# ALGA – Advances in Lie Group Analysis

## International research centre ALGA

**Lie group analysis is a growing field of mathematics with numerous applications. Ideas of symmetry and invariance that lie at the core of Lie’s theory, permeate all mathematical models in natural and engineering sciences. ALGA is an international research centre aimed at producing new knowledge and enhancing the understanding of classical and modern group analysis.**

## Objectives

The main objectives of ALGA are:

- To advance studies in modern group analysis, differential equations and mathematical modelling.
- To develop and teach courses in Lie group analysis and its applications targeted towards producing expertise and skilled researchers.
- To enhance the understanding of the classical heritage in group analysis.
- To provide rapid publication of significant results in group analysis.
- To coordinate efforts of world experts in implementing a database containing all the latest information on theoretical group analysis and applications to mathematical models.

## Landmarks

- 1983 – A research laboratory for Lie group analysis was founded by N.H. Ibragimov at Ufa State Aviation and Technical University, Russia
- 1987 – Continued at the Institute of Applied Mathematics and Institute of Mathematical Modelling of the USSR Academy of Sciences in Moscow
- 1994 – International status is achieved as Centre for Symmetry Analysis and Differential Equations at the University of the Witwatersrand, Johannesburg, South Africa
- 1998 – Continued as International Institute for Symmetry Analysis and Mathematical Modelling (ISAMM) at the University of North-West, South Africa
- 2000 – Continued as an international research centre ALGA at the Department of Mathematics and Science, Blekinge Institute of Technology (BTH), Sweden
- 2004-2009 – ALGA was a university centre at BTH.
- 2012 – ALGA is reestablished as an international research centre ALGA at the Department of Mathematics and Natural Sciences, Blekinge Institute of Technology (BTH), Sweden for cooperation with the laboratory “Group Analysis of Mathematical Models in natural and engineering sciences”

## Research

ALGA carries out research and provides research projects in theory and applications of Lie group analysis in the following main areas:

## I. Theory:

- Perturbation methods and approximate symmetries
- Equivalence groups: Invariants and symmetry classification of families of differential equations
- Group analysis in theory of distributions and boundary value problems
- Invariance principles in nonlinear modelling
- Classical heritage of Lie group analysis: Developments in history and language

## II. Applications:

- Symmetries of differential and integro-differential equations and group methods of integration of systems of equations
- Symmetry and conservation laws
- Diffusion problems and wave propagation
- Group analysis of mathematical models in nonlinear physics and engineering
- Group analysis in biomathematics and ecology
- Database in applied group analysis

## Education

Formulation of fundamental natural laws and technological problems in the form of rigorous mathematical models is given prevalently in terms of differential equations. Therefore, differential equations is a most important discipline in mathematical education. Numerous ad hoc methods for solving various particular types of differential equations have been developed within 300 years. They are summarized in voluminous catalogues containing e.g. about 400 types of second-order ordinary differential equations together with special methods of their solution.

Sophus Lie showed that the old methods of integration could be deduced simply by means of his theory. In particular, Lie reduced the classical 400 types of equations to 4 types only!

However, students, teachers and engineers still use ad hoc methods presented in traditional texts instead of dealing with Lie’s four canonical equations. “Often the less there is to justify a traditional custom, the harder it is to get rid of it” (M. Twain).

ALGA develops courses based on Lie group analysis. We developed new mathematical programmes using Lie group methods in differential equations and mathematical modelling in nonlinear sciences. Introduction of these courses into curricula attracted more than 100 students instead of 10 at Moscow Institute of Physics and Technology, and 60 instead of 12 at Blekinge Institute of Technology.

New course on Canvas (English version)

New course on Canvas (Russian version)

## ALGA publications

ALGA publications include Archives of ALGA, textbooks, monographs and lecture notes on group analysis. When possible, a link has been provided to the online version of the publication. Requests for hard copies of these materials can be sent to

The following publications are available on the DIVA portal:

- Archives of ALGA
- Nail H. Ibragimov, Selected works
- Lie Group Analysis: Classical Heritage

Search for publications on the DIVA portal

Another possibility is to look for books and articles on ResearchGate

The textbook N.H. Ibragimov, A Practical Course in Differential Equations and Mathematical Modelling: Classical and new methods, nonlinear mathematical models, symmetry and invariance principles has been translated into Swedish (2009), Chinese (2010), Russian (2012) and recently also appeared in German (2018).

## Director of ALGA

Nail H. Ibragimov was educated at Moscow Institute of Physics and Technology and Novosibirsk University, and worked in the USSR Academy of Sciences from 1965 to 1993. Since 1976 he lectured intensely all over the world. During 1994-2000 he held a professorial position in South Africa. In 2000 he became Professor of Mathematics and director of ALGA at the Blekinge Institute of Technology (BTH), Karlskrona, Sweden. Today, Nail Ibragimov is Professor Emeritus at the Department of Mathematics and Natural Sciences at BTH and director of ALGA .

Professor Ibragimov is widely regarded as one of the world’s foremost experts in the field of symmetry analysis of differential equations. He initiated and conducted major developments in theory and applications of modern group analysis. His contributions are: theory of generalized motions in Riemannian spaces containing Killing’s equations as a particular case (1969); extension of Pauli’s group for the Dirac equations (1969); differential algebraic approach to conservation theorems and proof of the inverse Noether theorem (1969); discovery of a group theoretic nature of the Huygens principle in wave propagation and solution of Hadamard’s problem in space-times with non-trivial conformal group (1970); new conservation laws in fluid dynamics (1973); new theories on Lie-Bäcklund transformation groups (1979) and approximate symmetries (1987); nonlocal symmetries in mechanics (1987); symmetry approach to fundamental solutions and invariance principles in initial value problems (1992); derivation of Laplace type invariants for parabolic equations (2000) and solution of Laplace problem on invariants of hyperbolic equations (2004).

Professor Ibragimov initiated and edited publication of the most authoritative source in modern group analysis – the multivolume Handbook of Lie group analysis of differential equations (CRC Press, 1996). His monograph Transformation groups applied to mathematical physics (Nauka, 1983) was awarded the USSR State Prize in Science and Technology. His recent book Elementary Lie group analysis and ordinary differential equations (Wiley, 1999) is the first modern university text where the basic integration methods are derived from invariance principles.

Apart from research, Nail Ibragimov took a keen interest in popularization of the philosophy of Lie groups. In 1983 he started a series of annual conferences MOGRAN (Modern Group Analysis) which is a world wide known international forum in this field.