# Guide A Course in Functional Analysis (Graduate Texts in Mathematics, Volume 96) Contents:

The development of the latter will be self-contained. The Weierstrass-Enneper representation of minimal surfaces. Many more examples of minimal surfaces. Surfaces that locally maximise area in Lorenztian space maximal surfaces. A lot of examples and analogous results, as in minimal surface theory, for maximal surfaces. The purpose of this course will be to understand to an extent and appreciate the symbiotic relationship that exists between mathematics and physics. This course will focus on the structure as well as on finite dimensional complex representations of the following classical groups: General and special Linear groups, Symplectic groups, Orthogonal and Unitary groups.

No prior knowledge of combinatorics or algebra is expected, but we will assume a familiarity with linear algebra and basics of group theory. Discrete parameter martingales, branching process, percolation on graphs, random graphs, random walks on graphs, interacting particle systems. Erdos - Renyi random graphs, graphs with power law degree distributions, Ising Potts and contact process, voter model, epidemic models.

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Real trees, the Brownian continuum random tree, phase transition in random graphs, scaling limits of discrete combinatorial structures, random maps, the Brownian map and its geometry. We shall illustrate some important techniques in studying discrete random structures through a number of examples. The techniques we shall focus on will include if time permits. We shall discuss applications of these techniques in various fields such as Markov chains, percolation, interacting particle systems and random graphs.

Binomial no- arbitrage pricing model: single period and multi-period models. Trading in continuous time: geometric Brownian motion model. Option pricing: Black-Scholes-Merton theory. Hedging in continuous time: the Greeks. One-variable Calculus: Real and Complex numbers; Convergence of sequences and series; Continuity, intermediate value theorem, existence of maxima and minima; Differentiation, mean value theorem,Taylor series; Integration, fundamental theorem of Calculus, improper integrals. Linear Algebra: Vector spaces over real and complex numbers , basis and dimension; Linear transformations and matrices.

Linear Algebra continued: Inner products and Orthogonality; Determinants; Eigenvalues and Eigenvectors; Diagonalisation of symmetric matrices. Basic notions from set theory, countable and uncountable sets. Metric spaces: definition and examples, basic topological notions. Sequences and series: essential definitions, absolute versus conditional convergence of series, some tests of convergence of series. Continuous functions: properties, the sequential and the open- set characterizations of continuity, uniform continuity.

Differentiation in one variable. The Riemann integral: formal definitions and properties, continuous functions and integration, the Fundamental Theorem of Calculus. Uniform convergence: definition, motivations and examples, uniform convergence and integration, the Weierstrass Approximation Theorem. Integrated Ph. Interdisciplinary Ph. Programme Integrated Ph. Limaye and S. Spivak, M. Benjamin, co. Some background in algebra and topology will be assumed. It will be useful to have some familiarity with programming. Topics: Basic type theory: terms and types, function types, dependent types, inductive types.

Most of the material will be developed using the dependently typed language Idris. Connections with programming in functional languages will be explored. Manin, Yu. Srivastava, S. Suggested books : Artin, M. Hoffman, K and Kunze R. Halmos, P. Greub, W.

Suggested books : Artin, Algebra , M. Prentice-Hall of India, Dummit, D. Herstein, I. Lang, S. Atiyah, M.

## Series: Graduate Texts in Mathematics

Suggested books : T. Becker and V. Adams and P. Sturmfels, Grobner bases and convex polytopes , American Mathematical Society Suggested books : Serre, J. Koblitz, N. Iwaniec, H. Diamond, F. Suggested books : Adrian Bondy and U.

Springer-Verlag, Berlin, ISBN: Douglas B. West, Introduction to graph theory , Prentice Hall, Inc. Suggested books : Bondy, J. Burton, D. Clark, J. Polya G. Suggested books : Narasimham, R. Niven, I. Apostol, T. Ireland, K. Hoffman, K. American Mathematical Society. Serre, Linear representations of finite groups , Graduate Texts in Mathematics. New York-Heidelberg. Suggested books : Rudin, W. Suggested books : Royden, H. Folland, G. Hewitt, E. Suggested books : Rudin, Functional Anaysis 2nd Ed. Yosida, K. Goffman, C. Suggested books : Ahlfors, L.

Conway, J. Suggested books : Narasimhan, R. Nievergelt , Birkhauser 2nd ed. Greene, R. Suggested books : Spivak, M. Hirsh, M. Suggested books : Armstrong, M. Munkres, K. Viro, O. MA Introduction to algebraic topology Prerequisite courses: MA and MA The fundamental group: Homotopy of maps, multiplication of paths, the fundamental group, induced homomorphisms, the fundamental group of the circle, covering spaces, lifting theorems, the universal covering space, Seifert-van Kampen theorem, applications.

Hatcher, A. Kosniowski, C. Press, Croom, F. Suggested books : do Carmo, M. Thorpe, J. O'Neill, B. Gray, A. MA Metric Geometry of Spaces and Groups Prerequisite courses: MA Pre-requisites : A first course in Topology can be taken concurrently Metric geometry is the study of geometric properties such as curvature and dimensions in terms of distances, especially in contexts where the methods of calculus are unavailable, An important instance of this is the study of groups viewed as geometric objects, which constitutes the field of geometric group theory.

MA Ordinary Differential Equations Prerequisite courses: MA Basics concepts:Introduction and examples through physical models, First and second order equations, general and particular solutions, linear and nonlinear systems, linear independence, solution techniques. Birkhaeuser, Coddington, E. Perko, L. Suggested books : Garabedian, P. Renardy, M. Suggested books : Hoffman, K. Simmons G. Churchill, R. MA Numerical Methods Numerical solution of algebraic and transcendental equations, Iterative algorithms, Convergence, Newton Raphson procedure, Solutions of polynomial and simultaneous linear equations, Gauss method, Relaxation procedure, Error estimates, Numerical integration, Euler-Maclaurin formula.

Suggested books : Gupta, A. 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.

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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. 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.

## Curriculum

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. 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. 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.

## A course in functional analysis (Graduate texts in mathematics) by John B Conway

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. 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.

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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. 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. We have a dedicated site for Germany. Functional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call themselves functional analysts, who have great difficulty understanding the work of the other.

The common thread is the existence of a linear space with a topology or two or more. Here the paths diverge in the choice of how that topology is defined and in whether to study the geometry of the linear space, or the linear operators on the space, or both. In this book I have tried to follow the common thread rather than any special topic. I have included some topics that a few years ago might have been thought of as specialized but which impress me as interesting and basic. Near the end of this work I gave into my natural temptation and included some operator theory that, though basic for operator theory, might be considered specialized by some functional analysts.

Many interesting and important applications are included. This book is a fine piece of work. It includes an abundance of exercises, and is written in the engaging and lucid style which we have come to expect from the author. This book is a comprehensive introduction to functional analysis. JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser.

Mathematics Analysis.