Lectures on Algebraic Topology: Reprint of the 1972 Edition


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Reprint of the 1972 Edition

Continuity properties. Markov and strong Markov property and applications. Further sample path properties. 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. Optimal Control of PDE:Optimal control problems governed by elliptic equations and linear parabolic and hyperbolic equations with distributed and boundary controls, Computational methods.

Homogenization:Examples of periodic composites and layered materials. Various methods of homogenization. Applications and Extensions:Control in coefficients of elliptic equations, Controllability and Stabilization of Infinite Dimensional Systems, Hamilton- Jacobi-Bellman equations and Riccati equations, Optimal control and stabilization of flow related models.

Algebraic Topology Introduction (Peter May)

Topological groups, locally compact groups, Haar measure, Modular function, Convolutions, homogeneous spaces, unitary representations, Gelfand-Raikov Theorem. Introduction and examples, optimal control problems governed by elliptic and parabolic systems, adjoint systems, optimality conditions, optimal control and optimality systems for other PDEs like Stokes systems.

The dynamics alluded to by the title of the course refers to dynamical systems that arise from iterating a holomorphic self-map of a complex manifold. In this course, the manifolds underlying these dynamical systems will be of complex dimension 1. The foundations of complex dynamics are best introduced in the setting of compact spaces. Iterative dynamical systems on compact Riemann surfaces other than the Riemann sphere — viewed here as the one-point compactification of the complex plane — are relatively simple. We shall study what this means. Thereafter, the focus will shift to rational functions: these are the holomorphic self-maps of the Riemann sphere.

Along the way, some of the local theory of fixed points will be presented. In the case of rational maps, some ergodic-theoretic properties of the orbits under iteration will be studied. 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.


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Erdos - Renyi random graphs, graphs with power law degree distributions, Ising Potts and contact process, voter model, epidemic models. 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.

Topology (book list)

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.

Lectures on Algebraic Topology

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.

Group Theory

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. 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.


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Apostol, T. Ireland, K.

Infinitesimal 2-braidings and differential crossed modules - INSPIRE-HEP

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.

Lectures on Algebraic Topology: Reprint of the 1972 Edition Lectures on Algebraic Topology: Reprint of the 1972 Edition
Lectures on Algebraic Topology: Reprint of the 1972 Edition Lectures on Algebraic Topology: Reprint of the 1972 Edition
Lectures on Algebraic Topology: Reprint of the 1972 Edition Lectures on Algebraic Topology: Reprint of the 1972 Edition
Lectures on Algebraic Topology: Reprint of the 1972 Edition Lectures on Algebraic Topology: Reprint of the 1972 Edition
Lectures on Algebraic Topology: Reprint of the 1972 Edition Lectures on Algebraic Topology: Reprint of the 1972 Edition
Lectures on Algebraic Topology: Reprint of the 1972 Edition Lectures on Algebraic Topology: Reprint of the 1972 Edition
Lectures on Algebraic Topology: Reprint of the 1972 Edition Lectures on Algebraic Topology: Reprint of the 1972 Edition
Lectures on Algebraic Topology: Reprint of the 1972 Edition Lectures on Algebraic Topology: Reprint of the 1972 Edition
Lectures on Algebraic Topology: Reprint of the 1972 Edition Lectures on Algebraic Topology: Reprint of the 1972 Edition

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