PRESENTAZIONE CORSI INTERDOTTORATO

IN MATEMATICA 2015/2016

Politecnico di Milano, Università di Milano Bicocca, Università degli Studi di Milano,

Università di Pavia

15 GENNAIO 2016

ore 14.00, Sala di Rappresentanza,

Dipartimento di Matematica "F. Enriques",

Università degli Studi di Milano, Via Saldini 50, 20133 Milano

Titolo: Analisi di Fourier e Discrepanza geometrica

Docente: G. Travaglini (Dipartimento di Matematica e Applicazioni, Università di Milano-

Bicocca)

Periodo: Aprile-Maggio 2016

Durata: 30 ore

Esame: Sarà chiesto di spiegare e discutere alcuni degli argomenti introdotti nel corso.

Programma: In questo corso sarà presentata la discrepanza geometrica, che misura in vari

modi l’errore commesso quando si fissa una successione di punti e la si utilizza per stimare

volumi di figure geometriche. In parallelo saranno introdotti gli argomenti dell’Analisi di

Fourier necessari.

1. Medie di funzioni aritmetiche e punti interi.

2. Distribuzione uniforme. Criterio di Weyl e completezza del sistema trigonometrico. Numeri

normali e legge di Benford.

3. Polinomi trigonometrici e approssimazione di integrali mediante somme di Riemann.

Disuguaglianze di Erdos-Turan, Koksma e Koksma-Hlawka.

4. Punti interi in corpi convessi e formula di sommazione di Poisson. Variazioni sul problema

del cerchio di Gauss.

5. Punti interi e somme esponenziali. Il metodo di van der Corput.

6. Discrepanza geometrica e decadimento di trasformate di Fourier. La teoria nata dal teorema

di Roth.

7. Discrepanza in molte variabili e funzioni di Bessel. Situazioni intermedie tra le successioni

esplicite e il metodo Monte Carlo.

Riferimento: G. Travaglini, Number Theory, Fourier Analysis and Geometric Discrepancy,

Cambridge University Press, 2014.

Title: Variational methods for semilinear elliptic equations

Lecturers: V. Felli, S. Secchi (Università di Milano-Bicocca)

Period: March-April 2016

Duration: 30 hours

Exam: oral exam.

Description: This course aims to be an introduction to the study of semilinear elliptic problems

through the application of variational methods and critical point theory. Variational methods

find solutions of PDEs by interpreting a solution as a critical point of some associated

functional. The course will present some classical tools from critical point theory, such as

constrained minimization, deformation lemmas, mountain pass and linking theorems. As

applications, some existence and nonexistence results of solutions to semilinear elliptic

problems will be discussed.

A crucial assumption for the use of many classical results of critical point theory is the Palais-

Smale condition which provides the good compactness property to solve nonlinear elliptic

equations by finding critical points of the associated functional. In the case of failure of the

Palais-Smale condition at certain levels the application of variational methods is more

delicate; important examples of failure of the Palais-Smale condition can be found in

situations involving the critical exponent in the Sobolev embedding theorems or in the cases

of the action of a noncompact group. The last part of the course will be devoted to the study

of semilinear elliptic problems with lack of compactness, by discussing the P. L. Lions

Concentration- Compactness principle and the Brezis-Nirenberg problem.

References:

[1] Ambrosetti-Prodi: A primer of Nonlinear Analysis, Cambridge University Press, 1993.

[2] Ambrosetti-Malchiodi: Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge

University Press, 2007.

[3] Kavian, Introduction à la théorie des points critiques et applications aux

problèmes elliptiques, Mathématiques & Applications, Springer, 1993.

[4] Struwe: Variational Methods, Springer, 2008.

[5] Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their

Applications, 24, Birkhäuser, Boston (1996).

