Kerckhoff, Steve
Overview
Works:  11 works in 18 publications in 1 language and 55 library holdings 

Roles:  Thesis advisor, Author 
Classifications:  QA613.2, 514.3 
Publication Timeline
.
Most widely held works by
Steve Kerckhoff
Threedimensional orbifolds and conemanifolds by
Daryl Cooper(
Book
)
6 editions published in 2000 in English and held by 35 WorldCat member libraries worldwide
6 editions published in 2000 in English and held by 35 WorldCat member libraries worldwide
The geometry and topology of threemanifolds by
William P Thurston(
Book
)
3 editions published in 1997 in English and held by 6 WorldCat member libraries worldwide
3 editions published in 1997 in English and held by 6 WorldCat member libraries worldwide
Threedimensional orbifolds and conemanifolds by
Daryl Cooper(
)
in English and held by 2 WorldCat member libraries worldwide
in English and held by 2 WorldCat member libraries worldwide
Geometric transitions : from hyperbolic to Ads geometry by Jeffrey Edward Danciger(
)
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
We introduce a geometric transition between two homogeneous threedimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of threedimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds generated on the "other side" of the transition have tachyon singularities. The method involves the study of a new transitional geometry called halfpipe geometry. We also discuss combinatorial/algebraic tools for constructing transitions using ideal tetrahedra. Using these tools we prove that transitions can always be constructed when the underlying manifold is a punctured torus bundle
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
We introduce a geometric transition between two homogeneous threedimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of threedimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds generated on the "other side" of the transition have tachyon singularities. The method involves the study of a new transitional geometry called halfpipe geometry. We also discuss combinatorial/algebraic tools for constructing transitions using ideal tetrahedra. Using these tools we prove that transitions can always be constructed when the underlying manifold is a punctured torus bundle
Tropical coordinates on the space of persistence barcodes by Sara Kališnik Verovšek(
)
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
In the last two decades applied topologists have developed numerous methods for `measuring' and building combinatorial representations of the shape of the data. The most famous example of the former is persistent homology. This adaptation of classical homology assigns a barcode, i.e. a collection of intervals with endpoints on the real line, to a finite metric space. Unfortunately, barcodes are not welladapted for use by practitioners in machine learning tasks. In this dissertation, I identify classes of maxplus polynomials and tropical rational functions that can be used as coordinates on the space of barcodes. All of these are stable with respect to standard distance functions (bottleneck distance, Wasserstein distances) used on the barcode space. I demonstrate how these coordinates can be used by combining persistent homology with SVM to classify numbers from the MNIST dataset. In order to identify functions on the barcode space, I find generators for the semirings of tropical polynomials, maxplus polynomials and tropical rational functions invariant under the action of the symmetric group. The fundamental theorem of ordinary symmetric polynomials has an equivalent in the tropical and maxplus semirings. There are interesting differences if we consider the tropical polynomial semiring with nr variables that come in n blocks of r variables each and are permuted by the symmetric group. As opposed to the ordinary polynomial case, the semiring of rsymmetric tropical polynomials is not finitely generated, but the semiring of rsymmetric tropical rational functions is
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
In the last two decades applied topologists have developed numerous methods for `measuring' and building combinatorial representations of the shape of the data. The most famous example of the former is persistent homology. This adaptation of classical homology assigns a barcode, i.e. a collection of intervals with endpoints on the real line, to a finite metric space. Unfortunately, barcodes are not welladapted for use by practitioners in machine learning tasks. In this dissertation, I identify classes of maxplus polynomials and tropical rational functions that can be used as coordinates on the space of barcodes. All of these are stable with respect to standard distance functions (bottleneck distance, Wasserstein distances) used on the barcode space. I demonstrate how these coordinates can be used by combining persistent homology with SVM to classify numbers from the MNIST dataset. In order to identify functions on the barcode space, I find generators for the semirings of tropical polynomials, maxplus polynomials and tropical rational functions invariant under the action of the symmetric group. The fundamental theorem of ordinary symmetric polynomials has an equivalent in the tropical and maxplus semirings. There are interesting differences if we consider the tropical polynomial semiring with nr variables that come in n blocks of r variables each and are permuted by the symmetric group. As opposed to the ordinary polynomial case, the semiring of rsymmetric tropical polynomials is not finitely generated, but the semiring of rsymmetric tropical rational functions is
Threedimensional orbifolds and conemanifolds by
Daryl Cooper(
)
1 edition published in 2000 in English and held by 1 WorldCat member library worldwide
1 edition published in 2000 in English and held by 1 WorldCat member library worldwide
Nonsimple geodesics on surfaces by Jenya Sapir(
)
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
In this thesis, we study geodesics on pairs of pants, and then generalize our results to all surfaces. We get bounds on the number of closed geodesics in a hyperbolic surface given certain upper bounds on both length and selfintersection number. Then, we study the image on a pair of pants of certain sets of complete geodesics, which have restrictions for how the selfintersection number of their subarcs grows with subarc length. We show that such sets have low Hausdorff dimension, and in some cases are nowhere dense
