Mar 072024
 

Quantum computing: foundations and state of the art

Prof. Michele Marchesi
Dipartimento di Matematica e Informatica
Università degli Studi di Cagliari

Abstract

The seminar is about the current status of Quantum Computing. It will present the math fundamentals of quantum computing, and then will cover Quantum Annealing, which is the current most consolidated application field, featuring working machines with 5000 qbits. The seminar will then cover the current status and possible applications of Quantum Computing, including Shor’s algorithm and the quest for quantum resistant cryptographic algorithms.

Schedule

March, Monday, 11st and Tuesday, 12nd – 17:30-19.00 (Palazzo delle Scienze, Room B) 

 Scritto da in 7 Marzo 2024  Senza categoria  Commenti disabilitati su Seminar: Quantum computing: foundations and state of the art
Feb 172024
 

Isometric Immersions and Harmonic Maps

Prof. Cezar Oniciuc
Universitatea “Alexandru Ioan Cuza” Iași

Outline

1. Generalities on Riemannian Geometry
2. Isometric immersions (submanifolds) – generalities
3. Special isometric immersions: umbilicals, minimal, CMC
4. Operators on vector bundle
5. Harmonic maps between Riemannian manifolds: first and second variation; fundamental examples

Schedule

May 21st, 16.00-18.00 Aula II
May 22nd, 16.00-18.00 Aula II
May 23rf, 16.00-18.00 Aula II

May 28th, 16.00-18.00 Aula II
May 29th, 16.00-18.00 Aula II
May 30th, 16.00-18.00 Aula II

June 4th, 16.00-18.00 Aula II
June 6th, 16.00-18.00 Aula II

 

 Scritto da in 17 Febbraio 2024  Senza categoria  Commenti disabilitati su PhD Course: Isometric Immersions and Harmonic Maps
Feb 102024
 

MAIN PhD Seminars 2024

Date Speaker(s)
March, 6th Marco Casula
March, 13rd Luca Zedda
March, 20th Filippo Maria Cassanello
March, 27th Alessandro Iannella
April, 3rd Elisa Crabu
April, 17th Jacopo Mereu
April, 24th Alessandra Perniciano
May, 8th Antonio Sanna
May, 15th Giuseppe Demuru
May, 22nd Massimiliano Fadda
Federico Meloni
May, 29th Andrea Cabriolu
Giorgia Nieddu

All the seminars start at 5 PM.

 

Marco Casula: Bochner-Euclidean volume

We will start with examples of calculating the volume of objects in three-dimensional space and then extend the definition to any manifold. Therefore we will introduce a new and different volume on complex manifolds, with particular attention to cases of finite and infinite volumes. The work is based on the article by Loi-Placini.

Luca Zedda: Self-Supervised Learning: The Dark Matter of Artificial Intelligence

In this seminar, we shall delve into the concept of Self-Supervised Learning, an intriguing and rapidly expanding branch of artificial intelligence. Fundamental concepts of this innovative approach will be introduced, demonstrating how it is possible to connect the process of human cognitive development to that of artificial within the context of deep learning. Through the analysis of self-supervised models, it will be explained how AI can autonomously learn, addressing the challenges posed by the lack of explicit annotations in data and the application of these technologies to real-world scenarios.

Filippo Maria Cassanello: An alternative approach to the Hölder continuity of solution of the fractional p-laplacian

In this seminar we will define the non-local operator “fractional p-laplacian” by also talking about his biological interpretation for describing the movement of population in hostile habitat. Then we will give a different proof of the Hölder continuity of weak solution of this operator by extending the approach that DiBenedetto developped for the p-laplacian. This work is based on the paper “An alternative approach to the Hölder continuity of solution of some elliptic equations” of Duzgun, Marcellini, Vespri and is in collaboration with Prof. Antonio Iannizzotto.

Alessandro Iannella: The Transitional Space: Generative Artificial Intelligence as an Opportunity for Professional Growth for Teachers
This seminar aims to illustrate the benefits, risks, and challenges of using Generative Artificial Intelligence in teaching, also drawing on concepts and metaphors from psychology and sociology. Particular attention will be paid to the different phases of the teaching process, from design to evaluation.

