1) Algebraic preliminaries: Noetherian rings, K-algebras and finiteness conditions, modules and localizations. Graded rings and homogeneous ideals.
2) Affine space, affine closed subsets and the Zariski topology. Radical ideals. Hilbert Nullstellensatz. Irreducibility
and irreducible components. Affine and quasi-affine varieties: examples. Coordinate ring and field of rational
functions of an affine variety. Structural sheaf.
3) Projective space and projective closed subsets. Affine and projective cones. Homogeneous Hilbert Nullstellensatz. Projective varieties. Quasi-projective varieties and locally closed
subset. Structural sheaf.
4) Algebraic varieties. Morphisms between algebraic varieties. Constructible sets. Examples: Veronese embedding. Dominant morphisms. Rational maps,
birational maps. Rational varieties. Examples: linear system of hypersurfaces in a projective space, projections, stereographical projection of the smooth quadric, blow-up at a point,
resolution of singularities of some singular plane curves.
5) Products of algebraic varieties. Segre embedding and Segre variety. Diagonals and graph of a morphism. Main theorem of elimination theory: completeness of projective varieties.
6) Trascendence degree of an integral K-algebra of finite type. Dimension of an algebraic variety.
7) Embedded tangent spaces and non-singularity. Zarisky tangent space.
8) Further topics (either if time permits or for seminars/thesis)
- Hilbert function and Hilbert polynomial of a projective variety. Degree and arithmetic genus of a projective variety. Examples.
- Finite morphisms. Ramification.
- Semi-continuity of the fibre-dimension of a dominant morphism.
- Other examples of projective varieties: Grassmannians and Pluecker embedding.
Projective curves in the plane and their families. Resolution of singularities of plane curves.
Parameter spaces.
Recommended readings:
Lecture Notes: drafts. Free download from F. Flamini web-page.
Further books:
- I. Dolgachev, Introduction to Algebraic Geometry,
http://www.math.lsa.umich.edu ~idolga/631.pdf
- J. Harris, Algebraic geometry (a first course) Graduate Texts in Math.
No. 133. Springer-Verlag, New York-Heidelberg, 1977.
- R. Hartshorne, Algebraic geometry Graduate Texts in Math. No. 52.
Springer-Verlag, New York-Heidelberg, 1977.
- M. Reid, Undergradutae Algebraic Geometry, London Math. Soc. Student
Texts, vol. 12, 1988.
- I. Shafarevich, Basic algebraic geometry vol. 1 Springer-Verlag, New
York-Heidelberg, 1977.
Recollections on homology; cohomology, homotopy groups,
Eilenberg-MacLane spaces, fibrations, loop spaces, rational homotopy,
geometric applications.
Recommended readings:
Griffiths and Morgan, Rational homotopy theory and differential
forms.
Brisk review of linear algebra.
Historical notes on methods for solving systems of linear equations.
The sparse matrices. Storing sparse matrices and parallel computing.
Direct methods for sparse matrices. Iterative methods of stationary type.
Projective methods: description, analysis, and performance algorithms CG, GMRES, BiCGstab (l).
Preconditioning techniques. Preconditioning with incomplete factorization.
Incomplete inverse factorizations and inexact updates of matrices.
Preconditioning with algebra of matrices.
Methods for eigenvalue problems of large dimension.
Applications to fluid dynamics problems, image restoration, mathematical finance.
Recommended readings:
D. Bertaccini, C. Di Fiore, P. Zellini, Complessitá e Iterazione, Boringhieri, 2013.
The course deals with Classical and Celestial Mechanics.
The detailed program of the course is the following:
- basics of Classical Mechanics: canonical transformations, canonical criteria, Poisson brackets, first integrals;
- Integrable systems;
- Theorem on local integrability;
- Arnold-Liouville Theorem and action-angle variables;
- Examples of integrable systems: harmonic oscillators, motion in a central field, gyroscopic motion;
- Regular and chaotic motions;
- Conservative and dissipative systems;
- Continuous and discrete systems, Poincaré map, standard map;
- Lyapunov exponents;
- The two body problem;
- Kepler's laws
- Action-angle Delaunay variables and the three-body problem;
- Lagrangian equilibrium points;
- The restricted three body problem;
- Rotational dynamics;
- Spin-orbit resonances: derivation of the model and construction of invariant surfaces;
- Perturbation theory
- Applications of perturbation theory;
- KAM Theorem: proof, interval arithmetic, continued fractions;
- Classical and superconvergent techniques;
- Collisions and regularization theory;
- Levi-Civita transformation.
