Algebra

Group theory, normal subgroups, quotient groups, homomorphisms, class equation, Sylow's theorems nilpotent and solvable groups.

Ring/module theory, including ideals, quotients, homomorphisms, domains (unique factorization, euclidean, principal ideal), Jordan canonical form. Introduction to field theory, including finite fields, Galois theory. Vector spaces, linear transformations over abstract fields.

Topology

Metric spaces: continuity, compactness, Ascoli-Arzela' Theorem, completeness and completion, Baire Category Theorem. General topological spaces: bases and subbases, products, quotients, subspaces, continuity, topologies generated by sets of functions, homeomorphisms. Convergence (inclusing the inadequacy of sequences). Separation: Hausdorff spaces, regular spaces, normal spaces, Urysohn's Lemma, Tietze's Extension Theorem. Connectedness. Countability conditions: first and second countability, separability, Lindelof property. Compactness: Tychonoff's Theorem, local compactness.

Lebesgue Integration and Measure Theory

Measurable spaces and functions, positive measures, Lebesgue integral, properties of the integral, the monotone convergence theorem, Fatou's lemma, the dominated convergence theorem, Hahn's theorem, product measures and integration in product spaces, Fubini's theorem, Introduction to Lp spaces: Holder and Minkowski inequalities, convergence and completeness, continuous linear functionals.

Functional Analysis

General concepts: linear spaces, bases, norms, completeness. Linear mappings: continuity, Hahn-Banach theorem and separation of convex sets, uniform boundedness, open-mapping theorem, compact operators, unbounded operators, closed operators. Duality: weak and weak* topologies, reflexivity, convexity. Adjoints: basic properties, null spaces and ranges. Sequences of bounded linear operators: weak, strong and uniform convergence. Hilbert spaces: geometry, projections, Riesz representation theorem, bilinear and quadratic forms, orthonormal sets and Fourier series. Elementary spectral theory in Banach spaces: spectra and resolvents of bounded operators, spectral theory of compact operators, Fredholm alternative.

Partial Differential Equations (PDEs)

Basic linear PDE: Transport equation,  Laplace equation (fundamental solution, mean value property, Green's function, energy estimates, maximum principle, uniqueness, regularity), Heat equation (fundamental solution, mean value property, Duhamel's principle, energy estimates, maximum principle, uniqueness, regularity), Wave equation (spherical means, Duhamel's principle, energy methods, uniqueness).

Nonlinear first order PDE: (method of characteristics), Hamilton-Jacobi equations Conservation laws (weak solution, shocks, rarefaction waves, Rankine-Hugoniot condition, entropy conditions, existence and uniqueness of weak solutions). Variational principles.

Differential Geometry

Foundations of Differential Manifolds: Manifolds, partitions of unity, tangent space. Submersions, imersions, submanifolds, Whitney Theorem. Foliations.

Lie Theory: Vector fileds, Lie brackets, Lie derivative. Distributions and Frobenius Theorem. Lie groups, Lie algebras, actions.

Differential Forms: Tensor and exterior algebras, differential forms. Cartan's formula, de Rham cohomology, Poincare''s lemma. Orientation, integration over manifolds, homotopy. Stokes Theorem.

Fiber Bundles: Vector bundles, connections, curvature, metrics. Parallel transport, Riemannian manifolds, geodesics.  Gauss-Bonnet Theorem. Principal bundles.

Riemannian Geometry

Differentiable Manifolds: Tangent space; differentiable maps; immersions and embeddings; vector fields; differential forms.

Riemannian Manifolds: isometries; affine connections, Levi-Civita connection; geodesics, minimizing properties of geodesics; Hopf-Rinow theorem.

Curvature: curvature tensor, sectional curvature, Ricci tensor, scalar curvature; connection and curvature forms, Cartan's structure equations; isometric immersions of surfaces in euclidean space, Gauss map, mean and Gaussian curvature; Gauss Theorem; first and second fundamental forms.

Stochastic Calculus in Finance

Part I. Basic Concepts from Probability Theory. Conditional expectation. Discrete parameter martingales. Stochastic processes in continuous time. Brownian motion. Stochastic Integral: construction; properties; Itô´s formula. Stochastic differential equations: Existence and
Uniqueness Theorem; Markov property of solutions. Martingale Representation Theorem. Girsanov Theorem. The Feynman-Kac formula.

Part II. Discrete time models. Optimal stopping and American options. The Black-Scholes Model.

Ordinary Differential Equations (ODEs)

Dynamical systems and differential equations: fixed point theorems, existence, uniqueness, regularity and extension of solutions, continuous dependence with respect to initial conditions and the Arzelá-Ascoli theorem.

Geometric theory: phase portraits and orbits, invariant and limit sets, transverse sections, Jordan's curve theorem, Poincaré-Bendixson's theorem, linear equations, hyperbolicity, Grobman-Hartman theorem, stability and Lyapunov functions, index theory, bifurcation theory, stable and unstable manifold theorems