MANIFESTO DEGLI STUDI A.A. 2017/2018 … di Matematica Corso di Dottorato in Matematica Approvato dal Collegio dei Docenti nella seduta del 24 ottobre 2017 Manifesto degli studi Il

Dipartimento di Matematica Corso di Dottorato in Matematica

Approvato dal Collegio dei Docenti nella seduta del 24 ottobre 2017

MANIFESTO DEGLI STUDI

A.A. 2017/2018

CORSO DI DOTTORATO IN MATEMATICA



Manifesto degli studi Il Corso di Dottorato in Matematica ha come finalità l’acquisizione di una qualificata preparazione per una formazione specialistica di giovani ricercatori, avviandoli a svolgere una futura attività presso Università, Enti di ricerca pubblici e privati, ed industrie. Il Corso si configura come naturale completamento della formazione scientifica conseguita con le lauree di primo e secondo livello, con le quali è coordinata. Il Corso di Dottorato di ricerca in Matematica è istituito ai sensi del D.M. n. 45/2013. Il Corso di Dottorato viene proposto dal Dipartimento di Matematica. Alcune borse di studio sono offerte da enti esterni, sulla base di convenzioni approvate. Al corso di Dottorato collaborano, mettendo a disposizione degli allievi le proprie competenze e le proprie strutture, i seguenti enti di ricerca nazionali e locali:

Università di Verona

FBK - Fondazione Bruno Kessler, Trento

COSBI

FEM

INdAM–COFUND-DP-2015 FAIRMAT

Indirizzi di ricerca Il Corso di Dottorato di ricerca in Matematica, per l’anno accademico 2017/2018 è articolato nei seguenti indirizzi di ricerca:

a. Indirizzo generale L’indirizzo riguarda una o più delle seguenti aree scientifiche caratterizzanti: - Calcolo delle variazioni: analisi in spazi metrici, teoria geometrica della misura, convergenze

variazionali (Gamma-convergenza), trasporto ottimo - Analisi Geometrica, Geometria Riemanniana, Flussi Geometrici. - Equazioni alle derivate parziali (EDP) non lineari: problemi a frontiera libera, modelli di isteresi,

comportamento asintotico ed omogeneizzazione di EDP, metodi variazionali e topologici, equazioni di Ginzburg-Landau e Schrödinger non lineari

- Geometria analitica e geometria algebrica: Curve algebriche e spazi di moduli. Superfici di tipo generale e spazi dei moduli Varietà di dimensione alta: teoria di Mori, varietà di Fano. Geometria algebrica reale, analisi complessa e ipercomplessa.

- Fisica matematica: aspetti fondazionali, analitici e geometrici delle teorie quantistiche e relativistiche. Tecniche geometriche in meccanica analitica; applicazioni alla teoria del controllo.

- Sistemi Dinamici e Teoria del Controllo: esistenza, molteplicità, stabilità di soluzioni periodiche di equazioni differenziali, sistemi lagrangiani e hamiltoniani; giochi differenziali e problemi di controllo ottimo, soluzioni di viscosità di equazioni di Hamilton-Jacobi; ottimizzazione di sistemi ibridi.



- Processi stocastici: equazioni differenziali stocastiche alle derivate parziali, Integrazione funzionale ed applicazioni.

- Statistica: Statistica Robusta, Misure statistiche di profondità dei dati Statistica Computazionale. - Logica matematica ed informatica teorica: applicazione di tecniche non standard (alla A. Robinson)

in analisi funzionale, sviluppo di logiche non classiche, teoria dei linguaggi di programmazione, sistemi di tipi, analisi statica, aspetti generali e filosofici, fondamenti, matematica costruttiva e programma di Hilbert.

- Teoria di Lie, teoria dei gruppi, rappresentazioni di algebre, algebra omologica, algebra computazionale.

b. Indirizzo "Modellizzazione Matematica e Calcolo Scientifico (MOMACS)" L’indirizzo è trasversale alle seguenti aree scientifiche caratterizzanti: - Processi Stocastici: equazioni integro-differenziali e EDP stocastiche per modellizzazione di fenomeni

fisici, biologici e finanziari. - Metodi Numerici per Equazioni a Derivate Parziali: modellizzazione di fenomeni elettromagnetici, e

di problemi di fluidodinamica (classica e quantistica), metodi di approssimazione basati su elementi finiti, elementi di frontiera, differenze o volumi finiti.

