Primitive roots. Algebraic Number Theory: Algebraic numbers and algebraic integers, Class groups, Groups of units, Quadratic fields, Quadratic reciprocity law, Class number formula. Analytic Number Theory: Fundamental theorem of arithmetic, Arithmetical functions, Some elementary theorems on the distribution of prime numbers, Congruences, Finite Abelian groups and their characters, Dirichlet theorem on primes in arithmetic progression.
Vector spaces: Definition, Basis and dimension, Direct sums. Linear transformations: Definition, Rank-nullity theorem, Algebra of linear transformations, Dual spaces, Matrices. Eigenvalues and eigenvectors, the characteristic polynomial, the Cayley- Hamilton Theorem, the minimal polynomial, algebraic and geometric multiplicities, Diagonalization, The Jordan canonical form.
Symmetry: Group of motions of the plane, Discrete groups of motion, Finite groups of S0 3.
ISBN 13: 9780486474175
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.
Linear Algebra -- from Wolfram MathWorld
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.
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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.
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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. Quadratic number fields. Cyclotomic number fields. Algebraic integers: Integral extensions: Algebraic number fields and algebraic integers. Norms and traces. Resultants and discriminants.
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