PhD students cycle XXXIX:
PhD students cycle XXXVIII:
PhD students cycles XXXV-XXXVII (Università di Parma)
PhD students cycle XXXIV (Università di Modena e Reggio Emilia)
PhD students cycle XXXIII (Università di Modena e Reggio Emilia)
PhD students cycle XXXII (Università di Modena e Reggio Emilia)
PhD students cycle XXIX-XXXI (Università di Ferrara)
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Title and Credits: Numerical approximation of contact and friction problems, 6 CFU
Teacher: Franz Chouly (Université de Bourgogne, INdAM visiting)
Syllabus:
1-Introduction
2-Fractional Sobolev spaces
3-Signorini problem and a naïve finite element approximation
4-New approximation techniques and optimal error bounds
5-Extension to friction (Tresca, Coulomb)
6-Extension to elastodynamics and impact of elastic bodies
Bibliography
Chouly, P. Hild, and Y. Renard, Finite element approximation of contact and friction in elasticity, no. 48 inAdvances in Continuum Mechanics, Birkhäuser, Springer, 2023. ISBN 978-3-031-31422-3
Dates 7-9-14-16-21-23-28-30 november 2023: 14:30-16:30; 16h lectures + home assignment; the students who are interested to follow lectures online, are asked to contact in advance alessandra.aimi@unipr.it
Title and Credits: Morse Theory 4 CFU
Teacher: Leonardo Biliotti
Syllabus:non-degenerate smooth functions on manifold, homotopy thype in terms of critical values, morse inequalies, the existence of non-degenerate functions, the Lefschets Theorem on Hyperplena sections, the calculus of variations applied to geodesic.
References: ''Morse Theory '', Milnor, ''An invitation to Morse Theory'', Nicolaescu
Dates 2023/2024: reading course.
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Title and Credits: Complex surfaces and their automorphisms 5 CFU
Teacher: Andrea Cattaneo
Syllabus: the aim of the course is to introduce the main techniques used in complex and algebraic geometry to study the geometry of compact complex surfaces. We will review the main results of the theory: the Riemann—Roch theorem with its implications, the effects of a blow up on the topology and the cohomology of a surface and Castelnuovo’s contraction theorem, the concept of minimal surface and the problem of classification (Castelnuovo’s rationality criterion and Enriques—Kodaira classification). Finally, we will focus on automorphisms of surfaces, focussing on what is known about their automorphism groups in particular for surfaces of Kodaira dimension 0.
Dates: 8 weeks, starting mid January 2024
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Title and credits: Topics in Representation Theory 6 CFU
Teacher: Andrea Appel
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Title and Credits: Normal Families Format 6 CFU
Teacher: Anna Miriam Benini
Syllabus: Consider a family of holomorphic maps from a complex manifold M to a compact complex manifold X. It is natural to ask oneself under which hypothesis such a family is precompact in Hol(M, X), i.e. every sequence has a converging subsequence. A precompact family is called normal. When M=C the complex plane, and X is the Riemann sphere, there is a famous criterion by Montel which claims that precompactness follows if the family omits 3 points in the Riemann sphere. The theory of normal families plays a crucial role in holomorphic iteration and more generally in functional analysis. We will study normal families of holomorphic/meromorhpic maps from C to the sphere and also give some information on normal families in higher dimensions.
Bibliography:
1)Schiff, Joel L. Normal families. Springer-Verlag, New York, 1993. xii+236 pp.ISBN: 0-387-97967-0
2)Lyubich, Mikhail, Conformal Geometry and Dynamics of Quadratic Polynomials, vol I-II
Dates will be decided with students; Individual Reading + Presentations + Exercises + 12 hours lectures.
Title and Credits: Numerical methods for Boundary Integral Equations, 6 CFU
Teacher: Alessandra Aimi
Syllabus: The course is principally focused on Boundary Element Methods (BEMs).
Lectures involve: Boundary integral formulation of elliptic, parabolic and hyperbolic problems - Integral operators with weakly singular, strongly singular and hyper-singular kernels - Approximation techniques: collocation and Galerkin BEMs - Quadrature formulas for weakly singular integrals, Cauchy principal value integrals and Hadamard finite part integrals -
Convergence results - Numerical schemes for the generation of the linear system coming from Galerkin BEM discretization.
Knowledge of basic notions in Numerical Analysis and in particular in numerical approximation of partial differential equations is required.
References will be provided directly during the course.
Dates Lectures will take place in II semester at the University of Parma for an amount of 24 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance
Title and Credits: Fourier and Laplace transforms and some applications, 4 CFU
Teacher: Marzia Bisi
Syllabus: Fourier transform: from Fourier series to Fourier transform, definition of inverse transform, transformation properties, convolution theorem, explicit computation of some transforms, applications to ODEs and PDEs of some physical problems. Laplace transform: definition, region of convergence, transformation properties, Laplace transform of Gaussian distribution, applications to some Cauchy problems. Definite integrals by means of residue theorem: integrals of real functions, and integrals of Fourier and Laplace useful to evaluate inverse transforms; theorems (with proofs) and examples.
Dates: February-March 2024 reading course; pdf slides and videos of all lectures are available on-line, number of expected hours: 24 + individual project.
Title and Credits: Extended kinetic theory and recent applications, 6 CFU
Teachers: Marzia Bisi, Maria Groppi
Syllabus: The course is intended to provide an introduction to classical kinetic Boltzmann approach to rarefied gas dynamics, and some recent advances including the generalization of kinetic models to reactive gas mixtures and to socio-economic problems.
