Vakil, Ravi
Overview
Works:  25 works in 61 publications in 1 language and 1,324 library holdings 

Genres:  Conference papers and proceedings 
Roles:  Author, Editor, Thesis advisor 
Classifications:  QA43, 510.76 
Publication Timeline
.
Most widely held works by
Ravi Vakil
The William Lowell Putnam Mathematical Competition 19852000 : problems, solutions, and commentary by
Kiran Sridhara Kedlaya(
Book
)
8 editions published between 2002 and 2011 in English and held by 553 WorldCat member libraries worldwide
Publisher description: The William Lowell Putnam Mathematical Competition is the premier undergraduate mathematical competition in North America. This volume contains problems from the years 19852000, with solutions and extensive commentary. It is unlike the first two Putnam volumes and unlike virtually every other problembased book, in that it places the problems in the context of important mathematical themes. The authors highlight connections to other problems, to the curriculum, and to more advanced topics. The best problems contain kernels of sophisticated ideas related to important current research, and yet the problems are accessible to undergraduates. The heart of the book is in the solutions, which have been compiled through extensive research. In editing the solutions, the authors have kept a student audience in mind, explaining techniques that have relevance to more than the problem at hand, suggesting references for further reading, and mentioning related problems, some of which are unsolved
8 editions published between 2002 and 2011 in English and held by 553 WorldCat member libraries worldwide
Publisher description: The William Lowell Putnam Mathematical Competition is the premier undergraduate mathematical competition in North America. This volume contains problems from the years 19852000, with solutions and extensive commentary. It is unlike the first two Putnam volumes and unlike virtually every other problembased book, in that it places the problems in the context of important mathematical themes. The authors highlight connections to other problems, to the curriculum, and to more advanced topics. The best problems contain kernels of sophisticated ideas related to important current research, and yet the problems are accessible to undergraduates. The heart of the book is in the solutions, which have been compiled through extensive research. In editing the solutions, the authors have kept a student audience in mind, explaining techniques that have relevance to more than the problem at hand, suggesting references for further reading, and mentioning related problems, some of which are unsolved
A mathematical mosaic : patterns & problem solving by
Ravi Vakil(
Book
)
11 editions published between 1996 and 2008 in English and held by 430 WorldCat member libraries worldwide
Ravi Vakil, a preeminent winner of International Mathematics Olympiads, develops some powerful problemsolving ideas underpinning the major branches of mathematics and weaves them into a mosaic that reveals their interconnections. The mathematics is presented at the level of the capable high school mathematics student, but there is much substance for the advanced undergraduate and the intelligent lay reader. You will find this book an invaluable source of enrichment problems and ideas. The style is informal, friendly, and often humorous. In this book, Professor Vakil profiles seven other mathematics olympiad winners including Noam Elkies, the youngest professor to achieve tenure at Harvard.Publisher
11 editions published between 1996 and 2008 in English and held by 430 WorldCat member libraries worldwide
Ravi Vakil, a preeminent winner of International Mathematics Olympiads, develops some powerful problemsolving ideas underpinning the major branches of mathematics and weaves them into a mosaic that reveals their interconnections. The mathematics is presented at the level of the capable high school mathematics student, but there is much substance for the advanced undergraduate and the intelligent lay reader. You will find this book an invaluable source of enrichment problems and ideas. The style is informal, friendly, and often humorous. In this book, Professor Vakil profiles seven other mathematics olympiad winners including Noam Elkies, the youngest professor to achieve tenure at Harvard.Publisher
Snowbird lectures in algebraic geometry : proceedings of an AMSIMSSIAM Joint Summer Research Conference on Algebraic GeometryPresentations
by Young Researchers, July 48, 2004, Snowbird, Utah by AMSIMSSIAM Joint Summer Research Conference on Algebraic Geometry: Presentations by Young Researchers(
Book
)
13 editions published in 2005 in English and held by 196 WorldCat member libraries worldwide
13 editions published in 2005 in English and held by 196 WorldCat member libraries worldwide
A celebration of algebraic geometry : a conference in honor of Joe Harris' 60th birthday, Harvard University, Cambridge, Massachusetts,
