# Download PDF Algebraic Structures and Operator Calculus: Volume III: Representations of Lie Groups

All other elements of Ln are obtained by taking adjoints, commutators, and linear combinations. We thus arrive to the following: Definition 3. Outline of the Fock space construction method. Chapter 5 of [18] , equation 3. P P Take the product of 3. Lemma 2. Proposition 3. Corollary 1. The functions hn,k appearing in 3. The proof follows from Proposition 3 and 3. Corollary 2. The functions Gn appearing in 3. The proof follows from the fact that for Gn given by 3. The proof follows directly from Corollary 2.

The proof follows from 3. The function G1 of 3. The functions Gn of 3. Definition 4.

## Representation theory - Wikipedia

P P Proposition 4. To prove 3. Corollary 5. Classical stochastic processes on Fn. Definition 5. Moreover, by 1. In the theory of Bernoulli systems and the Fock representation of finite-dimensional Lie algebras cf. Chapter 5 of [18] the Riccati equation 4. Chapter 5 of [18] and also [20] with moment generating function sec s t and associated special functions the Meixner-Pollaczek polynomials.

In the infinite-dimensional T RHP W N case the underlying classical probability distributions are given in the following. Proposition 6 Moment generating functions.

The continuous Binomial and Beta Processes. See Proposition 4. Corollary 6. The detailed proofs of all results presented in this section can be found in [6]. Basic concepts. A central extension is trivial if L e is the direct sum of a subalgebra M and C E as Lie algebras, where the subalgebra M is isomorphic to L. Two extensions are equivalent if each one of them is a trivial extension of the other, i. Notation 2. Central extensions of the Heisenberg algebra. Central extensions of the Oscillator algebra.

Central extensions of the Square of White Noise algebra. Proposition The centrally extended Heisenberg algebra. In this section a continuation of subsec- tion 6. Detailed proofs can be found in [7]. Real four—dimensional solvable Lie algebras are fully classified. There are exactly fifteen isomor- phism classes and they are listed, for example, in proposition 2. But it is the first time that its connection with the non-trivial central extensions of an important Lie algebra such as the Heisenberg one has been pointed out.

Random variables in CEHeis. All proofs can be found in [7]. Lemma 4. Remark 2. The centrally extended Heisenberg Lie group. By exponentiating the elements of CEHeis we obtain the elements of the centrally extended Heisenberg Lie group.

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In this section we provide the group law. Construction of the field of real numbers and the least upper-bound property. Metric Spaces: topological properties, the topology of Euclidean space. Sequences and series. Continuity: definition and basic theorems, uniform continuity, the Intermediate Value Theorem. Differentiability on the real line: definition, the Mean Value Theorem. Sequences and series of functions, uniform convergence, the Weierstrass Approximation Theorem.

Differentiability in higher dimensions: motivations, the total derivative, and basic theorems. Partial derivatives, characterization of continuously-differentiable functions. The Inverse and Implicit Function Theorems. Higher-order derivatives. The Banach- Steinhaus theorem. The maximum-modulus theorem. Isolated singularities, residue theorem, the Argument Principle, real integrals via contour integration. Mobius transformations, conformal mappings.

The Schwarz lemma, automorphisms of the dis. The Riemann mapping theorem. The inverse function and implicit function theorems, extreme of functions of several variables and Lagrange multipliers. Manifolds: Definitions and examples, vector fields and differential forms on manifolds, Stokes theorem. 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. Simplicial Homology: Simplicial complexes, chain complexes, definitions of the simplicial homology groups, properties of homology groups, applications.

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. This course will introduce concepts, examples and basic results of Metric Geometry and Geometric Group theory.

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. Linear system:The fundamental matrix, stability of equilibrium points, Phase- plane analysis, Sturm-Liouvile theory. 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.

Newton-Cotes formulae, Error estimates, Gaussian quadratures, Extensions to multiple integrals. Numerical integration of ordinary differential equations: Methods of Euler, Adams, Runge-Kutta and predictor - corrector procedures, Stability of solution. Solution of stiff equations. Solution of boundary value problems: Shooting method with least square convergence criterion, Quasilinearization method, Parametric differentiation technique and invariant imbedding technique. Solution of partial differential equations: Finite-difference techniques, Stability and convergence of the solution, Method of characteristics.

Finite element and boundary element methods. Finite difference methods for two point boundary value problems, Laplace equation on the square, heat equation and symmetric hyperbolic systems in 1 D. Lax equivalence theorem for abstract initial value problems. Introduction to variational formulation and the Lax-Milgram lemma. Finite element methods for elliptic and parabolic equations.

Financial market. Financial instruments: bonds, stocks, derivatives. Binomial no-arbitrage pricing model: single period and multi-period models. Martingale methods for pricing. American options: the Snell envelope. Capital asset pricing model CAPM. Utility theory. Conservative Systems: Hamiltonians, canonical transformations, nonlinear pendulum, perturbative methods, standard map, Lyapunov exponents, chaos, KAM theorem, Chirikov criterion.

