The research center of Applied Analysis focuses on fundamental, applicative and interdisciplinary research in the following fields: nonlinear functional analysis, complex analysis, optimization, ordinary and partial differential equations, theoretical mechanics and mathematical modeling.
Our research concentrates on:

• Fixed point and variational methods for nonlinear equations and systems;
• Loewner theory and open problems of univalent functions;
• Nonlinear multicriteria optimization;
• The Poincare problem and periodic solutions for differential systems;
• Geometry of Banach spaces and generalized metric spaces;
• Nonlinear elliptic boundary value problems in Lipschitz domains and different spaces, using methods of potential theory and variational and topological techniques;
• Numerical and approximation methods for nonlinear equations;
• Analysis of some mathematical models from biology and medicine.

Research directions: Category theory and Homological algebra, Universal algebras and multialgebras, Group graded rings, Hopf algebra actions,Abelian groups and Modules, Representations of finite groups and finite dimensional algebras, Applications of algebra in Coding theory

Research directions: Artificial Intelligence, Computational Intelligence, Machine Learning, Soft Computing, Real Applications

Software Engineering research group interests are focused on the following directions:

• Program Analysis and Verification: formal mechanisms (such as Session Logic, K-framework) applied to specify and verify different program properties;
• Software Quality: studies related to evaluation and estimation of software quality factors in large applications and in different versions, and their relation to OO metrics;
• Model Driven Engineering: studies related to Executable Models and their specification language (fUML) and their impact on software development;
• Component Based Software Engineering: deals with developing software as a composition of various third party components. Our studies address problems such as: component selection, constraint based configuration of components and different optimizations.

Research directions: Program Analysis and Verification, Software Quality, Model Driven Engineering, Component Based Software Engineering

The research center focuses on research activities in the fields of modeling, optimization, and simulation, supported by the existence of a high-performance computing infrastructure – Kotys (https://hpc.cs.ubbcluj.ro/).
The research targets the following areas:

• Mathematical modeling for various processes and phenomena / Iterative methods;
• Numerical and statistical simulations / Simulations of natural phenomena;
• Image visualization and processing;
• Parallel and distributed programming / Models for parallel and distributed computing;
• Data analysis and processing / Data engineering;
• Domain-Specific Languages;
• Optimization → Mathematical models / Artificial intelligence;
• Software development models focused on performance and productivity.

The equilibrium problem and its applications (investigating the existence of solutions, stability and sensitivity of solutions, algorithms for solving the problem), monotone operators, variational inequalities, scalar, vector and set-valued optimization.

The research group focuses on the following research directions:

1. Critical sets and critical values
   1. 1.1 Functions with a finite number of critical points/components
   2. 1.2 The minimum number of critical points of functions between two manifolds
   3. 1.3 Functions with large critical sets (of dimension at least 1)
2. Rodrigues-type formulas for the general linear group and its subgroups
   1. 2.1 Applications in geometric mechanics
3. Study of properties in generalized geometry
4. New approaches to the study of Blundon-type configurations
5. Convexity and truncated convexity
   1. 5.1 The Gauss map and the spherical image of a closed convex set
   2. 5.2 Truncated convex functions
   3. 5.3 Analysis of real-valued C² functions through the associated Hess⁺ region
   4. 5.4 Properties of convex sets preserved under Minkowski addition
   5. 5.5 Metric projection onto a closed convex set
6. Geometry of the hyperbolic plane
7. Differentiable supermanifolds
8. Geometry of black holes
9. Arithmetic geometry and algebraic points: the study of algebraic points on curves and varieties. In particular, the group focuses on results supporting the principle that “geometry controls arithmetic”. One such result, in the case of curves, is Faltings’ Theorem, as well as much more recent theorems involving covers of curves of genus zero and 1. The group is also interested in investigating the distribution of the number fields in which these points occur, in terms of their discriminant.
10. Arithmetic of hyperelliptic curves and their Jacobians: quantitative and density results concerning the number of curves of genus g ≥ 2 whose Jacobians have positive Mordell-Weil rank, the construction of explicit families of such curves, and the study of implications for hyperelliptic curve-based cryptography and quantum algorithms for computing discrete logarithms.
11. Post-quantum cryptography and homomorphic cryptography: the study of the algebraic and arithmetic structures underlying modern cryptographic schemes, with an emphasis on lattice-based cryptography and homomorphic cryptography. The group is particularly interested in the use of number fields and cyclotomic fields in the design and optimization of cryptographic schemes, as well as questions concerning their security against classical and quantum algorithms. A complementary research direction concerns the connections between number theory, additive combinatorics in finite groups, and improving the efficiency of cryptographic operations.

1. Loewner chains and the Loewner differential equation in Cⁿ. Analytic and geometric approaches.
2. Modern applications of Loewner’s theory and control theory to study of univalent mappings with parametric representation on the unit ball in Cⁿ.
3. Variational methods in the study of extremal problems for univalent mappings on the unit ball in Cⁿ.
4. Approximation results for families of univalent mappings in Cⁿ.
5. Harmonic and pluriharmonic univalent mappings.
6. Global injectivity results on finite and infinite dimensional complex Banach spaces.
7. Applications of geometric function theory in the study of free boundary problems (Hele-Shaw flows).