- Approssimazione/interpolazione numerica di funzioni multivariate: metodi efficienti ed applicazioni. - Matematica discreta: modellizzazione in ricerca operativa, teoria dei grafi, ottimizzazione

combinatoria e applicazioni in biologia computazionale. - Controllo ottimo, ottimizzazione: applicazioni alla scienza delle decisioni, trattamento delle

immagini, patrimonio culturale. - Modelli matematici e computazionali in medicina: simulazione dei meccanismi fisiologici e

patologici dell'organismo umano, con particolare attenzione ai sistemi circolatorio e linfatico e alle loro interazioni con il sistema nervoso centrale.

c. Indirizzo "Metodi algebrici e geometrici in crittografia e teoria dei codici" L’indirizzo si propone di avviare la ricerca nel campo di una varietà di metodi matematici impiegati in crittografia, e nella teoria dei codici a correzione d'errore, in particolare: Metodi algebrici: algebra lineare, algebra commutativa, algebra computazionale, basi di Groebner, campi numerici, teoria dei gruppi; Metodi geometrici: geometria algebrica, curve ellittiche. I problemi di ricerca proposti spaziano da classificazioni puramente teoriche a problemi vicini alla ricerca industriale, questi ultimi complementati da stage presso aziende leader del settore.

d. Indirizzo “Biologia matematica e computazionale” L’indirizzo si propone di avviare alla ricerca nel vasto campo dell’applicazione di modelli matematici e computazionali nella biologia.



I metodi che verranno approfonditi possono andare dalla teoria delle equazioni differenziali ordinarie, parziali e con ritardo o dei processi stocastici, alla statistica computazionale, alla bioinformatica, ai metodi formali per la descrizione di sistemi complessi. Anche le aree della biologia coinvolte sono molteplici, dall’epidemiologia ed ecologia, alle reti molecolari con applicazioni alla farmacologia sistemica e alla nutrigenomica. Il lavoro di tesi consisterà nella modellizzazione di un problema biologico, con l’ausilio di metodi matematici, statistici, computazionali, informatici. Esso potrà coinvolgere altri centri di ricerca in provincia di Trento (COSBI, FBK, FEM) o altrove. Corsi attivati per l’anno accademico 2017/2018: I corsi attivati sono contenuti nell’allegato documento “Manifesto dei corsi”. In accordo con l’art. 6 del regolamento interno del dottorato, uno studente del I anno di corso dovrà scegliere tre corsi del manifesto:

- Algebra e logica matematica

- Analisi Matematica

- Analisi Numerica

- Geometria

- Fisica Matematica

- Probabilità e Statistica/Metodi matematici dell’economia

- Ricerca Operativa/Informatica



Allegato n.

Courses of the PhD School in Mathematics, a.y. 2017-18

Advanced Coding Theory and Cryptography, Modulo 2 (borrowed from the master degree in mathematics at Trento)

- Lecturer: Edoardo Ballico (University of Trento) - Period and Venue: February-May 2018 in Trento - Contents: Abelian group Cryptography, affine-variety codes, n-th root codes, Order-Domain codes. Abelian groups Cryptography. Elliptic curves. Elliptic Curve Cryptography and Hyperelliptic Curve Cryptography. Quantum Elliptic Curve.

Cryptography (if required by at least one student). Curves over finite fields; Hasse-Weil. Goppa codes and their generalizations. Quantum Cryptography (if people are interested).

Algebraic Geometry II: Rational Homogeneous Varieties (borrowed from the master degree in mathematics at Trento)

- Lecturer: Eduardo Luis Sola Conde (University of Trento) - Period and Venue: February-May 2018 in Trento - Contents: Within the framework of Complex Algebraic Geometry, the class of rational homogeneous

spaces contains some of the most important varieties in the general theory, including projective spaces, Grassmannians, and quadrics. Their geometric properties are determined by the existence of certain semisimple linear algebraic groups acting transitively on them, and this fact turns the Representation Theory of these type of groups into a fundamental tool to study –and classify– the class of rational homogeneous spaces. In this course we will begin by introduce the basic concepts of algebraic geometry (affine and projective varieties, morphisms, etc), presenting projective spaces, quadric and Grassmannians under a clasic point of view. Then we will focus on the study and classification of semisimple groups, and their parabolic subgroups, and we will finish by studying geometric properties of the rational homogeneous spaces, particularly line bundles and their cohomology.