Possible list of topics:
● distribution function and Boltzmann equation for a single gas: collision operator, collision invariants, Maxwellian equilibrium distributions;
● entropy functionals and second law of thermodynamics;
● hydrodynamic limit, Euler and Navier-Stokes equations;
● kinetic theory for gas mixtures: extended Boltzmann equations and BGK models;
● kinetic models for reacting and/or polyatomic particles;
● Boltzmann and Fokker-Planck equations for socio-economic phenomena, as wealth distribution or opinion formation.
Bibliography:
● C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York, 1988.
● M. Bisi, M. Groppi, G. Spiga, Kinetic Modelling of Bimolecular Chemical Reactions, in “Kinetic Methods for Nonconservative and Reacting Systems” edited by G. Toscani, Quaderni di Matematica 16, Dip. di Matematica, Seconda Università di Napoli, Aracne Editrice, Roma, 2005.
● L. Pareschi, G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2014.
Dates February 2024; the interested students are asked to contact the teachers in advance to define the calendar.
Title and Credits: Numerical methods for option pricing, 2 CFU (50 ore)
Teacher: Chiara Guardasoni
Syllabus:
● Introduction to differential model problems for option pricing in the Black-Scholes framework
● Analysis of peculiar troubles and advantages in application of standard numerical methods for partial differential problems: Finite Difference Methods, Finite Element Methods, Boundary Element Method
Dates 2023/24: reading course always available.
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Title and Credits: Basic theory of the Riemann zeta-function, 3 CFU
Teacher: Alessandro Zaccagnini
Syllabus: Elementary results on prime numbers. The Riemann zeta-function and its basic properties: analytic continuation, functional equation, Euler product and connection with prime numbers, the Riemann-von Mangoldt formula, the explicit formula, the Prime Number Theorem. Prime numbers in all and "almost all" short intetvals.
Dates 2023/2024;: the interested students are asked to contact the teachers in advance to define the calendar.
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Title and Credits: Semigroups of bounded linear operators and applications to PDEs, 6 CFU.
Teacher: Luca Lorenzi
Syllabus: In this course we present the theory of semigroups of bounded operators in Banach spaces, paying particolar attention to analytic semigroups. Applications are given to the analysis of partial differential equations of parabolic type.
Dates: February-May 2024.
Title and Credits: Introduction to Geometric Measure Theory, 6 CFU
Teacher: Massimiliano Morini
Syllabus: The course covers the following topics: review and complements of Measure Theory; covering theorems and their application to the proof of the Lebesgue and Besicovitch Differentiation Theorems; rectifiable sets and rectifiability criteria; the theory of sets of finite perimeter; applications to geometric variational problems; the isoperimetric problem; the partial regularity theory for quasi-minimiser of the perimeter.
Hand-written notes of the whole course are available in Italian on the Elly platform.
Further references:
Dates 2023/2024: reading course.
Title and Credits: Several complex variables, 6CFU
Teacher: Alberto Saracco
Syllabus: Theory of several complex variables. Hartogs theorem, Cartan-Thullen theorem, Kontinuitatsatz. Domains of holomorphy, Levi convexity and plurisubharmonic functions. Cauchy-Riemann equation. Sheaves and cohomology (Cech cohomology). The course will be mainly based on Chapters 1-6 of the book by Giuseppe Della Sala, Alberto Saracco, Alexandru Simioniuc and Giuseppe Tomassini: "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).
Dates 2023/2024: reading course.
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Title and Credits: Introduction to Elliptic partial differential equations 6 CFU
Teacher: Paolo Baroni
Syllabus: the course aims at introducing basic problems and classic existence and regularity techniques for uniformly elliptic PDEs with linear growth. In particular, the following topics will be analyzed, more or less in detail according to the students’ interest:
- Harmonic functions, weak formulation and Weyl's Lemma.
- Second order regularity for the Poisson equation via Calderon-Zygmund decomposition and singular integrals.
- Second order Sobolev regularity for equations with constant coefficients.
- Linear equations with variable coefficients: W^{1,q} estimates for continuous coefficients.
- Campanato spaces and Schauder theory for linear equations with Holder coefficients and data.
- De Giorgi theory (Holder regularity of solutions for measurable coefficients).
- Harnack inequalities, expansion of positivity for equations with measurable coefficients.
- Gehring theory (higher integrability of the gradient).
Dates: The course is intended to span approximatively two consecutive months (to be decided) between October 2023 and February 2024. Students are encouraged to contact the lecturer as soon as possible in order to fix a timetable.
Title and Credits: Plane Cremona transformations, 4 CFU
Teacher: Alberto Calabri
Syllabus: Rational and birational maps of the complex projective plane. Fundamental points and exceptional curves of a plane Cremona transformation. Quadratic transformations and De Jonquières maps. Properties like equations of conditions and Noether's inequality. Factorization of transformations and proofs of Noether-Castelnuovo theorem, Cremona equivalence of plane curves. Properties of the varieties parametrizing plane Cremona maps of fixed degree. Lengths in the plane Cremona group.
Dates 2023/2024: 24h lectures + home assignments, February-March 2024.