Elisa Crabu: Mathematical tools for Computer Vision

Photometric Stereo is a Computer Vision tecnique that leads to reconstructing the digital shape of an object from a set of images, obtained by lighting the object with a light source placed at different positions around it. The method, by estimating the surface normals, computes an approximation of the surface. In this talk we will describe the main steps of the solution method, presenting the mathematical tools that underlie it, including the singular value decomposition, least square problems and the numerical solution of partial differential equations.

Jacopo Mereu: AI-supported End User Development in VR

End-User Development (EUD) is a research field that aims to design and develop software or hardware technology (digital artifacts) such that their consumers (end users) should be able to adapt such artifacts according to their needs. End users are not a static category; the unique context of the application determines their identity, skills, and experience. In the context of this seminar, the end users are proficient programmers in Unity but lack expertise in constructing Extended Reality environments. The research aims to assist these end users in using a XR Development toolkit, the Mixed Reality Toolkit (MRTK), whose latest version has recently been released. Large Language Models (LLMs) have been chosen as the method to support the end users. These models are trained with extensive documents, allowing them to acquire knowledge across various domains. However, their knowledge has a temporal limitation, as the models lack information about events or developments occurring after a certain date. Consequently, an LLM may lack information about the MRTK3 library. This seminar thus presents a practical case of enhancing the performance of an LLM in a domain where it possesses limited or no prior knowledge.

Alessandra Perniciano: Radiomics: the issue of high dimensional data

Radiomics, a branch of Computer Vision, involves the extraction and analysis of quantitative features from medical imaging modalities such as MRI, PET, and CT scans. The central idea behind Radiomics is that imaging features specific to various diseases may offer valuable insights into predicting prognosis and treatment outcomes across different types of pathologies. Notably, these characteristics remain elusive through traditional visual inspection methods employed in current radiologic practice, yet they provide insights into the underlying biological processes. However, the quantitative extraction of features leads to a situation of high dimensionality where not all the extracted features are necessarily relevant. During this seminar, I will present the challenges related to high dimensionality in Radiomics, providing an analysis of the current state of knowledge and discussing some future development directions.

Antonio Sanna: Harmonic and Biharmonic maps between Riemannian Manifolds

The object of this seminar is the definition of harmonic maps and biharmonic maps between Riemannian manifolds. During the exposition we will introduce the energy functional for smooth maps between two Riemannian manifolds,  and, deriving the corresponding Euler-Lagrange equation — in order to find its critical maps, we will define a certain vector field, called tension field, which is identically zero when the map is harmonic, i.e. critical. We will extend the notion of harmonic maps to that of biharmonic maps which are the critical points of the bienergy functional. We will see that harmonic maps are trivially biharmonic. Thus a crucial problem is to understand when the converse is also true, that is: under what conditions biharmonic maps are harmonic. Beyond this theoretical exploration, we will give some examples of biharmonic maps which are not harmonic. In particular, we will consider the geometrically interesting case of biharmonic isometric immersions.

Giuseppe Demuru: An Introduction to Causal Inference

Causal inference involves the study of cause-and-effect relationships among variables, based on experimental or observational data. Understanding these relationships in depth is essential for making informed decisions and solving complex problems. The well-known statement “Correlation does not imply causation” underscores that simple associations do not necessarily imply causality. Causal inference utilizes methods such as Potential Outcomes (PO) and Directed Acyclic Graphs (DAGs) to identify and quantify the true causal relationships among variables.

Massimiliano Fadda: Translating HTML in proprietary JSON

Growens is an integrated industrial group that creates technologies for content creation, predictive marketing, and mobile messaging, aimed at organizations wishing to communicate effectively with their customers. The seminar will introduce the reasons that led the company to develop this project. An overview of the technologies and methodologies identified for its resolution will then be provided, introducing the architecture of the system that allows the conversion of generic HTML pages into proprietary Json.

Federico Meloni: Mesh generation in the volumetric domain

Representing an object in the virtual world is becoming a frequent practice in fields like industries, entertainment, medicine. To digitally represent an object, the space is discretized due to the inability of a computer to represent space continuously. Therefore, we utilize a series of primitives such as points, segments, polygons, and eventually polyhedra to represent an object, called in this context a mesh. A three-dimensional mesh can be superficial if only the exterior of the object is represented, or volumetric if it includes a description of the volume within. The latter unlocks the possibility of performing a variety of operations such as physical simulations, fluid dynamics, and many others. In this context, algorithms for automatic generation of volumetric meshes are becoming increasingly important and valuable. This seminar will review the basic concepts before proceeding to present high-level algorithms for generating volumetric meshes.