Introduzione alle equazioni alle derivate parziali Pierluigi Colli e Mauro Garavello (coordinatori);

Periodo: II semestre. Ore: 30 (da suddividere fra i docenti);

interdottorato: si

(presentazione sintetica)

Docenti: Pierluigi Colli, Rinaldo Colombo, Mauro Garavello, Graziano Guerra, Giuseppe

Savare', Giuseppe Toscani Il corso vorrebbe introdurre alla teoria delle equazioni a derivate parziali, tenendo presente sia

gli aspetti modellistici che quelli teorici.

Si presenteranno le equazioni note della fisica-matematica: in particolare equazioni di

diffusione, stazionarie ed evolutive, leggi di conservazione, equazioni e sistemi di

propagazione e trasporto, funzionali entropia e relative disuguaglianze.

Ci interesseremo alla risoluzione di alcuni problemi iniziali e/o al bordo e alla loro analisi con

i metodi sia dell'analisi funzionale che dell'analisi reale. Bibliografia

Bressan, Lecture Notes on Functional Analysis with applications to Linear Partial

Differential Equations, AMS

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations.

Universitext. Springer, New York, 2011

E. DiBenedetto, Partial Differential Equations, Birkhauser, 2010

L.C. Evans, Partial differential equations. Second edition. Graduate Studies in

Mathematics, 19. AMS

S. Kesavan, Topics in functional analysis and applications, Wiley Eastern Limited, 1989

S. Salsa, Partial Differential Equations in Action – From Modelling to Theory,

Universitext. Springer, New York, 2009

Title: Wavelets and Frames: Theory, Implementation and Applications

Lecturer: J. Stöckler (Fakultät für Mathematik, Technische Universität Dortmund, Germany)

Period: February 9-17, 2016

Duration: 28 hours

Exam: The students should carry out a project on a prescribed topic

Description:

The wavelet transform emerged in the 1980’s as a comprehensive method for the description of

nonstationary or transient features in signals, functions or solutions of differential equations. The

success of wavelets comes from the versatility in describing local and global features at various scales of

resolution, on the one hand, and the simple algorithms for their implementation on the other hand. The

course gives an overview on wavelets and frames from a theoretical and practical point of view and

consists of two parts. The first part describes the concept of multiresolution analysis (MRA) and general methods for the

construction of tight wavelet-frames. The essential ingredients of the constructions are known as

“unitary or oblique extension principles”. We first develop these principles based on matrix linear

algebra and formulas for the decomposition of MRA subspaces in the time-domain as in [1]. This

enables us to construct wavelet- frames for uniform and non-uniform MRA’s alike. We also describe

the Fourier transform techniques in [2] for the description of the same methods and for the case of

uniform (or shift-invariant) MRA’s. Our results are accompanied by examples of univariate splines and

the associated wavelet- frames. The second part deals with issues of implementation of wavelet-frames and presents two applications.

The implementation for uniform MRA’s is described by filter banks [3]. The first application uses

wavelet-frames for local analysis of Hölder regularity of functions [4]. The same wavelet-frames serve

in the second application for tasks of image processing such as image inpainting [3]. References:

[1] C. Chui, J. Stöckler, Recent developments of spline wavelet frames with compact support, in

“Beyond Wavelets”, G. Welland (Ed.), Academic Press, San Diego, 2003, pp. 151-214.

[2] B. Dong, Z. Shen, MRA-based wavelet frames and applications, in “Mathematics in Image

Processing,” H. Zhao (Ed.), IAS/Park City mathematics series, vol. 19, AMS, 2013, pp. 7-158.

[3] G. Strang, T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, 1997.

[4] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.

TITOLO: Harmonic maps between surfaces. DOCENTE: Francesco Bonsante

SUNTO:

The goal of the course is to give a general overview of the theory of harmonic maps between

Riemannian surfaces. Following the approach of Tromba and Wolf, we will derive a real analytic

parameterization of the Teichmüller space. Here a brief summary of the topics which will be treated: 1) Hermitian metrics on a Riemann surface and uniformization.

2) The energy of a map. Variations of a map and tension field. Harmonic maps

3) Classical harmonic maps between planar domains. 4) Existence of harmonic maps in a fixed homotopy class.