August 2528, 2011 by Algebraic geometry conference(
Book
)
4 editions published in 2013 in English and held by 59 WorldCat member libraries worldwide
4 editions published in 2013 in English and held by 59 WorldCat member libraries worldwide
Enumerative invariants in algebraic geometry and string theory : lectures given at the C.I.M.E. Summer School held in Cetraro,
Italy, June 611, 2005 by
D Abramovich(
Book
)
4 editions published in 2008 in English and held by 27 WorldCat member libraries worldwide
"Starting in the middle of the 80s, there has been a growing and fruitful interaction between algebraic geometry and certain areas of theoretical highenergy physics, especially the various versions of string theory. Physical heuristics have provided inspiration for new mathematical definitions (such as that of GromovWitten invariants) leading in turn to the solution of problems in enumerative geometry. Conversely, the availability of mathematically rigorous definitions and theorems has benefited the physics research by providing the required evidence in fields where experimental testing seems problematic. The aim of this volume, a result of the CIME Summer School held in Cetraro, Italy, in 2005, is to cover part of the most recent and interesting findings in this subject."Jacket
4 editions published in 2008 in English and held by 27 WorldCat member libraries worldwide
"Starting in the middle of the 80s, there has been a growing and fruitful interaction between algebraic geometry and certain areas of theoretical highenergy physics, especially the various versions of string theory. Physical heuristics have provided inspiration for new mathematical definitions (such as that of GromovWitten invariants) leading in turn to the solution of problems in enumerative geometry. Conversely, the availability of mathematically rigorous definitions and theorems has benefited the physics research by providing the required evidence in fields where experimental testing seems problematic. The aim of this volume, a result of the CIME Summer School held in Cetraro, Italy, in 2005, is to cover part of the most recent and interesting findings in this subject."Jacket
The William Lowell Putnam Mathematical Competition : problems, solutions, and commentary, 19852000 by
Kiran Sridhara Kedlaya(
Book
)
2 editions published in 2002 in English and held by 3 WorldCat member libraries worldwide
2 editions published in 2002 in English and held by 3 WorldCat member libraries worldwide
Enumerative geometry of curves via degeneration methods by
Ravi Vakil(
)
1 edition published in 1997 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 1997 in English and held by 2 WorldCat member libraries worldwide
Moduli spaces of PTstable objects by ChiehCheng Lo(
)
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
We develop techniques for performing semistable reduction on a flat family of objects in the heart of a tstructure on the bounded derived category of coherent sheaves of a smooth projective threefold. Then we show that, with respect to Bayer's PTstability function, the semistable objects in the heart of perverse sheaves form a proper Artin stack of finite type, provided the rank is nonzero, and the rank and degree are coprime
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
We develop techniques for performing semistable reduction on a flat family of objects in the heart of a tstructure on the bounded derived category of coherent sheaves of a smooth projective threefold. Then we show that, with respect to Bayer's PTstability function, the semistable objects in the heart of perverse sheaves form a proper Artin stack of finite type, provided the rank is nonzero, and the rank and degree are coprime
NonAbelian Lefschetz hyperplane theorems by Daniel Litt(
)
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
Let X be a smooth variety over the complex numbers, and let D be an ample divisor in X. For which spaces Y does every map from D to Y extend uniquely to a map X to Y? Our main result, proved via positive characteristic methods, is the following: if dim(X)> 2, Y is smooth, the cotangent bundle of Y is nef, and dim(Y) <dim(D), every such map extends. Taking Y to be the classifying space of a finite group BG, the moduli space of pointed curves M_g, n, the moduli space of principally polarized Abelian varieties A_g, certain period domains, and various other moduli spaces, one obtains many new and classical Lefschetz hyperplane theorems. We also prove many other such extension theorems, and develop general techniques for recognizing spaces Y for which these sorts of Lefschetz theorems hold
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
