Toro, Tatiana 1964
Overview
Works:  6 works in 19 publications in 1 language and 214 library holdings 

Genres:  Academic theses 
Roles:  Author 
Classifications:  QA312, 515.42 
Publication Timeline
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Most widely held works by
Tatiana Toro
Reifenberg parameterizations for sets with holes by
Guy David(
Book
)
14 editions published between 2011 and 2012 in English and held by 208 WorldCat member libraries worldwide
14 editions published between 2011 and 2012 in English and held by 208 WorldCat member libraries worldwide
Reifenberg parameterization of sets with holes by
Guy David(
Book
)
1 edition published in 2012 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 2012 in English and held by 2 WorldCat member libraries worldwide
On the geometry of rectifiable sets with Carleson and Poincarétype conditions by
Jessica Merhej(
)
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
A central question in geometric measure theory is whether geometric properties of a set translate into analytical ones. In 1960, E.R. Reifenberg proved that if an ndimensional subset M of Rn+d is well approximated by nplanes at every point and at every scale, then M is a locally biHölder image of an nplane. Since then, Reifenberg's theorem has been refined in several ways in order to ensure that M is a biLipschitz image of an nplane. In this thesis, we show that a Carleson condition on the oscillation of the tangent planes of an nAhlfors regular rectifiable subset M of Rn+d satisfying a Poincarétype inequality is sufficient to prove that M is contained inside a biLipschitz image of an ndimensional affine subspace of Rn+d. We also show that this Poincarétype inequality encodes geometrical information about M; namely it implies that M is quasiconvex
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
A central question in geometric measure theory is whether geometric properties of a set translate into analytical ones. In 1960, E.R. Reifenberg proved that if an ndimensional subset M of Rn+d is well approximated by nplanes at every point and at every scale, then M is a locally biHölder image of an nplane. Since then, Reifenberg's theorem has been refined in several ways in order to ensure that M is a biLipschitz image of an nplane. In this thesis, we show that a Carleson condition on the oscillation of the tangent planes of an nAhlfors regular rectifiable subset M of Rn+d satisfying a Poincarétype inequality is sufficient to prove that M is contained inside a biLipschitz image of an ndimensional affine subspace of Rn+d. We also show that this Poincarétype inequality encodes geometrical information about M; namely it implies that M is quasiconvex
Functions in W², ²(R²) have Lipschitz graphs by
Tatiana Toro(
)
1 edition published in 1992 in English and held by 1 WorldCat member library worldwide
1 edition published in 1992 in English and held by 1 WorldCat member library worldwide
The geometry of uniform measures by
Abdalla Dali Nimer(
)
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
Uniform measures have played a fundamental role in geometric measure theory since they naturally appear as tangent objects. They were first studied in the groundbreaking work of Preiss where he proved that a Radon measure is nrectifiable if and only if the ndensity at almost every point of its support is positive and finite. However, very little is understood about them: for instance the only known nuniform measures not supported on an affine nplane were constructed by Preiss in 1987. In this thesis, we prove that the Hausdorff dimension of the singular set of any $n$uniform measure is at most n3. Then we characterize 3uniform measures with dilation invariant support and construct an infinite family of 3uniform measures all distinct and nonisometric, one of which is the Preiss cone
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
Uniform measures have played a fundamental role in geometric measure theory since they naturally appear as tangent objects. They were first studied in the groundbreaking work of Preiss where he proved that a Radon measure is nrectifiable if and only if the ndensity at almost every point of its support is positive and finite. However, very little is understood about them: for instance the only known nuniform measures not supported on an affine nplane were constructed by Preiss in 1987. In this thesis, we prove that the Hausdorff dimension of the singular set of any $n$uniform measure is at most n3. Then we characterize 3uniform measures with dilation invariant support and construct an infinite family of 3uniform measures all distinct and nonisometric, one of which is the Preiss cone
Local set approximation : infinitesimal to local theorems and applications / Stephen Lewis by
Stephen Lewis(
)
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
In this thesis we develop the theory of Local Set Approximation (LSA), a framework which arises naturally from the study of sets with singularities. That is, we describe the local structure of a set A in Euclidean space through studying a class of sets S which approximates A well in small balls. We will give two interpretations LSA in Chapters 2 and 3. If in small balls B(x; r), our set A is close to some nice model set S, the approximation is unilateral. On the other hand, if in small balls B(x; r), our set A is close to S and S is close to A, the approximation is bilateral. Both of these models appear naturally in areas of geometric measure theory such as area minimizers, mass minimizers, free boundary, and regularity of measures. In Chapters 4 and 5, we give applications of local set approximation to the study of asymptotically optimally doubling measures
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
In this thesis we develop the theory of Local Set Approximation (LSA), a framework which arises naturally from the study of sets with singularities. That is, we describe the local structure of a set A in Euclidean space through studying a class of sets S which approximates A well in small balls. We will give two interpretations LSA in Chapters 2 and 3. If in small balls B(x; r), our set A is close to some nice model set S, the approximation is unilateral. On the other hand, if in small balls B(x; r), our set A is close to S and S is close to A, the approximation is bilateral. Both of these models appear naturally in areas of geometric measure theory such as area minimizers, mass minimizers, free boundary, and regularity of measures. In Chapters 4 and 5, we give applications of local set approximation to the study of asymptotically optimally doubling measures
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Related Identities
 David, Guy 1957 Author
 American Mathematical Society
 Lewis, Stephen 1987 Author
 Nimer, Abdalla Dali Author
 Merhej, Jessica Author
Useful Links
Alternative Names
Tatiana Toro Colombiaans wiskundige
Tatiana Toro Colombian–American mathematician
Tatiana Toro mathématicienne colombienneaméricaine
Languages