1
Rendiconto Scientifico dell'attività della Scuola
Matematica Interuniversitaria per il 2006
1 - Elenco dei Corsi estivi tenuti nell'estate del 2006
2 - Partecipanti al Corso Estivo di Matematica - Perugia (9 corsi)
3 - Partecipanti al Corso Estivo di Matematica - Cortona (4 corsi)
4 - Elenco dei partecipanti ai singoli Corsi - Perugia
5 - Programmi dei Corsi estivi di Perugia e Cortona 2006
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Rendiconto scientifico dell'attività della SMI per il 2006
Nell'estate 2006 la Scuola Matematica Interuniversitaria, con la collaborazione della Scuola
Normale Superiore di Pisa e del Dipartimento di Matematica dell'Università di Perugia, ha
organizzato corsi di base per laureandi e giovani laureati nella sede di Perugia e corsi più avanzati,
di avviamento alla ricerca a Cortona.
1 - ELENCO DEI CORSI ESTIVI TENUTI NELL'ESTATE DEL 2006
PERUGIA: (30 luglio –2 settembre)
Insegnamenti
- Algebra Alberto Facchini, Univ. Padova
- Analisi Complessa Edgar Lee Stout, Univ. Washington Seattle.
- Analisi Funzionale Eric T. Sawyer, McMaster Univ.
- Analisi Numerica Christian Lubich, Univ. Tuebingen
-Equazioni differenziali
della Fisica Matematica Guido Sweers, TU Delft
- Geometria Algebrica Marco Andreatta, Univ. Trento
- Geometria Differenziale Gudlaugur Thorbergsson, Univ. Koeln
- Probabilità Giovanni Pistone, Politecnico Torino
- Teoria dei Modelli Zachary Scott, East Carolina Univ.
CORTONA I:2 – 14 luglio 2006 - Syzygies, Hilbert functions generic initial ideals : Aldo Conca, Univ. Genova
Juan C. Migliore, Univ. Notre Dame-Indiana
CORTONA II: 2 – 23 luglio 2006 - A Geometrial Approach to Free Boundary Problems : Luis A. Caffarelli, Univ. Texas
Sandro Salsa, Politecnico Milano
CORTONA III:23 luglio – 12 agosto 2006
- Mathematical Finance Wolfgang Runggaldier, Univ. Padova
Uwe Schmock, Wien Univ. Technology
CORTONA IV:30 luglio – 19 agosto 2006 - Morse Theory Application to Diferential
Geometry and onedimensional variational Problems : Francesco Mercuri, Univ.Campinas,
Paolo Piccione, Univ.Camerino e Univ.
San Paolo
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2 – PARTECIPANTI AL CORSO ESTIVO DI MATEMATICA – PERUGIA (9 corsi)
July 30 – September 2, 2006 Studenti Italiani
Domande: 110
Studenti ammessi: 102
Partecipanti effettivi: 81
Studenti stranieri
Domande: 75
Studenti ammessi: 38
Partecipanti effettivi:31
3 –PARTECIPANTI AL CORSO ESTIVO DI MATEMATICA – CORTONA (4 CORSI)
3a) - CORTONA: 2 luglio – 14 luglio, 2006
Elenco dei partecipanti ai singoli corsi
- Syzygies, Hilbert functions generic initial ideals
Partecipanti Italiani
Domande : 10
Studenti ammessi: 9
Partecipanti effettivi: 8
BENEDETTI Beatrice Perugia
BERTELLA Valentina Genova
GRIECO Elena L’Aquila
GUERRINI Eleonora Pisa
LA BARBIERA Monica Messina
MALASPINA Francesco Torino
SOMONETTI Ilaria Pisa
SORRENTI Loredana Messina
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Partecipanti Stranieri
Domande : 13
Studenti ammessi : 13
Partecipanti effettivi : 11
CIMPOEAS Mircea Bucharest
CONSTANTINESCU Alexandru Bucharest
COOPER Susan Marie Queen’s (Canada)
KAMPF Gesa Osnabruck
OLLER MARCEN Antonio M. Zaragoza
OSTAFE Lavinia Bucharest
SECELEANU Alexandra Bucharest
SOGER Christof Osnabruck
STAMATE Dumitru Bucharest
STOKES Erik entucky
WIBMER Michael Innsbruck
