Cohomology of the algebraic varieties arising in the Birkhoff strata of the Sato Grasmanian is studied. Connection with the integrable systems of hydrodynamical type is analyzed.

Algebraic varieties and curves arizing in the Birkhoff strata of the Sato Grassmannian Gr^2 are studied.

Algebraic and geometric structures associated with Birkhoff stara of Sato Grassmannian are analysed.

Cohomological and Poisson structures associated with special tautological subbundles for Birkhoff strata of Sato Grassmannian are studied.

A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of Lauricelle type functions.

The structure and properties of families of critical points for classes of functions obeying the elliptic Euler-Poisson-Darboux equation E(1/2,1/2) are studied.

The geometry of hypersurfaces defined dy the relation which generalizes the classical formula for free energy in terms of microstates is studied.

Gradient catastrophe and flutter instability in the notion of a vortex filament within the localized induction approximation are analyzed.

Integrable flows on the Grassmannians are defined by the requirement of closedness of tghe differential forms associated with them.

A class of Hamiltonian deformations of plane curves is defined and studied.

It is shown that the hodograph solutions of the dispersionless coupled KdV hierarchies describe critical and degenerate critical points of a scalar function which obeys the Euler-Poisson-Darboux equation.

Discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type are presented.

Change of type transitionsfor two-component systems

It is shown that the celebrated Menelaus relation, Hirota-Miwa bilinear equation for the KP hierarchy and Fay's trisecant formula are associativity conditions for structure constants of certain three-dimensional quasi-algebra.

An algebraic scheme for constructing deformations of structure constants for associative algebras generated by deformation driving algebras is discussed.

The one-cut case of the Hermitian random matrix model in the large N limit is considered. Its singular sector in the space of coupling constants is analyzed from the point of view of the hodograph equations of the underlying dispersionless Toda hierarchy.

General setting for multidimensional dispersionless integrable hierarchies in terms of differential forms in projective spaces is discussed.

Quasiclassical approximation in the intrinsic description of the vortex filament dynamics is discussed.

Singular sector of the one-layer Beeey system and dispersionless Toda system are compeletely described. The associated Euler-Poisson-Darboux equation is the main toll of the analysis.

Moduli space of Cachazo, Dijkgraaf, Intriligator and Vafa hyperelliptic spectarl curves is studied. A characterization of such spectral curves in terms of critical points of a family of polynomial solutions of the Euler-Poisson-Darboux equations is provided.

Tropical limit for macroscopic systems in equilibrium defined as the formal limit of Boltzmann cosntant tends to infinity is discussed.

Interrelation between Thom's catastrophes and differential equations revisited.