Title and Credits: Geometry of principal frequencies
Teacher: Lorenzo Brasco
Syllabus: The first eigenvalue of the Laplacian on an open set, and more generally of a second order elliptic operator, is an important object both from an applied and theoretical point of view. In Mathematical Physics, it usually plays the role of the ground state energy of a physical system. Despite its importance, for general sets it is not easy to explicitly compute it: thus, we aim at finding estimates in terms of simple geometric quantities of the sets, which are the sharpest possible. The most celebrated instance of this kind of problems is the so-called Faber-Krahn inequality.
This course offers an overview of the methods and results on sharp geometric estimates for the first eigenvalue of the Laplacian and more generally of sharp Poincaré-Sobolev embedding constants (sometimes called "generalized principal frequencies"). In particular, we will present: supersolutions methods, symmetrization techniques, convex duality methods, the method of interior parallels, conformal transplantation techniques.
Dates 2023/24: 24h lectures + home assignments; precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.
Title and Credits: (Modal) Symbolic Learning, 2CFU+2CFU (optional, for some research work)
Teacher: Guido Sciavicco
Syllabus: Symbolic learning is the sub-discipline of machine learning that is focused on symbolic (that is, logic-based) methods. As such, it contributes to the foundations of modern Artificial Intelligence. Symbolic learning is usually based on propositional logic, and in part, on first-order logic. Modal symbolic learning is the extension of symbolic learning to modal (and therefore, temporal, spatial, spatio-temporal) logics, and it deals with dimensional data. In this course we shall lay down the logical foundations of symbolic learning, prove some basic properties, and present the modal extensions of classical learning algorithms, highlighting which ones of those properties are preserved, and which ones are not.
Dates : January 2024, 4 lectures, 8 hours
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Title and Credits: An introduction to uncertainty quantification for PDEs 4CFU
Teacher: Giulia Bertaglia, Elisa Iacomini
Title and Credits: Embodiment, Discourse and Technology in Mathematics Education Research (4CFU + 2CFU optional, for some research work)
Coordinator: Federica Ferretti
Syllabus: The aim of the course will be to acquire specific knowledge - and their use in research practices - of some of the most internationally popular theoretical frameworks in mathematics education. There will be plenary sessions, led by Ferdinando Arzarello (University of Turin), Paul Drijvers (Utrecht University), Anna Sfard (University of Haifa) and Osama Swidan (Ben-Gurion University of Negev), a round table discussion led by Ornella Robutti (University of Turin) and workshops - held by Federica Ferretti and other Italian researchers - in which doctoral students will be able to engage with the theoretical lenses discussed.
Dates: 18-19-20 April, 2024 :
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Title: internet Seminar on Harmonic Analysis (ISEM27), edition 27
Dates: october 2023- february 2024, local coordinator Prof. Michele Miranda
At the end of the course (project phase (3CFU)), in-depth topics are proposed. Small groups of students from different universities are formed with the aim to study these topics and prepare a communication for the Final Workshop to be held in Luminy (Marseille, France).
]]>The program is structured in three years and it includes disciplinary activities (courses, workshops, seminars, conferences, schools), interdisciplinary activities (courses, seminars) and research activity.
Each PhD student is required to acquire 60 credits (CFU) per year.
At least 20 credits should be earned through interdisciplinary activities offered by IUSS 1391 Ferrara (http://iuss.unife.it/scuole-en/attivita-trasversali), by analogous course offered by the Universities of Modena-Reggio and Parma and by tutoring. At least 10 credits should be acquired during the first year.
At least 30 credits should be earned through disciplinary activities during the first year. The list of disciplinary activities offered by the PhD program is reported below in this page. Additional disciplinary activities are reported in the home page
At least 35 (resp.50) credits should be earned through research activities during the second (resp. third) year.
At the beginning of each year the PhD-student writes a plan of activities with her/his Supervisor. At the end of each year he/she writes the annual activities report. The forms can be found at https://www.unife.it/studenti/dottorato/it/pratiche/piani
The above rules apply only to PhD students of the 38th cycle or higher (see Regolamento).
Courses offered to students enrolled in the Ph.D. in Mathematics include:
Courses offered to students enrolled in the Ph.D. in Mathematics include:
Syllabus: The course will present the duality approach to the study of Markov processes. This will combine, in a joint effort, probabilistic and algebraic tools. In particular we will consider several interacting particle systems that are used in (non-equilibrium) statistical mechanics, we will discuss "integrable probability", we will show how (stochastic) PDE arise by taking scaling limits.
Dates: 2023/24 the interested. students are asked to contact the teachers in advance.
Title and credits: Topological and comparison-type methods for the study of boundary value problems in differential equations, 4CFU
Teacher: Luisa Malaguti
Syllabus: the course deals with some important methods for the study of boundary value problems to ordinary and partial differential equations. The Leray-Schauder topological degree will be briefly introduced, and its applications discussed in the study of periodic solutions and solutions satisfying Cauchy multi-point conditions in parabolic equations. The upper and lower solutions technique for ordinary differential equations will be then proposed and its application given to the study of traveling wave solutions of reaction-diffusion equations with degenerate diffusivities.
Dates February-March 2024 for an amount of 18 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.
Title and credits: Selected topics on algebraic curves over finite fields (20 hours)
Teacher: Giovanni Zini
Syllabus:the course will consider some selected topics in the theory of algebraic curves over finite fields. Useful previous knowledge: elementary theory of algebraic curves. The topics will be selected among the following ones.
● Maximal curves over finite fields: properties, classical examples (Hermitian, Suzuki and Ree curves), recent families (GK curve, GGS curve, BM curve, Skabelund curves).