3b) – CORTONA: 2 luglio – 22 luglio
A Geometrial Approach to Free Boundary Problems
Studenti Italiani
Domande : 14
Studenti ammessi: 14
Partecipanti effettivi: 13
Elenco dei partecipanti ai singoli corsi
ANTONANGELI Giorgio Roma La Sapienza
ANTONELLI Paolo L’Aquila
ARGIOLAS Roberto Cagliari
AROSIO Leandro Pisa
CASTELPIETRA Marco Roma To Vergata
CECCHINI Simone Firenze
CESERI Maurizio Firenze
CIRAOLO Giulio Firenze
DI NARDO Rosaria Napoli
GAVITONE Nunzia Napoli
NORIS Benedetta Milano Bicocca
PATRIZI Stefania Perugia
PERROTTA Adamaria Napoli
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Studenti stranieri
Domande: 14
Studenti ammessi: 14
Partecipanti effettivi: 12
ARAMA Danut Al. I. Cuza
CIOMAGA Adina G. Al. I. Cuza
GRIGORIU Andreea G. Al. I. Cuza
HILLERMAA Kadri Tartu
HITZAZIS Iasonas Patras
IBRAHIM Hassan Liban
MARTINEZ Sandra Buenos Aires
MILBERS Zoja Koeln
OLECH Michal Wroclaw
PAVLICEK Libor Praga
SYLWESTRZAK Ewa Zielona Gora
VARVARUCA Eugen Al. I. Cuza
3c) – CORTONA: 23 luglio – 12 agosto 2006
Mathematical Finance
Studenti Italiani
Domande : 18
Studenti ammessi : 18
Partecipanti effettivi: 16
Elenco dei partecipanti ai singoli corsi
ACCIAIO Beatrice Perugia
BLASI Francesco Roma La Sapienza
D’AMICO Guglielmo Chieti
D’IPPOLITI Fernanda L’Aquila
D’URZO Eleonora Perugia
FEDELE Mariagrazia Bari
FEDERICO Salvatore Pisa
FERRETTI Camilla Firenze
GIULIETTI Paolo Pisa
GOBBI Fabio Firenze
LOMBARDI Luana L’Aquila
MASTROLEO Marcello Perugia
MERCURI Lorenzo Ancona
PREZIOSO Valentina L’Aquila
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RUSSO Emilio Calabria
VALENTE Carla L’Aquila
Studenti stranieri
Domande : 14
Studenti ammessi: 13
Partecipanti effettivi: 11
DENIZ Asli Izmir Institute Technology
DENGLER Barbara Vienna
GEVEILERS Vjaceslavs Hamburg
HUNT Julien Catholique de Louvaine
MAKAR Nadyia Lviv
RAFLER Mathias Postdam
RENZ Norbert Ulm
VAJDA Istvan Corvinus Budapest
VANDAELE Nele Gent
ZAKHAROVA Anastasia Mosca
XU Ling Leipzig
3d) – CORTONA: 30 luglio – 19 agosto 2006
Morse Theory, Application to Differential Geometry and One-dimensional variational
Problems
Studenti Italiani
Domande: 7
Studenti ammessi: 7
Partecipanti effettivi: 7
Elenco dei partecipanti ai singoli corsi
DE LEO Barbara Lecce
GAZZINI Marita Milano
MACIOCCO Giovanni Cagliari
RINALDELLI Mauro Firenze
ROSATI Lilia Firenze
SANTI Andrea Firenze
SICILIANO Gaetano Bari
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Studenti stranieri
Domande: 6
Studenti ammessi: 6
Partecipanti effettivi: 4
ABARDIA BOCHACA Judit Autonoma Barcelona
BALMUS Adina Al. I. Cuza
CEBANU Radu Bucharest
POCOVNICU Oana Al. I. Cuza
4 - ELENCO DEI PARTECIPANTI DEI SINGOLI CORSI DI PERUGIA
Algebra – (27)
Studenti Italiani
BENEDETTI Bruno Genova
BOVENZI Michele Napoli
CASPANI Luigi Como Insuria
CIGOLI Alan Stefano Milano
DI MARIA Giovanni Napoli
FINOCCHIARO Carmelo Antonio Catania
GALETTO Federico Torino
GENTILE Tommaso Calabria
IMPERATORE Diana Salerno
MESSINA Simona Catania
MORINI Francesco Messina
POVERO Masismiliano Torino Politecnico
RAGUSA Giorgio Catania
REDUZZI Davide Milano
TARASCA Nicola Roma Tor Vergata
TEDESCO Giovanna Napoli
TERRAGNI Tommaso Milano
VENEZIANO Francesco Pisa
Studenti stranieri
COPIL Vlad Alexandru Bucharest
CRONIN Anthony National Univ. Ireland
DEMIRCI Yilmaz Mehmet Izmir Inst.