5) Convexity of the energy and uniqueness of harmonic maps

6) Bochner formulas and the geometry of harmonic diffeomorphisms.

7) Wolf parameterization of the Teichmüller space.

The last lectures will be devoted to some more recent development. The content of this last part will be

decided considering the interests of the participants.

Aspetti analitici e geometrici delle varietà complesse Costantino Medori, Adriano Tomassini I. Funzioni olomorfe di più variabili complesse. 1.1. La formula di Cauchy. Equazioni di Cauchy-Riemann. 1.2. Gli operatori ∂ e ∂ e il complesso di Dolbeault. 1.3. Singolarità eliminabili. Teorema di Hartogs. 1.4. Nozioni di convessità. II. Varietà complesse e gruppi di automorfismi. 2.1. Atlanti complessi. Varietà quoziente. 2.2. Strutture quasi complesse e il Teorema di Newlander-Nirenberg. 2.3. Limiti di automorfismi. Automorfismi di domini limitati. 2.4. Automorfismi del disco e del polidisco. III. Varietà di Cauchy-Riemann. 3.1. Sottovarietà reali di varietà complesse. Forma di Levi e pseudoconvessità. 3.2. Immergibilità di varietà CR (cenni). 3.3. Tracce di funzioni olomorfe e problemi di estensione (cenni). IV Coomologia di varietà complesse e Kähleriane 4.1. Metriche Hermitiane. Metriche Kähleriane. 4.2. Teoria di Hodge complessa. Il ∂∂-Lemma. 4.3. Formalità di varietà Kähleriane. 4.4. Coomologia di Bott-Chern e di Aeppli e teoria di Hodge. V Fibrati vettoriali Hermitiani 5.1. Fibrati vettoriali complessi, olomorfi ed Hermitiani. 5.2. Connessioni. Curvatura di una connessione. 5.3. La connessione di Chern. Classi caratteristiche. Riferimenti bibliografici [1] Boggess A.: CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1991. xviii+364 pp. [2] Della Sala G., Saracco A., Simioniuc A., Tomassini G.: Lectures on complex analysis and analytic geometry, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 3. Edizioni della Normale, Pisa, 2006. xx+430 pp. [3] Griffiths P., Harris J.: Principles of algebraic geometry, Reprint of the 1978 original. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994. xiv+813 pp.. [4] Gunning R.C., Rossi H.: Analytic functions of several complex variables, Prentice-Hall, 1965. [5] Hormander L.: An introduction to complex analysis in several variables, D. Van Nostrand Co., 1966. [6] Huybrechts D.: Complex geometry An introduction, Universitext. Springer-Verlag, Berlin, 2005. xii+309 pp.

[7] Kobayashi S.: Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15. Kan Memorial Lectures, 5. Princeton University Press, Princeton, NJ; Iwanami Shoten, Tokyo, 1987. xii+305 pp. [8] Kobayashi S., Nomizu N.: Foundations of differential geometry, Vol. II. Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney 1969 xv+470 pp. [9] Morrow J., Kodaira K.: Complex manifolds, Reprint of the 1971 edition with errata. AMS Chelsea Publishing, Providence, RI, 2006. x+194 pp.. [10] Narasimhan R.: Several complex variables, Reprint of the 1971 original. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1995. x+174 pp.

Bayesian inference: foundational, modelling and computational aspects PhD program in Mathematics

Instructors: Federico Bassetti and Antonio Lijoi (University of Pavia)

Description The course is composed of two parts. The first part provides an introduction to some foundational

issues of the Bayesian approach and extensively lingers on a wide variety of parametric models, most

notably those that belong to the exponential family. This is supplemented by a discussion of some

computational methods that are adopted for an approximate evaluation of Bayesian inferences. The

second part introduces some simple models that are very popular in Bayesian nonparametrics and

whose construction relies on the use of random measures. The Dirichlet process will be the starting

point of a broad overview of some remarkable nonparametric priors that are used in problems of

survival analysis, density estimation and prediction with species sampling data. Syllabus Part 1 (15 hours) - F. Bassetti

• Bayes-Laplace paradigm

• Exchangeable sequences. De Finetti's representation theorem and de Finetti's measure.

