# Singularities

Saturday, June 2, 2012 - 3:30pm - 4:30pm

Malabika Pramanik (University of British Columbia)

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.

Tuesday, December 8, 2009 - 9:40am - 10:20am

Jens Eggers (University of Bristol)

*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.

Monday, March 5, 2007 - 10:40am - 11:30am

Mathias Drton (University of Chicago)

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.

Tuesday, July 22, 2008 - 2:00pm - 2:10pm

Thomas Pence (Michigan State University)

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

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

Monday, July 14, 2008 - 11:10am - 12:00pm

Jens Eggers (University of Bristol)

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,

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,

Tuesday, July 22, 2008 - 10:30am - 11:20am

Itamar Procaccia (Weizmann Institute of Science)

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,

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,

Thursday, January 12, 2006 - 9:45am - 10:30am

Alexander Katsevich (University of Central Florida)

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

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