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.

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.

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.

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.

From random walk to Brownian motion, quadratic variation and volatility, stochastic integrals, martingale property, Ito formula, geometric Brownian motion, solution of Black-Scholes equation, stochastic differentialequations, Feynman-Kac theorem, Cox-Ingersoll-Ross and Vasicek term structure models, Girsanov's theorem and risk neutral measures, Heath-Jarrow-Morton term structure model, exchange-rate instruments.

Differentiable manifolds; differentiable forms; integration on manifolds; de Rhamm cohomology; connections and curvature

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.

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.

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.