FERAGEN Aasa Helsinki
LICHIARDOPOL Elena-Raluca Bucharest
PETRISAN Daniela-Luana Bucharest
PORUMBEL Daniel Cosmin Bucharest
TOP Serpil Izmir Inst.
WALTON Chelsea Michigan State Univ.
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Analisi Complessa – (22)
Studenti Italiani
ARLOTTO Alessandro Torino
BOCCIA Serena Salerno
BOCHICCHIO Ivana Salerno
CERREIA VIOGLIO Simone Milano Bocconi
GALETTO Federico Torino
GRANDI Stefania Bologna
MAININI Edoardo Milano Politecnico
PANICCIA Irene Roma La Sapienza
Studenti stranieri
BLAGA Camelia-Elena Bucharest
BLANCO Ivan Complutense Madrid
BLASZKE Malgorzata Silesian
CAGATAY Filiz Izmir Inst.
COPIL Vlad-Alexandru Bucharest
CRONIN Anthony National Univ. Ireland
DUMITRU Dan Bucharest
LICHIARDOPOL Elena-Raluca Bucharest
MAINKA Ewelina Silesian
NOVIKOVA Anna Voronezh
PUMPERLA Max Kaiserslautern
RAICU Claudiu Cristian Bucharest
TACHE Alexandru-Petre Bucharest
TYC Katarzyna Silesian
Analisi Funzionale - (16)
Studenti Italiani
BOCCIA Serena Salerno
CERREIA VIOGLIO Simone Milano Bocconi
CHIEPPA Loredana Bari
DE FUSCO Rossella Napoli
DI MICHELE Federica L’Aquila
MAININI Edoardo Milano Politecnico
MERCURI Carlo Milano
PANICCIA Irene Roma La Sapienza
ROSSARO Pier Cristoforo Torino Politecnico
SELVITELLA Alessandro Milano
TAVERNISE Marianna Calabria
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Studenti stranieri
DUMITRU Dan Bucharest
NESIC Svetozar Belgrado
REMUS Radu Bucharest
TACHE Alexandru-Petre Bucharest
TANASE Raluca Bucharest
Analisi Numerica - (17)
Studenti Italiani
BERNARDI Mauro Venezia
CHIEPPA Loredana Bari
DE ANGELIS Guido Perugia
FELACO Elisabetta L’Aquila
FLERES Mirko Bologna
GAETANO Raffaele Napoli
GRANDI Stefania Bologna
LABITA Marzia Como Insubria
TAVERNISE Marianna Calabria
UBERTINI Filippo Perugia
Studenti stranieri
CARDENAS PRIETO Ernesto Adolfo Externado Colombia
DE KORT Johan Peter Delft
GALAN Ioana-Catalina Al.I. Cuza
GOUIN Cindy Debureaux
NOVIKOVA Anna Voronezh
PRYER Tristan Sussex
SAVA Ecaterina Al.I. Cuza
Equazioni Differenziali della Fisica Matematica – (17)
Studenti Italiani
BOCHICCHIO Ivana Salerno
CAVALETTI Fabio Roma La Sapienza
DE ANGELIS Guido Perugia
DI MICHELE Federica L’Aquila
FELACO Elisabetta L’Aquila
MARI Luciano Milano
MERCURI Carlo Milano
SELVITELLA Alessandro Milano
UBERTINI Filippo Perugia
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Studenti stranieri
BEREZOVSKA Ganna Kyiv National
CAGATAY Filiz Izmir Inst.
