Linear algebra: Vector and inner product spaces, linear operators, eigenvalue problems; Vector calculus: Review of differential and integral calculus, divergence and Stokes' theorems. Ordinary differential equations: Linear equations, Sturm-Liouville theory and orthogonal functions, system of linear equations; Methods of mathematics for science and engineering students.
A graduate level introduction to matrix-based computing. Stable and efficient algorithms for linear equations, least squares and eigenvalue problems. Both direct and iterative methods are considered and MATLAB is used as a computing environment.
Calculus of variations; Partial differential equations: First order linear equations and the method of characteristics; Solution of Laplace, wave, and diffusion equations; Special functions; Integral equations.
Development and analysis of numerical methods for ODEs, an introduction to numerical optimization methods, and an introduction to random numbers and Monte Carlo simulations. The course starts with a short survey of numerical methods for ODEs. The related topics include stability, consistency, convergence and the issue of stiffness. Then it moves to computational techniques for optimization problems arising in science and engineering. Finally, it discusses random numbers and Monte Carlo simulations. The course combines the theory and applications (such as programming in MATLAB) with the emphasis on algorithms and their mathematical analysis.
Convergence of sequences in Rn, multivariate Taylor's theorem. Optimality conditions for unconstrained optimization. Newton's and quasi-Newton methods for unconstrained optimization. Equality-constrained optimization, Karush-Kuhn-Tucker theorem for constrained optimization. Inequality-constrained optimization. Interior point methods for constrained optimization. Linear and quadratic programs, their numerical solution.
Existence and uniqueness theorems; continuation of solutions; continuous dependence and stability, Lyapunovs direct method; differential inequalities and their applications; boundary-value problems and Sturm-Liouville theory.
First order equations, method of characteristics; the Cauchy-Kovalevskaya theorem; Laplace's equation: potential theory and Greens?s function, properties of harmonic functions, the Dirichlet problem on a ball; heat equation: the Cauchy problem, initial boundary-value problem, the maximum principle; wave equation: the Cauchy problem, the domain of dependence, initial boundary-value problem.
Review of functional spaces and embedding theorems; existence and regularity of solutions of boundary-value problems for second-order elliptic equations; maximum principles for elliptic and parabolic equations; comparison theorems; existence, uniqueness and regularity theorems for solutions of initial boundary-value problems for second-order parabolic and hyperbolic equations.
Basic notions of commutative algebra and homological algebra: category of modules over a ring, flatness, Ext and Tor. General properties of schemes: affine schemes. projective schemes, dimension, projective and proper morphisms. Normal and regular schemes. Flat and smooth morphisms. Zariski's main theorem and applications. Coherent sheaves and Cech Cohomology.
Free groups, group actions, group with operators, Sylow theorems, Jordan-Hölder theorem, nilpotent and solvable groups. Polynomial and power series rings, Gauss?s lemma, PID and UFD, localization and local rings, chain conditions, Jacobson radical.
Galois theory, solvability of equations by radicals, separable extensions, normal basis theorem, norm and trace, cyclic and cyclotomic extensions, Kummer extensions. Modules, direct sums, free modules, sums and products, exact sequences, morphisms, Hom and tensor functors, duality, projective, injective and flat modules, simplicity and semisimplicity, density theorem, Wedderburn-Artin theorem, finitely generated modules over a principal ideal domain, basis theorem for finite abelian groups.
Valuations of a field, local fields, ramification index and degree, places of global fields, theory of divisors, ideal theory, adeles and ideles, Minkowski's theory, extensions of global fields, the Artin symbol.
Method of descent, unique factorization, basic algebraic number theory, diophantine equations, elliptic equations, p-adic numbers, Riemann zeta function, elliptic curves, modular forms, zeta and L-functions, ABC-conjecture, heights, class numbers for quadratic fields, a sketch of Wiles? proof.
Primes in arithmetic progressions, Gauss' sum, primitive characters, class number formula, distribution of primes, properties of the Riemann zeta function and Dirichlet L-functions, the prime number theorem, Polya- Vinogradov inequality, the large sieve, average results on the distribution of primes.
Lebesgue measure and Lebesgue integration on Rn, general measure and integration, decomposition of measures, Radon-Nikodym theorem, extension of measures, Fubini's theorem.
Normed and Banach spaces, Lp-spaces and duals, Hahn-Banach theorem, Baire category and uniform boundedness theorems, strong, weak and weak*-convergence, open mapping theorem, closed graph theorem.
Review of the complex number system and the topology of C, elementary properties and examples of analytic functions, complex integration, singularities, maximum modulus theorem, compactness and convergence in the space of analytic functions.
Runge's theorem, analytic continuation, Riemann surfaces, harmonic functions, entire functions, the range of an analytic function.
Topological vector spaces, locally convex spaces, weak and weak* topologies, duality, Alaoglu's theorem, Krein-Milman theorem and applications, Schauder fixed point theorem, Krein-Shmulian theorem, Eberlein-Shmulian theorem, linear operators on Banach spaces.
Review of linear operators in Banach spaces and Hilbert spaces; Riesz ·Schauder theory; fixed point theprems of Banach and Schauder; semigroups of linear operators; Sobolev spaces and basic embedding theorems; boundary - value problems for elliptic equations; eigenvalues and eigenvectors of second order elliptic operators; initial boundary-value problems for parabolic and hyperbolic equations.
Existence and uniqueness of solutions of abstract evolutionary equations. Global non-existence and blow up theorems. Applications to the study of the solvability and asymptotic behavior of solutions of initial boundary-value problems for reaction diffusion equations, Navier-Stokes equations, nonlinear Klein-Gordon equations and nonlinear Schrödinger equations.
Differentiable manifolds; differentiable forms; integration on manifolds; de Rhamm cohomology; connections and curvature
An introduction to measure theory, Kolmogorov axioms, independence, random variables, product measures and joint probability, distribution laws, expectation, modes of convergence for sequences of random variables, moments of a random variable, generating functions, characteristic functions, distribution laws, conditional expectations, strong and weak law of large numbers, convergence theorems for probability measures, central limit theorems.