Mathematics

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 103: Calculus IFirst semester of calculus. Topics include limits, continuity, the derivative, basic differentiation formulas and applications (curvesketching, optimization, related rates), definite and indefinite integrals, the fundamental theorem of calculus.

MAT 104: Calculus IIContinuation 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.

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.

APC 199/MAT 199: Math AliveMathematics 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. compression, animation and using statistics to explain, or hide, facts). The emphasis is on ideas, not on sophisticated mathematical techniques, but there will be substantial problemset requirements. Students will learn by doing simple examples.

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 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.

MAT 204: Advanced Linear Algebra with ApplicationsCompanion 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.

MAT 215: Honors Analysis (Single Variable)An introduction to the mathematical discipline of analysis, to prepare for higherlevel course work in the department. Topics include rigorous epsilondelta treatment of limits, convergence, and uniform convergence of sequences and series. Continuity, uniform continuity, and differentiability of functions. The HeineBorel Theorem. The Riemann integral, conditions for integrability of functions and term by term differentiation and integration of series of functions, Taylor's Theorem.

MAT 217: Honors Linear AlgebraA 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 CayleyHamilton theorem, Jordan form, the spectral theorem for normal transformations, bilinear and quadratic forms.

MAT 218: Accelerated Honors Analysis IIContinuation of the rigorous introduction to analysis in MAT216

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 problems in this field.

MAE 306/MAT 392: Mathematics in Engineering IIThis 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, numerical integration, analytical and numerical solution of ordinary and partial differential equations, harmonic functions and conformal maps, and timeseries data. The course synthesizes descriptive observations, mathematical theories, numerical methods, and engineering consequences.

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

MAT 325: Analysis I: Fourier Series and Partial Differential EquationsBasic 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.

MAT 330: Complex Analysis with ApplicationsThe 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.

MAT 346: Algebra IIThis course is a continuation of MAT 345 and its introduction to representation theory. We will cover semisimple algebras, application to group theory, Artin's and Brauer's theorems characterizing representations over the complex numbers, rationality questions, and Brauer's theory of representations mod p. This will lead to one or more advanced topics in finite groups or Lie algebras.

APC 350/CEE 350/MAT 322: Introduction to Differential EquationsThis 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.

MAT 355: Introduction to Differential GeometryIntroduction to geometry of surfaces. Surfaces in Euclidean space, second fundamental form, minimal surfaces, geodescis, Gauss curvature, GaussBonnet formula. Then differential forms and the higherdimensional GaussBonnet, as time permits.

MAT 375/COS 342: Introduction to Graph TheoryThe fundamental theorems and algorithms of graph theory. Topics include: connectivity, matchings, graph coloring, planarity, the fourcolor theorem, extremal problems, network flows, and related algorithms.

MAT 378: Theory of GamesGames in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristicfunction form, imputations, solution concepts; related topics and applications.

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.

PHY 403/MAT 493: Mathematical Methods of PhysicsMathematical 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, nonlinear 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.

MAT 419: Topics in Number Theory: Algebraic Number TheoryCourse on algebraic number theory. Topics covered include number fields and their integer rings, class groups, zeta and Lfunctions.

MAT 429: Topics in Analysis: Distribution Theory, PDE & Basic Inequalities of AnalysisIntroduction 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.

COS 433/MAT 473: CryptographyAn 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.

MAT 478: Topics In Combinatorics: The Probabilistic MethodThis course will cover probabilistic methods in combinatorics and their applications in theoretical computer science. The topics include linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, geometry, the VC dimension of a range space and its applications, and possibly more as time permits.

COS 488/MAT 474: Introduction to Analytic CombinatoricsAnalytic 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.

MAT 515: Topics in Number Theory and Related Analysis: Diophantine Analysis on Homogeneous and Affine VarietiesWe discuss some basic problems, techniques and applications in studying certain diophantine equations. The emphasis is on producing solutions. After presenting basic tools to address such questions such as the circle method and its variats, automorphic forms and the Schmidt subspace theorem, we turn to some recent special techniques and some applications.

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.

