· 2014 (summer school), block-lecture course on Computational Geodynamics, University of the Chinese Academy of Sciences, Beijing, China
· 2012 (winter semester), lecture course “Modeling of Geodynamic Processes”, Karlsruhe Institute of Technology, Germany.
· 2007-2012, block lecture course “Data Assimilation and Inverse Problems in Geodynamics”, Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy.
· 2005-2007 (winter semesters), lecture course “Introduction to Tectonic Stress Analysis and Modeling”, Karlsruhe University, Germany.
· 2004-2005 (winter semester), curriculum “Physics of the Earth” and lecture course “Computational Fluid Dynamics of the Earth”, Karlsruhe University, Germany.
· 2001-2005, block lecture course “Numerical Modeling of Nonlinear Dynamics of the Lithosphere”, Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy.
· 1998-2000 (summer semesters), lecture courses “Foundations of Geophysics and Geodynamics” and “Computational Geodynamics”, M. Gubkin Russian State University of Oil and Gas in Moscow, Russia.
· 1997-2001, chair of colloquium “Mathematical and Numerical Problems in Geodynamics”, Russian Academy of Sciences, Moscow, Russia.
My teaching philosophy is as follows. A professor should
(i) be a qualified expert in a main topic of lecture courses;
(ii) have broad and deep knowledge of the subject of the courses as well as a basic knowledge in wider scientific areas;
(iii) employ both approaches from reductionism to holism and vice versa in an educational process;
(iv) link theoretical and experimental approaches, lectures and practical courses;
(v) look for solutions of scientific puzzles together with students;
(vi) bring a humor in a teaching process and a freedom of expressions; and
(vii) teach students to think, but not just to collect new knowledge without a subsequent analysis.
I have been popularizing a scientific knowledge on mantle and lithosphere dynamics writing articles for the Russian science popular journal “Nauka i Zhizn’”(Science and Life). Since 1998 I am chairing discussions in EuroScience working group on problems of science and urgent problems of society. I have been an organizer of the two workshops on topic of science, risk and sustainability during European Science Open Forums (in Budapest 2002, Stockholm 2004, Munich 2006, Barcelona, 2008, Turin 2010).
At present I am a scientific adviser of a PhD student of the International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences in Moscow. The topic of his research is numerical modeling of heat transfer in the crust and mantle. He develops new methods for accurate computations of the heat transfer and applies the methodology to investigate the influence of heat diffusion on the evolution of mantle plumes and sedimentary layers.
“It is impossible to explain honestly the
beauties of the laws of nature in a way that people can feel,
without their having some deep understanding of mathematics” – R. Feynman.
Great advances in understanding of the Earth as well as in experimental techniques, allowing for high-precision laboratory analysis, and in computational tools, permitting accurate numerical modeling, are transforming the geoscience. Characteristic of this new intellectual landscape is the need for strong interaction across traditional disciplinary boundaries: mathematics, computer science and Earth sciences. Earth science curricula at many universities have not kept pace with these developments. Even though most geoscience students take several semesters of prerequisite courses in mathematics, the physical and/or computer sciences, these students have too little education and experience in quantitative thinking and computation to prepare them to participate in the new world of quantitative geosciences.
During the XXth century, the educational path leading to professional degrees in Earth sciences in developed countries became rather standard. Undergraduates interested in geosciences begin their studies with a set of “prerequisite” courses, typically one or two semesters of mathematics and a few semesters of physics. For most geologists and geophysicists, this early university experience, most of it preceding serious engagement with geology itself, is the end of their education in mathematics. For reasons of history, this prerequisite mathematical education is delivered by departments as a service to students who take them because it is required for a degree in geosciences or for entry into Earth science school. Many of the students taking these courses have no real enthusiasm for mathematics and perceive these courses simply as obstacles to be overcome on the way to a career in Earth sciences. Not surprisingly, the faculty who teach these service courses are ill-prepared to make connections between what is presented in the prerequisites and what is exciting in the geosciences. The difference in sophistication (and difficulty) of the quantitative content of these separate tracks can be startling.
These traditions have resulted in a deep bifurcation in culture and quantitative competence among the scientific disciplines. On one branch are mathematics, the physical and computer sciences. “Scientists educated along this branch achieve a high level of quantitative expertise. They generally have some mastery over and comfort with not only multivariate calculus and differential equations, but also linear algebra, Fourier analysis, probability, and statistics. All scientists in these areas are expected to be able to program as well as to use computers themselves. Beyond textbook knowledge of mathematical and computational methods, quantitative thinking is the daily essence of the science to which this educational path leads, and this is expressed in a rich interplay of theory, experiment, and computation” .
