Ionel, Eleny
Overview
Works:  13 works in 13 publications in 1 language and 13 library holdings 

Roles:  Thesis advisor 
Publication Timeline
.
Most widely held works by
Eleny Ionel
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
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 new construction of virtual fundamental cycles in symplectic geometry by John Vincent Pardon(
)
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
We develop techniques for defining and working with virtual fundamental cycles on moduli spaces of pseudoholomorphic curves which are not necessarily cut out transversally. Such techniques have the potential for applications as foundations for invariants in symplectic topology arising from "counting" pseudoholomorphic curves. We introduce the notion of an implicit atlas on a moduli space, which is (roughly) a convenient system of local finitedimensional reductions. We present a general intrinsic strategy for constructing a canonical implicit atlas on any moduli space of pseudoholomorphic curves. The main technical step in applying this strategy in any particular setting is to prove appropriate gluing theorems. We require only topological gluing theorems, that is, smoothness of the transition maps between gluing charts need not be addressed. Our approach to virtual fundamental cycles is algebraic rather than geometric (in particular, we do not use perturbation). Sheaftheoretic tools play an important role in setting up our functorial algebraic "VFC package". We illustrate the methods we introduce by giving definitions of GromovWitten invariants and Hamiltonian Floer homology over $\QQ$ for general symplectic manifolds. Our framework generalizes to the $S^1$equivariant setting, and we use $S^1$localization to calculate Hamiltonian Floer homology. The Arnold conjecture (as treated by Floer, HoferSalamon, Ono, LiuTian, Ruan, and FukayaOno) is a wellknown corollary of this calculation. We give a construction of contact homology in the sense of EliashbergGiventalHofer. Specifically, we use implicit atlases to construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of holomorphic curves
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
We develop techniques for defining and working with virtual fundamental cycles on moduli spaces of pseudoholomorphic curves which are not necessarily cut out transversally. Such techniques have the potential for applications as foundations for invariants in symplectic topology arising from "counting" pseudoholomorphic curves. We introduce the notion of an implicit atlas on a moduli space, which is (roughly) a convenient system of local finitedimensional reductions. We present a general intrinsic strategy for constructing a canonical implicit atlas on any moduli space of pseudoholomorphic curves. The main technical step in applying this strategy in any particular setting is to prove appropriate gluing theorems. We require only topological gluing theorems, that is, smoothness of the transition maps between gluing charts need not be addressed. Our approach to virtual fundamental cycles is algebraic rather than geometric (in particular, we do not use perturbation). Sheaftheoretic tools play an important role in setting up our functorial algebraic "VFC package". We illustrate the methods we introduce by giving definitions of GromovWitten invariants and Hamiltonian Floer homology over $\QQ$ for general symplectic manifolds. Our framework generalizes to the $S^1$equivariant setting, and we use $S^1$localization to calculate Hamiltonian Floer homology. The Arnold conjecture (as treated by Floer, HoferSalamon, Ono, LiuTian, Ruan, and FukayaOno) is a wellknown corollary of this calculation. We give a construction of contact homology in the sense of EliashbergGiventalHofer. Specifically, we use implicit atlases to construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of holomorphic curves
GopakumarVafa conjecture for genus 0 real GromovWitten invariants by Alexandr ZamorzaevOrleanschii(
)
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
The GopakumarVafa conjecture arising from string theory states that BPS counts of a CalabiYau 3fold (obtained from GromovWitten invariants using a power series formula) are integers. A symplectic version of the conjecture was proved by IonelParker. In this thesis we prove a generalization of the symplectic GopakumarVafa conjecture to the case of real GromovWitten invariants. Namely, real genus 0 BPS states of a CalabiYau 3fold with an antisymplectic involution (which are obtained from real genus 0 GromovWitten invariants) are integers. The proof combines topological methods from IonelParker's proof of the original conjecture and KatzLiu's computation of local BPS counts
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
The GopakumarVafa conjecture arising from string theory states that BPS counts of a CalabiYau 3fold (obtained from GromovWitten invariants using a power series formula) are integers. A symplectic version of the conjecture was proved by IonelParker. In this thesis we prove a generalization of the symplectic GopakumarVafa conjecture to the case of real GromovWitten invariants. Namely, real genus 0 BPS states of a CalabiYau 3fold with an antisymplectic involution (which are obtained from real genus 0 GromovWitten invariants) are integers. The proof combines topological methods from IonelParker's proof of the original conjecture and KatzLiu's computation of local BPS counts
Characteristic numbers of genus one space curves by Dung Hoang Nguyen(
)
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
The purpose of this thesis is to develop an algorithm to compute all the characteristic numbers of genus one curves in projective spaces of arbitrary dimension. The characteristic numbers of genus zero curves, genus zero curves with an ordinary node, genus zero curves with an ordinary cusp are also computed en route
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