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
In this thesis, we study geodesics on pairs of pants, and then generalize our results to all surfaces. We get bounds on the number of closed geodesics in a hyperbolic surface given certain upper bounds on both length and selfintersection number. Then, we study the image on a pair of pants of certain sets of complete geodesics, which have restrictions for how the selfintersection number of their subarcs grows with subarc length. We show that such sets have low Hausdorff dimension, and in some cases are nowhere dense
Singular hyperbolic structures on pseudoAnosov mapping tori by
Kenji Kozai(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We study threemanifolds that are constructed as mapping tori of surfaces with pseudoAnosov monodromy. We use Danciger's halfpipe geometry to regenerate hyperbolic structures when the monodromy has orientable foliations and its induced action on cohomology does not have 1 as an eigenvalue. We also include some discussion on Agol's veering triangulation construction
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We study threemanifolds that are constructed as mapping tori of surfaces with pseudoAnosov monodromy. We use Danciger's halfpipe geometry to regenerate hyperbolic structures when the monodromy has orientable foliations and its induced action on cohomology does not have 1 as an eigenvalue. We also include some discussion on Agol's veering triangulation construction
The asymptotic geometry of Teichmuller space by
Steve Kerckhoff(
)
1 edition published in 1978 in English and held by 1 WorldCat member library worldwide
1 edition published in 1978 in English and held by 1 WorldCat member library worldwide
String topology and the based loop space by Eric James Malm(
)
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
We relate the BatalinVilkovisky (BV) algebra structure of the string topology of a manifold to the homological algebra of the singular chains of the based loop space of that manifold, showing that its Hochschild cohomology carries a BV algebra structure isomorphic to that of string topology. Furthermore, this structure is compatible with the usual cup product and Lie bracket on Hochschild cohomology. This isomorphism arises from a derived form of Poincare duality using modules over the based loop space as local coefficient systems. This derived Poincare duality also comes from a form of fibrewise Atiyah duality on the level of fibrewise spectra, and we use this perspective to connect the algebraic constructions to the ChasSullivan loop product
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
We relate the BatalinVilkovisky (BV) algebra structure of the string topology of a manifold to the homological algebra of the singular chains of the based loop space of that manifold, showing that its Hochschild cohomology carries a BV algebra structure isomorphic to that of string topology. Furthermore, this structure is compatible with the usual cup product and Lie bracket on Hochschild cohomology. This isomorphism arises from a derived form of Poincare duality using modules over the based loop space as local coefficient systems. This derived Poincare duality also comes from a form of fibrewise Atiyah duality on the level of fibrewise spectra, and we use this perspective to connect the algebraic constructions to the ChasSullivan loop product
Methods and applications of topological data analysis by Jennifer Novak Kloke(
)
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
The focus of this dissertation is the development of methods for topological analysis as well as the application of topological tools to real world problems. The first half of the dissertation focuses on an algorithm for denoising highdimensional data for topological data analysis. This method significantly extends the applicability of many topological data analysis methods. In particular, this method extends the use of persistent homology, a generalized notion of homology for discrete data points, to data sets that were previously inaccessible because of noise. The second half of this dissertation focuses on a method for using topology to simplify complex chemical structures and to define a metric to quantify similarity for use in screening large databases of chemical compounds. This method has shown very promising initial results in locating new materials for efficiently separating carbon dioxide from the exhaust of coalburning power plants
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
The focus of this dissertation is the development of methods for topological analysis as well as the application of topological tools to real world problems. The first half of the dissertation focuses on an algorithm for denoising highdimensional data for topological data analysis. This method significantly extends the applicability of many topological data analysis methods. In particular, this method extends the use of persistent homology, a generalized notion of homology for discrete data points, to data sets that were previously inaccessible because of noise. The second half of this dissertation focuses on a method for using topology to simplify complex chemical structures and to define a metric to quantify similarity for use in screening large databases of chemical compounds. This method has shown very promising initial results in locating new materials for efficiently separating carbon dioxide from the exhaust of coalburning power plants
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Related Identities
 Hodgson, Craig David
 Cooper, Daryl Author
 Stanford University Department of Mathematics
 Thurston, William P. 19462012 Author
 Floyd, W. J.
 Milnor, John W. (John Willard) 1931
 Carlsson, G. (Gunnar) 1952 Thesis advisor
 Church, Thomas (Thomas Franklin) Thesis advisor
 Mirzakhani, Maryam Thesis advisor
 Malm, Eric James Author
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Alternative Names
Kerckhoff, Steve
Kerckhoff, Steven P.
Steven Kerckhoff American mathematician
Steven Kerckhoff Amerikaans wiskundige
Steven Kerckhoff amerikansk matematikar
Steven Kerckhoff amerikansk matematiker
Steven Kerckhoff matemático estadounidense
Steven Kerckhoff USamerikanischer Mathematiker
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