Let X be a smooth variety over the complex numbers, and let D be an ample divisor in X. For which spaces Y does every map from D to Y extend uniquely to a map X to Y? Our main result, proved via positive characteristic methods, is the following: if dim(X)> 2, Y is smooth, the cotangent bundle of Y is nef, and dim(Y) <dim(D), every such map extends. Taking Y to be the classifying space of a finite group BG, the moduli space of pointed curves M_g, n, the moduli space of principally polarized Abelian varieties A_g, certain period domains, and various other moduli spaces, one obtains many new and classical Lefschetz hyperplane theorems. We also prove many other such extension theorems, and develop general techniques for recognizing spaces Y for which these sorts of Lefschetz theorems hold
Gvalued flat deformations and local models by Brandon William Allen Levin(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We construct resolutions of Gvalued local Galois deformation rings by moduli spaces of Kisin modules with Gstructure when l = p. This generalizes Mark Kisin's work on potentially semistable deformation rings. In the case of flat deformations, we prove a structural result about these resolutions which relates the connected components of Gvalued flat deformation rings to the connected components of projective varieties in characteristic p, which are moduli spaces of linear algebra data. As a key step in the study of these resolutions, we prove a full faithfulness result in integral padic Hodge theory. We also generalize results of Pappas and Zhu on local models of Shimura varieties to groups arising from Weil restrictions
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We construct resolutions of Gvalued local Galois deformation rings by moduli spaces of Kisin modules with Gstructure when l = p. This generalizes Mark Kisin's work on potentially semistable deformation rings. In the case of flat deformations, we prove a structural result about these resolutions which relates the connected components of Gvalued flat deformation rings to the connected components of projective varieties in characteristic p, which are moduli spaces of linear algebra data. As a key step in the study of these resolutions, we prove a full faithfulness result in integral padic Hodge theory. We also generalize results of Pappas and Zhu on local models of Shimura varieties to groups arising from Weil restrictions
Relations among characteristic classes of manifold bundles by Ilya Grigoriev(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We study a generalization of the tautological subring of the cohomology of the moduli space of Riemann surfaces to manifold bundles. The infinitely many "generalized MillerMoritaMumford classes" determine a map R from a free polynomial algebra to the cohomology of the classifying space of manifold bundles. In the case when M is the connected sum of g copies of the product of spheres (S^d times S^d), with d odd, we find numerous polynomials in the kernel of the map R and show that the image of R is a finitely generated ring. Some of the elements in the kernel do not depend on d. Our results contrast with the fact that the map R is an isomorphism in a range of cohomological degrees that grows linearly with g. This is known from theorems of MadsenWeiss and Harer for the case of surfaces (d=1) and from the recent work of Soren Galatius and Oscar RandalWilliams in higher dimensions. For surfaces, the image of the map R coincides with the classical tautological ring, as introduced by Mumford
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We study a generalization of the tautological subring of the cohomology of the moduli space of Riemann surfaces to manifold bundles. The infinitely many "generalized MillerMoritaMumford classes" determine a map R from a free polynomial algebra to the cohomology of the classifying space of manifold bundles. In the case when M is the connected sum of g copies of the product of spheres (S^d times S^d), with d odd, we find numerous polynomials in the kernel of the map R and show that the image of R is a finitely generated ring. Some of the elements in the kernel do not depend on d. Our results contrast with the fact that the map R is an isomorphism in a range of cohomological degrees that grows linearly with g. This is known from theorems of MadsenWeiss and Harer for the case of surfaces (d=1) and from the recent work of Soren Galatius and Oscar RandalWilliams in higher dimensions. For surfaces, the image of the map R coincides with the classical tautological ring, as introduced by Mumford
Counting curves of any genus on rational ruled surfaces by
Ravi Vakil(
Book
)
1 edition published in 1997 in English and held by 1 WorldCat member library worldwide
1 edition published in 1997 in English and held by 1 WorldCat member library worldwide
Stratifications and equivariant cohomology of spaces of uppertriangular squarezero matrices by Jonathan Wayne Lee(
)
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
