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Partial differential equations, optimization, dynamical systems, probability theory, statistical physics, numerical methods.
In the mid-1990s it was discovered (P Martin & J. Piasecki 1994; W. E et al 1996; Y. Brenier & E. Grenier 1998) that certain degenerate quasilinear systems of conservation laws (the "zero-pressure gas dynamics") in 1D have weak solutiuons of an unusual type: delta-like singularities of one unknown coincide with jumps of another unknown, rendering the very definition of these solutions nontrivial. The physical meaning of this is related to modelling the large-scale structure in the Universe (S.F. Shandarin, Ya.B. Zel'dovich 1989). I obtained a construction of such solutions in the vanishing viscosity limit (Sobolevskiĭ 1997) and shed some light on why it is so difficult to extend these results to the multidimensional case (Andrievsky et al 2007).
Another interesting mathematical model of the large-scale structure can be formulated in terms of optimal transport. An optimal-transport based efficient numerical method for reconstruction of the peculiar velocities of galaxies from catalogues of their positions, which we implemented with the French colleagues (U. Frisch et al 2002; Y. Brenier et al 2003), led to improved estimates of the mean mass density and masses of certain clusters (see the review R. Mohayaee & A. Sobolevskiĭ 2008).
The Burgers equation (or the Hamilton-Jacobi equation) is a simple mathematical model of matter transport, which has many features common to the above optimization problems and bears considerable interest per se, especially in the forced case. A periodic forcing leads to stationary regimes described in the framework of the "weak KAM theory" (my results in this direction are published in Sobolevskiĭ 2000); an arbitrary forcing, even modest, can lead to catastrophes rendering the solution non-continuable in time even in a weak sense (K. Khanin et al 2004).
The nonlinear optimization probles listed above, as well as many others, can be formulated as linear if the algebaic operations are taken to be min, + in lieu of addition and multiplication respectively. Problems of (min, +)-linear optimization with interval uncertainty were considered in a series of joint works with G. Litvinov (G. Litvinov et al 2000).
An (almost) complete list of things to be read and learned (time permitting…) is in my library on CiteULike.
See lists of my refereed publications and arXiv preprints at CiteULike.
"Optimal Transport: Theory and Applications to cosmological Reconstruction and Image processing" (OTARIE) supported by the French National Research Agency.
I co-organized or was a member of scientific committee for the following international conferences and workshops:
Refereeing, editing, translating: