended, though still self contained, discussion of quantum systems.