Given an irreducible component X of the variety of squarezero uppertriangular matrices, a combinatorial formula developed by Rothbach gives a stratification of X into orbits of the Borel group. Specializing to the complex numbers and imposing a rank condition motivated by the HalperinCarlsson conjecture on the free ranks of products of spheres, we consider a coarser stratification into orbits of the parabolic group. After illustrating the use of the singular value decomposition theorem to describe the topology of the strata, we then compute their equivariant cohomology. We conclude with applications to the HerzogKühl equations and their role in obstruction theory arguments motivated by the aforementioned conjecture
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
Given an irreducible component X of the variety of squarezero uppertriangular matrices, a combinatorial formula developed by Rothbach gives a stratification of X into orbits of the Borel group. Specializing to the complex numbers and imposing a rank condition motivated by the HalperinCarlsson conjecture on the free ranks of products of spheres, we consider a coarser stratification into orbits of the parabolic group. After illustrating the use of the singular value decomposition theorem to describe the topology of the strata, we then compute their equivariant cohomology. We conclude with applications to the HerzogKühl equations and their role in obstruction theory arguments motivated by the aforementioned conjecture
The motivic cohomology of varieties of long exact sequences by Thomas Benedict Williams(
)
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
A perfect field, $k$, is fixed throughout. The aim of the present work is to compute the equivariant motivic cohomology of a certain variety, $X$, which represents the space of long exact sequences of given length and with prescribed ranks, all in the category of finite dimensional $k$vector spaces, equivariant with respect to an action of the multiplicative group scheme of $k$. In order to calculate this, it is necessary to establish a number of results pertaining to the motivic cohomology of the general linear group scheme, and to introduce a spectral sequence for deriving the motivic cohomology of homogeneous varieties. We then present the variety $X$ as a homogeneous variety, and so obtain the cohomology. As an application, we show that the cohomology furnishes obstructions to equivariant maps from punctured affine $n$spaces to $X$, which amounts to the same thing as obstructions to the existence of certain differential graded modules over polynomial rings over $k$. The obstructions so found are a generalization of the HerzogK\"uhl equations, which are wellknown in the particular case where the differential graded module is in fact a resolution of an Artinian module
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
A perfect field, $k$, is fixed throughout. The aim of the present work is to compute the equivariant motivic cohomology of a certain variety, $X$, which represents the space of long exact sequences of given length and with prescribed ranks, all in the category of finite dimensional $k$vector spaces, equivariant with respect to an action of the multiplicative group scheme of $k$. In order to calculate this, it is necessary to establish a number of results pertaining to the motivic cohomology of the general linear group scheme, and to introduce a spectral sequence for deriving the motivic cohomology of homogeneous varieties. We then present the variety $X$ as a homogeneous variety, and so obtain the cohomology. As an application, we show that the cohomology furnishes obstructions to equivariant maps from punctured affine $n$spaces to $X$, which amounts to the same thing as obstructions to the existence of certain differential graded modules over polynomial rings over $k$. The obstructions so found are a generalization of the HerzogK\"uhl equations, which are wellknown in the particular case where the differential graded module is in fact a resolution of an Artinian module
The moduli space of curves, double Hurwitz numbers, and Faber's intersection number conjecture by
I. P Goulden(
Book
)
1 edition published in 2007 in English and held by 1 WorldCat member library worldwide
1 edition published in 2007 in English and held by 1 WorldCat member library worldwide
On higher q, tCatalan numbers by
Yuncheng Lin(
)
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
The purpose of this thesis is twofold: to give a selfcontained evaluation of the symmetric function version of higher $q, t$Catalan numbers $SC_n^{(m)}(q, t) = \langle \nabla^m e_n, e_n \rangle$ as a summation of rational functions of $q$ and $t$ indexed by partitions, and to prove the conjecture that the combinatorial version $WC_n^{(m)}(q, t)$ and the symmetric function version $SC_n^{(m)}(q, t)$ of higher $q, t$Catalan numbers are equivalent for $n$ up to 6, thus strengthening a result of Lee, Li, and Loehr (see~\cite[Section 5]{LLL14}) which shows the equivalence for $n$ up to 4
