## Mathematics

- 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 problem-set requirements. Students will learn by doing simple examples.
- APC 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.
- 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.
- 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; de-randomization and the hardness vs randomness paradigm; quantum computing.
- 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 time-series data. The course synthesizes descriptive observations, mathematical theories, numerical methods, and engineering consequences.
- 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 (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 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: 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 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 Cayley-Hamilton theorem, Jordan form, the spectral theorem for normal transformations, bilinear and quadratic forms.
- MAT 218: Multivariable Analysis and Linear Algebra IIContinuation of the rigorous introduction to analysis in MAT 216
- MAT 321/APC 321: Numerical MethodsIntroduction 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.
- 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 IILocal Fields and the Galois theory of Local Fields.
- MAT 355: Introduction to Differential GeometryIntroduction to geometry of surfaces. Surfaces in Euclidean space: first fundamental form, second fundamental form, geodesics, Gauss curvature, Gauss-Bonnet Theorem. Minimal surfaces in the Euclidean space.
- MAT 375/COS 342: Introduction to Graph TheoryThe 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.
- MAT 378: Theory of GamesGames in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations, solution concepts; related topics and applications.
- MAT 419: Topics in Number Theory: A Journey Towards the Herbrand-Ribet TheoremA course in algebraic number theory, consisting of two topics. The first topic is the algebraic number theory behind the statement of the Herbrand-Ribet theorem: number fields and their integer rings, local fields, global fields, the statement of class field theory. The second topic is one of the following five: The proof of the Herbrand-Ribet theorem, a proof of class field theory, the theory of cyclotomic fields, quadratic forms and higher composition laws (Bhargava 2001), or the basic theory of elliptic curves.
- MAT 425: Analysis III: Integration Theory and Hilbert SpacesThe 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.
- MAT 449: Topics in Algebra: Representation TheoryAn introduction to representation theory of Lie groups and semisimple Lie algebras.
- MAT 478: Topics In Combinatorics: Extremal CombinatoricsThis 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.
- 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 528: Topics in Nonlinear Analysis: Topics in General RelativityThis course covers the following topics: formation of trapped surfaces and Mots, rigidity of black holes, and stability of Kerr.
- MAT 547: Topics in Algebraic Geometry: Introduction to the theory of schemesA gentle introduction to the theory of schemes, emphasizing examples. We assume familiarity with quasi-projective varieties, maps between quasi-projective varieties, some basic awareness about manifolds and their homology, and the language of commutative algebra. Time permitting, we formulate the main results of intersection theory and discuss their applications. The primary references are Vakil's note 'The Rising Sea' and the book of Hartshorne 'Algebraic geometry'.
- 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 569: Topics in Topology: Characteristic Classes and ApplicationsIn this course, we study real and complex vector bundles, their characteristic classes and various applications. These include Stiefel-Whitney classes, Chern classes, Pontrjagin classes, the splitting principle, classifying spaces, Grassmanians, Hirzebruch's signature theorem, and the 7-dimensional exotic spheres of Milnor. Additional topics may include Morse Theory, the h-cobordism Theorem, and the Poincare Conjecture in higher dimensions.
- MAT 577: Topics in Combinatorics: Extremal CombinatoricsThis course covers topics in Extremal Combinatorics including ones motivated by questions in other areas like computer science, information theory, number theory and geometry. The subjects that are 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.
- MAT 579: Topics in Discrete Mathematics: Induced Subgraphs and Tree DecompositionsTree decompositions are a powerful tool in structural graph theory that is traditionally used in the context of forbidden graph minors. Connecting tree decompositions and forbidden induced subgraphs is a new research area that has gained momentum over the last several years. This course will introduce the subject and discuss the recent developments.
- MAT 586/APC 511/MOL 511/QCB 513: Computational Methods in Cryo-Electron MicroscopyThis course focuses on computational methods in cryo-EM, including three-dimensional ab-initio 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, optimization, and dimensionality reduction. The software packages RELION and ASPIRE are routinely used for class demonstration on both simulated and publicly available experimental datasets.
- MAT 589: Topics in Probability, Statistics and Dynamics: First Passage PercolationThis course covers a range of topics in first and last passage percolation. First passage percolation is a model of the shortest path through a random environment and is conjectured to be in the KPZ universality class. However, many of the simplest questions remain open. The tools to study it come mainly from percolation theory. By contrast, several variants of last passage percolation (roughly the longest oriented path in a random environment) are exactly solvable meaning many quantities can be calculated explicitly. We examine recent developments in both settings.
- 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.