University of Georgia
Department of Mathematics

Seminar Schedule
December 1, 2003 - December 5, 2003

All Seminars are held in Boyd Graduate Studies Bldg. unless otherwise noted.

MONDAY, December 1, 2003

Numerical Analysis
1:30p.m., Room 524
Speaker: TBA
Title of talk: TBA

Geometry
1:30p.m., Room 410
Speaker: TBA
Title of talk: TBA

Topology
2:30p.m., Room 322
Speaker: TBA
Title of talk: TBA


Faculty and Graduate Social
3:00p.m., Room 409
Coffee, Tea, Cookies

Lie Theory
3:30p.m., Room 303
Speaker: TBA
Title of talk: TBA

TUESDAY, December 2, 2003

VIGRE Graduate Student Seminar
2:00-3:15pm, Room 304
Speaker: Alla Balueva, University of Georgia
Title of talk: Integro-differential Equations and Cracks: why are they closely related?
Abstract: Integro-differential equations will be constructed to model how cracks expand. This model can be used to simulate catastrophic events that may take place due to the presence of cracks in structures (e.g., pipeline ruptures, unexpected failure of buildings or bridges). Another application of the model is to describe the development of hydraulic fractures in rocks, which is essential for applications in geophysics and petroleum industry.

The problems about crack occupying some domain in the plane in 3-dimensional elastic medium may be viewed as the mixed boundary-value problem for the displacements of the elastic medium: Laplace’s equation of three variables + the derivative of the displacements given in the domain of the crack and the displacements given = 0 outside the domain. The solution to this mixed Dirichlet/Neiman’s problem possesses the singularity on the boundary of the domain, so the use of the Finite Element Methods for the solution is not applicable. Besides, we need the solution not for a whole 3D space, but only in the domain of the crack (we are interested only in the crack aperture and are not interested in the other displacements of the body around).

The 3D boundary-value problem will be reduced to an integro-differential equation in this domain using the Green function theory. This will reduce the dimensions of the problem by 1 (and instead of the equations in three variables now we have just one equation and only of two variables, which is already a 2D problem!). Integro-differential equations for cracks of some symmetrical forms, for example circular, can be sometimes solved analytically (e.g. Sneddon, Fourier Transforms). But who has ever seen a real circular crack? In practice, the crack is never circular!
The numerical-analytical method will be developed for the solution of the integro-differential equations for cracks of any arbitrary shape. The method will use Fourier transformation and the property of convolution of the functions under the integral.

If the software runs during the presentation (usually because of “stubborn” habits, it runs only after presentation…), the 3D software package will be demonstrated to show how cracks expand.

Analysis
3:30p.m., Room 326
Speaker: TBA
Title of talk: TBA

WEDNESDAY, December 3, 2003

VIGRE - Algebra Seminar
2:30p.m., Room 410
Speaker: Markus Hunziker, University of Georgia
Title of talk: How to construct a Chevalley basis

Algebraic Geometry
2:30pm, Room 303
Speaker: TBA
Title of talk: TBA

Faculty and Graduate Social
3:00pm, Room 409
Coffee, Cookies, Tea

Number Theory/Arithmetic Geometry
3:45pm, Room 304 **Note permanent time change**
Speaker: TBA
Title of talk: TBA

THURSDAY, December 4, 2003

VIGRE - Contact Topology
9:00a.m., Room 326

VIGRE Quantum Mechanics Seminar
2:00 p.m., Room 303
Speaker: Mukul Patel, University of Georgia
Title of talk: "Where do Hilbert spaces come from?" A quantum mystery, and a 'logical' answer
Abstract: Starting with some very general and "reasonable" assumptions on logic of propositions about outcomes of physical experiments, we are inexorably lead to noncommutative algebras, and thenceforth to a Hilbert space formalism. This is a very classic approach to foundations of quantum theory going back to von Neumann. The idea is that the order in which various physical measurements are conducted matters, and hence the propositions about the outcomes of combinations of such measurements follow a "nonboolean logic". The lattice of such "quantum propositions" is an orthomodular lattice, a sort of 'noncommutative' generalization of Boolean algebras (lattices). Boolean algebras are distributive, whereas orthomodular lattices are not in general. The 'quantum-ness' of the system is reflected in the departure from distributivity. To wit, in an orthomodular lattice, we can define a product using the usual lattice operations, and then, the orthomodular lattice is distributive (i.e. is a boolean lattice), if and only if this product is commutative. Orthomodular lattices naturally lead to noncommutative C*-algebras, and hence to Hilbert space formalism.

Student Number Theory
3:30p.m., Room 304
Speaker: Jim Blair, University of Georgia
Title of talk: TBA

FRIDAY, December 5, 2003

Wavelet Analysis
2:30p.m., Room 524
Speaker: TBA
Title of talk: TBA