The purpose of this thesis is to develop an algorithm to compute all the characteristic numbers of genus one curves in projective spaces of arbitrary dimension. The characteristic numbers of genus zero curves, genus zero curves with an ordinary node, genus zero curves with an ordinary cusp are also computed en route
Loose legendrian embeddings in high dimensional contact manifolds by Maxwell Le Murphy(
)
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
We give an hprinciple type result for a class of Legendrian embeddings in contact manifolds of dimension at least $5$. These Legendrians, referred to as loose, have trivial pseudoholomorphic invariants. We demonstrate they are classified up to ambient contact isotopy by smooth embedding class equipped with an almost complex framing. This result is inherently high dimensional: analogous results in dimension $3$ are false
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
We give an hprinciple type result for a class of Legendrian embeddings in contact manifolds of dimension at least $5$. These Legendrians, referred to as loose, have trivial pseudoholomorphic invariants. We demonstrate they are classified up to ambient contact isotopy by smooth embedding class equipped with an almost complex framing. This result is inherently high dimensional: analogous results in dimension $3$ are false
Filtered floer and symplectic homology via GromovWitten theory by Luís Miguel Pereira De Matos Geraldes Diogo(
)
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
We describe a procedure for computing Floer and symplectic homology groups, with action filtration and algebraic operations, in a class of examples. Namely, we consider closed monotone symplectic manifolds with smooth symplectic divisors, Poincaré dual to a positive multiple of the symplectic form. We express the Floer homology of the manifold and the symplectic homology of the complement of the divisor, for a special class of Hamiltonians, in terms of absolute and relative GromovWitten invariants, and some additional Morsetheoretic information. As an application, we compute the symplectic homology rings of cotangent bundles of spheres, and compare our results with an earlier computation in string topology
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
We describe a procedure for computing Floer and symplectic homology groups, with action filtration and algebraic operations, in a class of examples. Namely, we consider closed monotone symplectic manifolds with smooth symplectic divisors, Poincaré dual to a positive multiple of the symplectic form. We express the Floer homology of the manifold and the symplectic homology of the complement of the divisor, for a special class of Hamiltonians, in terms of absolute and relative GromovWitten invariants, and some additional Morsetheoretic information. As an application, we compute the symplectic homology rings of cotangent bundles of spheres, and compare our results with an earlier computation in string topology
The topology of spaces of Jholomorphic maps to CP2 by Jeremy Kenneth Miller(
)
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
In [Seg79], Graeme Segal proved that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding continuous mapping space through a range of dimensions increasing with degree. I will address if a similar result holds when other almost complex structures are put on projective space. For any compatible almost complex structure J on CP^2, I prove that the inclusion map from the space of Jholomorphic maps to the space of continuous maps induces a homology surjection through a range of dimensions tending to infinity with degree. The proof involves comparing the scanning map of topological chiral homology ([Sal01], [Lur09], [And10]) with gluing of Jholomorphic curves ([MS94], [Sik03])
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
In [Seg79], Graeme Segal proved that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding continuous mapping space through a range of dimensions increasing with degree. I will address if a similar result holds when other almost complex structures are put on projective space. For any compatible almost complex structure J on CP^2, I prove that the inclusion map from the space of Jholomorphic maps to the space of continuous maps induces a homology surjection through a range of dimensions tending to infinity with degree. The proof involves comparing the scanning map of topological chiral homology ([Sal01], [Lur09], [And10]) with gluing of Jholomorphic curves ([MS94], [Sik03])
Orientability of moduli spaces and open GromovWitten invariants by Penka Vasileva Georgieva(
)
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
We show that the local system of orientations on the moduli space of Jholomorphic maps from a bordered Riemann surface is isomorphic to the pullback of a local system defined on the product of the Lagrangian and its free loop space. The latter is defined using only the first and second StiefelWhitney classes of the Lagrangian. In the presence of an antisymplectic involution, whose fixed locus is a relatively spin Lagrangian, we define open GromovWitten type invariants in genus zero
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
We show that the local system of orientations on the moduli space of Jholomorphic maps from a bordered Riemann surface is isomorphic to the pullback of a local system defined on the product of the Lagrangian and its free loop space. The latter is defined using only the first and second StiefelWhitney classes of the Lagrangian. In the presence of an antisymplectic involution, whose fixed locus is a relatively spin Lagrangian, we define open GromovWitten type invariants in genus zero
Moduli spaces of pseudoholomorphic disks and floer theory of cleanly intersecting immersed lagrangians by Yin Kwan Chan(
)
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