DE KORT Johan Peter Delft
GOUIN Cindy Duberaux
NESIC Svetozar Belgrado
PILARCZYC Dominika Wroclaw
PRYER Tristan Sussex
TACHE Alexandru-Petre Bucharest
Geometria Algebrica - (27)
Studenti Italiani
CASPANI Luigi Como Insuria
CAVALLETTI Fabio Roma La Sapienza
CHIECCHIO Alberto Torino
GENTILE Maria Napoli
GENTILE Tommaso Calabria
IMPERATORE Diana Salerno
MAGGIOLO Stefano Ferrara
POMA Flavia Pisa
POVERO Massimiliano Torino Politecnico
TARASCA Nicola Roma Tor Vergata
TERRAGNI Tommaso Milano
VENEZIANO Francesco Pisa
VERONELLI Giona Como Insubria
VISCO COMANDINI Filippo Roma La Sapienza
Studenti stranieri
BLANCO Ivan Complutense Madrid
DE BALLE PIGEM Borja Catalogna
DEMIRCI Yilamz Mehmet Izmir Inst.
FERAGEN Aasa Helsinki
FLUCH Martin Ruprecht-Karls
PETRISAN Daniela Luana Bucharest
PUMPERLA Max Kaiserslautern
RAICU Claudiu Cristian Bucharest
REMUS Radu Bucharest
TANASE Raluca Bucharest
TOP Serpil Izmir Inst.
TYC Katarzyna Silesian
WALTON Chelsea Michigan State
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Geometria Differenziale – (10)
Studenti Italiani
BOCHICCHIO Ivana Salerno
FLERES Mirko Bologna
GENTILE Maria Napoli
MAGGIOLO Stefano Ferrara
MARI Luciano Milano
POMA Flavia Pisa
ROSSARO Pier Cristoforo Torino Politecnico
VERONELLI Giona Como Insubria
VISCO COMANDINI Filippo Roma La Sapienza
Studenti stranieri
PILARCZYK Dominika Wroclaw
Probabilità – (10)
Studenti Italiani
ARLOTTO Alessandro Torino
BERNARDI Mauro Venezia
GAETANO Raffaele Napoli
LABITA Marzia Como Insubria
Studenti stranieri
BEREZOVSKA Ganna Kyiv National
BLAGA Camelia-Elena Bucharest
CARDENAS PRIETO Ernesto Adolfo Externado Colombia
FLUCH Martin Ruprecht Karls
GALAN Ioana-Catalina Al. I. Cuza
SAVA Ecaterina Al. I. Cuza
Teoria dei Modelli – (15)
Studenti Italiani
BENEDETTI Bruno Genova
BOVENZI Michele Napoli
CIGOLI Alan Stefano Milano
DE FUSCO Rossella Napoli
DI MARIA Giovanni Napoli
FINOCCHIARO Carmelo Antonio Catania
MESSINA Simona Catania
MORINI Francesco Messina
RAGUSA Giorgio Catania
REDUZZI Davide Milano
TEDESCO Giovanna Napoli
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Studenti Stranieri
BLASZKE Malgorzata Silesian
DE BALLE PIGEM Borja Catalogna
MAINKA Ewelina Silesian
PORUMBEL Daniel Cosmin Bucharest
5 - PROGRAMMI DEI CORSI DI PERUGIA E CORTONA
Programmi Corso Estivo Perugia : 30 luglio agosto-2 settembre 2006 ALGEBRA Docente: Prof. Alberto Facchini, Univ. Padova
Course contents Rings and ring homomorpisms. Simple rings, division rings. Modules and module homomorphisms. Direct sums, quotient modules. Isomorphism theorems. Cyclic modules. Zorn's lemma. Exact sequences. Maximal submodules. Free modules, IBN rings. Projective modules and their properties. Group rings. Simple modules, semisimple modules. Composition series. Jordan-H\"older theorem. Artinian/noetherian modules and rings. Semisimple artinian rings. Schur's lemma. The theorem of Artin-Wedderburn. Simple artinian rings. Faithful modules, primitive rings, Chevalley-Jacobson theorem. Group representations. Maschke's theorem. Hopkins-Levitzki's theorem. Jacobson radical. Hereditary rings. Dedekind domains. Local rings. Injective modules. Baer's criterion. Every module can be embedded in an injective module. Essential extensions. Lezioni in Inglese Prerequisites: The basic definitions and the first elements of the theory of groups, rings, modules, and linear algebra. Any student of Mathematics at the University, after three years of study, should know them. Textbook: Donald S.~Passman, ``A Course in Ring Theory'', AMS Chelsea Publishing, 2004. ANALISI COMPLESSA Docente: Prof. Edgar Lee Stout, Univ. of Washington Seattle, Washington Programma: 1. The arithmetic and geometry of the complex plane. 2. Complex differentiation and the Cauchy-Riemann equations. 3. Elementary functions. Power series. 4. Complex integration. Cauchy's Theorem-the simplest case. 5. Elementary properties of holomorphic functions. 6. Infinite Products. The Blaschke condition. 7. More general versions of Cauchy's Theorem.