Andrea Cabriolu: A Bayesian approach to an optimization algorithm for the dynamic scheduling of astronomical observations

In the context of the dynamic scheduling of observations with Sardinia Radio Telescope, a key role is played by Optimizer, a set of algorithms to optimize the sequence of the astronomical observations. The calculations are based on several parameters, like weather conditions, device availability, operator’s availability and others. In this talk I’ll introduce the architecture which allows the communication between Optimizer and the whole scheduling system, consisting of a central database and a bunch of other components. The core concepts of the Bayesian statistics will be introduced as well, since this is the main pillar of the computing performed by the algorithm, to optimize the parameters set regargind the observations to be scheduled.

Giorgia Nieddu: State of art on the use of A.I. in mathematics education

In this seminar the most recent results on the use of A.I. in mathematics education, its areas of application, limits and possibilities will be presented.

 Scritto da in 10 Febbraio 2024  Senza categoria  Commenti disabilitati su MAIN PhD Seminars 2024
Gen 142024
 

Foliations

Prof. Gianluca Bande
Dipartimento di Matematica e Informatica
Università degli Studi di Cagliari

Abstract

The course is an introduction to the Theory of Foliations. Basic knowledge of Differential Geometry is required and the basics of Fundamental Group.

Outline

– Definition(s) and examples of Foliations. Dynamical systems. Frobenius’ Theorem.
– Holonomy of a leaf and the Reeb Stability Theorem. Basic and foliated Cohomology. Godbillion-Vey class for a codimension 1 foliation on a 3-manifold.
– The Reeb foliation: definition and a 3D-printer model. Novikov and Likorisch Theorem.

Schedule

The course spans over 3 lectures of 2 hours each (6 hours total). The lectures will be given on February 5, February 12, February 15 – 2024 at 4:30 p.m. in Room B of the Department of Mathematics and Computer Science.

Exam

The final exam consists in a presentation.

References

1. C. Camacho and A. Lins Neto, Geometric theory of foliations, Birkhäuser, 1985.
2.
A. Candel; L. Conlon, Foliations I, Grad. Stud. Math. 23, American Mathematical Society, Providence, 2000.
3.
P. Tondeur, Geometry of Foliations, Monogr. Math 90, Birkhäuser Verlag, Basel, 1997.

Set 212023
 

Geometric Analysis

Prof. Antonio Greco
Dipartimento di Matematica e Informatica
Università degli Studi di Cagliari

Abstract

The course is an introduction to the problem of determining the shape of solutions to boundary-value problems for second-order partial differential equations, mainly of elliptic type, occasionally parabolic.

Outline

– Review of the weak maximum principle, the strong maximum principle, and the Hopf lemma.
– Some motivations for Geometric Analysis and some characteristic results: the soap bubble theorem (Aleksandrov’s theorem), Serrin’s overdetermined problem, the Gidas-Ni-Nirenberg symmetry result.
– Convexity of solutions to the Dirichlet problem. Quasiconvexity.
– The Morse index of a solution and its role in Geometric Analysis. Work in progress.

Schedule

The course spans over 8 lectures of 2 hours each (16 hours total), one lecture per week. Details will be specified on the occasion of the first lecture, which will be given on October 20, 2023 at 4 p.m. in room B of the Department of Mathematics and Computer Science

Exam

The final exam consists in a presentation, and it can be recognized as 3.2 CFR.

References

1. Gidas, B.; Ni, Wei-Ming; Nirenberg, L. Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209-243 (1979).
2. Berestycki, H.; Nirenberg, L. On the method of moving planes and the sliding method. Bol. Soc. Bras. Mat., Nova Sér. 22, No. 1, 1-37 (1991).
3. Fraenkel, L. E. An introduction to maximum principles and symmetry in elliptic problems. Cambridge University Press. x, 340 p. (2011).
4. Protter, Murray H.; Weinberger, Hans F. Maximum principles in differential equations. Prentice-Hall, Inc. X, 261 p. (1967).
5. Serrin, James. A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304-318 (1971).
6. Sperb, Rene P. Maximum principles and their applications. Academic Press. IX, 224 p. (1981).