• Assessment of de Finetti's measure for parametric models. Predictive and posterior distributions

• Conjugate priors and the exponential family.

• Markov Chain Monte Carlo procedures for approximating Bayesian inferences Part 2 (15 hours) - A. Lijoi

• A Bayesian nonparametric approach to inference: the Dirichlet process and its distinctive properties.

• Completely random measures (CRMs) and priors for survival analysis.

• Discrete nonparametric priors: stick-breaking construction and normalized CRMs.

• Predictive distributions and posterior characterizations. Examples: the Pitman-Yor and the

normalized inverse-Gaussian processes.

• Markov Chain Monte Carlo algorithms for Bayesian density estimation. References • Ghosh, J.K., Delampady, M. and Samanta, T. (2006). An introduction to Bayesian

analysis. Springer, New York.

• Ghosh, J.K. and Ramamoorthi, R.V. (2003). Bayesian nonparametrics. Springer, New York.

• Further detailed references will be provided during the lectures Dates Lectures will be delivered in Spring 2016, at the University of Pavia (Departments of Mathematics

and of Economics & Management) Exam Students will deliver a seminar on a research topic to be agreed on with the instructors.

History of Probability in the First Half of XX Century PhD program in Mathematics

Instructor: Eugenio Regazzini (University of Pavia)

Description: The first half of the last century has been a formidable period for the

development of probability and its applications. In point of fact, during its course a number of

open problems found a definitive answer and, at the same time, sound bases towards new important

achievements were established.

The method adopted in the present course in order to illustrate the advance of probability

consists in analyzing the most original and streamlined lines of reasoning to prove a certain number

of theorems generally seen as determining the magnificence of modern probability. Since the aforesaid progress has been made possible even by the axiomatization of probability, the

first part of the course will be devoted to this subject, drawing particular attention to the foundational

work of A. N. Kolmogorov (1933) and B. de Finetti (1931). The second part will deal with distinguished versions of the strong law of large number, starting

from E. Borel (1909) and F.P. Cantelli (1917), to get at the general formulation from the

Soviet School (Kolmogorov, A. Khinchin, etc.). In the case of stochastic independence, their

extensions to exchangeable random elements (de Finetti), and more general stationary sequences

(ergodic theorem of von Neumann and Birkhoff) will be displayed and discussed. In this very same

part, the role played by stable and infinitely divisible laws to solve the central limit problem (P.

Levy, Khinchin, W. Doeblin, B. Gnedenko) will be emphasized.

The last part will be devoted to the birth of the theory of stochastic processes, starting from the works

of Kolmogorov (1929) on Markov processes, de Finetti (1929 -1933) and Levy (1934-1935) on

random functions with independent increments, Levy (1935-1937) and J. Doob (1940) on

martingales. This last part will end with an outline to the methods devised in the fifties to study

convergence in law of stochastic processes.

References: Reading of original writings by some of the aforesaid Authors, suggested by the

instructor, with the aid of some good treatise on modern probability.

Duration:30 hours

Dates: Lectures will be delivered in Spring 2016, at the University of Pavia (Departments of

Mathematics)

Bayesian Analysis of Stochastic Process Models

LECTURER: Dr. Fabrizio Ruggeri, CNR IMATI, Milano, Italy

Email Address: fabrizio@mi.imati.cnr.it

CLASS SCHEDULE: Two 90 minutes classes per week (or 2 hours classes per week,

reducing the number of classes to 8).

PREREQUISITES: The course is designed for Ph.D. students in Decision Sciences, Finance,

Statistics and Engineering Management. An introductory course on stochastic processes

(especially Markov chains and Poisson processes) is not re- quired but it could

be helpful. Earlier exposure to Bayesian methods would be helpful as well.

