**Faculty of Mathematics and computer science**

**Faculty of Mathematics and computer science**

The Faculty of Mathematics and Computer Science has more than 100 research staff including many internationally renowned specialists, including four Humboldt Foundation Fellows and one Fulbright Foundation Fellow. Faculty members conduct research jointly with scholars from such renowned universities as Princeton University, the Sorbonne and the University of California, Los Angeles. Publications by our staff can be found in prestigious scientific journals and at top conferences. The Faculty is involved in more than a dozen multi-source funded research projects. The scope of interests of individual departments and laboratories of the Faculty is described below.

## INSTITUTE OF COMPUTER SCIENCE

### DEPARTMENT OF SOFTWARE ENGINEERING

Deals with practical aspects of IT technologies. Research topics include:

- programming languages: low-level programming, object-oriented programming, aspect-oriented programming, security policies
- methods and tools of software development: enterprise platforms, frameworks, integrated IDEs, continuous integration & delivery
- software development: multitier and distributed software architecture, design patterns, architecture patterns, integration patterns,
- project management: principles for software management.

### DEPARTMENT OF PROGRAMMING LANGUAGES

Conducts research on formal semantics of programming languages and their theoretical foundations with particular focus on:

- functional languages,
- Curry-Howard isomorphism,
- formal proofs,
- logical systems.

### DEPARTMENT OF NUMERICAL METHODS

Investigates theory and applications of computational methods, among others:

- algorithms for curves and surfaces ,
- dual bases,
- convergence acceleration,
- mathematical methods in computer graphics ,
- optimalisation methods ,
- computational statistics
- approximation theory
- orthogonal polynomials and special function

### DEPARTMENT OF COMBINATORIAL OPTIMISATION

The research focuses on discrete optimisation problems arising in network design, scheduling, logistics, planning, and graph theory. We design efficient algorithms along with provable guarantees on the quality of constructed solutions. Areas of expertise include:

- approximation algorithms for NP-hard graph problems
- algorithms for travelling salesman problems
- structure of graphs and matchings
- online algorithms
- algorithmic game theory

### DEPARTMENT OF COMPUTER SCIENCE AND DATABASE THEORY

The research concerns mathematical foundations of computer science and practical and theoretical aspects of the application of logic in various fields of computer science, in particular:

- theory of automata
- hardware and software verification
- model checking
- automated deduction
- modal and temporal logic
- two-variable fragment and guarded fragment
- finite model theory
- databases theory

### DEPARMENT OF COMPUTATIONAL COMPLEXITY AND ALGORITHMS

Research aims at seeking effective algorithms solving variety of problems, among others:

- algorithms on words
- information security
- distributed computing
- automata theory
- computational complexity and formal languages

### LABORATORY OF COMPUTER GRAPHICS

The Laboratory has been created for research, education, popularisation of omputer graphics, and co-operation with industry. It also hosts local Wrocław ACM SIGGRAPH Chapter as well as Wrocław Khronos Chapter. Research includes:

- photorealistic Image Synthesis and Monte Carlo methods
- GPU for visualizations and computations (OpenGL/CUDA/Vulkan
- 3D Scanning, photogrammetry, 3D filming, MOCAP
- image processing and computational photography

### LABORATORY OF COMPUTATIONAL INTELLIGENCE

Research focuses on intelligent approaches to data analysis and developing intelligent decision support systems, with particular focus on:

- evolutionary algorithms,
- neural networks
- text processing and analysis

### HARMONIC ANALYSIS

The research interests of the group concentrate on real analysis, harmonic analysis, and applications of analysis in probability theory. Probabilistic research concerns stochastic equations, branching processes, and branching random walks. Analytic studies focus on discrete harmonic analysis, analytic number theory, real analysis on Lie groups and spaces of homogenuous type, Littlewood-Paley theory, singular and oscillating integrals, functional spaces and orthogonal polynomials.

### PARTIAL DIFFERENTIAL EQUATIONS

Research is focused on the asymptotics of non-linear evolution equations with dissipation in continuum mechanics (including Navier-Stokes system and Boltzmann equation). The group maintains long-standing scientific contacts with French and Austrian mathamaticians.

### APPLICATIONS OF PROBABILITY THEORY

Research topics include:

- Gaussian processes; extreme value theory of Gaussian processes; exact asymptotic for extrema; Pickand’s constants
- Levy processes; extreme value theory of Levy processes; fluctuation theory of Levy processes; reflected Levy processes; refracted Levy processes; heavy tailed distributions
- Markov processes; stochastic networks, queueing networks; fluid queues; Kendall walks; speed of convergence for finite Markov chains; cut-off phenomena; Markov chains in cryptography; MCMC; dual Markov chains
- point processes; Boolean models in stochastic geometry, determinantal and permanental point processes, Gibbs point processes; correlation inequalities; FKG inequalities; particle systems
- generalized convolutions; Kendall convolutions; non-commutative convolutions; sequences of moments for convolutions
- financial and actuarial mathematics; Parisian ruin models; mortality rate models; optimal detection in Levy models; copulas and Levy copulas

### NONCOMMUTATIVE PROBABILITY, HARMONIC ANALYSIS AND QUANTUM FUNCTIONAL ANALYSIS

Generalised noncommutative stochastic processes and connections with deformed commutation relations and with classical probability, and properties of the associate von Neumann algebras. Brownian Motions related to Coxeter groups of type B and D. Simultaneous infinite divisibility of probability measures in both classical and free probability and their Bargmann representations. Properties of deformations of operators corresponding to deformations of measures in noncommutative probability. Combinatorial aspects of noncommutative probability. New models of independence in noncommutative probability. Levy processes on quantum groups. Orthogonal polynomials and the problem of moments.

### GEOMETRY

Research interests focus on geometric group theory, symplectic topology, and contact topology.

### MODEL THEORY

Research interests of the group focus on classical and algebraic model theory with applications to differential algebra, and o-minimality.

### TOPOLOGY AND SET THEORY

Research is focused on Banach space theory, forcing axioms, descriptive set theory, continuum theory and applications of infinitary combinatorics in topology, functional analysis, and measure theory.