 Scritto da in 21 Settembre 2023  Senza categoria  Commenti disabilitati su PhD Course: Geometric Analysis
Giu 232023
 

Boundary value problems with nonsmooth and multivalued terms

Prof. Vasile Staicu
University of Aveiro, Portugal

Abstract

The aim of this course is to present some methods and tools used in the study of nonlinear boundary value problems involving multivalued maps and nonsmooth functions and illustrate those methods with some recent results concerning existence and multiplicity of solutions. The course will be given during the period April 26 – May 16, 2023, under the support of Visiting Professor Program of INdAM.

Outline

We start by introducing some notions of multivalued analysis: multivalued maps and their continuity. Then we introduce some elements of nonsmooth analysis that we use to develop nonsmooth critical point theory succeeded in extending a big part of the smooth theory to nonsmooth functionals: the main tool in the variational method for solving boundary value problems, which consists of trying to find solutions for a given boundary value problem by looking for stationary points of a real functional defined on a space of functions in which the solution of the boundary value problem is assumed to lie.

Degree theory is a basic tool of nonlinear analysis and produces powerful existence and multiplicity results for nonlinear boundary value problems. We present a generalization of Brouwer degree theory to multivalued perturbations of monotone type maps, developed in a joint paper with Aizicovici and Papageorgiou (Mem. Amer. Math. Soc., 196, 2008). Then we consider a nonlinear Dirichlet problem with nonsmooth potential (hemivariational inequality) and use variational method to prove two constant sign solutions (one positive and another negative) and then by using degree theoretical approach we prove the existence of a third one, nontrivial solution, distinct from the previous two.

We present several existence and multiplicity results, with sign information for the solutions of nonlinear boundary value problems driven by the Laplacian, the p-Laplacian and then try to extend such results to nonlinear boundary value problems driven by fractional order differential operators and to nonlocal pseudo-differential inclusions, possibly including obstacles or constraints. It is reasonable to expect that the pairing of nonlocal diffusion operators and set-valued reaction will provide a more realistic model for applications to such problems as quantum mechanics, image restoration, and financial mathematics, which typically present a high degree of uncertainty, rather than elliptic equations with smooth, single-valued reactions.

Schedule

27/4 11-13 28/4 9-11 2/5 11-13 5/5 9-11 9/5 11-13 11/5 11-13 12/5 9-11 16/5 11-13 (room F)

References

Vasile Staicu, Lecture Notes, 2023

Open questions will be considered and analyzed with the Ph. D. students.

 

Gen 042023
 

Matrix functions: Computation and application to Complex Networks analysis

Dott. Caterina Fenu
Dipartimento di Matematica e Informatica
Università degli Studi di Cagliari

Abstract

The aim of the course is to introduce basic concepts related to the computation of matrix functions and their use in the analysis of complex networks. Half of the course will be dedicated to the implementation of some algorithms by using the MATLAB software.

Outline 

– Linear algebra: eigenvalue decomposition, Jordan decomposition, singular value decomposition;
– Introduction to matrix functions and bilinear/quadratic forms: orthogonal polynomials, Lanczos algorithms, quadrature formulas;
– Introduction to complex networks theory: graphs, centrality/communicability indices, network indices;
– Use of the MATLAB software.

Exam

The oral exam will consist of a discussion of a research paper concerning the topics of the course.

References

1. G. H. Golub and G. Meurant, Matrices, Moments and Quadrature with Applications, Princeton University Press, Princeton, 2010.
2. N. J. Higham, Functions of matrices: theory and computation, SIAM, 2008.
3. W. Gautschi, Orthogonal polynomials: computation and approximation, Oxford University Press, 2004.