COURSE OUTLINE:

• Markov chains (approx. 3 classes)

– Inference and prediction for discrete time Markov chains

– Inference and prediction for continuous time Markov chains

– Examples (predator-prey, hardware and software reliability)

• Poisson processes (approx. 3 classes)

– Homogeneous Poisson processes

– Nonhomogeneous Poisson processes

– Example (earthquake data)

• Reliability (approx. 3 classes)

– Basic notions

– Models for repairable and non repairable systems

– Examples (gas escapes, train doors’ failures)

• Queues (approx. 1 class)

– Inference for M/M/1 queues

– Other queues

– Examples (Bed occupancy in hospital)

TEXTBOOK:

Rios Insua, D., Ruggeri, F., Wiper, M.P. (2012). Bayesian Analysis of Stochastic

Process Models, Wiley, Chichester, UK.

Finite simple groups (the course will be given in English)

28 hours, summer term 2016

Contact person: Thomas Weigel, University of Milano-Bicocca

In 1980 D. Gorenstein announced the “Classification of Finite Simple Groups”, the largest collaborative piece of pure mathematics ever attempted. This classification theorem - which proof occupies more than 10000 pages of textbooks and articles - states that a finite simple group is either cyclic of prime order, or isomorphic to (a) an alternating group, (b) a finite group of Lie type,

(c) or one of 26 sporadic simple groups.

The course intends to give a brief discussion of the examples of finite simple groups. The main

focus will be laid on finite groups of Lie type which arise as the fixed point group of a

Frobenius map defined on a simple algebraic group. In the final lectures we will discuss some

sporadic simple groups which come along with a “pathological” combinatorial structure they

act on. E.g., the Mathieu groups act naturally on the Witt designs or Golay codes, while the

Conway groups act naturally on the Leech lattice, the “pathological” even unimodular lattice

of degree 24.

The course will take place at the “Dipartimento di Matematica e Applicazioni” of the University of Milano-Bicocca. In order to avoid unnecessary commuting the intention is to proceed on 7 days of 2+2 hours of lectures (late morning+early afternoon). PhD students which are interested to participate are invited to write an email to thomas.weigel@unimib.it

before February 15th, 2016. The final schedule of the lectures will be fixed in accordance with

the participants.

Shape Optimization and Optimal Control Problems meet Polygonal Methods for

PDEs Lecturers: P. F. Antonietti, M. Verani (MOX- Dipartimento di Matematica, Politecnico di Milano)

Period: September-October 2016

Duration: 20 hours

Exam: The students should carry out a project on a prescribed topic and give a lecture on it. Description: In recent years, there has been extensive research on the numerical approximation of

shape optimization and optimal control problems governed by partial differential equations.

As these problems are ubiquitous in practical and industrial applications, the need of efficient and

accurate numerical methods dealing with complicated geometries is of paramount importance.

The aim of this course is twofold. On one hand, we provide basic analytical and numerical

tools to deal with shape optimization and optimal control problems governed by partial

differential problems.

On the other hand, we introduce and discuss some very recent polygonal methods to efficiently solve

partial differential equations on complex geometries by employi ng very general meshes made of

polygonal/polyedral elements

In particular, in the first part of the course we will introduce and discuss the Lagrange multiplier

theory (see, e.g, [1]) which provides a powerful framework to analyse general constrained

optimization problems (including shape optimization problems) and to develop efficient and powerful

iterative methods (e.g., Newton, SQP and primal-dual active set methods).

In the second part of the course, we will introduce and discuss two specific polygonal methods,

namely the Virtual Element Method (see [2]) and the polygonal Discontinuous Galerkin method (see

[3]). In particular, we will focus on the discretization of some paradigmatic problems (mainly of

elliptic type) and on the development of efficient solution techniques on (aggloberated) polytopal

elements.

A brief overview on other polygonal discretizations techniques as hybridizable discontinuous Galerkin

[4] and hybrid high-order [5] methods will be also presented. References:

[1] K. Ito, K. Kunisch, Lagrange multiplier approach to variational problems and

applications, SIAM, 2008

[2] L. Beirao Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, D. Marini, A. Russo, Basic

Principles of Virtual Element Methods, Math. Models Methods Appl. Sci. 23, 199, 2013.