- Tentative description of the contents of the course: (1) Affine and projective varieties. (2) Projective spaces, smooth quadrics and Grassmannians. (3) Linear algebraic groups. Lie algebras. Examples. (4) Abelian and solvable subgroups. Borel subgroups. (5) Reductive and semisimple groups. (6) Root systems and Weyl groups. Classification of semisimple groups.



(7) Parabolic subgroups and their Levi decompositions. Homogeneous spaces. (8) Line bundles. Homogeneous line bundles. Weights. (9) The Borel–Weyl–Bott Theorem.

- References: - W. Fulton, J. Harris: Representation Theory, A First Course, Graduate Texts in Mathematics 129,

Springer-Verlag, Berlin, New York (1991). - G. Malle, D. Testerman: Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge studies in

mathematics, 133, Cambridge (2011). - J.E. Humphreys: Linear algebraic groups. Graduate Texts in Mathematics 21. Springer-Verlag, New

York-Heidelberg (1975). - J.E. Humphreys: Introduction to Lie algebras and representation theory, Graduate Texts in

Mathematics, 9, Springer-Verlag, New York (1978). - J. Harris: Algebraic geometry: a first course, Springer, New York, 1992. - R. Hartshorne: Algebraic geometry, Graduate Text in Mathematics 77, Springer, New York (1977). - A. Kirillov Jr: An Introduction to Lie Groups and Lie Algebras, Cambridge studies in mathematics,

113, Cambridge (2008). - T.A. Springer: Linear algebraic groups. Progress in Mathematics 9. Birkh ̈auser, Boston, Mass. (1981). - E. Arrondo: Subvarieties of Grassmannians, Lecture Note Series Dipartimento di Matematica Univ.

Trento 10, Trento (1996).

Biological Networks MOD 2 (borrowed from the master degree in quantitative and computational biology at Trento)

- Lecturer: Ozan Kahramanogullari (University of Trento) - Period and Venue: February-May 2018 in Trento - Contents: The course will cover various methods for modeling, simulating and analyzing biological

systems with an emphasis on dynamical properties. Published results with biological significance will be discussed as illustrative case studies for the methods and techniques that will be introduced throughout the course.

Some of the topics that will be covered are: - Systems point of view to biology. - Transcription factors and gene regulation. - Separation of time scales in formal models. - Transcription network motifs. - Dynamic properties of simple regulation and auto-regulation. - Feed-forward loop network motif. - Single-input modules, densely overlapping regulons, and other graph patterns and their roles in

sensory and developmental networks. - Signal transduction in biological systems. - Signal transduction and robustness in bacterial chemotaxis. - Diffusion and morphogen formation. - Biochemical proof reading. - Chemical reaction networks and stochasticity. - Modeling complex biological structures such as branching polymers and other forms of self-

assembly.



- Flux analysis in chemical reaction networks. - DNA computing.

Bayesian Statistics (borrowed from the master degree in mathematics at Trento)

- Lecturers: Pier Luigi Novi Inverardi (University of Trento) and Claudio Agostinelli (University of Trento) - Period and Venue: February-May 2018 in Trento - Contents: Likelihoods, Priors and Bayesian rule; Frequentist and Bayesian Approaches: Differences;

How to specify the priors: conjugate, non informative, improper, Jeffreys priors. Regular Exponential Family: univariate and multivariate. Point estimation, Credibility intervals. Examples: binomial-beta model, normal-inversegamma, etc.; Exchangeability. Bayes factor and model comparison. Models Averaging. Predictive Distributions. Examples: regression model (also with G-priors). Hierarchical models. Laplace (Normal) Approximations. Pseudo Random number generator for a given distribution. Acceptance/Rejection and Sampling Resampling Techniques. Markov Chain Monte Carlo. Metropolis-Hasting. Gibbs sampler. Several Examples.

- References: - A. Gelman, J.B. Carlin, H.S. Stern and D.B. Rubin (2004) Bayesian Data Analysis, Chapman and

Hall/CRC. - J. Albert (2009) Bayesian Computation with R, Springer - C.P. Robert and George Casella (2010) Introducing Monte Carlo Methods with R, Springer - D. Gamerman and H.F. Lopes (2006) Markov Chain Monte Carlo. Stochastic Simulation for Bayesian

Inference. Chapman and Hall/CRC

Discrete Fourier Analysis (borrowed from the master degree in mathematics at Trento)

- Lecturers: Giancarlo Rinaldo (University of Trento) - Period and Venue: February-May 2018 in Trento - Contents: The Discrete Fourier Analysis course is an introductory one that starting from subjects that

are studied in Analysis and therefore under Continuum hypothesis soon is specialized on concrete, discrete and often finite, cases. The main arguments studied are : Representation Theory of Finite Abelian Groups, the group of characters to them related and Finite Fourier Transform. Such algebraic tools after being described, are applied in an effective way to different fields : Coding Theory, Algebraic Combinatorics and Statistics. To better appreciate the course, when possible, the student can verify the obtained results and find conjectures by symbolic/numerical computation Softwares like Magma and MatLab.

