The group deals with the development of analysis and algorithms for linear algebra and optimization problems, especially in the case of big dimensions. Emphasis is given to structures also of geometric type, as in the case of optimization on differentiable manifolds. One of the main fields of application of this research is the "data science", which includes, as subject of study, the analysis of complex networks and machine learning.

**SSD sectors:**

**ERC sectors:**

PE1_17 - Numerical analysis

PE1_18 - Scientific computing and data processing

**Representative:**

The research group deals with the study of methods of Real Analysis and Approximation Theory aimed at signals and images reconstruction and processing. It is now evident, in fact, the strong need, in almost all the applied sectors in the medical and industrial fields, for the mathematical modeling of processes and their consequent algorithmic application. Science, even experimental science, needs very refined and rigorous scientific tools that require mathematical skills aimed at the applications. In particular, the theoretical models developed within this research topic, which mainly concern with the approximation properties of families of operators in various functional spaces, are implemented to provide mathematical and numerical algorithms for digital image processing aimed at the solution of concrete problems in particular in medical and engineering fields. The group refers to the "Imaging and Computer Vision (ICV)" laboratory, at the Department of Mathematics and Computer Science, created also with the purpose of developing research in this area. The same group of researchers is among the founding members of the Working Group of the Unione Matematica Italiana called "Theory of Approximation and Applications" which brings together Italian researchers who are experts in these research topics. Many of the researchers in this group are lecturers in the Laurea Magistrale in Mathematics for Biomedical and Industrial Applications. In addition to the various international scientific collaborations, the group has a strong interdisciplinary character and has developed strong collaborations and research in the medical field (mathematical models and algorithms for the study of aneurysmatic pathologies, retinal pathologies, brain pathologies for the search of biomarkers for Alzheimer's disease) and in the engineering field (mathematical models for the seismic vulnerability of buildings, for the study of thermal bridges and acoustic bridges), making use of an intense collaboration with the Departments of Civil and Environmental Engineering (DICA), of Engineering (DI) and of Medicine and Surgery of the University of Perugia, and with the Radiology and Medical Physics section of the Santa Maria della Misericordia Hospital in Perugia. The mentioned collaborations have led to the production of scientific publications in ISI journals, as well as very intense project collaborations. On these studies members of the research group have two patents. There are also active collaborations with local and non-local companies. In many of these areas, research is still being developed: new areas of application are opening up (for example the study of thrombotic pathologies in cancer patients) and others will open in the future. A line of research in continuous ferment, where both theoretical and applied research will surely destined to spark a deep development and interest within the international scientific community.

**SSD sectors:**

MAT/05 - Mathematical Analysis

MAT/08 - Numerical Analysis

**ERC sectors:**

PE1_8 - Analysis

PE1_20 - Application of mathematics in sciences

PE1_21 - Application of mathematics in industry and society

**Representative:**

(PhD student)

(Collaborators)

The main focus of such research field are: High performance Computing (GPGPU Computing, Cloud Computing) e Quantum Computing; Artificial Intelligence (Neural Networks, Machine Learning, Signal Processing); Virtual/Augmented/Mixed and Extended Reality; Computational Science; Elearning, EAssessment.

**SSD sectors:**

INF/01 – Computer Science

ING-INF/05 - Information processing systems

**ERC sectors:**

PE6_11 - Machine learning, statistical data processing and applications using signal processing (e.g. speech, image, video)

PE6_12 - Scientific computing, simulation and modelling tools

PE6_4 - Theoretical computer science, formal methods, and quantum computing

PE6_5 - Cryptology, security, privacy, quantum crypto

PE6_9 - Human computer interaction and interface, visualization and natural language processing

**Representative:**

The focus of the research group are: Artificial Intelligence, Knowledge Representation (Argumentation), Automated Reasoning (Constraint Solving and Programming), Cybersecurity, Distributed Ledgers technology, protocols and algorithms, Fintech (blockchain, smart contract, cryptocurrencies, stochastic control problems with applications to finance and insurance, modeling of investor sentiment, evaluation of financial derivatives), explainable AI, logica reasoning, trustable AI, interpretable Machine Learning.