[3] A. Cangiani, E. H. Georgoulis, P. Houston. hp-Version discontinuous Galerkin methods on

polygonal and polyhedral meshes. Math. Models Methods Appl. Sci., 24, 2009–2041, 2014.

[4] B. Cockburn, J. Gopalakrishnan, R. Lazarov, Unified hybridization of discontinuous Galerkin,

mixed, and conforming Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal.

47, 1319–1365, 2009

[5] D. A. Di Pietro, A. Ern, S. Lemaire, An arbitrary-order and compact-stencil discretization of di usion on general meshes based on local reconstruction operators, Comput. Methods Appl. Math., 14, 461–472, 2014.

Mathematical Models for Suspension Bridges Lecturer: prof. Filippo Gazzola (Politecnico di Milano).

Period: Mid March-End April

Abstract:

Most people have seen the videos of the spectacular collapse of the Tacoma Narrows Bridge

(TNB). The torsional oscillations were considered the main cause of the collapse. But the

appearance of torsional oscillations is not an isolated event, they have been observed in several

other cases where suspension bridges manifested aerodynamic instability and uncontrolled

oscillations leading to collapses. Most attempts of explanations of the TNB collapse are based on

aeroelastic effects such as the frequency of the vortex shedding, parametric resonance, and flutter

theory. We will survey the existing literature. Then we will focus our attention on possible

alternative responses, based on the nonlinear behavior of structures. We will introduce and discuss

several possible models aiming to describe the instability in bridges. The Euler-Lagrange equations

are derived from variational principles and tools from the theory of both ODE's and PDE's will be

discussed. In particular, a survey of the Floquet theory will be given.

Onde lineari e non lineari

Docenti: Davide Ambrosi e Pasquale Ciarletta, (MOX, Dipartimento di Matematica del

Politecnico di Milano).

Periodo: settembre-ottobre 2016

Ore di lezione: 30

Finalità: Fornire agli studenti di dottorato che seguiranno il corso i fondamenti fisico- matematici

per la descrizione dei fenomeni ondosi di propagazione, con un’enfasi sul caso non-lineare sul caso

dispersivo. Le applicazioni riguarderanno la fluidodinamica a superficie libera, le onde lineari e

nonlineari nei solidi, la gasdinamica.

Programma:

• Elementi di teoria lineare: propagazione e dispersione, trasformata di Fourier, velocità di fase e

di gruppo.

• Elastodinamica lineare. Onde volumetriche. Onde omogenee in mezzi isotropi e anisotropi.

Fronti d'onda. Onde di superficie. Onde di Rayleigh, onde di Love. Onde all'interfaccia tra

materiali, onde di Stoneley.

• Elastodinamica nonlineare. Statica incrementale e dinamica dei materiali elastici nonlineari.

Propagazione ondosa nonlineare in un semispazio. Onde di interfaccia in materiali pre-stressati.

• Idrodinamica a superficie libera: tipiche scale spaziali e temporali e regimi relativi, numeri

caratteristici. Onde lineari: relazione di dispersione e limiti in acque basse e acque profonde.

Tsunami.

• Regime non-lineare non-dispersivo: equazioni per acque basse. Moto fluviale:

invarianti di Riemann, velocità di un'onda di piena.

• Onde non lineari dispersive: equazione di Korteweg de Vries, equazioni di

Boussinesq, solitoni e onde cnoidali.

Bibliografia:

Whitham, G.B., Linear and Non Linear Waves, John Wiley and Sons, (1974). Mei, C.C., The

Applied Dynamics of Ocean Surface Waves, World Scientific, (1989).

Destrade, M. and Saccomandi, G. (Eds.), Waves in Nonlinear Pre-Stressed

Materials, CISM International Centre for Mechanical Sciences (2007).

Titolo: Moduli of $p$-divisible groups

Fabrizio Andreatta, Unimi

Descrizione: Lo scopo del corso è capire l'articolo "Moduli of $p$-divisible groups" di P. Scholze e

J. Weinstein. Questo offre una delle prime applicazioni della teoria degli spazi perfettoidi introdotta

da Scolze nel 2011.