8. Residue theory and the evaluation of real integrals. 9. Conformal mapping. The automorphisms of the disc, the plane and the sphere. Examples of mappings by elementary functions. 10. Normal families. 11. The Riemann Mapping Theorem. 12. Runge's Theorem. Applications. Lezioni in Inglese Prerequisiti: Elementary analysis at the level of Rudin's,Principals of Mathematical Analysis. Testo: John B. Conway Functions of one Complex Variable I, Springer-Verlag
ANALISI FUNZIONALE Course contents Part I of the Text - especially chapters 2, 3, 4 and 5 – including Banach-Steinhaus theorem, open mapping theorem, closed graph theorem Hahn-Banach theorem, Banach-Alaoglu theorem, Krein-Milman theorem, holomorphic functions. Duality, compact operators Various applications to closed subspace of $L^p$ spaces, range of a vector valued measure, Bishop's theorem, interpolation theorems, fixed point theorems, Haar measure, and complemented subspaces. Time permitting, brief introductions to distributions and partial differential equations (Part II) and spectral theory of Banach algebras (Part III) will be given. Lectures in English Prerequisites: Lebesgue integration, completeness of $L^p$ spaces, elementary properties of holomorphic functions. A reference for the prerequisites topics is chapters 2, 3 and 10 of "Real and Complex Analysis" by Walter Rudin, McGraw Hill, Inc. Textbook: "Functional Analysis" by Walter Rudin, McGraw Hill, Inc. 1991
ANALISI NUMERICA Docente: Prof. Christian Lubich, Univ. Tuebingen Course contents The course will introduce into basic techniques and methods of Numerical Analysis. It will cover the following topics: 1. Interpolation and approximation (Polynomial interpolation by Newton's formula, Errors in polynomial interpolation, Chebyshev interpolation, spline interpolation, Numerical differentiation 2. Numerical integration (Quadrature formulas, order and error, Gaussian quadrature, adaptive quadrature) 3. Numerical solution of ordinary differential equations (Basics, Runge-Kutta methods, extrapolation methods, multistep methods) Lezioni in Inglese Testo : W. Gautschi, Numerical Analysis: An Introduction, Birkhaeuser 1997. EQUAZIONI DIFFERENZIALI DELLA FISICA MATEMATICA Docente : Prof. Guido Sweers, Universitaett zu Koeln and Delft University of Technology Initial programme: 1. From models to differential equations - Laundry on a line: a linear and a nonlinear model - Flow through area and more 2d - Problems involving time: Wave equation, Heat equation - Differential equations from calculus of variations - Mathematical solutions and `real life' 2. Spaces, Traces and Imbeddings - Function spaces: Hoelder spaces, Sobolev spaces - Restricting and extending, traces and corresponding Sobolev spaces
- Inequalities by Gagliardo, Nirenberg, Sobolev and Morrey 3. Some new and old solution methods I - Direct methods in the calculus of variations - Solutions in flavours - Characteristics and local solutions by Cauchy-Kowalevski: 4. Some old and new solution methods II - Special domains and almost explicit formula - Weak solutions by Lax-Milgram - The wave equation in 3 and 2 space dimensions 5. Some classics for a unique solution - Energy methods - Maximum principles Lezioni in Inglese Prerequisites: Analysis, Ordinary Differential Equations, and preferably some elementary knowledge of Functional Analysis or Partial Differential Equations Testo: Lawrence C. Evans, Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. GEOMETRIA ALGEBRICA Docente: Prof. Marco Andreatta, Univ. Trento Course contents The course will introduce into the study of Riemann surfaces (Rs) and algebraic curves. The prerequisites are some basic definitions of general topology and the first elements of the theory of holomorphic functions of one complex variable. 