Differential and Integro-Differential Equations for Evolutionary Dynamics of Structured Populations: Qualitative Analysis and Oncology Applications

- Lecturers: Tommaso Lorenzi (University of St. Andrews, UK) and Andrea Pugliese (University of Trento)

- Period and Venue: 1) December 11th -15th 2017 8 hours in Verona (Lorenzi) 2) February 12th-16th 2018 8 hours in Trento (Lorenzi)



3) April, 8 hours to be decided (Lorenzi and Pugliese) 4) o be decided, possible lessons on required backgrounds (Pugliese)

- Sketched program. 1) Mathematical Models for the study of the dynamics of tumor cells populations. 2) Qualitative analysis of local and non-local parabolic equations arising in the study of evolutionary

dynamics for structured populations. 3) Teachers-students interactions and developing of assigned projects)

HA Homological Algebra (borrowed from the master degree in mathematics at Trento

- Lecturers: Edoardo Ballico (University of Trento) - Period and Venue: February-May 2018 in Trento - Contents: A short introduction to Category Theory. Additive categories and abelian categories.

Complexes and homology. Homotopy. Derived functors. Examples: The Tor functors and the Ext-functors. Examples and applications in Geometry and Commutative Algebra (Koszul complexes, Hilbert’s Syzygies Theorem, minimal free resolutions).

Mathematical Programming

- Lecturers: Romeo Rizzi (University of Verona) - Period and Venue: January-February 2018 in Trento

- Contents: The course offers an introduction to Linear Programming (LP) andCombinatorial Optimization (CO) also exploring some of the links between the two. The approach adopted is algorithmic The two parts comprising the course (LP and CO) will not be treated strictly one after the other but rather in parallel where most convenient.

• Introduction to Linear Programming (LP)

1 what is an LP problem

2 modeling your problem as a linear program

3 the simplex method (description and analysis)

4 duality theory

5 complementary slackness

6 economic interpretation

7 sensitivity analysis

8 geometric interpretation

• Introduction to graphs and Combinatorial Optimization (CO)

1 graphs and digraphs as models.

2 a few good characterizations (bipartite graphs, eulerian graphs, Planar Graphs).

3 shortest paths.

4 minimum spanning trees.

5 max flows and min cuts.

6 bipartite matching.



- Reference Material: - Linear Programming: Foundations and Extensions. Robert J. Vanderbei, Kluwer Academic Publishers

(2001) www: http://www.princeton.edu/∼rvdb/LPbook/ - notes and manuals borrowed from the web or elaborated by the teacher www: profs.sci.univr.it/~

rrizzi/classes/MathProg - wwwpage of the course profs.sci.univr.it/~ rrizzi/classes/MathProg

Mathematical Physics: Quantum and Relativistic Theories (borrowed from the master degree in mathematics at Trento)

- Lecturers: Valter Moretti (University of Trento) - Period and Venue: February-May 2018 in Trento

- Contents: Elements of Hilbert space and operator theory with applications to quantum relativistic theories. Tensor caluculs, riemannian and pseudo riemannian differential geometry, applications to continuous mechanics and relativistic theories.

Introduction to Quantum Information (borrowed from the Doctoral School of Information andCommunication Technology

- Lecturers: Davide Pastorello (University of Trento) - Period and Venue: February 2018 (twenty hours) - Contents: The first part of the course will be an introduction about the structure of general quantum

theories and the required mathematical formalism. Then foundations of quantum mechanics will be reconsidered within an information theoretical context discussing concepts like entanglement, quantum channels, no-cloning, quantum circuits in order to describe some communication schemes like quantum teleportation and superdense coding, quantum algorithms and quantum cryptographic protocols. For example Shor and Grover algorithms will be described and entanglement-based quantum key distribution will be discussed.

Model Theory (borrowed from the master degree in mathematics at Trento)

- Lecturers: Stefano Baratella (University of Trento) - Period and Venue: February-May 2018 in Trento - Contents: The main course topics are the following. Basic notions from model theory. Categoricity.