In this thesis we investigate moduli spaces of pseudoholomorphic disks with Lagrangian boundary conditions, in which the Lagrangians are immersed with clean selfintersections. We then discuss the compactification of these moduli spaces, and show that under specific assumptions, the moduli spaces can be oriented. Finally, we use these moduli spaces to construct and compute Lagrangian Floer cohomology for sphere and orientation covers of the real projective space embedded as a Lagrangian submanifold of the complex projective space
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
In this thesis we investigate moduli spaces of pseudoholomorphic disks with Lagrangian boundary conditions, in which the Lagrangians are immersed with clean selfintersections. We then discuss the compactification of these moduli spaces, and show that under specific assumptions, the moduli spaces can be oriented. Finally, we use these moduli spaces to construct and compute Lagrangian Floer cohomology for sphere and orientation covers of the real projective space embedded as a Lagrangian submanifold of the complex projective space
A homotopytheoretic view of BottTaubes integrals and knot spaces by Robin Michael John Koytcheff(
)
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
We construct cohomology classes in the space of knots by considering a bundle over this space and "integrating along the fiber'' classes coming from the cohomology of configuration spaces using a PontrjaginThom construction. The bundle we consider is essentially the one considered by Bott and Taubes, who integrated differential forms along the fiber to get knot invariants. By doing this "integration'' homotopytheoretically, we are able to produce integral cohomology classes. Inspired by results of Budney and Cohen, we study how this integration is compatible with homology operations on the space of long knots. In particular we derive a product formula for evaluations of cohomology classes on homology classes, with respect to connectsum of knots. We then adapt the construction to be compatible with tools coming from the GoodwillieWeiss embedding calculus, in particular Sinha's cosimplicial model for the space of knots
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
We construct cohomology classes in the space of knots by considering a bundle over this space and "integrating along the fiber'' classes coming from the cohomology of configuration spaces using a PontrjaginThom construction. The bundle we consider is essentially the one considered by Bott and Taubes, who integrated differential forms along the fiber to get knot invariants. By doing this "integration'' homotopytheoretically, we are able to produce integral cohomology classes. Inspired by results of Budney and Cohen, we study how this integration is compatible with homology operations on the space of long knots. In particular we derive a product formula for evaluations of cohomology classes on homology classes, with respect to connectsum of knots. We then adapt the construction to be compatible with tools coming from the GoodwillieWeiss embedding calculus, in particular Sinha's cosimplicial model for the space of knots
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
On symplectic homology of the complement of a positive normal crossing divisor in a projective variety by Khoa Lu Nguyen(
)
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
This dissertation studies how symplectic homology of the complement of the smooth zero set $D$ of a section of a positive line bundle $\cL$ over a projective variety $(X, J)$ changes as $D$ degenerates to a normal crossing divisor $D_{\mbox{sing}}$ with two smooth connected components. By analyzing the plurisubharmonic functions obtained from a metric on $\cL$, we show that the change in the Weinstein structure of the complement is characterized by handle removals along the unstable submanifolds of critical points in a small neighborhood of the set of singular points of $D_{\mbox{sing}}.$ Parallel to work by BourgeoisEckholmEliashberg (\cite{BEE}), we construct a chain complex from the removed unstable submanifolds such that its homology completes the Viterbo transfer map in a long exact sequence. The effect of divisor degeneration on symplectic homology of the complement is then essentially reflected by the $A_\infty$ structure of a collection of Lagrangian spheres on $D$, which are the boundary at infinity of the removed unstable submanifolds
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
This dissertation studies how symplectic homology of the complement of the smooth zero set $D$ of a section of a positive line bundle $\cL$ over a projective variety $(X, J)$ changes as $D$ degenerates to a normal crossing divisor $D_{\mbox{sing}}$ with two smooth connected components. By analyzing the plurisubharmonic functions obtained from a metric on $\cL$, we show that the change in the Weinstein structure of the complement is characterized by handle removals along the unstable submanifolds of critical points in a small neighborhood of the set of singular points of $D_{\mbox{sing}}.$ Parallel to work by BourgeoisEckholmEliashberg (\cite{BEE}), we construct a chain complex from the removed unstable submanifolds such that its homology completes the Viterbo transfer map in a long exact sequence. The effect of divisor degeneration on symplectic homology of the complement is then essentially reflected by the $A_\infty$ structure of a collection of Lagrangian spheres on $D$, which are the boundary at infinity of the removed unstable submanifolds
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Related Identities
 Stanford University Department of Mathematics
 Eliashberg, Y. 1946 Thesis advisor
 Galatius, Søren 1976 Thesis advisor
 Li, Jun Thesis advisor
 Vakil, Ravi Thesis advisor
 Cohen, Ralph L. 1952 Thesis advisor
 Goodman, Elizabeth Sarah Quirk Author
 De Matos Geraldes Diogo, Luís Miguel Pereira Author
 Pardon, John Vincent Author
 Georgieva, Penka Vasileva Author
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