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Ph.D. candidate in Mathematics
Centre of Mathematics for Applications, CMA
University of Oslo, Norway Thesis advisor: Ragni Piene
Recruited by SAGA in September 2009 2007: M.Sc. Pure Mathematics
Erasmus Mundus Master. Algebra, Geometry and Number Theory (ALGANT)
Leiden University, The Netherlands. 2005-2006
Université Bordeaux 1, France. 2006-20072005: B.A. Mathematics
Universidad Nacional de Colombia The fellow's interests span:
Applied Algebraic Geometry - Algebraic Spline Geometry E-mail:
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Project Details:
Algebraic Spline Geometry
All the current approximations to CAD from algebraic geometry rely on mathematical foundations based on the use of complex numbers and projective geometry while, in practice, all the considered practical questions are presented over the real numbers and in an affine setting. This obvious remark implies the need of analyzing, from the point of view of real algebraic geometry, the consideration of curves, surfaces and solids as semi-algebraic sets (i.e., sets defined by means of polynomial equalities and inequalities), together with their manipulation techniques such as quantifier elimination algorithms.
By replacing an (affine real) algebraic variety in Rn by a convex polyhedral domain ∆, and regular functions by algebraic spline functions (piecewise polynomial functions), one would like to develop a new theory: algebraic spline geometry, guided by concrete modeling problems provided by the SAGA industrial or institution partners. To bring real algebraic geometry and spline surface representation in CAD closer together, one needs to develop and understand certain aspects of the “classical“ real (semi)algebraic geometry so that it can extend to the theory of multivariate algebraic splines.
Given a polyhedral subdivision of a region ∆⊂R^n , one wants to study the ring C^r (∆) of algebraic spline functions on ∆, and the vector spaces C^r _k (∆) of such functions of degree ≤ k. One wants to determine the dimension of this space and study its Hilbert series (or generating function of this problem). This could possibly be related to the Eckhardt function for convex lattice polytopes and toric geometry. Planned Visits:
November 2010 and two months in 2011 : INRIA, France
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