Thu, 20 Feb 2003

14:00 - 15:00
Comlab

Improving spectral methods with optimized rational interpolation

Prof Jean-Paul Berrut
(University of Fribourg)
Abstract

The pseudospectral method for solving boundary value problems on the interval

consists in replacing the solution by an interpolating polynomial in Lagrangian

form between well-chosen points and collocating at those same points.

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Due to its globality, the method cannot handle steep gradients well (Markov's inequality).

We will present and discuss two means of improving upon this: the attachment of poles to

the ansatz polynomial, on one hand, and conformal point shifts on the other hand, both

optimally adapted to the problem to be solved.

Mon, 31 Jan 2011

15:45 - 16:45
L3

Surfaces of large genus

Hugo Parlier
(University of Fribourg)
Abstract

Surfaces of large genus are intriguing objects. Their geometry

has been studied by finding geometric properties that hold for all

surfaces of the same genus, and by finding families of surfaces with

unexpected or extreme geometric behavior. A classical example of this is

the size of systoles where on the one hand Gromov showed that there exists

a universal constant $C$ such that any (orientable) surface of genus $g$

with area normalized to $g$ has a homotopically non-trivial loop (a

systole) of length less than $C log(g)$. On the other hand, Buser and

Sarnak constructed a family of hyperbolic surfaces where the systole

roughly grows like $log(g)$. Another important example, in particular for

the study of hyperbolic surfaces and the related study of Teichmüller

spaces, is the study of short pants decompositions, first studied by Bers.

The talk will discuss two ideas on how to further the understanding of

surfaces of large genus. The first part will be about joint results with

F. Balacheff and S. Sabourau on upper bounds on the sums of lengths of

pants decompositions and related questions. In particular we investigate

how to find short pants decompositions on punctured spheres, and how to

find families of homologically independent short curves. The second part,

joint with L. Guth and R. Young, will be about how to construct surfaces

with large pants decompositions using random constructions.

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