● Automorphism groups of curves, and quotient curves: bounds on the size, examples. Automorphism and quotients of the Hermitian curve: classification.
● Rational points of curves over finite fields: criteria and methods for the analysis of absolutely irreducible rational components of curves, in particular for what concerns plane curves.
● Applications of the study of rational points to some remarkable families of polynomials over finite fields which are of interest in cryptography.
Dates: Lectures will take place from March 2024 to May 2024; the interested students, are encouraged to contact the teacher in advance.
Title and credits: Hodge Theory (20hours, 6CFU)
Teacher: Camilla Felisetti
Syllabus: Hodge theory is a powerful tool to understand the topology of Riemannian Manifolds. The aim of this course is to give an introduction to the theory together with applications to complex algebraic geometry. In particular we will treat the following topics:
● Recap on differentiable manifolds, differential forms and vector bundles
● Compact Kähler manifolds
● Hodge theory for Kähler manifolds
● Hodge theory for algebraic projective varieties
Dates: Lectures will take place from 15 November 2023 to 15 December 2023 at the University of Modena and Reggio Emilia for an amount of 20 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.
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Syllabus: Design Theory is a rich branch of Combinatorics that deals with the existence and construction of discrete structures
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Title: Variational Methods and Data-Driven Approaches for Imaging (6 CFU 16h)
Teachers: Alessandro Benfenati, Simone Rebegoldi
Syllabus: In the first part of the course, we introduce first order iterative methods suited for the minimization of the sum of a differentiable (possibly non-convex) function plus a convex (possibly non-differentiable) term, with particular attention to forward-backward methods. Convergence of these methods will be analyzed, both in the convex and non-convex setting. Furthermore, we discuss some popular acceleration strategies, such as the introduction of variable metrics, or the use of inertial steps in the iterative procedure. Finally, we present numerical results obtained on some image restoration problems, assessing the impact of acceleration techniques and hinting at possible future developments in this field.
The second part of the course is devoted to presenting data-driven approaches for solving imaging problems; we work in the MatLab environment, using the Deep Learning Toolbox. We address classification problems using deep learning techniques under supervised frameworks, i.e. when large datasets containing the ground truths for neural network training are available. Under the Deep Image Prior (DIP) framework we employ neural network architectures for solving image restoration problems, where one has to recover images corrupted by linear operators (e.g. Gaussian or motion blur) and by the presence of noise (e.g. Poisson noise). We also address semantic segmentation tasks, both in a supervised and unsupervised fashion.
Course plan:
References:
Dates: February 2024 ; 16 h lectures; the interested. students are asked to contact the teachers in advance.
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Title: Inexact and stochastic optimization methods for big data applications (6 CFU, 16h)
Teachers: Giorgia Franchini, Federica Porta
Syllabus: Over the past few years, machine learning and deep learning techniques have emerged as cutting-edge methodologies in several domains, representing promising alternatives to traditional approaches. Learning techniques usually require solving minimization problems which are characterized by both large scale datasets and many parameters to be optimized. Due to their very low cost per iteration, inexact and stochastic gradient-based methods represent effective tools to address these problems. The aim of this course is to introduce the optimization models which typically arise in machine learning and deep learning applications and several inexact and stochastic optimization methods suitable to deal with these models. Convergence results and practical implementation aspects of such algorithms will be provided.
Course plan:
References:
Dates: March 2024; 16 h lectures; the interested. students are asked to contact the teachers in advance
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Title: Hypoelliptic Partial Differential Equations (6 CFU)
Teachers: Sergio Polidoro, Maria Manfredini
Syllabus: The subject of the course is linear second order Partial Differential Equations with non-negative characteristic form satisfying the Hormander's hypoellipticity condition. Maximum principle, local regularity, boundary value problem will be discussed for several examples of equations. Some open research problems will be described. The course will focus on the following topics:
-Bony's maximum principle for degenerate second order PDEs, propagation set and Hormander's hypoellipticity condition.
-Perron method for the boundary value problem in a bounded open set of the Euclidean space.
-Boundary regularity, barrier functions. Boundary measure, Green function.
-Fundamental solution. Mean value formulas. Harnack inequalities.
-Degenerate Kolmogorov equations. Applications to some financial problems.
The program may be modified in accordance with the requirements of the students.
Reference text: A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians (Springer Monographs in Mathematics - 2007). Further references and lecture notes will be given during the course..
Dates: May-June 2024, 15h lectures + 3 h of autonomous work; the interested. students are asked to contact the teachers in advance
Lectures take place at “Edificio Matematica”, via Campi 213/c, 41125 Modena.
Title and Credits: Duality Theory of Markov Processes, 3 CFU
Teacher: Cristian Giardinà, Gioia Carinci
Syllabus: The course will present the duality approach to the study of Markov processes. This will combine, in a joint effort, probabilistic and algebraic tools. In particular we will consider several interacting particle systems that are used in (non-equilibrium) statistical mechanics, we will discuss "integrable probability", we will show how (stochastic) PDE arise by taking scaling limits.
Dates 2022/2023: Reading course, beginning of 2023 (precise schedule will be decided together with the students).
Title and credits: Optimization methods for machine learning
Teachers: Federica Porta, Giorgia Franchini, Alessandro Benfenati
Syllabus: Introduction to machine learning. Supervised Learning: loss functions, empirical risk minimization, regularization approaches. Gradient descent approaches: deterministic and stochastic frameworks, also with reduced variance. Hyperparameters choice in the stochastic gradient method. Stochastic Optimization in learning methodologies: topics and perspectives in constrained and
unconstrained fields. Implementation issues for large-scale learning. Link between stochastic gradient and Deep learning. Neural architecture search techniques.