Recommended readings:
- V.I. Arnold, "Metodi Matematici della Meccanica Classica", Editori Riuniti (1979);
- A. Celletti, "Stability and Chaos in Celestial Mechanics", Springer-Praxis, XVI, 264 p., Hardcover ISBN: 978-3-540-85145-5 (2010);
- H. Goldstein - Meccanica Classica, Zanichelli (2005).
Rings, ideal, morphisms; basic notions and properties. Modules over a ring, morphisms (of modules), free modules; tensor product of modules; algebras over a ring.
Rings and modules of fractions.
Notherian or Artinian modules and rings. Modules over principal ideal domains. The prime spectrum of a ring.
Recommended readings:
- M. F. Atiyah, I. G. Macdonald, "Introduzione a l'algebra commutativa", Feltrinelli, Milano, 1981 (or the original english version);
- M. F. Atiyah, I. G. Macdonald, "Introduction to Commutative Algebra",
Addison-Wesley Publishing Company, Inc., 1969 (also available on-line);
- C. A. Finocchiaro, "Lo spettro primo di un anello", available on-line;
- S. Lang, "Algebra", revised Third Edition, Graduate Texts in Mathematics 211, Springer-Verlag New York, Inc, 2002;
- J. S. Milne, "A Primer of Commutative Algebra", freely available at
http://www.jmilne.org/math/xnotes/ca.html
(2009).
Introduction to numerical solving of ordinary and partial differential equations and to their graphical representation:
finite elements, stochastic simulation, eigenfunction developments. Introduction to the use of mathematical software: scilab, C e Freefem.
Recommended readings:
Lecture notes.
The theme of the course will be the difference between smooth, piecewise
linear, and topological manifolds, The principal theorem is the h-cobordism theorem of Smale,
and its demonstration alongside its
corollaries will be the principle goal of the course. Nevertheless, other
phenomenon particular to smooth manifolds,e.g. Morse theory, Thom transversality,
will be considered.
Recommended readings:
- Milnor, John, Lectures on the h-cobordism theorem, notes by L.
Siebenmann and J. Sondow.
- Milnor, John Morse theory.
- Milnor, John Differentil topology.
- Hirsch, W & Mazur B Smoothings of Piecewise Linear Manifolds
Elements of theory of ordinary differential equations: existence and
uniqueness of the solutions for C^1 vector fields. Floquet Theory.
Poincare' sections. Smooth dependence from initial conditions and
parameters.
Qualitative study of the solutions of an O.D.E. on the plane.
Flow box theorem. Stability and Lyapunov functions. Grobman-Hartmann
theorem. Stable and unstable manifolds: Hadamard-Perron, central manifold.
Generic families of vector fields. Generic bifurcations: saddle-node,
Hopf.
Omega-limit set and PoincarBendixon theorem.
O.D.E. on the two dimensional torus and reduction to the study of circle
diffeomorphism. Rotation number. KAM theory.
Hamiltonian systems and symplectic geometry. Canonical transformations.
Connection with Lagrangian systems. Completely integrable systems.
Averaging. Melnikov integral and horseshoes.
Measurable dynamical systems (definitions and simple examples).
Krylov-Bogoliubov theorem. Ergodic theory (Birkhoff, Von Neumann and
Poincare theorems; ergodicity, mixing, ..).
Recommended readings:
- HIRSCH Morris W., SMALE Stephen, DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND LINEAR ALGEBRA;
- Lecture notes.
Bernoulli polynomials and numbers, Eulero-Mclaurin formula, Romberg
extrapolation tecniques. Methods for the computation of the
eigenvalues and eigenvectors of a matrix, Perron-Frobenius theory for
non-negative matrices, computing the pagerank eigenvector, the power
method. Low complexity matrix algebras and applications. Numerical
solution of ordinary and partial differential problems,
the finite difference method.
