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Conte, S. Hildebrand, F. Froberg, C. MA Numerical Methods for Partial Differential Equations Finite difference methods for two point boundary value problems, Laplace equation on the square, heat equation and symmetric hyperbolic systems in 1 D. Suggested books : Smith, G. Evans, G. Blackledge, J. Suggested books : Faires, J. Stoer, J. Iserlas, A. Suggested books : Ross, S. Taylor, H. Suggested books : Luenberger, D. Shiryaev, A. Shreve, S. Suggested books : Lichtenberg, A.

Guckenheimer, J. Prerequisites, if any: familiarity with linear algebra - matrices, and ordinary differential equations Desirable: ability to write codes for solving simple problems. Suggested books : S. Alligood, T. Tabor, Chaos and Integrability in Non-linear Dynamics , Rudin, W.

Rudin, Functional Anaysis 2nd Ed. House, New Delhi, Munkres, J. Milnor, Morse Theory , Ann. Press, Princeton, Zariski spectrum as a frame Refresher on categories : Categories, functors, Yoneda Lemma, equivalence of categories, adjoints. Presheaves and Sheaves Locally ringed spaces and schemes Separated schemes, proper schemes, irreducible schemes, reduced schemes, integral schemes, noetherian schemes. Morphisms : separated, proper, finite morphisms, finite type morphisms, affine morphisms Sheaves of algebras : affine morphisms as sheaves of algebras Sheaves of modules over a scheme, Quasi-coherent and coherent sheaves Divisors and Line Bundles, Weil divisors, Cartier divisors, Line bundles on Projective spaces, Serre sheaves.

Springer-Verlag, New York-Heidelberg, Robin Hartshorne, Residues and duality , Lecture notes of a seminar on the work of A. With an appendix by P. Lecture Notes in Mathematics, No. Triangulated categories, Derived categories of abelian categories. Injective and flasque resolutions. Suggested books : Artin, E.

Borevich, Z. Cassels, J. Hasse, H. Hecke, E. Samuel, P. Suggested books : Shafarevich, I. Smith, K. Reid, M. Fulton, W. Holme, A.

  • Exercises 7.2.
  • Department of Mathematics;
  • CALCULUS 1 and 2: RESOURCES !!!.
  • Works of Harriet Beecher Stowe;

MA Introduction to Homological Algebra Polynomial ring, Projective modules, injective modules, flat modules, additive category, abelian category, exact functor, adjoint functors, co limits, category of complexes, snake lemma, derived functor, resolutions, Tor and Ext, dimension, local cohomology,group co homology, sheaf cohomology, Cech cohomology, Grothendieck spectral sequence, Leray spectral sequence. Suggested books : Cartan and Eilenberg, Homological Algebra. Weibel, Introduction to Homological Algebra. Rotman, Introduction to Homological Algebra. Suggested books : Apostol, T.

Davenport, H. MA Combinatorics Pre-requisites : Calculus, Linear algebra and some exposure to proofs and abstract mathematics. Programming in Sage will be a part of every lecture. Richard P. Suggested books : Stanley, R. Sagan, B. Prasad, A. Stanley, R. Suggested books : V. Varadarajan, Lie groups, Lie algebras and their representations , Sringer Hall, Lie groups, Lie algebras and representations , Springer Knapp, Representation theory of semismiple lie groups. An overview based on examples , Princeton university press Kesavan, S.

Evans, L. Schwartz, L. Hermann, Theorie des Distributions , Suggested books : Conway, J. Berberian, S. MA Topics in Complex Analysis The general theory of holomorphic mappings between bounded domains, automorphisms of bounded domains, discussions on the non-existence of a classical Riemann Mapping Theorem in several variables, discussion of the various forms of the one-variable Riemann Mapping Theorem, the Rosay-Wong Theorem, other Riemann-Rosay-Wong-type results e.

Suggested books : Krantz, S. Suggested books : John B. Conway, A course in Functional Analysis , Springer, Suggested books : Dym, H. Stein, E. Sadosky, C. Suggested books : Korner, I. Robert Ash. Serre, J. Thangavelu, S. Rudin W. Students who have not seen any one-dimensional complex dynamics earlier but are highly interested in this course are encouraged to speak to the instructor.

