Differential forms on RDifferential forms, introduction to Lie groups, the DeRham theorem, Riemannian manifolds, curvature, the Hodge theory. Topics include matrix calculus for finite and infinite matrices (e.g., Wigner's semi-circle and Marcenko-Pastur laws), free probability, random graphs, combinatorial methods, matrix statistics, stochastic operators, passage to the continuum limit, moment methods, and compressed sensing. Includes classroom and laboratory demonstrations. Enables students to understand the role of mathematics in the research and development of efficient statistical methods. Definite integral; fundamental theorem of calculus. Additional topics may include sonic booms, Mach cone, caustics, lattices, dispersion and group velocity.

Students in Course 18 must register for the undergraduate version, Initial value problems: finite difference methods, accuracy and stability, heat equation, wave equations, conservation laws and shocks, level sets, Navier-Stokes.
Enrollment limited.Topics vary from year to year. Deduction and proof. Representations of SL(2). Solution of first-order ODEs by analytical, graphical, and numerical methods. Seminar in combinatorics, graph theory, and discrete mathematics in general. Frequent use of MATLAB in a wide range of scientific and engineering applications. Differentiation and integration of functions of one variable, with applications. Infinite series: geometric, p-harmonic, simple comparison tests, power series for some elementary functions. The partial fractions decomposition.

Recent research papers presented from leading conferences such as the International Conference on Computational Molecular Biology (RECOMB) and the Conference on Intelligent Systems for Molecular Biology (ISMB). Six-week review of one-variable calculus, emphasizing material not on the high-school AB syllabus: integration techniques and applications, improper integrals, infinite series, applications to other topics, such as probability and statistics, as time permits. Subject matter illustrated using natural fluid and solid systems found, for example, in geophysics and biology. Differentiation: definition, rules, application to graphing, rates, approximations, and extremum problems. Introduction to graph theory. Readings from current research papers in combinatorics.

Im Mai feiert mit18band ihren 30. The basis for understanding optical phenomena such as photonic crystals and band gaps, anomalous diffraction, mechanisms for optical confinement, optical fibers (new and old), nonlinearities, and integrated optical devices.

More demanding than Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. The mathematics of inverse problems involving waves, with examples taken from reflection seismology, synthetic aperture radar, and computerized tomography. Introduces new and significant developments in algebraic topology with the focus on homotopy theory and related areas. Covers functions of a complex variable; calculus of residues. Singularities, residues and computation of integrals. Basics of stochastic processes. Harmonic functions and Dirichlet's problem for the Laplace equation. Polar coordinates. Students in Course 18 must register for the undergraduate version, Multilinear algebra: tensors and exterior forms. Covers scientific computing topics (numerical differential equations, dense and sparse linear algebra, Fourier transformations, parallelization of large-scale scientific simulation) simultaneously with modern data science (machine learning, deep neural networks, automatic differentiation), focusing on the emerging techniques at the connection between these areas, such as neural differential equations and physics-informed deep learning. Lorenz and Rossler systems. Fundamental concepts illustrated through problems drawn from a variety of areas, including geophysics, biology, and the dynamics of sport. CALC I Credit cannot also be received for 18.01, ES.1801, ES.181A. Topics include Turan-type problems, Szemeredi's regularity lemma and applications, pseudorandom graphs, spectral graph theory, graph limits, arithmetic progressions (Roth, Szemeredi, Green-Tao), discrete Fourier analysis, Freiman's theorem on sumsets and structure.


Introduction to differential geometry, centered on notions of curvature. Recent developments in quantum field theory require mathematical techniques not usually covered in standard graduate subjects. Normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators. Topics vary from year to year. Lattice methods, finiteness of the class group, Dirichlet's unit theorem. Closed curves and immersed curves. Techniques of integration. Compared with Reviews linear algebra with applications to life sciences, finance, engineering, and big data. Instruction and practice in written and oral communication provided. Enrollment limited.Study of illustrative topics in discrete applied mathematics, including probability theory, information theory, coding theory, secret codes, generating functions, and linear programming. In person not required. Some programming required for homework and final project. More rigorous treatment of basic mathematical logic, Godel's theorems, and Zermelo-Fraenkel set theory. Vector algebra in 3-space, determinants, matrices. Uses linear algebra software. Includes ordinary differential equations; Bessel and Legendre functions; Sturm-Liouville theory; partial differential equations; heat equation; and wave equations. Improper integrals. Chebyshev inequality, law of large numbers, and central limit theorem.


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