Dates: 13-14-15 February 2023, all day long (9:30-13:00 and 14:00-17:30).
Title and Credits: Geometry of Deep Learning, 4 CFU
Teacher: Rita Fioresi
Syllabus:
This course will be self-contained as much as possible. The necessary differential geometric concepts (manifolds, Frobenius theorem, Cartan formalism) will be introduced and explained. The necessary programming skills will NOT be assumed, but a part of the course will be "hands on" illustrating key examples on colab.
The exam will consist in a brief exposition of some concepts and the students can choose the part of the
program they like the most and present a focused exposition on one argument.
Six lectures of approximately 2.5 hours each.
Dates 2022/2023: 15h lectures + reading course + home assignments
Title and Credits: New Directions in Designs and Graphs, 6 CFU
Teachers: Simona Bonvicini, Giuseppe Mazzuoccolo, Anita Pasotti, Tommaso Traetta.
Syllabus: The course will present some advanced topics and recent results in Design Theory and in Graph Theory.They include (but are not limited to) decompositions and factorizations of graphs and their symmetries, and the study of several classes of arrays.Open problems and applications will be presented too. Interested students are requested to send an email to: anita.pasotti@unibs.it
Dates 2022/2023: Reading course, beginning of 2023 (the precise schedule will be decided together with the students).
Title and credits: Topological and comparison-type methods for the study of boundary value problems in differential equations, 4CFU
Teacher: Luisa Malaguti
Syllabus: The Course deals with some important methods for the study of boundary value problems to ordinary and partial differential equations. The Leray-Schauder topological degree will be briefly introduced, and its applications discussed in the study of periodic solutions and solutions satisfying Cauchy multi-point conditions in parabolic equations. The upper and lower solutions technique for ordinary differential equations will be then proposed and its application given to the study of traveling wave solutions of reaction-diffusion equations with degenerate diffusivities.
Dates 2022/2023: Lectures will take place from March to June 2023 at the University of Modena and Reggio Emilia for an amount of 18 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.
Title and credits: Selected topics on algebraic curves over finite fields (20 hours)
Teacher: Giovanni Zini
The course will consider some selected topics in the theory of algebraic curves over finite fields. Useful previous knowledge: elementary theory of algebraic curves. The topics will be selected among the following ones.
Title and credits: Hodge Theory (20hours, 6CFU)
Teacher: Camilla Felisetti
Hodge theory is a powerful tool to understand the topology of Riemannian Manifolds. The aim of this course is to give an introduction to the theory together with applications to complex algebraic geometry. In particular we will treat the following topics:
Dates 2023: Lectures will take place from September to October 2023 at the University of Modena and Reggio Emilia for an amount of 20 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.
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Prof. Valeria Ruggiero, Università di Ferrara
Alessandra Aimi, Università di Parma
Andrea Bandini, Università di Pisa
Leonardo Biliotti, Università di Parma
Cinzia Bisi, Università di Ferrara
Marzia Bisi, Università di Parma
Simona Bonvicini, Università di Modena e Reggio Emilia
Walter Boscheri, Università di Ferrara
Maria Rita Casali, Università di Modena e Reggio Emilia
Andrea Corli, Università di Ferrara
Vincenzo Coscia, Università di Ferrara
Paola Cristofori, Università di Modena e Reggio Emilia
Alessandra Fiocca, Università di Ferrara
Stefania Gatti, Università di Modena e Reggio Emilia
Claudio Giberti, Università di Modena e Reggio Emilia
Maria Groppi, Università di Parma
Claudia Landi, Università di Modena e Reggio Emilia
Lucaﾠ Francesco Giuseppe Lorenzi, Università di Parma
Maria Giulia Lugaresi, Università di Ferrara
Alessandra Lunardi, Università di Parma
Massimiliano Mella, Università di Ferrara
Claudia Menini, Università di Ferrara
Carlo Mercuri, Università di Modena e Reggio Emilia
Michele Miranda, Università di Ferrara
Massimiliano Morini, Università di Parma
Lorenzo Nicolodi, Università di Parma
Cristina Patria, Università di Ferrara
Sergio Polidoro, Università di Modena e Reggio Emilia
Marco Prato, Università di Modena e Reggio Emilia
Gloria Rinaldi, Università di Modena e Reggio Emilia
Valeria Ruggiero, Università di Ferrara
Andrea Sacchetti, Università di Modena e Reggio Emilia
Alberto Saracco, Università di Parma
Valentina Taddei, Università di Modena e Reggio Emilia
Cecilia Vernia, Università di Modena e Reggio Emilia
Alessandro Zaccagnini, Università di Parma
Luca Zanni, Università di Modena e Reggio Emilia
Title and Credits: Numerical methods for Boundary Integral Equations, 6 CFU
Teacher: Alessandra Aimi
Syllabus: The course is principally focused on Boundary Element Methods (BEMs).
Lectures involve: Boundary integral formulation of elliptic, parabolic and hyperbolic problems - Integral operators with weakly singular, strongly singular and hyper-singular kernels - Approximation techniques: collocation and Galerkin BEMs - Quadrature formulas for weakly singular integrals, Cauchy principal value integrals and Hadamard finite part integrals -
Convergence results - Numerical schemes for the generation of the linear system coming from Galerkin BEM discretization.