**SSD sectors:**

INF/01 – Computer Science

SECS-S/06 – Applied Economy

**ERC sectors:**

PE1_1 - Logic and foundations

PE1_13 - Probability

PE1_14 - Statistics

PE1_16 - Mathematical aspects of computer science

PE6_4 - Theoretical computer science, formal methods, and quantum computing

PE6_5 - Cryptology, security, privacy, quantum cryptography

PE6_6 - Algorithms, distributed, parallel and network algorithms, algorithmic game theory

PE6_7 - Artificial intelligence, intelligent systems, multi agent systems

**Representative:**

(Collaborators)

The research group deals with models and technologies for e-learning and teaching support, with particular reference to models of user behavior and human-machine interaction, adaptive interactive interfaces, technologies to support mobility and disabilities, distributed architectures for elearning.

**SSD sectors:**

INF/01 – Computer Science

**ERC sectors:**

PE6_10 - Web and information systems, database systems, information retrieval and digital libraries, data fusion

PE6_9 - Human computer interaction and interface, visualization and natural language processing

PE6_7 - Artificial intelligence, intelligent systems, multi agent systems

**Representative:**

Poisson structures are geometric structures that play a decisive role in explaining the relationships between quantum mechanics and classical Hamiltonian mechanics and are the basis for some of the most popular approaches to quantization procedures such as "deformation quantization" and geometric quantization. The study of Poisson manifold quantization using symplectic integration and producing as output the C * -algebra of convolution of a groupoid has opened new study scenarios on the relationships between geometric invariants and quantum properties. In particular, the attempt to understand the functorial properties of this construction naturally leads to consider the notion of differentiable stack and the analysis of homotopy properties, both in the proper sense and in categorial sense of the Poisson invariants.

**SSD sectors:**

MAT/03 – Geometry

**ERC sectors:**

PE1_5 - Lie groups, Lie algebras

PE1_6 - Geometry and Global Analysis

PE1_7 - Topology

PE1_9 - Operator algebras and functional analysis

PE1_12 - Mathematical physics

**Representative:**

The core themes of this research field are Galois Geometries, Algebraic Curves in positive characteristic and Combinatorial Graphs. In their study, several tools developed reveal to be particularly performing in the interaction both with classical mathematics (Number Theory, Algebraic Geometry, Group Theory), and with the more recent ones connected with applications to Code Theory and Cryptography, with particular reference to correcting and covering codes, to secret sharing schemes and to functions on highly non-linear finite fields. One of the main objectives is to create infinite new classes of remarkable objects in Galois spaces and / or algebraic curves and / or combinatorial graphs with many automorphisms. There are several reasons that justify this choice. First, this is consistent with the discrete analog of Felix Klein's "Erlangen Program", considered by many to be the starting point of modern geometry: organizing geometric knowledge (in our case combinatorial knowledge) in terms of group theory . Secondly, algebra is able to capture / enlighten structures that would otherwise remain hidden. Finally, combinatorial objects with a high degree of symmetry are particularly relevant in the applied field also because they can be stored more efficiently in terms of memory space. The research carried on by the group is also developing in the direction of cryptography applied to Cloud Encryption. Cloud encryption uses advanced encryption techniques to protect the data that will be used or stored in the cloud. It allows users to conveniently and securely access shared services, so that all data hosted by providers are protected with encryption. The mathematical primitives, more precisely algebraic, used in cloud encryption are intended to encrypt sensitive data without delaying the exchange of information, thus protecting critical data beyond the business IT environment.