Ultraproducts. Los' theorem and the Löwenheim-Skolem theorems. Compactness. Ultraproducts in algebra and in analysis. The Ax-Grothendieck theorem. Quantifier elimination. Quantifier elimination for algebraically closed fields and for real closed fields. Model completeness. Hilbert's Nullstellensatz. Hilbert's 17th problem. o-minimality. o-minimality in real analysis. Types. Saturated models. Elements of stability theory.

Numerical methods for PDE: an introduction to finite elements in electromagnetism (borrowed from the master degree in mathematics at Trento)



- Lecturer: Ana Maria Alonso Rodriguez (University of Trento) and Francesca Rapetti (University of Nice, France)

- Period and venue: December 2017-April 2018 in Trento - Contents:

12 December 2017 (two hours) and 14 December 2017 (two hours) - Mathematical models in electromagnetism. - Topics in interpolation - Whitney differential forms

From 19 February 2018 to 2 March 2018 (four hours a week) - Elliptic partial differential equations: variational form of boundary value problems. Lax-Milgram

lemma and its consequences. Galerkin methods. - The finite element method. Lagrange finite elements: the interpolation operator and the interpolation

error.

6 March 2018 (two hours) and 8 March 2018 (two hours) - Other finite element spaces in computational electromagnetism (Raviart-Thomas and N\'ed\'elec

finite elements). - An overview of applications and some open problems in computational elctromagnetism.

From 19 March 2018 to 6 April 2018 (four hours a week) - Convergence of the finite element methods. - Parabolic problems: semidiscrete approximation and time advancing by finite differences.

The aim of this course is to give a brief description of the finite element methods, in particular those used in computational electromagnetism, and to present some applications and open problems. These could be the argument of a PhD thesis within a double degree agreement Trento - Nice.

Partial Differential Equations (borrowed from the master degree in mathematics at Trento)

- Lecturer: Augusto Visintin (University of Trento) - Period and venue: February-May 2018 in Trento - Contents:

1. Basic Linear Second-Order PDEs Review of ordinary differential equations (ODEs). Existence and uniqueness of the solution of initial-value problems. Linear systems of ODEs. Fundamental solution and matrix function. Formula of variation of parameters. Gronwall's lemma. Laplace/Poisson equation: boundary conditions. Green identities. Variational formulation of the Laplace/Poisson equation. Fundamental lemma of the calculus of variations. Maximum principle. Heat equation: boundary and initial conditions. Backward heat equation. Energy inequality. Duhamel principle for evolutionary PDEs. Variational formulation. Energy inequality. Maximum principle. Wave equation: boundary and initial conditions. Solution by separation of variables. D'Alembert solution of the wave equation. Domain of dependence and domain of influence. Comparison between the qualitative properties of the heat and wave equations.

2. Characteristics



Multi-indices. Principal part and symbol of a linear differential operator. Classiffication of nonlinear PDEs: quasilinear, semilinear, fully-nonlinear equations. Classiffication of linear second-order PDEs: elliptic, parabolic, hyperbolic. Characteristics of linear and quasilinear equations. Statement of the Cauchy-Kovalevskaya and Holmgren theorems (without proofs).

3. Sobolev Spaces Spaces of Holder class. Euclidean domains of Holder class. Cone property. Sobolev spaces of positive order. Characterization of the dual. Extension operators. Theorem of Calderon-Stein. The method of extension by reflection. Density results: interior approximation (Mayer-Serrin theorem) and exterior approximation. Sobolev inequality and imbedding between Sobolev spaces. Morrey theorem. Sobolev and Morrey indices. Rellich-Kondrachov compactness theorem. Lp-spaces on manifolds. Traces.

4. Weak Formulation of Second-Order PDEs Elliptic operators. Classical, strong and weak formulation. Operators in divergence or nondivergence form. Conormal derivative. Strong and weak formulation of second-order elliptic equations in divergence form, coupled with either Dirichlet or Neumann boundary-conditions. Theorem of Lax-Milgram. Application to the analysis of PDEs in divergence form.

- Reference textbook of the course: M. Renardy, R. Rogers: An introduction to partial di_erential equations. Springer-Verlag, New York, 2004. Teacher's lecture notes (partly already available on the web).

Regularity Theory for Elliptic Partial Differential Equations

- Lecturer: Annalisa Massaccesi (University of Verona) - Period and venue: February-March 2018 in Verona - Contents: In this course we will address the problem of well-posedness and regularity for elliptic

partial differential equations. Analysing the classical tools and literature on the subject, we will address the proof of the Hilbert's XIX problem given by De Giorgi.