Knowledge of basic notions in Numerical Analysis and in particular in numerical approximation of partial differential equations is required.
References will be provided directly during the course.
Dates 2022/2023: Lectures will take place in Spring 2023 at the University of Parma for an amount of 24 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.
Title and Credits: Infinite Dimensional Analysis, 4 CFU
Teacher: Alessandra Lunardi
Syllabus: This is an introductory course about analysis in abstract Wiener spaces, namely separable Banach or Hilbert spaces endowed with non-degenerate Gaussian measures. Sobolev spaces and spaces of continuous functions will be considered. The basic differential operators (gradient and divergence) will be studied, as well as the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup, that are the Gaussian analogues of the Laplacian and the heat semigroup. The most important functional inequalities in this context, such as Poincaré and logarithmic Sobolev inequalities, will be proved. Hermite polynomials and the Wiener chaos will be described.
The reference book is "Gaussian Measures" by V. Bogachev (Mathematical Surveys and Monographs 62, AMS 1998).
According to the interests of the audience, it is possible to consider only the Hilbert space setting, in which case the reference book is "Second Order Partial Differential Equations in Hilbert Spaces" by G. Da Prato and J. Zabczyk (Cambridge Univ. Press 2002).
In any case, lecture notes prepared by the teacher will be available.
Dates 2022/2023: Lectures will take place in Spring 2023 at the University of Parma for an amount of 24 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.
Title and credits: Topics in Representation Theory, 6 CFU
Teacher: Andrea Appel
Syllabus: The course will provide an introduction to an advanced topic in representation theory and quantum groups. These include:
1) Categorified quantum groups.
2) Quantum symmetric pairs.
3) Cluster algebras and quantum groups.
4) Yangians and quantum affine algebras.
5) Geometric representation theory of quantum affine algebras.
The topics of the course will be chosen during an organizational meeting in December 2022 according to the interests of the participants and their background. The interested students are invited to contact the instructor in due time.
The course will be held in hybrid format. Lectures will take place in January-March 2023. 30 hours.
Title and credits: Normal Families, 6 CFU
Teacher: Anna (Miriam) Benini
Syllabus: Consider a family of holomorphic maps from a complex manifold M to a compact complex manifold X. It is natural to ask oneself under which hypothesis such family is precompact in Hol(M,X), i.e. every sequence has a converging subsequence. A precompact family is called normal. When M=\C the complex plane, and X is the Riemann sphere, there is a famous criterion by Montel which claims that precompactness follows if the family omits 3 points in the Riemann sphere.
The theory of normal families plays a crucial role in holomorphic iteration and more generally in functional analysis. We will study normal families of holomorphic/meromorhpic maps from \C to the sphere and also give some informations on normal families in higher dimensions.
Bibliography:
1) Schiff, Joel L.
Normal families.
Springer-Verlag, New York, 1993. xii+236 pp. ISBN: 0-387-97967-0
2) Lyubich, Mikhail
Conformal Geometry and Dynamics of Quadratic Polynomials, vol I-II
Manuscript
3) Additional references for several complex variables to be decided later
Format: Individual Reading + Presentations + Exercises + 12 hours
Dates: November 2022 - January 2023
Title and Credits: Fourier and Laplace transforms and some applications, 4 CFU
Teacher: Marzia Bisi
Syllabus: Fourier transform: from Fourier series to Fourier transform, definition of inverse transform, transformation properties, convolution theorem, explicit computation of some transforms, applications to ODEs and PDEs of some physical problems. Laplace transform: definition, region of convergence, transformation properties, Laplace transform of Gaussian distribution, applications to some Cauchy problems. Definite integrals by means of residue theorem: integrals of real functions, and integrals of Fourier and Laplace useful to evaluate inverse transforms; theorems (with proofs) and examples.
Dates 2022/2023: reading course; pdf slides and videos of all lectures are available on-line, number of expected hours: 24 + individual project.
Title and Credits: Extended kinetic theory and recent applications, 4 CFU
Teachers: Marzia Bisi, Maria Groppi
Syllabus: The course is intended to provide an introduction to classical kinetic Boltzmann approach to rarefied gas dynamics, and some recent advances including the generalization of kinetic models to reactive gas mixtures and to socio-economic problems.
Possible list of topics:
Bibliography:
Dates 2022/2023: About 18 hours in January - February 2023 (flexible). The interested Ph.D. students are asked to contact the teachers in advance to define the calendar.
Title and Credits: Numerical methods for option pricing, 2 CFU (50 ore)
Teacher: Chiara Guardasoni
Syllabus:
Dates 2022/2023: reading course always available.
Title and Credits: Semigroups of bounded linear operators and applications to PDEs, 6 CFU.
Teacher: Luca Lorenzi
Syllabus: In this course we present the theory of semigroups of bounded operators in Banach spaces, paying particolar attention to analytic semigroups. Applications are given to the analysis of partial differential equations of parabolic type.
February-May 2023.
Title and Credits: Introduction to Geometric Measure Theory, 6 CFU
Teacher: Massimiliano Morini
Syllabus: The course covers the following topics: review and complements of Measure Theory; covering theorems and their application to the proof of the Lebesgue and Besicovitch Differentiation Theorems; rectifiable sets and rectifiability criteria; the theory of sets of finite perimeter; applications to geometric variational problems; the isoperimetric problem; the partial regularity theory for quasi-minimiser of the perimeter.
