## The program

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.

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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.

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Introduction to variational formulation and the Lax-Milgram lemma. Finite element methods for elliptic and parabolic equations. Financial market.

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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. 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. 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.

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.

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4. Resultants and discriminants. Integral bases. Class numbers:Lattices and Minkowski theory. Finiteness of class number. Ramification Theory: Discriminants. Applications to cryptography. Schemes examples. Finite dimensional Lie algebras, Ideals, Homomorphisms, Solvable and Nilpotent Lie algebras, Semisimple Lie algebras, Jordan decomposition, Kiling form, root space decomposition, root systems, classification of complex semisimple Lie algebras Representations Complete reducibility, weight spaces, Weyl character formula, Kostant, steinberg and Freudenthal formulas.

Polynomial ring, Projective modules, injective modules, flat modules, additive category, abelian category, exact functor, adjoint functors, co limits, category of complexes, snake lemma, derived functor, resolutions, Tor and Ext, dimension, local cohomology,group co homology, sheaf cohomology, Cech cohomology, Grothendieck spectral sequence, Leray spectral sequence.

Time permitting: advanced topics like sieves, bounds on exponential sums, zeros of functions. Counting problems in sets, multisets, permutations, partitions, trees, tableaux; ordinary and exponential generating functions; posets and principle of inclusion-exclusion, the transfer matrix method; the exponential formula, Polya theory; bijections, combinatorial identities and the WZ method. No prior knowledge of combinatorics is expected, but a familiarity with linear algebra and finite groubs will be assumed. Fourier transform and Sobolev-spaces:Definitions, Extension operators, Continuum and Compact imbeddings, Trace results.

Elliptic boundary value problems: Variational formulation, Weak solutions, Maximum Principle, Regularity results. Unbound linear operators on Hilbert spaces: Symmetric and self adjoint operators, Spectral theory. Banach algebras Gelfand representation theorem. The general theory of holomorphic mappings between bounded domains, automorphisms of bounded domains, discussions on the non-existence of a classical Riemann Mapping Theorem in several variables, discussion of the various forms of the one-variable Riemann Mapping Theorem, the Rosay-Wong Theorem, other Riemann-Rosay-Wong-type results e.

Creation operators on the full Fock space and the symmetric Fock space. Operators spaces. Completely positive and completely bounded maps. Towards dilation of completely positive maps. Unbounded operators: Basic theory of unbounded self-adjoint operators. Introduction to Fourier transform; Plancherel theorem, Wiener-Tauberian theorems, Interpolation of operators, Maximal functions, Lebesgue differentiation theorem, Poisson representation of harmonic functions, introduction to singular integral operators.

In this course we begin by stating many wonderful theorems in analysis and proceed to prove them one by one. We take a somewhat experimental approach in stating the results and then exploring the techniques to prove them. The theorems themselves have the common feature that the statements are easy to understand but the proofs are non-trivial and instructive.

And the techniques involve analysis. Convexity theory: Analytic continuation: the role of convexity, holomorphic convexity, plurisub-harmonic functions, the Levi problem and the role of the d-bar equation. This topics course is being run as an experiment in approaching the basic concepts in several complex variables with the eventual aim of studying some topics in multi-variable complex dynamics. The course will begin with a complete and rigorous introduction to holomorphic functions in several variables and their basic properties.

This will pave the way to motivating and studying a concept that is, perhaps, entirely indigenous to several complex variables: the notion of plurisubharmonicity.

## Value Distribution Theory Of The Gauss Map Of Minimal Surfaces In Rm

Next, we shall look at some of the motivations behind the study of complex dynamics in several variables. Using the tools developed, we shall undertake a crash-course in currents, which are objects central to the study of some aspects of complex dynamics. We shall then cover as much of the following topics as time permits:. Fundamental Groups: 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.

Homology : Singular homology, excision, Mayer-Vietoris theorem, acyclic models, CW-complexes, simplicial and cellular homology, homology with coefficients. Review of differentiable manifolds and tensors, Riemannian metrics, Levi-Civita connection, geodesics, exponential map, curvature tensor, first and second variation formulas, Jacobi fields, conjugate points and cut locus, Cartan-Hadamard and Bonnet Myers theorems. This course introduces homotopy type theory, which provides alternative foundations for mathematics based on deep connections between type theory, from logic and computer science, and homotopy theory, from topology.

Progress on existence of minimal surfaces - Andre Neves

This connection is based on interpreting types as spaces, terms as points and equalities as paths. Many homotopical notions have type-theoretic counterparts which are very useful for foundations. Such foundations are far closer to actual mathematics than the traditional ones based on set theory and logic, and are very well suited for use in computer-based proof systems, especially formal verification systems.

The course will also include background material in Algebraic Topology beyond a second course in Algebraic Topology. This is an introduction to hyperbolic surfaces and 3-manifolds, which played a key role in the development of geometric topology in the preceding few decades.