1. Definitions, examples and constructions of Rs. 2. Functions and morphisms between Rs. Differential forms and integration on Rs. 3. Meromorphic functions and divisors on Rs. Morphisms and linear sysitems. 4. Riemann Roch theorem, Serre duality and applications.
Lectures in Italian/English Textbook: Rick Miranda, Algebraic Curves and Riemann Surfaces, Am.Math.Soc.GSM vol.5 (1997) GEOMETRIA DIFFERENZIALE Docente: Prof. Gudlaugur Thorbergsson, Univ. Koeln The aim of the course is to give an introduction to basic notions and results of Riemannian Geometry. Program: Differentiable manifolds, Riemannian metrics, covariant derivatives, geodesics, the curvature tensor, first and second variation formulas, Jacobi fields, conjugate points, completeness, the theorem of Hopf-Rinow, the Theorems of Hadamard and Bonnet-Myers Lezioni in Inglese Prerequisites: Good knowledge of Multivariable Calculus and Linear Algebra will be assumed. Some familiarity with the notion of a differentiable manifold will be helpful Testo: Manfredo do Carmo, Birkheuser, 1992, Riemannian Geometry PROBABILITA' Docente: Prof. Giovanni Pistone, Politecnico di Torino The plan is to cover as much as possible of the material contained in the textbook J. Jacod & Ph. Protter, {Probability Essentials} 2nd Ed. Springer, - Elementary probability (Ch. 2--5) - Probability measures and random variables (Ch. 6--10) - Probability distributions on real vector spaces (Ch. 11--16) - Convergences and limit theorems (Ch. 17--21) - Conditional expectation and martingales (Ch. 22--28) The precise choice of topics will depend on the actual interests and background of the students. Much room will be left to examples and
exercises.
TEORIA DEI MODELLI Docente: Prof. Zachary Robinson, East Carolina University Programma: This is an introduction to model theory with applications to algebra and algebraic geometry. The model theory is developed beginning with first-order languages and structures, theories and models, definability and interpretability. Fundamental general techniques such as model-theoretic compactness, completeness and back-and-forth constructions are introduced. The final segment covers quantifier elimination for algebraically closed and real closed fields. Along the way, applications to algebra and algebraic geometry will be discussed. These include Ax's theorem that injective endomorphisms of complex algebraic varieties are surjective, Artin's solution to Hilbert's 17th Problem, Milnor's Curve Selection Theorem, and cell decomposition for real semi-algebraic sets. Lezioni in Inglese Prerequisiti: A course in abstract algebra (properties of integers, polynomials, groups, rings, fields) and minimal familiarity with logic (propositional calculus, predicate calculus, proof). To gain some familiarity with logic, students with no prior experience might want to first look through a basic logic text such as: the first half of "A Mathematical Introduction to Logic," by Herbert Enderton, or the first quarter of "Mathematical Logic," by Ebbinghaus, Flum and Thomas. Testo : "Model Theory: An Introduction," David Marker, Graduate Texts in Mathematics 217, Springer-Verlag, New York, 2002 (ISBN: 0-387-98760-6).