Hand-written notes of the whole course are available in Italian on the Elly platform.
Further references:
Dates 2022/2023: reading course.
Title and Credits: Several complex variables, 6CFU
Teacher: Alberto Saracco
Syllabus: Theory of several complex variables. Hartogs theorem, Cartan-Thullen theorem, Kontinuitatsatz. Domains of holomorphy, Levi convexity and plurisubharmonic functions. Cauchy-Riemann equation. Sheaves and cohomology (Cech cohomology). The course will be mainly based on Chapters 1-6 of the book by Giuseppe Della Sala, Alberto Saracco, Alexandru Simioniuc and Giuseppe Tomassini: "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).
Dates 2022/2023: reading course.
Title and Credits: Basic theory of the Riemann zeta-function, 3 CFU
Teacher: Alessandro Zaccagnini
Syllabus: Elementary results on prime numbers. The Riemann zeta-function and its basic properties: analytic continuation, functional equation, Euler product and connection with prime numbers, the Riemann-von Mangoldt formula, the explicit formula, the Prime Number Theorem. Prime numbers in all and "almost all" short intetvals.
Dates 2022/2023: late Winter, early Spring 2023.
Title and Credits: Decision theory for artificial intelligence, 6 CFU
Teachers: Federico Bergenti
Syllabus: The course introduces students to the topics of Decision Theory that are relevant for Artificial Intelligence. In particular, the couse discusses decision-theoretic planning and learning through the following agenda: brief review of random variables and stochastic processes (if needed), discrete-time Markov chains, Markov decision processes, base algorithms for automated planning using Markov decision processes (e.g., value iteration and policy iteration), base algorithms for machine learning using Markov decision processes (e.g., Q-learning and SARSA), brief overview of additional topics (e.g., partially-observable Markov decision processes, game-theoretic planning). The course is delivered as a set of classes and exercize sessions tailored to the specific needs of attending students.
Dates 2022/2023: Two weeks in March or April 2023.
Title and Credits: Geometry of principal frequencies
Teachers: Lorenzo Brasco
Syllabus: The first eigenvalue of the Laplacian on an open set, and more generally of a second order elliptic operator, is an important object both from an applied and theoretical point of view. In Mathematical Physics, it usually plays the role of the ground state energy of a physical system. Despite its importance, for general sets it is not easy to explicitly compute it: thus, we aim at finding estimates in terms of simple geometric quantities of the sets, which are the sharpest possible. The most celebrated instance of this kind of problems is the so-called Faber-Krahn inequality.
This course offers an overview of the methods and results on sharp geometric estimates for the first eigenvalue of the Laplacian and more generally of sharp Poincaré-Sobolev embedding constants (sometimes called "generalized principal frequencies"). In particular, we will present: supersolutions methods, symmetrization techniques, convex duality methods, the method of interior parallels, conformal transplantation techniques.
Dates 2022/2023: 24h lectures + home assignments.
Title and Credits: An introduction to uncertainty quantification for PDEs, 4 CFU
Teacher: Lorenzo Pareschi, Giulia Bertaglia
Syllabus: The course aims to provide an introduction to numerical methods for uncertainty quantification with specific reference to PDEs. After defining the main concepts in the field of uncertainty quantification, including some references to probability theory, the course focuses on two main approaches. The Monte Carlo method, in its variants characterized by multi-fidelity techniques, and the methods based on generalized polynomial chaos expansions, both in intrusive and non-intrusive form. Specific applications to the case of hyperbolic systems with relaxation terms and reaction-diffusion equations will be considered. In-depth study by students through specific reading of articles will also be suggested.
Dates 2022/2023: 12h lectures + reading course + home assignments
Title and Credits: Recent topics in numerical methods for hyperbolic and kinetic equations, 4 CFU
Teacher: Lorenzo Pareschi, Giacomo Dimarco, Walter Boscheri
Hyperbolic and kinetic partial differential equations arise in a large number of models in physics and engineering. Examples of the applications area range from classical gas dynamics and plasma physics to semiconductor design and granular gases. Recent studies employ these models to describe the collective motion of many particles such as pedestrian and traffic flows, epidemiology and other dynamics driven by social forces. This course will cover the mathematical foundations behind some of the most important numerical methods for these types of problems. To this goal, the first part of the course will be devoted to hyperbolic balance laws where we will introduce the notions of finite-difference, finite volume, and semi-Lagrangian schemes. In the second part we will focus on kinetic equations
where, due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods requires a careful balance between accuracy and computational complexity. Finally, we will consider some recent developments related to the construction of asymptotic preserving methods.
Dates 2022/2023: 12h lectures + reading course + home assignments.
Title and credits: Computational intelligence and gradient-free optimization, 6 CFU
Teachers: Filippo Poltronieri, Mauro Tortonesi and Lorenzo Pareschi
Syllabus: This course provides an introductory overview of key concepts in computational intelligence with a focus on metaheuristic methods for global optimization. These include Genetic Algorithms (bitstring and integer vector genotype representations) and Particle Swarm Optimization (constrained PSO, quantum-inspired PSO, and a multi-swarm version of quantum-inspired PSO), extended with adaptation mechanisms to provide support for dynamic optimization problems. The main algorithms will be illustrated with the help of simple implementations in Matlab and/or R language. In the last part of the course, using a mean-field approach, rigorous convergence results for some of the methods will be presented.