On the other branch are geosciences. With insignificant exceptions, Earth scientists today rarely achieve mathematical competence beyond elementary calculus and maybe a few statistical formulae. Although everybody uses a computer, geologists rarely use anything but standard commercial software. Virtually all geologists today must use some sophisticated programs (e.g., ABAQUS, CONMAN, FLAC, MARC, PDE2D for modeling, AVS, AMIRA, IDL for visualization), yet distressingly few academic geologists feel comfortable teaching the underlying principles to their students, and fewer are able to program even a rudimentary software implementation of such an algorithm themselves. Most geologists require consultations with mathematical geoscientists and geostatisticians in order to do anything but the simplest statistics, and all too often mathematical analysis in published geological papers is inadequate or omitted entirely.
Moreover, the increasing speed and ease of use of computer enables an increasingly large number of Earth scientists who have no substantial background in mathematics to explore mathematical models and draw conclusions about them. In some sense it becomes a fashion today when geologists make numerical modeling as their primary research method. Such activity usually consists of representing sensible and evidence-based assumptions as the starting point for a complicated Earth system, assigning particular parameters (often in an arbitrary way), and then letting this complicated system run. How we can trust geoscientists whose knowledge in applied mathematics leaves hope for improvement, when they present, let say, numerical results of a model of dynamics of the Earth lithosphere and try to explain some natural observations by their model which employs a few dozen (!) of turning parameters? Sometimes they do not even clarify an influence of these parameters on the model results. Perhaps such geoscientists did not hear a dictum by famous physicist E. Fermi: “With four exponents I can fit an elephant”. Lord Robert May of Oxford writes : “This represents a revolutionary change in such theoretical studies. Until only a decade or two ago, anyone pursuing this kind of activity had to have a solid grounding in mathematics. And that meant that such studies were done by people who had some idea of how the original assumptions related to the emerging graphical display or other conclusions on their computer. Removing this link means that we are seeing arguably an increasingly large body of work in which sweeping conclusions are drawn from the alleged working of a mathematical model, without clear understanding of what is actually going on”. Nevertheless, a hundred of research proposals and articles on quantitative geosciences are emerging every year where authors are not competent in applied mathematics and numerical modeling.
The cultures of students following the two paths of quantitative competence, not surprisingly, are also different. Whereas the students (and their teachers) on the mathematical and physical science branch are focused on principles and reasoning as the goal of their education, students (and teachers) on the geology branch find themselves focused more on mastering huge arrays of facts. Although this characterization is partly a stereotype and good teaching can help bridge these cultures, undergraduates are strongly influenced by these ideas.
In order to participate fully in the research of the future, it will be essential for scientists to be conversant not only with the language of geosciences but also with the languages of mathematics and computers. It is important to recognize that the problem cannot be solved by specifying minimal mathematical expertise for future geologists and geophysicists and assigning our colleagues in the mathematics department the task of inculcating this expertise in our students. Today we have two cultures within science itself, one mathematical and the other not. If Earth Science is to assimilate into the world of quantitative science, geologists and non-geologists alike will need a different kind of education than we provide today.
What can be done? I suggest to develop an integrated curriculum in which mathematics, computer science, and geosciences are introduced together. Students interested in a research career in the quantitative Earth sciences (whether in academia or in industry) should get the integrated study course immediately after the prerequisite courses in mathematics, physics, computation, and geosciences. Initially (during the first two-three semesters of their university’s study) the students receive background knowledge in each scientific discipline, and then they enter into the world of quantitative geosciences. The basic ideas of each of these disciplines should be introduced during this integrated course at a high level of sophistication in context with relevant geological problems.
A major goal of the course is to show the students how each discipline contributes to understanding the Earth’s phenomena, how these phenomena illustrate and reinforce our quantitative understanding of phenomena in the world, and how the boundaries between disciplines are becoming arbitrary and irrelevant. Integration will allow students to learn the languages of the different disciplines in context. Scientists educated in this way, regardless of their ultimate professional speciality, would share a common scientific language, facilitating both cross-disciplinary understanding and collaboration.
To summarize, I would use the words of R. Feynman, who said that “two cultures separate people who have and people who have not had this experience of understanding mathematics well enough to appreciate nature once”. Mathematics provides the best way of understanding complex natural systems, and a good mathematical education for geoscientists is the best route for enabling the most able people to address really important problems in Earth sciences.
1. Bialek, W., and Botstein, D., 2004. Introductory science and mathematics education for 21st-century biologists. Science, 303: 788-790.
2. May, R., 2004. Uses and abuses of mathematics in biology. Science, 303: 790-793.