Research directions: Complex analysis in one and higher dimensions, Geometric function theory, Loewner theory in Cⁿ, Univalent mappings, Applications of complex analysis in fluid mechanics

The research group on Nonlinear Operators and Differential Equations (NODE) focuses on fundamental, applicative and interdisciplinary research in the field of Nonlinear Analysis and the Theory of Integral, Differential and Partial Differential Equations. Our research concentrates on:

• Development of operator methods (Fixed Point Theory, Critical Point Theory, Coincidence Point Theory, Spectral Operator Theory) for equations and system of nonlinear equations;
• Applications of the operator methods for the study of various integral, differential and partial differential equations and systems;
• Poincare problem and periodic solutions for systems of differential equations;
• Banach spaces and generalized metric spaces geometry;
• Real-life applications.

The scientific activity of the research group takes into account institutional and personal international cooperation with researchers from Spain, France, Italy, Russia, Hungary, Serbia, China, Taiwan, Thailand, and so on. An important role of the research group scientific life is the research seminar on Nonlinear Operators and Differential Equations (NODE) which has activities every Thursday from 10.30 to 11.30, in room e, Mathematica building.

Research directions:

• Development of various operator methods for the study of different nonlinear equations and systems of equations, at least from the following perspectives: existence, uniqueness, qualitative properties of the solutions;
• Applications of the abstract operator methods to nonlinear integral, differential and partial differential equations;
• Modelling and simulation for real-life problems from economy, biology, medicine, etc.

Wavelets-based collocation methods for integral equations and equations arising in fluid flow problems; applications of symbolic-numerical algorithms; linear approximation processes; interpolation operators for curved domains; computer aided geometric design by using non-negative normalized B-basis functions of extended Chebyshev spaces; long-time behavior of the solutions of certain stochastic PDE.

Research directions: Numerical methods, Approximation operators, Computer aided geometric design

• Nonlinear elliptic boundary value problems in fluid mechanics based on potential theory, variational methods, and topological methods.
• Boundary value problems on domains with low regularity in the n-dimensional Euclidean setting or compact Riemannian manifolds.
• Partial differential equations, boundary integral equations, integro-differential equations, and their applications in fluid mechanics.
• Pseudodifferential elliptic operators.
• Applications of complex analysis in fluid mechanics.
• Numerical analysis in fluid mechanics and heat and mass transfer theory.
• Dynamical systems in celestial mechanics and astrononmy.

Research directions: Potential theory, Fluid mechanics, Partial differential equations, Computational analysis, Celestial mechanics

The research group on Machine Learning (ML) focuses on fundamental, applicative and interdisciplinary research on Machine Learning. Our research concentrates on pure theoretical and algorithmic contributions to machine learning (supervised, unsupervised and reinforcement learning models, classification and regression models based on relational association rule mining, hybrid and dynamic ML models, ) as well as on interdisciplinary applications of machine learning in various domains, like: software engineering (SBSE – Search-Based Software Engineering), bioinformatics and computational biology, bioarchaeology, computer vision, natural language processing, social media, etc. We aim to explore new domains in which machine learning solutions are applicable.

Research directions: Supervised, unsupervised and reinforcement learning, Hybrid and dynamic machine learning models, Relational association rule mining, Search-based software engineering, Applications of machine learning in various domains, like: software engineering, bioinformatics and computational biology, bioarchaeology, computer vision, natural language processing, social media, etc.

Research directions: Distributed Systems, Computer Networks, Web Communications and Optimization, Web Services Architectures, Cloud systems, Multimedia Streaming, Distributed Virtual Environments, Virtual Reality

The “High Performance Interdisciplinary Applications” research group functions within the Center of Modeling, Optimization and Simulation (MOS) and targets fundamental research for interdisciplinary application domains. Using HPC-UBB infrastructure capable of 40TFlops Rmax, the research includes modeling and simulation of different natural phenomena, optimization, visualization, data analysis, parallel and distributed programming, big data analytics, and high performance computing.

Research directions: Distributed and parallel programming, Models optimization, Simulation and visualization, Image processing and virtual reality, Big Data Analytics, Numerical and statistical simulations

Formal concept analysis (FCA) is a prominent data analysis method for extracting, processing and knowledge representation. We are interested (but not restricted to) the following topics:

1. Theory development: Triadic and temporal FCA (3FCA, TCA);
2. Applications: weblog analysis, business intelligence, using VR and AR to visualize conceptual structures.

Research directions: Formal Concept Analysis and Applications

The Metaheuristics for Complex Systems (MECO) research group is multidisciplinary bringing together expertise in metaheuristic computing, computational intelligence, machine learning and complex systems. The research interests of the group focus on modeling, analysis and development of metaheuristic computing techniques from theoretical studies and empirical investigations to real-world applications. Members of the MECO group have vast experience in the development of various computational intelligence models, hybrid algorithms and machine learning techniques and their application in the context of complex systems and problems.

Research group type: fundamental and applied, multi-, inter- and trans-disciplinary research in the field of software testing.

The group investigates research problems at the intersection of Software Testing, Software Metrics, Software Teams, Software Project Management, Artificial Intelligence and Computer Science Education (learning/teaching).