  1. Undergraduate Course Descriptions | Department of Mathematics | NYU Courant;
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  4. Suggested books : L. Amsterdam, Janich, K. Suggested books : Brickell, F. Guillemin, V. Milnor, John W. Suggested books : Hatcher, A. Press, Indian edition is available. Munkres, I. Shastri, A. MA Riemannian Geometry Review of differentiable manifolds and tensors, Riemannian metrics, Levi-Civita connection, geodesics, exponential map, curvature tensor, first and second variation formulas, Jacobi fields, conjugate points and cut locus, Cartan-Hadamard and Bonnet Myers theorems.

    Springer-Verlag, New York, MA Introduction to Homotopy Type Theory Prerequisite courses: MA , MA , MA Pre-requisites : Algebraic Topology, Dependent Type Theory This course introduces homotopy type theory, which provides alternative foundations for mathematics based on deep connections between type theory, from logic and computer science, and homotopy theory, from topology. Tutorial for the Lean Theorem Prover. Benedetti-Petronio, Lectures on Hyperbolic Geometry.

    Introduction to Calculus and Analysis: Volume II

    Martelli, Introduction to Geometric Topology. A first course in algebraic topology is helpful but not necessary. Real analysis in more than one variable. Linear algebra.

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    Suggested books : Spivak M. Kumaresan S. Hindustan Book Agency, New Delhi, Warner F. Springer-Verlag, New York-Berlin, Lee J. Analysis multivariable calculus, some measure theory, function spaces. Ideally, the spectral theory of compact self-adjoint operators too, but we will recall the statement if not the proof Basics of Riemannian geometry Metrics, Levi-Civita connection, curvature, Geodesics, Normal coordinates, Riemannian Volume form , The Laplace equation on compact manifolds Existence, Uniqueness, Sobolev spaces, Schauder estimates , Hodge theory, more general elliptic equations Fredholmness etc , Uniformization theorem.

    Suggested books : Do Carmo, Riemannian Geometry. Griffiths and Harris, Principles of Algebraic Geometry. Nicolaescu, Lectures on the Geometry of Manifolds.

    Aubin, Some nonlinear problems in geometry. Evans, Partial differential equations. Gilbarg and Trudinger, Elliptic partial differential equations of the second order. Szekelyhidi, Extremal Kahler metrics. Douglas, R. List of topics time permitting : 1. Suggested books : Rajendra Bhatia, Matrix Analysis , vol. Roger A. Horn and Charles R. Johnson, Matrix analysis , Cambridge University Press, Johnson, Topics in matrix analysis , Cambridge University Press, Semi group theory:Hille-Yosida theorem, Applications to heat, Schroedinger and wave equations.

    Suggested books : Evans, L. Pazy, A. Prasad, P. Treves, J. Ando, On a pair of commutative contractions , Acta Sci.

    Introduction to Calculus and Analysis II/1 by Richard Courant, Fritz John | Waterstones

    Szeged 24 88— Das, B. Krishna and Sarkar, Jaydeb, Ando dilations, von Neumann inequality, and distinguished varieties. Bensoussan, J. Jikov, S. Kozlov, and O. Suggested books : Rabinowitz, Minimax methods in critical point theory with applications to differential equations , C. Ghoussoub, N. Struwe, M. Suggested books : Porter, D.

    Gakhov, F. Muskhelishvilli, N. Nobe, B. Schwartz, Theorie des Distributions , Hermann, MA Topics around the Grothendieck inequality Top. Suggested books : H. Iwaniec and E. Suggested books : J.

    Calculus: Single Variable Part 1 - Functions

    Diamond and J. Other models of random matrices - Wishart and Jacobi ensembles. Fluctuation behaviour of eigenvalues if time permits. Suggested books : Durrett, R. Billingsley, P. Kallenberg, O. Walsh, J. Suggested books : P. Billingsley, Convergence of probability measures. Karatzas and Shreve, Brownian motion and stochastic calculus. Revuz and Yor, Continuous martingales and Brownian motion. Oksendal, Introduction to stochastic differential equations. Suggested books : Box, G. Jenkins, G. Efron, B.