- Prerequisites: basic knowledge of functional analysis and measure theory (Sobolev spaces and some harmonic analysis will be recalled anyway). Bibliography: I will mainly follow Ambrosio's lecture notes on PDE (draft currently available in CVGMT), first 3 chapters

Stochastic Analysis: From Population Dynamics to Finance

- Lecturers: Luca Di Persio (Verona), Simone Scotti (University Paris Diderot, France) and Zenghu Li (Beijing Normal University, China)

- Period and venue: March-May 2018 in Verona - Contents: The course will be mainly focused on a s,elf-contained presentation of the modern theory

of stochastic processes and related applications, spanning from Biology, to Demography, from Economy to Mathematical Finance, etc. In particular, we will consider stochastic processes in continuous time, starting from the definition of infinitely divisible ones, e.g., the Brownian motion and the Lévy process. Concerning financial applications, we shall present those related to



derivatives pricing methods, taking into account the statistical properties behind them. Therefore, filtering and calibration techniques will be taken into account to enlighten their statistical peculiarities as well as characterizing their operative limits. Within the same framework, issues associated to financial Enterprises dynamics will be also considered, while, both biological and demographic applications will be mainly studied exploiting infinitely divisible processes with respect to the initial data. It is worth to mention that such models, known as branching processes, constitute the milestone for the modern description of population dynamics. Along this line of study, it will be defined the integral representation by Dawson-Li. Ample room will be dedicated to related practical aspects, as well as to the applications, in dependence of the characterizing parameters as, e.g., extinction rate, Eva effect, etc. At the end of the course, a link between ecology and financial models, will be provided aiming at building a rigorous bridge between exact sciences and social sciences. The latter goal will serve as a proof of the, counterintuitive, small distance among worlds that too often over years, have been thought to be antipodal.

Stochastic Differential Equations (borrowed from the master degree in mathematics at Trento)

- Lecturer: Stefano Bonaccorsi (University of Trento) - Period and venue: February-May 2018 in Trento - Contents: Some complements about stochastic processes (martingale, Markov properties, stopping

times). Gaussian measures in finite and infinite dimensional spaces. Construction of Bronwian motion, Wiener and Ito type integrals. The Ito formula and the construction of Ito processes. Stochastic differential equations: existence and uniqueness under different sets of assumptions. Applications to diffusion problems: Dirichlet problem for general second order operators,; the heat equation.

Type Theory (borrowed from the master degree in mathematics at Trento)

- Lecturer: Roberto Zunino (University of Trento) - Period and venue: February-May 2018 in Trento - Contents:

- The Untyped Lambda Calculus Syntax of lambda terms. Conversion/reduction rules: alpha, beta, eta. Encoding of booleans, naturals, pairs. Self-application and recursion. Lack of normalization

- The Simply Typed Lambda Calculus Algebraic data types: zero, one, sums, products. Function types (exponentials). Syntax of typed lambda terms. Type assignment rules: introduction and elimination. Conversion/reduction rules and normal forms. Normalization results (hints). A model based on sets. A model based on cartesian closed categories. The Curry-Howard isomorphism. Intuitionistic vs classical logic

Adding term-level recursion Recursive definitions and fixed points. Lack of normalization. Omega-CPOs. A model based on domain-theory. Induction principles

Practical examples Simple types and term-level recursion in programming languages.

- Recursive types (hints)



Equi- and iso-recursive types. Lack of normalization. Domains for recursive types (hints). Induction and catamorphisms (examples). Coinduction and anamorphisms (examples)

Practical examples Type-level recursion in programming languages

- Polymorphic types Universal quantification on types. Syntax and reduction semantics of terms. Natural transformations and parametricity.. The Yoneda Lemma (in a simple case).. Encoding algebraic data types. Encoding (strictly positive) recursive types

Practical examples Polymorphic types in programming languages.

- More complex types. Barendregt's lambda-cube. Values depending on values: functional abstraction. Values depending on types: polymorphic types. Types depending on types: parametric types. Types depending on values: dependent types

Practical examples Advanced types in programming languages.

Advanced topics (if time will allow for it) - Advanced type-level techniques in functional programming. Brief hints at Homotopy Type

Theory.

Documents

MANIFESTO DEGLI STUDI A.A. 2017/2018 … di Matematica Corso di Dottorato in Matematica Approvato dal Collegio dei Docenti nella seduta del 24 ottobre 2017 Manifesto degli studi Il