Dissipative Systems: logistics map, period doubling, chaos, strange attractors, fractal dimensions, Smale horseshoe, coupled maps, synchronization, control of chaos. Assignments will include numerical simulations. Prerequisites, if any: familiarity with linear algebra - matrices, and ordinary differential equations. The Lebesgue Integral:Riemann-Stieltjes integral, Measures and measurable sets, measurable functions, the abstract Lebesgue integral. Complex measures and the lebesgue - Radon - Nikodym theorem and its applications.

The conjugate spaces. Abstract Hilbert spaces. Differentiation:Basic definitions and theorems, Partial derivatives, Derivatives as linear maps , Inverse and Implicit function theorems. Integration:Basic definitions and theorems, Integrable functions, Partitions of unity, Change of variables. Transversality, Morse functions, stable and unstable manifolds, Morse-Smale moduli spaces, the space of gradient flows, compactification of the moduli spaces of flows, Morse homology, applications. Refresher on categories : Categories, functors, Yoneda Lemma, equivalence of categories, adjoints.

Separated schemes, proper schemes, irreducible schemes, reduced schemes, integral schemes, noetherian schemes.

Algebraic preliminaries: Algebraic field extensions: Normal, separable and Galois extensions. Euclidean rings, principal ideal domains and factorial rings.

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## Representation theory

Quadratic number fields. Cyclotomic number fields. Algebraic integers: Integral extensions: Algebraic number fields and algebraic integers. Norms and traces. Resultants and discriminants. Integral bases. Class numbers:Lattices and Minkowski theory. Finiteness of class number. Ramification Theory: Discriminants. Applications to cryptography.

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Schemes examples. Finite dimensional Lie algebras, Ideals, Homomorphisms, Solvable and Nilpotent Lie algebras, Semisimple Lie algebras, Jordan decomposition, Kiling form, root space decomposition, root systems, classification of complex semisimple Lie algebras Representations Complete reducibility, weight spaces, Weyl character formula, Kostant, steinberg and Freudenthal formulas.

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. Time permitting: advanced topics like sieves, bounds on exponential sums, zeros of functions. Counting problems in sets, multisets, permutations, partitions, trees, tableaux; ordinary and exponential generating functions; posets and principle of inclusion-exclusion, the transfer matrix method; the exponential formula, Polya theory; bijections, combinatorial identities and the WZ method.

No prior knowledge of combinatorics is expected, but a familiarity with linear algebra and finite groubs will be assumed. Fourier transform and Sobolev-spaces:Definitions, Extension operators, Continuum and Compact imbeddings, Trace results. Elliptic boundary value problems: Variational formulation, Weak solutions, Maximum Principle, Regularity results.

Unbound linear operators on Hilbert spaces: Symmetric and self adjoint operators, Spectral theory. Banach algebras Gelfand representation theorem. 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.

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Creation operators on the full Fock space and the symmetric Fock space. Operators spaces. Completely positive and completely bounded maps. Towards dilation of completely positive maps. Unbounded operators: Basic theory of unbounded self-adjoint operators. Introduction to Fourier transform; Plancherel theorem, Wiener-Tauberian theorems, Interpolation of operators, Maximal functions, Lebesgue differentiation theorem, Poisson representation of harmonic functions, introduction to singular integral operators.

In this course we begin by stating many wonderful theorems in analysis and proceed to prove them one by one. We take a somewhat experimental approach in stating the results and then exploring the techniques to prove them. The theorems themselves have the common feature that the statements are easy to understand but the proofs are non-trivial and instructive.

And the techniques involve analysis. Convexity theory: Analytic continuation: the role of convexity, holomorphic convexity, plurisub-harmonic functions, the Levi problem and the role of the d-bar equation. This topics course is being run as an experiment in approaching the basic concepts in several complex variables with the eventual aim of studying some topics in multi-variable complex dynamics. The course will begin with a complete and rigorous introduction to holomorphic functions in several variables and their basic properties. This will pave the way to motivating and studying a concept that is, perhaps, entirely indigenous to several complex variables: the notion of plurisubharmonicity.

Next, we shall look at some of the motivations behind the study of complex dynamics in several variables. Using the tools developed, we shall undertake a crash-course in currents, which are objects central to the study of some aspects of complex dynamics. We shall then cover as much of the following topics as time permits:. Fundamental Groups: 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.

Homology : Singular homology, excision, Mayer-Vietoris theorem, acyclic models, CW-complexes, simplicial and cellular homology, homology with coefficients. 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.

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. This connection is based on interpreting types as spaces, terms as points and equalities as paths. Many homotopical notions have type-theoretic counterparts which are very useful for foundations.

