Vakil, Ravi
Overview
Works:  23 works in 55 publications in 1 language and 1,281 library holdings 

Genres:  Conference proceedings 
Roles:  Author, Editor, Thesis advisor 
Classifications:  QA43, 516.35 
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 550 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 550 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
)
9 editions published between 1996 and 2008 in English and held by 427 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
9 editions published between 1996 and 2008 in English and held by 427 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 204 WorldCat member libraries worldwide
13 editions published in 2005 in English and held by 204 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
)
2 editions published in 2013 in English and held by 51 WorldCat member libraries worldwide
2 editions published in 2013 in English and held by 51 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."BOOK 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."BOOK JACKET
Enumerative geometry of curves via degeneration methods
by
Ravi Vakil(
)
2 editions published in 1997 in English and held by 3 WorldCat member libraries worldwide
2 editions published in 1997 in English and held by 3 WorldCat member libraries worldwide
A short proof of the [lamda]gconjecture without GromovWitten theory : Hurwitz theory and the moduli of curves
by
I. P Goulden(
Book
)
1 edition published in 2006 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 2006 in English and held by 2 WorldCat member libraries worldwide
The GromovWitten potential of a point, Hurwitz numbers, and Hodge integrals
by
I. P Goulden(
Book
)
1 edition published in 1999 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 1999 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
General existence theorems in moduli theory
by Jack Kingsbury Hall(
)
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
In this thesis, we prove that there is an algebraic stack parameterizing all curves. The curves that appear in this algebraic stack are allowed to be arbitrarily singular, nonreduced, disconnected, and reducible. We also prove the boundedness of the open substack parameterizing reduced and connected curves with fixed arithmetic genus g and at most e irreducible components. We also show that for essentially any algebraic stack, there is an algebraic stack, the Hilbert stack, parameterizing quasifinite maps to the stack. The technical heart of this result is a generalization of formal GAGA to a nonseparated morphism of algebraic stacks, something that was previously unknown for a morphism of schemes. We also employ derived algebraic geometry, in an essential way, to prove the algebraicity of the Hilbert stack. The Hilbert stack, for algebraic spaces, was claimed to exist by M. Artin (1974), but was left unproved due to a lack of foundational results for nonseparated algebraic spaces. Finally, we generalize the fundamental GAGA results of J.P. Serre (1956) in three waysto the nonseparated setting, to stacks, and to families. As an application of these results, we show that analytic compactifications of the moduli stack of smooth curves possessing modular interpretations are algebraizable
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
In this thesis, we prove that there is an algebraic stack parameterizing all curves. The curves that appear in this algebraic stack are allowed to be arbitrarily singular, nonreduced, disconnected, and reducible. We also prove the boundedness of the open substack parameterizing reduced and connected curves with fixed arithmetic genus g and at most e irreducible components. We also show that for essentially any algebraic stack, there is an algebraic stack, the Hilbert stack, parameterizing quasifinite maps to the stack. The technical heart of this result is a generalization of formal GAGA to a nonseparated morphism of algebraic stacks, something that was previously unknown for a morphism of schemes. We also employ derived algebraic geometry, in an essential way, to prove the algebraicity of the Hilbert stack. The Hilbert stack, for algebraic spaces, was claimed to exist by M. Artin (1974), but was left unproved due to a lack of foundational results for nonseparated algebraic spaces. Finally, we generalize the fundamental GAGA results of J.P. Serre (1956) in three waysto the nonseparated setting, to stacks, and to families. As an application of these results, we show that analytic compactifications of the moduli stack of smooth curves possessing modular interpretations are algebraizable
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
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
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
Enumerative Invariants in Algebraic Geometry and String Theory Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 6{u2013}11, 2005
by
D Abramovich(
)
1 edition published in 2008 in English and held by 1 WorldCat member library worldwide
1 edition published in 2008 in English and held by 1 WorldCat member library worldwide
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
The William Lowell Putnam Mathematical Competition : problems, solutions, and commentary, 19852000
by
Kiran Sridhara Kedlaya(
Book
)
1 edition published in 2002 in English and held by 1 WorldCat member library worldwide
1 edition published in 2002 in English and held by 1 WorldCat member library worldwide
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
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
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
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
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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 Polynomials Quantum field theory Quantum theory String models William Lowell Putnam Mathematical Competition
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Vakil, Ravi.
Vakil, Ravi Damodar
Vakil Ravi Damodar 1970....
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