For all these subjects both mathematical and algorithmic aspects are
investigated. (For a more detailed program see
www.mat.uniroma2.it/~difiore).
Recommended readings:
D. Bertaccini, C. Di Fiore, P. Zellini,
Complessitá e Iterazione - percorsi, matrici e algoritmi veloci
nel calcolo numerico, Bollati Boringhieri, Torino, 2013
Preliminaries on complex and differentiable manifolds; the sheaf of structure of a manifold; tangent space; regular submanifolds and the rank theorem;
immersions and embeddings; submersions, fibrations and fiber bundles; vector bundles; sections of vector bundles; operations on vector bundles;
metrics along the fibers;
line bundles and Picard's group; vector fields and foliations; Frobenius' theorem; the exterior differential operator; de Rham's cohomology;
complex structures and debar operator; metrics and Kaehler manifolds; sheaves of modules and Cech's cohomology; connections on vector bundles; Atiyah's class; different notions of
curvatures of a connection; Chern's classes; abstract de Rham theorem; Chern-Weil's theory.
Recommended readings:
- Lecture notes of the instructor.
Atomic structure. Periodic table of the elements. Chemical bonding (ionic, covalent, metallic). Intermolecular forces and hydrogen bonding. State of matter. Weight relations in chemical reactions. Oxidation number. Balance of chemical reactions. Thermodynamics. State functions. Equilibrium between phases. Homogeneous and heterogeneous chemical equilibria. The thermodynamic equilibrium constant. Solubility equilibria. Electrolytic dissociation. Solutions and colligative properties. Acid-base equilibria in aqueous solution: pH, hydrolysis, buffer solutions, indicators. Redox systems: electrode potentials, batteries, Nernst equation, electrolysis, Faraday's law Knowledge of the basic concepts and principles of chemistry, as concerns the comprehension of the general properties of the elements, of the chemical bonding defining compounds structure and of the fundamental laws that govern chemical and physical transformation of matter. Practical exercises aimed to a deeper understanding of the concepts presented during the lectures.
Lie algebras: definition and examples. Adjoint representation. Solvable and nilpotent algebras. Theorems of Lie and Engel. Killing form. Cartan criterion. Casimir element. Semisimple algebras.
Cartan subalgebras. Cartan decomposition. Cartan matrices. Root systems. Rank 2 examples.
Simple roots. Weyl chambers. Weyl group. Coxeter graphs. Dynkin diagrams. Classification of root systems. Automorhisms of semisimple Lie algebras. Groups of graph automorhisms of a Lie algebra. Isomorhism theorem for root systems. Weight spaces. Borel subalgebras. Cartan subalgebras. Engel subalgebras. The universal enveloping algebra. PBW theorem. Chevalley algebras.
Recommended book:
Humphreys J. "Introduction to Lie algebras and representation theory". Springer-Verlag, New York-Berlin.
Fourier series: L2 convergence, pointwise and uniform convergence. Rate of decay of Fourier coefficients. Gibbs phenomenon (if there is time). Approximate identities.
Convolutions and summation kernels (outline). Fourier transforms in L1 and L2 . Fourier transform of derivatives and of convolution. The inversion theorem and the Plancherel theorem.
The Schwartz class. Fourier transform on the Schwartz class. The Paley-Wiener class. Poisson summation formula. Tempered distributions and the Fourier transform
(in detail or outline according to time availability). Fourier transform of discrete periodic distributions and connection with Fourier series. Uniform sampling.
The Shannon sampling theorem. Aliasing.
The Discrete Fourier Transform and its properties. The Fast Fourier Transform. The Discrete Cosine Transform.
Recommended readings:
- M. Picardello, "Analisi armonica: aspetti classici e numerici" (available online at SMC/didattica/materiali_did/home_materiali_STM.html )
Differentiable functions on open sets of the euclidean space, implicit function
theorem, inverse function theorem, rank theorem. Notion of differentiable manifold,
examples, open sets in euclidean space, spheres, Grasmannians, projective space.