**SSD sectors:**

MAT/02 - Algebra

MAT/03 – Geometry

**ERC sectors:**

PE1_2 - Algebra

PE6_5 - Cryptology, security, privacy, quantum crypto

PE1_15 - Discrete mathematics and combinatorics

PE1_3 - Number theory

PE1_4 - Algebraic and complex geometry

PE1_16 - Mathematical aspects of computer science

**Representative:**

The group, referring to the Laboratory of Knowledge and Information Technology, successfully investigates in the field of artificial intelligence in general and with particular focus on evolutionary computing, complex networks and information retrieval, producing models, systems and computing techniques for the solution of problems ranging from combinatorial and function optimization to network immunization, complex network analysis, link prediction and sentiment analysis. The group is also specialized in the field of machine learning, data analysis and recommender systems with particular focus on neural networks, neuroevolution, adversarial machine learning and text data analysis, producing models and systems for the solution of problems ranging from network optimization, neural, object classification, adversarial machine learning, product and user recommendation systems, extrapolation of knowledge and information from texts.

**SSD sectors:**

INF/01 – Computer Science

**ERC sectors:**

PE6_10 - Web and information systems, database systems, information retrieval and digital libraries, data fusion

PE6_7 - Artificial intelligence, intelligent systems, multi agent systems

The topic deals with the study of correspondence between nineteenth-century mathematicians, deepening the analytical and geometric issues that arise from it. It is planned to use aspects of the history of mathematical sciences as a resource for teaching and museology.

**SSD sectors:**

MAT/04 - Complementary Mathematics

**ERC sectors:**

SH6_10 - Global history, transnational history, comparative history, entangled histories

PE1_1 - Logic and foundations

**Representative:**

**Staff member**:An interaction model between small amplitude acoustic waves and membrane-type extended reaction surfaces is studied. The specificity of the research is the mutual interaction between the physical-mathematical modeling aspect and the theoretical aspect, in the sense that the analytical results can provide indications on the appropriateness of the model. The problem is also related to hyperbolic evolution models for composite systems. In the field of stationary problems, we study some classes of elliptic-type equations or inequalities associated with problems governed by different non-linear operators of the p-Laplacian type or of the non-homogeneous type such as the (p, q) -Laplacian involving critical non-linearities modeling phenomena of interest in the physical, economic, biological and statistical fields. In particular, the p-Laplacian models both dilating and pseudoplastic non-Newtonian fluids such as paints, blood, asphalt or toothpaste. Finally, in the context of evolutionary problems, Fujita-type theorems are studied for quasilinear inequalities of parabolic type having nonlinearities involving nonlocal terms in the spirit of the Choquard equation that appears in various fields of quantum physics and in the theory of relativity, or non-linearities dependent on the gradient and with degenerate or singular weights that generalize models of population dynamics.

**SSD sectors:**

MAT/05 Mathematical Analysis

**ERC sectors:**

PE1_11 - Theoretical aspects of partial differential equations

PE1_12 - Mathematical physics

PE1_20 - Application of mathematics in sciences

PE1_21 - Application of mathematics in industry and society

PE1_8 - Analysis

PE1_9 - Operator algebras and functional analysis

**Representative:**

The research-action group intends to propose and test didactic paths, aimed at the formation of mathematical skills, for each level of school. The leading idea is that one of an education to modeling with elementary tools. The dynamics of modeling are adopted as an engine of didactic innovation. The project, by creating an unusual synergy between the world of applied mathematical research and school, intends to provide teachers with the opportunity for an important cultural and professional enrichment, a source of innovative teaching inspiring.

**SSD sectors:**

MAT/04 – Complementary Mathematics

**ERC sectors:**

PE1_20 - Application of mathematics in sciences

PE1_21 - Application of mathematics in industry and society

SH4_11 - Education: systems and institutions, teaching and learning

**Representative:**

**Staff member**:The group deals with the study of mathematical models of phenomena in the physical, biological, economic and techno-food fields through a combination of topological methods and functional analysis tools, as well as multivocal analysis. The mathematical formalization naturally leads to consider both differential and integro-differential equations or inclusions subject to various kinds of initial conditions (such as periodic, antiperiodic, multipoint and mean value problems), to pulses or with the presence of delay. The results obtained, that range from the determination of the existence and uniqueness of solutions, to their stability or continuous dependence, find applications to population dynamics, gas kinetic theory, portfolio theory, sterilization problems, heat treatments, microfiltration and clarification of fluids and study of rain infiltration in the soil.