COS 522/MAT 578: Computational ComplexityComputational complexity theory is a mathematical discipline that studies efficient computation. The course covers some of the truly beautiful ideas of modern complexity theory such as: approaches to the famous P vs NP question and why they are stuck; complexity classes and their relationship; circuit lower bounds; proof systems such as zero knowledge proofs, interactive proofs and probabilistically checkable proofs; hardness of approximation; derandomization and the hardness vs randomness paradigm; quantum computing.

MAT 528: Topics in Nonlinear Analysis: Recent Trends in Mathematical Fluid DynamicsThe aim of the course is to discuss recent developments in the field of mathematical fluid dynamics. Possible topics include nonuniqueness of weak solutions to equations arising from fluids, and singularity formation for strong solutions to fluid equations.

MAT 531: Introduction to Riemann SurfacesThis course is an introduction to the theory of compact Riemann surfaces, including some basic properties of the topology of surfaces, differential forms and the basic existence theorems, the general uniformization theorem, and the RiemannRoch theorem and some of its consequences.

MAT 560: Algebraic TopologyThe aim of the course is to study some of the basic algebraic techniques in Topology, such as homology groups, cohomology groups and homotopy groups of topological spaces.

MAT 567: Topics in Low Dimensional Topology: YangMills Equation and Instanton Floer HomologyThis is an introduction to the mathematical theory of the YangMills equation, with an emphasis on its applications to lowdimensional topology. The course consists of two parts. In the first half, we develop the analysis of the ASD equation on fourmanifolds and prove Donaldson's diagonalization theorem. Along the way, we introduce tools that are also useful in other fields of mathematics, such as principal bundles, spin structures, and the AtiyahSinger index theorem. In the second half, we develop the singular instanton Floer homology theory and discuss its applications to knots and links.

MAT 577: Topics in Combinatorics: The Probabilistic MethodThis course covers probabilistic methods in combinatorics and their applications in theoretical computer science. The topics include linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, geometry, the VC dimension of a range space and its applications, and possibly more as time permits.

MAT 579: Topics in Discrete Mathematics: Forbidding induced subgraphs: structure and propertiesOver the course of the semester, we discuss theorems and conjectures about graphs with certain induced subgraphs forbidden. Among others, we cover the following topics: perfect graphs, clawfree graphs, algorithms for detecting induced subgraphs, and the ErdoHajnal Conjecture.

MAT 586/APC 511/MOL 511/QCB 513: Computational Methods in CryoElectron MicroscopyThis course focuses on computational methods in cryoEM, including threedimensional abinitio modelling, structure refinement, resolving structural variability of heterogeneous populations, particle picking, model validation, and resolution determination. Special emphasis is given to methods that play a significant role in many other data science applications. These comprise of key elements of statistical inference, image processing, and linear and nonlinear dimensionality reduction. The software packages RELION and ASPIRE are routinely used for class demonstration on both simulated and publicly available experimental datasets.

MAT 588/APC 588: Topics in Numerical Analysis: Optimization On Smooth ManifoldsThe key player is the problem: minimize f(x), where x lives on a smooth manifold M, and f is a smooth cost function on M. Applications abound in scientific computing, signal processing, computer vision, machine learning and statistics. Manifolds arise in optimization as a result of constraints (e.g., lowrank, orthogonality) and as a result of symmetry (quotient spaces). By endowing the manifold with a Riemannian structure, we obtain meaningful notions of gradient and Hessian on the manifold. This enables us to generalize standard algorithms such as gradient descent and trustregions. The course mixes mathematical analysis and coding.

MAT 589: Topics in Probability, Statistics and Dynamics: Modern Discrete Probability TheoryThe aim of this course is to survey some of the fundamentals of modern discrete probability, particularly random processes on networks and discrete structures.

MAT 90: Topics in Discrete MathematicsNo description available

MAT 91: Optimization on Smooth ManifoldsNo description available

MAT 92: Langlands CorrespondenceNo description available

MAT 93: Topics in Low Dimensional TopologyNo description available

MAT 94: Rigid Analytic GeometryNo description available

MAT 95: Topics in Number Theory and Related AnalysisNo description available

MAT 96: Deformation TheoryNo description available

MAT 97: Mathematical Theory of Black HolesNo description available