Such foundations are far closer to actual mathematics than the traditional ones based on set theory and logic, and are very well suited for use in computer-based proof systems, especially formal verification systems. The course will also include background material in Algebraic Topology beyond a second course in Algebraic Topology. This is an introduction to hyperbolic surfaces and 3-manifolds, which played a key role in the development of geometric topology in the preceding few decades.

The first half of the course will focus on convergence theory of Riemannian manifolds. Gromov-Hausdorff convergence, Lipschitz convergence and collapsing theory will be discussed. The second half will be about the Ricci flow. The goal of this course is to use computers to address various questions in Topology and Geometry, with an emphasis on arriving at rigorous proofs.

The course will consist primarily of projects which will be contributions to open source software written in the scala programming language. Covariant differentiation, Riemannian metrics, Levi-Civita connection, Curvature and parallel transport, spaces of constant curvature. 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.

The L-infinity functional calculus for normal operators. This course explores matrix positivity and operations that preserve it. These involve fundamental questions that have been extensively studied over the past century, and are still being studied in the mathematics literature, including with additional motivation from modern applications to high-dimensional covariance estimation. The course will bring together techniques from different areas: analysis, linear algebra, combinatorics, and symmetric functions.

The cone of positive semidefinite matrices. Matrix identities Cauchy-Binet, Andreief. Generalized Rayleigh quotients and spectral radius. Schur complements. Positivity preservers. Schur product theorem. Polya-Szego observation. Positive definite functions to correlation matrices.

Herz, Christensen-Ressel. Fixed-dimension problem. Introduction and modern motivations. Characterization of Hankel total positivity preservers in the dimension-free setting. Which coefficients can be negative? Bounded and unbounded domains: Horn-type necessary conditions. Schur polynomials. Two definitions and properties. Specialization over fields and for real powers. First-order approximation. Sign patterns: The Horn-type necessary conditions are best possible. Sharp quantitative bound. Extension principle I: dimension increase. Entrywise maps preserving total positivity.

Variants for all TP matrices and for symmetric TP matrices. Matrix completion problems. Entrywise powers preserving positivity. Application of Extension principle I. Low-rank counterexamples. Extension principle III: pendant edges. The case of trees. Chordal graphs and their properties.

Non-chordal graphs. Introduction to distribution theory and Sobolev spaces, Fundamental solutions for Laplace, heat and wave operations. Second order elliptic equations: Boundary value problems, Regularity of weak solutions, Maximum principle, Eigenvalues. System of first order hyperbolic equations: Bicharacteristics, Shocks, Ray theory, symmetric hyperbolic systems. Review of Distributions, Sobolev spaces and Variational formulation.

Introduction to Homogenization. Homogenization of elliptic PDEs. Specific Cases: Periodic structures and layered materials. General Variational convergence: G -convergence and G- convergence, Compensated compactness. Study of specific examples and applications. Introduction to Calculus of Variations and Morse Theory. Further topics if time permits: Non-holomorphic modular forms overview , Siegel modular forms introduction , Elliptic curves and cusp forms, spectral theory, analytic questions related to modular forms.

Time permitting: More advanced topics like Sieves, bounds on exponential sums, zeros of zeta functions, circle method etc. Gaussian unitary and orthogonal ensembles: a Exact density of eigenvalues. Free probability: a Noncommutative probability space and free independence. Non-hemitian random matrices: a Ginibre ensemble.

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Linear time series analysis - modelling time series using stochastic processes, stationarity, autocovariance, auto correlation, multivariate analysis - AR, MA, ARMA, AIC criterion for order selection;. Spectral analysis - deterministic processes, concentration problem, stochastic spectral analysis, nonparametric spectral estimation periodogram, tapering, windowing , multitaper spectral estimation; parametric spectral estimation Yule-Walker equations, Levinson Durbin recursions ;.

Multivariate analysis - coherence, causality relations; bootstrap techniques for estimation of parameters;. Nonlinear time series analysis - Lyapunov exponents, correlation dimension, embedding methods, surrogate data analysis. A course in Gaussian processes. At first we shall study basic facts about Gaussian processes - isoperimetric inequality and concentration, comparison inequalities, boundedness and continuity of Gaussian processes, Gaussian series of functions, etc. Later we specialize to smooth Gaussian processes and their nodal sets , in particular expected length and number of nodal sets, persistence probability and other such results from recent papers of many authors.

Trading in continuous time : geometric Brownian motion model. Option pricing : Black-Scholes-Merton theory. American options. Exotic options. Market imperfections. Term-structure models. HJM model. LIBOR model. Credit derivatives. Discrete parameter martingales: Conditional expectation. Optional sampling theorems. Martingale convergence theorems. Brownian motion. 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. 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.

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. 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.

## Mathematics

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. 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.