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.
Prof. Lorenzo Pareschi, 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
Alessandra Lunardi, Università di Parma
Luisa Malaguti, Università di Modena e Reggio Emilia
Massimiliano Mella, Università di Ferrara
Claudia Menini, 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
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.
Doctoral Positions in Mathematics
Applications are invited for 12 doctoral positions within the PhD Program, starting
November 1, 2022
The program will provide doctoral researchers with a high-quality education in the fields of pure and applied mathematics, and their interactions with computer science. Carefully selected research projects will introduce doctoral researchers to relevant and challenging topics at the forefront of current mathematical research.
In addition to 9 fellowships with research theme open to all fields of mathematics, 3 thematic fellowships are available, two of which funded by the Ministry of Economy and Finance (RRP) at the University of Ferrara in the area of artificial intelligence and machine learning:
- Physics-informed neural networks with applications to socio-epidemiological dynamics
- Artificial intelligence and e-health techniques applied to proprietary platforms for aid and assistance in telemedicine
and a third one funded by EU (ERC advanced grant) at the University of Modena and Reggio-Emilia:
- Dynamic causal modeling of brain activity to uncover specific neurophenotypes of dementia associated with anosognosia and delusions
Applications
We are looking for highly motivated, excellent candidates with strong mathematical skills and interest in pure and applied mathematics, and computer science. Applicants must hold a master’s or equivalent degree in mathematics, computer science or a related field.
Doctoral researchers will work on a theoretical or applied research project, participate in the activities of the PhD program by attending some of the curses offered, and pursue a doctoral degree.
Employment will be for three years. Applications must be submitted by
July 25, 2022 at 12.00 (Italian time)
either in English or Italian at the following links:
http://www.unife.it/studenti/dottorato/concorsi/ordinario (Italian version)
http://www.unife.it/studenti/dottorato/it/concorsi/selection (English version)
Further Information
For further information, please contact the Ph.D. program coordinator: Prof. Lorenzo Pareschi of the University of Ferrara.
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)
]]>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
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: February 2023 - 20h lectures.
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: Geometry of Deep Learning
Teacher: Rita Fioresi
Syllabus:
- Introduction to Deep Learning, basic steps of the algorithm analogies with the human visual systems and its mathematical models.
- The geometry of the space of data and the space of parameters; KL divergence and its information geometry interpretation.
- Geometric Deep Learning: the algorithm of Deep Learning on Graphs.
- Message passing and GATs: a geometrical modelling via heat equation and laplacian on graphs.
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: 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, 3 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: Around 12h lectures + 4h assignments, February-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
February 24, 26, 2009, Department of Mathematics:
Erasmus Seminars
For additional information please contact Prof. Claudia Menini
The mailing list of the PhD students is available: dottorandi@dm.unife.it
January-March 2009, Scientific-Technological Campus:
Methodology of Pattern Recognition & Imaging
Macroarea mini-courses:
Downloads: announcement, contents. For additional details please contact Prof. Stefano Trillo.
October 2008, Scientific-Technological Campus:
start of the ISEM lectures on Ergodic Theory.
More details are available here. For additional information please contact Dr. Michele Miranda.
Meetings: every Friday, 5:00pm, at the prof. Miranda office, Dept. of Mathematics.
October 23, 2008, at the Department of Mathematics:
Prof. Claudia Menini is organizing a set of lectures. The first lesson will be given on
Tuesday, September 30, 2008, 4:00pm, Room 7 - Department of Mathematics
by Dr. Alessandro Ardizzoni (Department of Mathematocs, University of Ferrara):
CoMorita Equivalent Hopf Algebras
Summary: this set of lectures will highlights the link between bigalois extensions, the monoidal equivalences between the cathegories of comodules on Hopf algebras and the concept of cociclic deformation of a finite dimensional Hopf algebra.
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information and official documents about the Ph.D. course