Simple crypto-systems, public key cryptography, discrete logarithms and Diffie-Hellman key exchange, primality, factoring and RSA, elliptic curve crypto-systems, lattice based crypto-systems. Numerical solutions of initial value problems for ordinary differential equations (ODE), Picard-Lindelof theorem, single step methods including Runge-Kutta methods, examples and consistency, stability and convergence of multistep methods, numerical solution of boundary value problems for ODE’s, shooting, finite difference, and collocation methods, finite element methods, Riesz and Lax-Milgram lemmas, weak solutions, numerical solutions of partial differential equations, examples of finite difference methods and their consistency, stability, and convergence including Lax-Richtmeyer equivalence theorem, Courant-Friedrichs-Lewy condition, and von Neumann analysis, Galerkin methods, Galerkin orthogonality, Cea’s lemma, finite element methods for elliptic, parabolic and hyperbolic equations.ĭynamical systems with discrete and continuous time, differential equations on torus, invariant sets, topological dynamics, topological recurrence and entropy, expansive maps, homoemorphisms and diffeomorphisms of the circle, periodic orbits, hyperbolic dynamics, Grobman-Hartman and Hadamard-Perron theorems, geodesic flows, topological Markov chains, zeta functions, invariant measures and the ergodic theorem.įirst variation of a functional, necessary conditions for an extremum of a functional, Euler's equation, fixed and moving endpoint problems, isoperimetric problems, problems with constraints, Legendre transformation, Noether's theorem, Jacobi's theorem, second variation of a functional, weak and strong extremum, sufficient conditions for an extremum, direct methods in calculus of variations, the principle of least action, conservation laws, Hamilton-Jacobi equation. Numerical Solutions of Differential Equations Normed linear spaces, Hilbert spaces, least-squares estimation, dual spaces, geometric form of Hahn-Banach theorem, linear operators and their adjoints, optimization in Hilbert spaces, local and global theory of optimization of functionals, constrained and unconstrained cases. Review of vector spaces, normed vector spaces, lP and LP spaces, BanachĪnd Hilbert spaces, duality, bounded linear operators and functionals. pointwise and mean-square convergence, Weyl's equidistribution theorem, Fourier transform on the real line and Schwartz space, inversion, Plancherel formula, application to partial differential equations, Poisson summation formula.
Quadratic Forms, quadratic number fields, factorization of ideals in quadratic number fields, ramification theory, ideal classes and units in quadratic number fields, elliptic curves over rationals.Ĭonvergent series of meromorphic functions, entire functions, Weierstrass' product theorem, partial fraction expansion theorem of Mittag-Leffler, gamma function, normal families, theorems of Montel and Vitali, Riemann mapping theorem, conformal mapping of simply connected domains, Schwarz-Christoffel formula, applications.įourier series, Dirichlet and Poisson kernels, Cesàro and Abel summability.
Introduction to algebraic number theory and algebraic curves, geometric introduction to function fields of curves, affine and projective varieties, divisors on curves, Riemann-Roch theorem, basics of elliptic curves. Language and structure, theory, definable sets and interpretability, compactnees theorem, complete theories, Löwenheim-Skolem theorems, quantifier elimination, algebraic examples.Īffine varieties, Hilbert’s Nullstellensatz, projective varieties, rational functions and morphisms, smooth points, dimension of a variety. Sets, relations, functions, order, set-theoretical paradoxes, axiom systems for set theory, axiom of choice and its consequences, transfinite induction, recursion, cardinal and ordinal numbers. Propositional and quantificational logic, formal grammar, semantical interpretation, formal deduction, completeness theorems, selected topics from model theory and proof theory.
(Math 202 and Math 221) or consent of the instructor
Introduction to computational mathematics, basics of a mathematics software (Sage, Mathematica, Maple, MATLAB), solving systems of linear equations, interpolation, locating roots of equations, least squares problems, numerical integration, numerical differentiation and solution of ordinary differential equations. Selected topics in the history of mathematics and related fields. Courses Offered by Mathematics Department