Singularities

The structure of the zero set of a multivariate polynomial is a topic of wide interest, in view of its ubiquity in problems of analysis, algebra, partial differential equations, probability and geometry. The study of such sets, known in algebraic geometry literature as resolution of singularities, originated in the pioneering work of Jung, Abhyankar and Hironaka and has seen substantial recent advances, albeit in an algebraic setting.

Keywords:
Singularities, Free surface flows

Abstract: Viscous flow is extremely effective in deforming a free
surface into very sharp features such as tips or cups. Under increased
driving such free surface singularities may turn unstable, and
give way to secondary structures. In particular, this may be turned
into a method to manufacture small things by using the nonlinear
character of the equations of hydrodynamics.

Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semi-algebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions. We show that the assumptions of Chernoff's theorem hold for semi-algebraic sets such that the asymptotics are determined by the tangent cone at the true parameter point.

This talk will discuss certain singularities that
arise in the solution to boundary value problems involving the
swelling of otherwise hyperelastic solids. In this setting,
both non-uniform swelling and constrained swelling give rise to
nonhomogeneous deformation in the absence of externally applied
load. The standard singularities that are encountered in
nonlinear elasticity may occur, such as cavitation. Additional
singularities also arise, such as loss of smoothness associated

We survey rigorous, formal, and numerical results on the formation of
point-like singularities (or blow-up) for a wide range of evolution
equations. We use a similarity transformation of the original equation with
respect to the blow-up point, such that self-similar behaviour is mapped to
the fixed point of an infinite dimensional dynamical system. We
point out that analysing the dynamics close to the fixed point is a useful
way of classifying the structure of the singularity. As far as we are aware,

A free material surface which supports surface diffusion becomes
unstable when put under external non-hydrostatic stress. Since the chemical
potential on a stressed surface is larger inside an indentation, small shape
fluctuations develop because material preferentially diffuses out of
indentations. When the bulk of the material is purely elastic one expects
this instability to run into a finite-time cusp singularity. It is shown
here that this singularity is cured by plastic effects in the material,

A new local tomography function g is proposed. It is shown that g still
contains non-local artifacts, but their level is an order of magnitude
smaller than those of the previously known local tomography function. We
also investigate local tomography reconstruction in the dynamic case, i.e.
when the object f being scanned is undergoing some changes during the scan.
Properties of g are studied, the notion of visible singularities is suitably
generalized, and a relationship between the wave fronts of f and g is