Smooth maps, diffeomorphisms, immersions, embeddings, embedded submanifolds,
immersed submanifolds. Embedded submanifolds as zero loci. Partition of unity.
Tangent space, tangent bundle, notion of vector bundle. Group actions on manifolds.
Vector fields, local ows, one parameter subgroups of diffeomorphisms, brackets,
Frobenius theorem, foliations. Multilinear algebra. Tensor product, exterior product,
differential forms, exterior differential. Orientable manifolds, Riemannian metrics
volume forms. manifolds with boundary, Stokes theorem and corollaries in euclidean
space. Definition and few things on De Rham cohomology. If there is time left,
covering spaces, and according to the student interest few things on Riemannian
geometry or Lie groups.
Recommended readings:
-Boothby, An Introduction to differentiable manifolds and Riemannian Geometry.
-Frank Warner, Foundations of Differentiable Manifolds and Lie Groups.
1. BANACH SPACES
-Definitions and examples
-Finite-dimensional vector spaces
-Bounded Operators on normed spaces
-Dual Space, examples
-Quotients and direct sums of normed spaces, dual space
-Hahn-Banach theorem and its main consequences
2. HILBERT SPACES
-Orthonormal bases, examples
-Trigonometric system and Fourier series in L2(T)
3. TOPOLOGICAL VECTOR SPACES AND WEAK TOPOLOGIES
-Weak and weak*-topologies
-Banach-Alaoglu theorem
4. BOUNDED LINEAR OPERATORS ON BANACH SPACES
-Uniform boundedness principle
-Open mapping theorem and closed graph theorem
-Bounded operators on Hilbert spaces
-Adjoint of an operator
5. SPECTRAL THEORY AND COMPACT OPERATORS
-Spectrum of an operator
-Compact operators and finite rank operators
-Riesz-Schauder theory and Fredholm alternative
-Spectral theorem for compact self-adjoint operators
Recommended readings:
- J. B. Conway - A course in functional analysis (1990,2ed), Springer, New York.
- M. Reed, B. Simon - Methods of Modern Mathematical Physics 1 (1980,2ed), Academic Press, San Diego.
Definition of holomorphic functions in several complex variables, power
series, Cauchy formula, uniform convergence, Weierstrass preparation,
Analytic sets, Extension theorems, Hartogs theorem, domains of holomorphy,
holomorphic convexity, Cartan-Thullen theorem, Levi-convexity,
plurisubharmonic functions, pseudoconvexity, Levi problem and solution
with L^2 estimates.
Recommended readings:
-Demailly, Complex analytic and differential geometry;
-Krantz, Function theory of several complex variables;
-Range, Holomorphic functions and integral representations in several
complex variables;
-Hörmander: An introduction to complex analysis in several variables.
The course deals with the problems of the pricing and the hedging of European options when the underlying market model is set as a diffusion model.
Firstly, special topics in stochastic calculus are recalled and developed (Markov processes, Girsanov's theorem, diffusion processes and Feynman-Kac type representation formulas);
secondly, diffusion models are introduced for the study of the arbitrage and the completeness of the financial markets. A special emphasis is given to the Black and Scholes model.
A part of the course is devoted to Monte Carlo numerical methods in Finance. Students have to choose a free-choice part of the course among the following subjects:
American options, interest rate models, applications to finance of the Malliavin calculus.
Recommended readings:
- D. Lamberton, B. Lapeyre: Introduction to stochastic calculus applied to finance. Second Edition. Chapman&Hall, 2008.
- P. Baldi: Stochastic differential equations. Lecture notes, 2016.
- L. Caramellino: Monte Carlo methods in Finance. Lecture notes, 2016.
1. General measure theory Motivation (examples of "non measurable sets", based on the axiom of choice).
Algebras, Borel algebras, Borel sets, product algebra. Measures, measurable space, measure space, finite, in-finite e semi-finite measures, continuity of measures
wrt monotone sequences of sets, complete measures, completion. External measures inmeasurable sets. CarathTeodory's Theorem. Extension of a premeasure on an algebra to a measure on a algebra. Borel measures on R: Lebesgue-Stieltjes measures, approximation of Borel sets by compact sets from inside. Lebesgue measure in R, Cantor set.
2. General integration theory
Measurable functions and their properties, Borel functions, simple functions, approximation of measurable functions by simple functions. Integration of nonnegative functions, Monotone Convergence Theorem, Fatou's Lemma. Integrals of real and complex functions and their proprieties. Uniform, pointwise, almost everywhere, quasi-uniform convergence (Egoro's Theorem), convergence in measure and in L1.
Dominated Convergence Theorem. Product measures, monotone classes, Fubini's and Tonelli's Theorem. Lebesgue integral in Rn.
3. The Radon-Nikodym derivative; BV functions in R
Signed measures, Hahn decomposition, Jordan decomposition, absolutely continuous measures, absolute continuity of integrals, Lebesgue-Radon-Nikodym's Theorem,
Lebesgue's decomposition, the Radon-Nikodym derivative. Complex measures and their total variation. Lebesgue points and Lebesgue sets in Rn, Lebesgue's derivative theorem and its application to regular Borel measures in Rn. BV functions in R, their Jordan decomposition and total variation. Lebesgue-Stieltjes integrals, absolutely continuous functions, fundamental integration theorem for Lebesgue integrals, decomposition of regular Borel measures in Rn in discrete, absolutely continuous and singular parts.
4. Elements of topology
Topologies, topological spaces, (neighborhood-) base for a topology, countability and separability axioms, convergent sequences, separable topological spaces. Continuous maps, homeomorphisms, generation of weak topologies, product topologies and their properties, spaces of bounded functions, Urysohn's Lemma and Tietze's
Extension Theorem for normal spaces, Tychonov spaces. Nets and directed sets, limits and cluster points of nets, (pre)compact spaces and their properties, sequential compactness. Locally compact and LCH spaces, Urysohn's Lemma and Tietze's
Extension Theorem for LCH spaces, compactly supported functions, the spaces
Cc(X) and C0(X), topology of uniform convergence on compacts sets, compact spaces, partition of unity. Tychonov's Theorem and the Ascoli-Arzel_a Theorem.
5. Elements of functional analysis
Normed vector spaces, Banach spaces, linear operators, bounded linear operators and their norm. Bounded linear functionals, dual spaces, the Hahn-Banach
Theorem (real and complex version, without proof) and some elementary applications.
Bidual, reexive spaces. Topological vector spaces, locally convex spaces,
Cauchy nets, completeness, Frechet spaces, definition and examples of topologies generated by semi-norms, weak and weak. topologies, the Banach-Alaoglu Theorem for normed vector spaces.
6. The spaces Lp.
The case the p is finite: Hoelder's Inequality, triangular inequality, norm, completeness, density of simple functions.
The case that p is infinite.
A first interpolation theorem, the dual of Lp for finite vales of, Lp is reflexive if p is finite and greater than 1. Chebyshev's Inequality, a necessary e sufficient condition for the precompactness of subsets Lp (the Frechet-Kolmogorov Theorem).
7. Radon measures on LCH spaces.
Positive Linear Positive on Cc (X), Radon Measurement, Riesz Representation Theorem.
Regularity of Radon measurements, density of continuous functions with compact support in Lp if the reference measure is of Radon, the theorem of Lusin.
Radon measure with sign or complex. The duo of C0 (X).
The weak topology * on the space of complex Radon measurements.
Approximation of functions in Lp with smooth functions via softeners.
Recommended readings:
G.B. Folland, Real Analysis, 2nd edition, Wiley & Sons.
Computer graphics is widely used in the video game and movie industry. The goal of this course is to provide some basic techniques in computer graphics,
and to give an introduction to the programming language Java. The course consists of two parts.
Part 1. Introduction to Java as an object-oriented programming language.
Part 2. Principles of computer graphics, the basic rendering pipeline, and photo-realistic rendering by ray-tracing.
Recommended readings:
- Thinking in JAVA, by Bruce Eckel;
- Computer Graphics Using OpenGL, by Francis S. Hill and Stephen M. Kelley.