L'unico prerequisito è la conoscenza della geometria adica di Huber.

La cadenza degli incontri è bisettimanale (venerdì), a partire da venerdì 22 gennaio fino a maggio.

Mathematical aspects of the Bose-Einstein condensation Docente: Michele Correggi, Università Roma Tre

Presenta S. Ugolini (Unimi)

Main topics:

- Ideal Bose Gas: Bose-Einstein Condensation (BEC)

- Interacting Bose Gas: the Ground State Energy of a Dilute Gas

- Interacting Bose Gas: BEC in a Box for a Dilute Gas

- Inhomogeneous Bose Gas: BEC and Gross-Pitaevskii (GP) Theory

- Rotating BE Condensates: Superfluidity

- First Critical Speed: Renormalized Energy and Distribution of Vortices

- Second Critical Speed: Uniform Vorticity and Macroscopic Hole

- Third Critical Speed: Transition to Giant Vortex State

Vanishing theorems, generic vanishing theorems and applications Joan Carlos Naranjo, University of Barcelona

University of Milano, 25th January 2016 - 3th February 2016

The lessons and the seminars will be held at Aula Dottorato (I floor) from 14.00

1. Classical vanishing theorems: Kodaira, Nakano, Kawamata-Viehweg

2. Generic vanishing theorems (with preliminars on Abelian Varieties)

3. Derived categories, Fourier-Mukai transform and derived vanishing theorems (PhD seminars by

Martí Lahoz)

4. Applications to AV and irregular varities: racionality of the singularities of the theta divisor

representing a principal polarization (Ein-Lazarsfeld), derived approach to Torelli Theorems,

bicanonical map for varieties of maximal Albanese dimension, characterization of theta divisors,

higher Castelnuovo-Severi inequality,...

6 sessions of 3 hours lessons (Naranjo) + 2 sections of 2 hours of Phd seminars (Lahoz)

Aproximate schedule:

Week 1:

Part 1 (2 sessions: 25-1 and 26-1 ), Part 2 (2 sessions: 27-1 and 28-1), Part 3 (1 session: 29-1)

Week 2:

Part 3 (1 session: 1-2) and Part 4 (2 sessions: 2-2 and 3-2)

References:

Lazarsfeld's book on positivity (first volume)

Birkenhake-Lange: Complex Abelian Varieties

Griffiths-Harris: Principles of Algebraic Geometry

[GL87] M. Green and R. Lazarsfeld, Deformation theory, generic vanishing theorems, and some

conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), no. 2, 389–407.

[GL91] , Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc.

4 (1991), no. 1, 87–103.

[Hac04] C. D. Hacon, A derived category approach to generic vanishing, J. Reine Angew. Math.

575 (2004), 173–187.

[PS13] M. Popa and C. Schnell, Generic vanishing theory via mixed Hodge modules, Forum Math.

Sigma 1 (2013), e1, 60. The following articles contain various examples and applications of the

theory. We will be discussing most of them over the course of the semester.

[EL97] L. Ein and R. Lazarsfeld, Singularities of theta divisors and the birational geometry of

irregular varieties, J. Amer. Math. Soc. 10 (1997), no. 1, 243258. [CH01] J. A. Chen and C. D.

Hacon, Characterization of abelian varieties, Invent. Math. 143 (2001), no. 2, 435–447.

[Par12] G. Pareschi, Basic results on irregular varieties via Fourier-Mukai methods, Current

developments in algebraic geometry, Math. Sci. Res. Inst. Publ., vol. 59, Cambridge Univ. Press,

Cambridge, 2012, pp. 379–403.

Mukai paper on Fourier transform

Beilinson Polishchuck: Torelli

Polishchuck book: Abelian varieties

Pareschi Popa: Higher Castelnuovo

Barja Lahoz Naranjo Pareschi: Bicanonical map of irregular varieties