 

 Scritto da in 4 Gennaio 2023  Senza categoria  Commenti disabilitati su PhD Course: Matrix functions: Computation and application to Complex Networks analysis
Nov 292022
 

Factorizations in monoids and rings

Laura Cossu
University of Graz

Outline

Many problems in algebra involve the decomposition of certain elements of a ring (or more generally of a monoid) into a product of certain other elements (hereinafter generically referred to as building blocks) that are in some sense minimal. The classical theory of factorization investigates factorizations in which the building blocks are atoms, i.e., non-unit elements of a monoid that are not products of two non-units. For example, it is well known that every non-zero non-unit of a Dedekind domain (more generally, of a Noetherian domain) can be written as a finite product of atoms and that in general such decompositions are not unique. On the other hand, examples of factorizations that lie beyond the scope of the classical theory include additive decompositions into multiplicative units in rings; cyclic decompositions of permutations in the symmetric group of degree n; idempotent factorizations of the “singular elements” of a monoid; and so on.

After an introductory overview of the history and main results in the classical framework, in this course, we combine the language of monoids and preorders (partial orders that are not necessarily antisymmetric) to make first steps towards the construction of a “unified theory of factorization”. In particular, we prove an abstract existence theorem that recover, among others, a classical theorem of Cohn on atomic factorizations in cancellative monoids, a classical result by Anderson and Valdes-Leon on “irreducible factorizations” in commutative monoids, but also a theorem by Erdos on idempotent factorizations of square singular matrices over fields. We also introduce a notion of “minimal factorization”, suitable for the new abstract setting, and we qualify and quantify the related non-uniqueness properties in specific cases but also in some generality. Examples will help to motivate and illustrate the theory.

Prerequisites: Standard knowledge of (basic) algebraic structures and (binary) relations.

Duration of the course: 10 hours.

Schedule

– November 28, 16:00-18:00 (Aula B)
– November 29, 16:00-18:00 (Aula A)
– November 30, 16:00-18:00 (Aula A)
– December 1, 16:00-18:00 (Aula A)
– December 2, 16:00-18:00 (Aula A)

Exam

Written. Writing of a short essay.

References

1. L. Cossu and S. Tringali, Abstract Factorization Theorems with Applications to Idempotent Factorizations, e-print (Aug. 2021), arXiv:2108.12379.
2. L. Cossu and S. Tringali, Factorization under Local Finiteness Conditions, e-print (Aug. 2022), arXiv:2208.05869.
3. A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math. 278, Chapman & Hall/CRC, Boca Raton (FL), 2006.
4. S. Tringali, An Abstract Factorization Theorem and Some Applications, J. Algebra 602 (2022), 352–380.

 Scritto da in 29 Novembre 2022  Senza categoria  Commenti disabilitati su PhD Course: Factorizations in monoids and rings
Mar 302021
 

Introduction to Finite Mixture Models and Latent Class Analysis

Dott.ssa Silvia Columbu
Postdoctoral researcher in Statistics
Department of Mathematics and Computer Science, University of Cagliari

Outline

This course is intended as both a theoretical and practical introduction to finite mixtures as a model based clustering tool. We will introduce the concept of finite mixture of distributions for categorical as well as continuous variables. We will study the problem of maximum likelihood via expectation-maximization (EM) algorithms. We will start from simple data structures and then move to the introduction of regressors and to more complex model structures. We will also explore the problem of model selection and choice of the number of clusters. We will see practical applications with the aid of devoted packages in the statistical software R. Students will learn how to implement an analysis and how to read and interpret outputs and graphical representations of results.

Schedule

Duration: 20 hrs, in the period going from 18th to 26th of February 2021, for a total of 4 CFR

Course schedule

 Scritto da in 30 Marzo 2021  Senza categoria  Commenti disabilitati su PhD Course: Introduction to Finite Mixture Models and Latent Class Analysis
Nov 162020
 

Corso di dottorato (20 ore con esame – 4 CFR)

Il dott. Mirko Marras  terrà, a partire dal 9 dicembre 2020, un corso per il dottorato dal titolo “Python for machine Learning Research”.

Le informazioni sono disponibili al seguente  link

Seminario dottorale (20 ore con esame – 4 CFR)

Il prof. Alessandro Buccini terrà, a partire dal 3 giugno 2020, un corso per il dottorato dal titolo “Regularization of ill-posed linear inverse problems”. Le lezioni si terranno online, su una piattaforma che verrà comunicata in seguito, e includeranno alcune sessioni di laboratorio Matlab.

L’abstract del corso è disponibile a questo link.
 Scritto da in 16 Novembre 2020  Senza categoria  Commenti disabilitati su PhD Courses: Python for machine Learning Research / Regularization of ill-posed linear inverse problems
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