Dates: 12h lectures + 4h assignments, May-June 2023
Title and Credits: (Modal) Symbolic Learning, 2CFU+2CFU (optional, for some research work)
Teacher: Guido Sciavicco
Syllabus: Symbolic learning is the sub-discipline of machine learning that is focused on symbolic (that is, logic-based) methods. As such, it contributes to the foundations of modern Artificial Intelligence. Symbolic learning is usually based on propositional logic, and in part, on first-order logic. Modal symbolic learning is the extension of symbolic learning to modal (and therefore, temporal, spatial, spatio-temporal) logics, and it deals with dimensional data. In this course we shall lay down the logical foundations of symbolic learning, prove some basic properties, and present the modal extensions of classical learning algorithms, highlighting which ones of those properties are preserved, and which ones are not.
Dates 2022/2023: September 2022, 4 lectures, 8 hours
Title and Credits: Plane Cremona transformations, 3 CFU
Teacher: Alberto Calabri
Syllabus: Rational and birational maps of the complex projective plane. Fundamental points and exceptional curves of a plane Cremona transformation. Quadratic transformations and De Jonquières maps. Properties like equations of conditions and Noether's inequality. Factorization of transformations and proofs of Noether-Castelnuovo theorem, Cremona equivalence of plane curves. Properties of the varieties parametrizing plane Cremona maps of fixed degree.
Dates 2022/2023: 20h-24h lectures + home assignments.
Title and Credits: Galois Theory: from groups and forms to descent theory and extensions
Teacher: Prof. Daniel Bulacu, University of Bucarest
Syllabus: The classical Galois theory dates back to 1830, but it took more than 100 years for it to be reformulated (by Artin) in the language of module theory. Artin's criterion that decides when a field extension K/k is Galois allows to extend the classical Galois theory to Hopf algebras. This was initiated by Chase and Sweedler in 1969 in the commutative case, the general case being considered by Kreimer and Takeuchi in 1981. Today Hopf-Galois extensions appear in various branches of mathematics and physics, being also known as dual algebraic versions of non-commutative fiber spaces (the notion of quantum fiber space can be introduced as a module associated with a Hopf-Galois extension).
The purpose of this course is to make the transition from the classical Galois theory to the Hopf-Galois theory and to present some directions of study for the latter. Briefly, the content of the course is as follows:
Dates 2022/2023: 4, 6, 11, and 13 July 2023, 16:00-18:00 (4 lectures, 8 hours).
The PhD program in Mathematics is organized within a close collaboration between the three universities of Ferrara, Modena-Reggio Emilia and Parma, and represents a consolidated reality at regional and national level since 2013. The students of PhD programs benefit both from the educational offers of the three campuses in the various fields of Mathematics, and from the plurality of skills in the teaching staff. Through a capillary network of national and international contacts and the high level qualification of its teaching staff, the program offers an educational path of the highest level in each specific sector of mathematical sciences.
The main objective of this program is to train highly qualified professionals in the field of Mathematics, with advanced skills on mathematical models and methodologies, which can also be used in application contexts of interdisciplinary type is achieved through series of lectures, seminars, participation in workshops and research periods in third-party institutions which, on the one hand, introduce PhD students into the academic and non-academic research mode and, on the other hand, allow them to establish business contacts that will accompany them in the years following graduation. Alongside the periods of study, presentations at conferences and workshops of international importance are organized to help the doctoral student in subsequent insertion into the working way. The training is therefore specifically aimed at the ability to recognize significant research problems in the field mathematician, to formulate solutions to such problems and to pass on the results to others through oral presentations or written work.
PhD candidates, at the end of the program, must be able to: carry out a research activity independently, produce original and relevant results and enter the international scientific community. Graduates will be in possession the skills necessary not only to work in universities and research institutes, but also in industry, in administration public and private commercial enterprises.
News
On Novembre 7-9-14-16-21-23-28-30, 2023: from 14:30 to 16:30; Prof. Franz Chouly from the Université de Bourgogne will hold the " Numerical approximation of contact and friction problems" course ((both in presence at the University of Parma and remote). The Prof. Chouly is an INdAM visiting. For information and enrolment please refer to the page describing the couses held at the University of Parma and/or contact Prof. Alessandra Aimi (alessandra.aimi@unipr.it)..
On July 4-13, 2023, Prof. Daniel Bulacu from the University of Bucarest will hold the "Galois Theory: from groups and forms to descent theory and extensions" course (both in presence at the University of Ferrara and remote). For information and enrolment please refer to the page describing the couses held at the University of Ferrara and/or contact Prof. Claudia Menini.
On December 2-3, 2022, Prof. Sebastien Motsch from University of Arizona will hold the "Around neural Network and tumor growth: modeling, analysis and numerics" course (both in presence at the University of Ferrara and remote). For information and enrolment please refer to the course description document.
From November 15 to December 13, 2022, Prof. Paolo Boero from University of Genova, Ferdinando Arzarello from University of Turin, Federica Ferretti from University of Ferrara, Andrea Maffia from University of Bologna, Francesca Martignone from University of Western Piedmont, and Eugenia Taranto from University of Catania will hold the "Primi approcci alla ricerca in didattica della matematica" course (in Italian; both in presence at the University of Ferrara and remote). For information and enrolment please refer to the course description document.
Further Information
For further information, please contact the Ph.D. program coordinator: Prof. Valeria Ruggiero of the University of Ferrara.