**SSD sectors:**

MAT/05 – Mathematical Analysis

**ERC sectors:**

PE1_8 - Analysis

PE1_6 - Topology

PE1_13 - Probability

PE1_20 - Application of mathematics in sciences

PE1_21 - Application of mathematics in industry and society

**Representative:**

The focus concerns the study of the computational complexity of optimization problems defined in various contexts, including communications and / or processing networks. The approach is to propose optimal, approximating and heuristic algorithms in various applicative contexts including the need to disseminate and / or retrieve data. Wireless networks, ad hoc networks and social networks are being established and discontinued within a short time in order to take advantage of local and instant information. Examples of such networks may concern the need to retrieve environmental information through networks of sensors, drones, or mobile devices such as robots or more commonly used ones such as laptops, smartphones, smartwatches and tablets. The diffusion of such networks is certainly promoted by technological advancement and by the spread of increasingly performing communication devices. These networks are now also used in agriculture where monitoring based on automatic data collection facilitates the automation of decision-making processes. The research group that deals with these issues also avails itself of the collaboration of various researchers from other Italian and foreign Universities. The Haly-id project is highlighted **haly-id****.**

**SSD sectors:**

INF/01 – Computer Science

**ERC sectors:**

PE6_6 - Algorithms, distributed, parallel and network algorithms, algorithmic game theory

PE6_2 - Computer systems, parallel/distributed systems, sensor networks, embedded systems, cyber-physical systems

PE7_8 - Networks (communication networks, sensor networks, networks of robots, etc.)

**Representative:**

(PhD student)

In this research group we deal with complex systems, that is, systems made up of a large number of elements that show a global behavior that is not easily understood by the rules with which the single entities interact with each other. In particular, we use techniques related to statistical physics and dynamical systems to study the formation of patterns and collective dynamics in reaction-diffusion systems both in continuous space and on a complex network. The applications, which are mainly focused in the biological field, make use of a highly multidisciplinary approach thanks to collaborations with various experimental groups.

**SSD sectors:**

MAT/07 – Mathematical Physics

**ERC sectors:**

PE1_12 - Mathematical physics

PE1_10 - ODE and dynamical systems

PE1_21 - Application of mathematics in sciences

PE3_15 - Statistical physics: phase transitions, condensed matter systems, models of complex systems, interdisciplinary applications

**Representative:**

Analysis of the contribution of the conditional probability approach consistent with merging techniques and aggregation of different information sources, in line with what has already been introduced in several recent contributions on the so-called Statistical Matching, on Probabilistic Databases and on Fuzzy aggregation operators. Search for areas of application of classification techniques, in particular under partial information, which allow the identification of risk conditions based on techniques of Fuzzy rules or aid to decisions with techniques of Rough Set. The group also studies techniques and results of algebraic geometry for the characterization and analysis of probabilistic models based on partial assignments of coherent conditional probabilities. This approach is based on what has been proposed in the literature for the selection of Bayesian network models, in particular the so-called "naive" ones, represented by varieties of the secants of a Segre variety, extending it to the case of more general models with the presence of logical constraints (structural zeros) and only partial or "extreme" assignments (conditioners of null probability). The translation of these constraints into algebraic-geometric properties could lead to properties and characterizations that have not yet been explored.

**SSD sectors:**

MAT/06 – Probability and Mathematical Statistics

SECS-S/06 – Applied Economy

MAT/02 - Algebra

MAT/03 - Geometry

**ERC sectors:**

PE1_13 - Probability

PE1_16 - Mathematical aspects of computer science

PE6_7 - Artificial intelligence, intelligent systems, multi agent systems

SH4_7 - Reasoning, decision-making; intelligence

PE1_2 - Algebra

PE1_6 - Geometry and Global Analysis

PE1_14 - Statistics

**Representative:**