    Parker, T. Isoperimetry and processes , Springer-Verlag, Berlin, Michel Ledoux, Isoperimetry and Gaussian analysis , St. Flour lecture notes Suggested books : Amman, M. Brigo, D and Mercurio, F. Hausdorff and Minkowski dimensions. Dimension computation of certain random fractals derived from Brownian motion range, graph and zero set. Random walks and discrete harmonic functions. Brownian motion and harmonic functions. Recurrence and transience. What sets does Brownian motion hit? Polar sets and Capacity. Brownian motion in the plane : Conformal invariance, Winding number.

    Gaussian free field : Definition and basic properties. Suggested books : I. Karatzas and S.

    CALCULUS 1 and 2: RESOURCES !!!

    Suggested books : Rick Durrett, Probability: theory and examples. Olav Kallenberg, Foundations of modern Probability. Second Edition , Springer-Verlag, MA Quantum Mechanics Origins, states, observables, interference, symmetries, uncertainty, wave and matrix mechanics, Measurement, scattering theory in 1 dimension, quantum computation and information, Prerequisites are analysis and linear algebra. Suggested books : Srinivas, M. Press Parthasarathy, K. MA Hermitian Analysis Top. MA Control and Homogenization Pre-requisites : Sobolev spaces Elliptic boundary value problems Heat and wave equations Variational formulation and semigroup theory Optimal Control of PDE:Optimal control problems governed by elliptic equations and linear parabolic and hyperbolic equations with distributed and boundary controls, Computational methods.

    Suggested books : B.

    Lee and L. Lions, Controlabilite exact et Stabilisation des systemes distribues, Vol. Bardi, I. Suggested books : Folland, G. Algebraic Geometry I David Mumford. Combinatorial Theory Martin Aigner. Probability in Banach Spaces Michel Ledoux. Lectures on Algebraic Topology Albrecht Dold. In the Nazi government dismissed Courant for being Jewish, and he emigrated to the United States.

    He found, in New York, what he called "a reservoir of talent" to be tapped. He built, at New York University, a new mathematical Sciences Institute that shares the philosophy of its illustrious predecessor and rivals it in worldwide influence. For Courant mathematics was an adventure, with applications forming a vital part.

    This spirit is reflected in his books, in particular in his influential calculus text, revised in collaboration with his brilliant younger colleague, Fritz John.

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    As he was half-Jewish and his bride Aryan, he had to flee Germany in He remained there until his death in New Rochelle on February 10, John's research and the books he wrote had a strong impact on the development of many fields of mathematics, foremost in partial differential equations. He also worked on Radon transforms, illposed problems, convex geometry, numerical analysis, elasticity theory.

    In connection with his work in latter field, he and Nirenberg introduced the space of the BMO-functions bounded mean oscillations. Fritz John's work exemplifies the unity of mathematics as well as its elegance and its beauty. Moser show more. Review Text From the reviews: " The mathematics are rigorous but the many examples that are given and the applications that are treated make the books extremely readable and the arguments easy to understand.

    These books are ideally suited for an undergraduate calculus course. Each chapter is followed by a number of interesting exercises. More difficult parts are marked with an asterisk. Concavity and inflection points 5. Optimization 2. Related Rates 3. Newton's Method 4. Linear Approximations 5. The Mean Value Theorem 7 Integration 1. Two examples 2. The Fundamental Theorem of Calculus 3. Some Properties of Integrals 8 Techniques of Integration 1. Substitution 2. Powers of sine and cosine 3. Trigonometric Substitutions 4. Integration by Parts 5.

    Rational Functions 6. Numerical Integration 7. Additional exercises 9 Applications of Integration 1. Area between curves 2. Distance, Velocity, Acceleration 3. Volume 4. Average value of a function 5. Work 6. Center of Mass 7. Kinetic energy; improper integrals 8. Probability 9. Arc Length Polar Coordinates 2. Slopes in polar coordinates 3. Areas in polar coordinates 4. Parametric Equations 5. Calculus with Parametric Equations 11 Sequences and Series 1. Sequences 2. Series 3. The Integral Test 4. Alternating Series 5.

    Comparison Tests 6. Absolute Convergence 7. The Ratio and Root Tests 8.

    http://graphql.muchmore.be/lustige-geschichten-in-einfachem-spanisch-2-el.php Power Series 9. Calculus with Power Series Taylor Series Taylor's Theorem Additional exercises 12 Three Dimensions 1. The Coordinate System 2. Vectors 3. The Dot Product 4. The Cross Product 5. Lines and Planes 6. Other Coordinate Systems 13 Vector Functions 1.