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

10

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

12

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, schmock@fam.tuwien.ac.at).

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.