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
The purpose of this thesis is twofold: to give a selfcontained evaluation of the symmetric function version of higher $q, t$Catalan numbers $SC_n^{(m)}(q, t) = \langle \nabla^m e_n, e_n \rangle$ as a summation of rational functions of $q$ and $t$ indexed by partitions, and to prove the conjecture that the combinatorial version $WC_n^{(m)}(q, t)$ and the symmetric function version $SC_n^{(m)}(q, t)$ of higher $q, t$Catalan numbers are equivalent for $n$ up to 6, thus strengthening a result of Lee, Li, and Loehr (see~\cite[Section 5]{LLL14}) which shows the equivalence for $n$ up to 4
Lagrangian Tori in R4 and S2xS2 by Elizabeth Sarah Quirk Goodman(
)
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
We study problems of classification of Lagrangian embeddings of a torus in a symplectic 4manifold. First we complete the proof of a claim that all Lagrangian tori in R^4 are isotopic. Next we present a construction of Lagrangian tori and Klein bottles in monotone S^2xS^2. Finally we show that all monotone tori may be produced from such a construction, and outline a new approach to the problem of finding the Hamiltonian isotopy classes of monotone tori in S^2xS^2
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
We study problems of classification of Lagrangian embeddings of a torus in a symplectic 4manifold. First we complete the proof of a claim that all Lagrangian tori in R^4 are isotopic. Next we present a construction of Lagrangian tori and Klein bottles in monotone S^2xS^2. Finally we show that all monotone tori may be produced from such a construction, and outline a new approach to the problem of finding the Hamiltonian isotopy classes of monotone tori in S^2xS^2
A proof of the GöttscheYauZaslow formula by Yujong Tzeng(
)
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
We prove the number of rnodal curves in [vertical line]L[vertical line] is a universal polynomial for all algebraic surface S and sufficiently ample line bundle L
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
We prove the number of rnodal curves in [vertical line]L[vertical line] is a universal polynomial for all algebraic surface S and sufficiently ample line bundle L
Compactifying picard stacks over degenerations of surfaces by Atoshi Chowdhury(
)
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
Moduli spaces of smooth varieties can be partially compactified by the addition of a boundary parametrizing reducible varieties. We address the question of partially compactifying the universal Picard stack (the moduli space of line bundles) over a moduli space of smooth varieties by extending it over such a partial compactification. We present a stability condition for line bundles on reducible varieties and use it to specify what boundary points should be added to the universal Picard stack to obtain a proper moduli space. Over surfaces with exactly two irreducible components, we give specific results on enumerating stable line bundles, which support the conjecture that these are the right boundary points to add. This generalizes work of Caporaso and others in the 1990s on compactifying the universal Picard variety over the moduli space of curves
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
Moduli spaces of smooth varieties can be partially compactified by the addition of a boundary parametrizing reducible varieties. We address the question of partially compactifying the universal Picard stack (the moduli space of line bundles) over a moduli space of smooth varieties by extending it over such a partial compactification. We present a stability condition for line bundles on reducible varieties and use it to specify what boundary points should be added to the universal Picard stack to obtain a proper moduli space. Over surfaces with exactly two irreducible components, we give specific results on enumerating stable line bundles, which support the conjecture that these are the right boundary points to add. This generalizes work of Caporaso and others in the 1990s on compactifying the universal Picard variety over the moduli space of curves
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Curves Curves, Algebraic Geometry, Algebraic Geometry, Differential Geometry, Enumerative Global differential geometry Intersection homology theory Mathematical recreations Mathematics MathematicsCompetitions Moduli theory Quantum theory String models William Lowell Putnam Mathematical Competition
Alternative Names
Ravi Vakil Canadees wiskundige
Ravi Vakil Canadian mathematician
Ravi Vakil canadisk matematiker
Ravi Vakil kanadensisk matematiker
Ravi Vakil kanadischer Mathematiker
Ravi Vakil kanadisk matematikar
Ravi Vakil kanadisk matematiker
Vakil, Ravi.
Vakil, Ravi Damodar
Vakil Ravi Damodar 1970....
Рави Вакил математик
바킬, 라비
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