Programmi Corso Estivo Cortona 2 luglio - 15 luglio 2006 Syzygies Hilbert Function and Generic Initial Ideas Docente: Prof. Aldo Conca,Univ. di Genova
1) Introduction to the basic invariants: Hilbert functions, Betti numbers, regularity. 2) Initial ideals and deformations. 3) Monomial ideals, stable ideals, strongly stable ideals, Borel fixed ideals, lex-segments and more generally tau-segments and their Betti numbers 4) Generic initial ideals: existence and main properties, 5) Polarizzation, distraction and gin. 6) Macaulay Theorem, Bigatti-Hullett and Pardue Theorem. 7) Rigidity: Herzog-Hibi-Aramova Theorem and extensions. 8) Froeberg conjcture, Gin of generic complete intersections. Anick's result. 9) Gin-lex 10) Simplical complexes, gin and shifting. 11) Regularity for powers Libri consigliati : - Bruns-Herzog "Cohen-Macaulay rings" Cambridge University Press, 1998. - D.Eisenbud, "Commutative Algebra : with a View Toward Algebraic Geometry" Springer 1999. Docente: Prof. Juan C. Migliore, Univ. Notre Dame, Indiana 1) Introduction (we will split the material ) 2) Deficiency modules 3) Gorenstein ideals and subvarieties 4) Liaison 1 5) Liaison 2 6) Liaison 3
7) Froeburg conjecture (preparation for Conca's talk \#7) 8) Weak Lefschetz property 9) Multiplicity conjectures 10) Fat points 11) Tetrahedral curves
Programmi Corso Estivo Cortona 2 luglio - 23 luglio 2006 A Geometrical Approach to Free Boundary Problems Docenti: Prof. Luis Caffarelli, Univ.Texas Austin-Prof. Sandro Salsa, Politecnico Milano
Course contents Caffarelli and Salsa will coordinate their lectures to cover simultaneously the following topics: Part I The obastacle problem and flux-discontinuity type free boundary problems (one and two phases). Introductory examples and problematic, the equations involved, variational and supersolution approach. Global optimal regularity of solutions. Regularity and stability of interphases. In the process, we develop the needed tools from geometric PDE: basic properties of solutions of second order elliptic equations in Lipschitz domains, interior and boundary harnack inequalities, monotonicity formulas (about I and 1/2 weeks). Part II Extension of ideas and methods to other problems: twophase parabolic problems (Stefan type), flow in porus media, problems involving fractional laplacians (thin obstacles, Levy process). Textbooks : L.A. Caffarelli, S.Salsa, A geometric approach to free boundary problems, A.M.S. Providence, 2005. The obstacle problem, Lezioni Fermiane, Pisa
Programmi Corso Estivo Cortona 23 luglio - 12 agosto 2006
Finanza Matematica Docenti: Prof. Wolfgang J. Runggaldier, Univ. di Padova
Program : [1.] Basic structure
1. Term structure of interest rates {\it (lectures and problem-solving sessions and seminars) 2. Hedging of general claims by martingale representation (mainly problem-solving sessions and seminars) [2] Specific structure Term structure of interest rates - Basic concepts and preliminaries; - Martingale models for the short rate and their calibration; Forward rate models {\it (HJM framework)}; - Change of numeraire techniques; - LIBOR and Swap market models Remarks: The basic theory will be presented in a Brownian framework. As the lectures on the general integration theory (Prof. Schmock) progress also settings beyond the Brownian framework will be envisaged. 2. Hedging After a short basic introduction during the lectures, this will be mainly a topic for the problem-solving sessions and seminars. As for the term structure, here too we shall start from a Brownian framework that will then be gradually generalized in line with the general integration theory (Prof. Schmock). Lezioni in Inglese/Italiano Testo : T. Bjoerk, Arbitrage Theory in Continuous Time. Oxford University Press 2004 (2nd edition). Letture consigliate : D.Brigo, F. Mercurio, Interest Rate Models – Theory and Practice. Springer Verlag 2005 (2nd edition). Possible additional material for lectures and problem-solving sessions and relating specific journal articles will be made available on site. Docente: Prof. Uwe Schmock, TU Wien Motivation:
Let S denote a stochastic process describing the evolution of the discounted price of an asset, and let H be the process describing the (possibly random) number of these assets at any given time in the investor's portfolio. The gains and losses of this investment strategy H is given by the stochastic integral of H with respect to S. It therefore lies at the heart of modern, continuous-time mathematical finance to clarify, for which investment strategies H and price processes S this stochastic integral is mathematically well defined and what its properties are. Contents: (I) We will follow the approach given in Ph. Protter's textbook, developing the theory of general stochastic integration with respect to semimartingales, which includes the cases of Brownian motion and Lévy processes. Applications of the theory, in particular to the modelling to the stochastic evolution of the term structure of interest rates, will be given in Prof. Runggaldier's part of the course. Ph. Protter's book contains an extensive list of exercises, which can be discussed in the problem-solving sessions. (II) Depending on time and interest of the course participants, (a) credit risk modelling with an emphasis on CreditRisk+ and its extensions, (b) properties of expected shortfall, and (c) allocation of risk capital by expected shortfall will be treated in the seminars. Lecture notes for preparing these seminars are available upon request. Prerequisites: Part (I) of the course requires familiarity with measure theoretic probability theory and basic results about martingales, because these will be used without proofs. The textbook by D. Williams and Chapter 2 of the textbook by S. Ethier and T. Kurtz are certainly a good source. Lezioni in Inglese Literature: - Philip E. Protter: Stochastic Integration and Differential Equations, (2nd edition), Applications of Mathematics: Stochastic Modelling and Applied Probability, Vol. 21, 2004, Springer-Verlag, ISBN 3-540-00313-4. - David Williams: Probability with Martingales, Cambridge Mathematical Textbooks, 1991, Cambridge University Press, ISBN 0-521-40605-6 - Stewart N. Ethier and Thomas G. Kurtz: Markov Processes, Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics, 1986, John Wiley \& Sons, ISBN 0-471-08186-8 - Uwe Schmock: Modelling Dependent Credit Risks with Extensions of CreditRisk+, An Implementation-Orientated Presentation, Lecture Notes, 2006 (latest version available upon request, [email protected]).
Programmi Corso Estivo Cortona 30 luglio - 19 agosto 2006
Morse theory, with applications to Differential Geometry Docenti: Prof. Francesco Mercuri (Unicamp) e Prof. Paolo Piccione (USP) Short program of the course First week: - Review of Algebraic Topology. - Ljusternik and Schnirelman theory. - Classical Morse Theory. Second week: - Applications of the finite dimensional Morse Theory to submanifold theory: Generalized Gauss-- Bonnet theorem, Chern-Lashof theorem, low codimensional submanifolds of positive curvature in $R^N$, hyperplane section theorem. - The Morse--Witten complex (in compact manifolds) and its homology. Dynamical formulation of the Morse inequalities. - Morse--Bott theory (critical submanifolds). Third week: - Some applications to Riemannian Geometry: The pinching Theorem, periodic geodesics, the Yamabe problem. - A strongly indefinite variational problem: Geodesics in Lorentzian manifolds, spectral flow, Maslov index. Basic bibliography: 1. Mercuri-Piccione-Tausk: {\it Morse Theory}, Published by I.M.P.A., Brazil, 2003. 2. Milnor: {\it Morse theory}, Annals of Math. Study, vol 51, Princeton University Press, 1963. 3. Palais-Terng: {\it Critical Point Theory and Submanifold Geometry}, Lectures Notes in Math., vol. 1353, Springer-Verlag, 1988.