Operator algebras:
- Banach algebras and C*-algebras.
Spectrum and functional calculus. Positive linear functionals, states and representations; representation of Gelfand-Naimark-Segal (GNS).
Structure of finite-dimensional C*-algebras.
Concrete operator algebras acting on Hilbert spaces: Bicommutant Theorem by John von Neumann and von Neumann Algebras (vNA).
W*-algebras, characterisation in terms of the predual: von Neumann Algebras as concrete W*-algebras.
Abelian operator algebras.
- Classification o W*-algebras:
Geometry of the proiections.
Normal semifinite and faithful traces.
Classification of the W*-algebras.
- Teoria modulare di Tomita:
Normal faithful states, and cyclic separating vectors on a vNA: Tomita's operator S.
Tomita's operator Delta and conjugation J.
One-parameter groups of normal automorphisms and Kubo-Martin-Schwinger (KMS) condition, Tomita's Theorem.
Normal semifinite and faithful weights: generalisation to the non sigma-finite case (shortly);
Standard rappresentation of a W*-algebra, examples: matrix algebras, the algebra of all bounded operators B(H) acting on the Hilbert space H, infinite tensor products.
- Applications:
Applications of the KMS condition to Quantum Statistical Mechanics.
Applications to Connes' classification of type III factors (shortly).
Normal faithful conditional expectations, Takesaki's existence theorem, generalisation by Accardi and Cecchini with applications to Quantum Probability (shortly).
Recommended readings:
(1) O. Bratteli, D. W. Robinson: Operator algebras and quantum statistical mechanics I,II,
Springer (sections 2.1-2.5, 5.3.1).
(2) M. Takesaki: Theory of operator algebras I, Springer (sections I.1-5, I.9, I11, II.1-4, III.1-3, V.1-2).
(3) S. Stratila, L. Zsido: Lectures on von Neumann algebras, Abacus press (sections 1, 3, 4, 5 and partially 10).
(4) S. Stratila: Modular theory in operator algebras, Abacus press (sections 1, 2, 9, 10).
(5) L. Accardi, C. Cecchini: Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal. 45 (1982), 245-273.
Introduction to optimal control problems: existence of solutions, Pontryagin's maximum principle, and dynamic programming. Hamilton-Jacobi equations:
characteristics, semiconcave solutions, and viscosity solutions.
Comparison theorems. Weak KAM theory. Study of the singular set: rectifiability theorems and propagation results. Application to homotopy equivalence.
Recommended readings:
- L.C. Evans, Partial differential equations. Second edition. American Mathematical Society, Providence, RI, 2010.
- P. Cannarsa & C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Birkhäuser, Boston, 2004.
- A. Fathi, Weak KAM theorem in Lagrangian dynamics (in press)
The complex plane and the Riemann sphere. Applications plane geometries.
Overview on holomorphic functions of a complex variable.
Conformal mappings.
The notion of a Riemann surface. Analysis on Riemann surfaces.
The uniformization theorem.
Parabolic surfaces.
The Moebius group and its subgroups. Hyperbolic Riemann surfaces.
Algebraic curves.
Divisors and Riemann-Roch theorem.
Abelian differentials. Abel and Jacobi theorems.
Hyperelliptic surfaces. Jacobi varieties. Torelli theorem.
Automorphisms and Hurwitz theorem.
Recommended readings:
- Hershel M. Farkas, Irwin Kra, "Riemann Surfaces" Springer-Verlag, NY USA , Graduate Texts in Mathematics 71,II ed., 1991,
pp. XIII+363;
- Olli Lehto, "Univalent functions and Teichmuller spaces" Springer-Verlag NY USA, Graduate Texts in Mathematics 109, 1987, pp. XII+257
Set theory. Intuitive set theory. Formal treatment. Language. Axioms. Classes. Well orderings. Ordinal and cardinal numbers. The cumulative hierarchy. Transfinite induction.
Ordinal and cardinal arithmetics.
The axiom of choice; weaker and equivalent forms. Brief description of independence results, large cardinals, and their import in algebra analysis and topology.
Recommended readings:
- T. Jech, Set Theory, any edition.
- Frank R. Drake, Set Theory: An Introduction to Large Cardinals, 1974.
- G. Gratzer, Universal algebra, any edition.
- H. P. Sankappanavar, S. Burris, A Course in Universal Algebra,
available at https://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html
Distributions. Sobolev spaces. The Sobolev-Gagliardo-Nirenberg and Morrey inequalities. Rellich's theorem. Poincaré's inequality. The Lax-Milgram lemma.
Variational formulation of elliptic boundary-value problems: existence, uniqueness, and regularity of weak solutions. Spectral theory for the Dirichlet problem.
Semigroups of bounded linear operators on Banach spaces.
Infinitesimal generator. The Hille-Yosida theorem. Contraction semigroups and compact semigroups. Perturbation theorems.
Asymptotic behaviour. Solution of the Cauchy problem. Maximal regularity. Application to the heat, wave, and Schrödinger equation.
Recommended readings:
- H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. Springer, New York, 2011.
- L.C. Evans, Partial differential equations. Second edition. American Mathematical Society, Providence, RI, 2010.
- A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York, 1983.
1. Prerequisites.
Banach spaces, dual space, strong, weak and weak* topology. Uniform boundedness principle. Analytic functions with values in a Banach space. Tychonoff theorem. Alaoglu theorem.
2. Banach algebras.
Resolvent set and spectrum of an element. Mazur theorem. Analyticity of the resolvent. The spectrum of an element is compact non-empty. Neumann series. Formula of the spectral radius.
3. Commutative Banach algebras.
Maximal ideals. Characters and spectrum of a commutative Banach algebra. Case of an algebra with identity generated by an element or by finitely many of elements, joint spectrum. The spectral mapping theorem for polynomials. Spectrum of an element in a minimal or maximal Abelian subalgebra.
4. Analytic functional calculus.
The spectral mapping theorem for analytic functions. Disconnected spectrum case. Perturbations of the spectrum, lower semicontinuity, examples. Perturbations of projectors.
5. Gelfand transformation.
Generalized nilpotent elements. Case of the l^1(Z) algebra. The spectral mapping theorem for analytic functions.
6. C*-algebras.
Involutive algebras and C* norms. The spectrum of a self-adjoint element is real; the spectral radius coincides with the norm. The spectrum does not depend on C* subalgebra.
7. Commutative C*-algebras.
Gelfand-Naimark theorem. Continuous functional calculus. The category of Abelian C*-algebras with identity is dual to the category of compact Haudorff spaces.
8. Borel functional calculus.
The spectral theorem for self-adjoint operators on a Hilbert space. Basic measures and L∞ functional calculation .
9. von Neumann algebras.
Density theorems of von Neumann and Kaplanski. Weak and strong operator topology. Weak compactness of the unit ball. Maximal Abelian subalgebras and their characterization on a separable Hilbert space.
10. States and representations of a C*-algebra.
Positive elements, extension of states. The GNS representation. Case of the algebra C(X). Intertwining operators, subrepresentations, equivalent representations and disjoint representations. Commutative case.
11. The spectral multiplicity theorem.
Cyclic representations and multiplicity free representations (separable Hilbert space) of an Abelian C*-algebra. Classification of self-adjoint operators (or representations of an Abelian C*-algebra) on a separable Hilbert space.
12. Unbounded Operators.
Closed operators, closable operators, adjoints. Symmetric and self-adjoint operators. Extensions of symmetric operators and Cayley transformation. The moment problem.
Recommended readings:
- G. Pedersen, Analysis Now;
- W. Arveson, An Invitation to C * -Algebras;
- J. B. Conway - A Course in Functional Analysis.
Stationary processes - existence, autocovariance function, linear filters,
ARMA. Herglotz-Bochner Theorem, stochastic integrals, Spectral
Representaion Theorem. Periodogram - asymptotic properties, spectral
density estimation, Whittle estimates. Hints on nonstationary processes
and random fields.
Recommended readings:
P.Brockwell e R.Davis, Time Series Models, 1991 Springer.