Galatius, Søren 1976
Overview
Works:  18 works in 30 publications in 1 language and 149 library holdings 

Genres:  Conference proceedings 
Roles:  Author, Thesis advisor, Editor 
Classifications:  QA612.14, 514.2 
Publication Timeline
.
Most widely held works by
Søren Galatius
Algebraic topology : applications and new directions : Stanford Symposium on Algebraic Topology: Applications and New Directions,
July 2327, 2012, Stanford University, Stanford, CA by Stanford Symposium on Algebraic Topology : Applications and New Directions(
Book
)
4 editions published in 2014 in English and held by 97 WorldCat member libraries worldwide
4 editions published in 2014 in English and held by 97 WorldCat member libraries worldwide
Impact of microvascular alterations in heart failure : effect of cardiac transplantation and ACE inhibition by
Sren Galatius(
Book
)
4 editions published in 2001 in English and held by 9 WorldCat member libraries worldwide
4 editions published in 2001 in English and held by 9 WorldCat member libraries worldwide
Divisibility of the stable MillerMoritaMumford class by
Sren Galatius(
Book
)
4 editions published in 2005 in English and held by 7 WorldCat member libraries worldwide
4 editions published in 2005 in English and held by 7 WorldCat member libraries worldwide
Mod p homology of the stable mapping class group by
Sren Galatius(
Book
)
4 editions published in 2002 in English and held by 7 WorldCat member libraries worldwide
4 editions published in 2002 in English and held by 7 WorldCat member libraries worldwide
String topology and twisted Ktheory by Anssi Sebastian Lahtinen(
)
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
The mathematics in this thesis is motivated by the desire to connect the twisted equivariant Ktheory of a compact Lie group G to the string topology of its classifying space BG. The thesis consists of two parts. In the first part, we prove a generalization of the AtiyahSegal completion theorem to twisted Ktheory, a result which Kriz, Westerland and Levin have subsequently used to connect the twisted equivariant Kgroups of G to the GruherSalvatore string topology of BG. In the second part, we present work towards the construction of a conjectural family of field theories which, on one hand, are closely related to Freed, Hopkins and Teleman's field theories featuring the twisted equivariant Ktheory of G, and which on the other hand contain the ChataurMenichi string topology of BG as a special case. The main result of the second part is the construction of the fieldtheory operation associated with a fixed cobordism
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
The mathematics in this thesis is motivated by the desire to connect the twisted equivariant Ktheory of a compact Lie group G to the string topology of its classifying space BG. The thesis consists of two parts. In the first part, we prove a generalization of the AtiyahSegal completion theorem to twisted Ktheory, a result which Kriz, Westerland and Levin have subsequently used to connect the twisted equivariant Kgroups of G to the GruherSalvatore string topology of BG. In the second part, we present work towards the construction of a conjectural family of field theories which, on one hand, are closely related to Freed, Hopkins and Teleman's field theories featuring the twisted equivariant Ktheory of G, and which on the other hand contain the ChataurMenichi string topology of BG as a special case. The main result of the second part is the construction of the fieldtheory operation associated with a fixed cobordism
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
Duality and linear approximations in Hochschild homology, Ktheory, and string topology by Cary Malkiewich(
)
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
This thesis encompasses at least three separate but related projects. The first project is a treatment of twisted Poincare duality for manifolds with coefficients in spectra, and is included as an appendix. The second project investigates the map from the stabilization of the gauge group of a certain principal bundle over a manifold M to the CohenJones string topology spectrum of M. This map is a linear approximation in the sense of Goodwillie and Weiss's embedding calculus. The third project is an ongoing and openended exploration of contravariant forms of algebraic Ktheory of spaces, with a focus on topological Hochschild homology
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
This thesis encompasses at least three separate but related projects. The first project is a treatment of twisted Poincare duality for manifolds with coefficients in spectra, and is included as an appendix. The second project investigates the map from the stabilization of the gauge group of a certain principal bundle over a manifold M to the CohenJones string topology spectrum of M. This map is a linear approximation in the sense of Goodwillie and Weiss's embedding calculus. The third project is an ongoing and openended exploration of contravariant forms of algebraic Ktheory of spaces, with a focus on topological Hochschild homology
A duality theorem for delignemumford stacks with respect to Morava Ktheory by Man Chuen Cheng(
)
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
In [7] Greenlees and Sadofsky used a transfer map to show that the classifying spaces of finite groups are selfdual with respect to Morava Ktheory K(n). By regarding these classifying spaces as the homotopy types of certain differentiable stacks, their construction can be viewed as a stack version of SpanierWhitehead type construction. From this point of view, we will extend their results and prove a K(n)version of Poincare duality for DeligneMumford stacks. A few examples of stacks defined by finite groups and moduli stack of Riemann surfaces will be discussed at the end
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
In [7] Greenlees and Sadofsky used a transfer map to show that the classifying spaces of finite groups are selfdual with respect to Morava Ktheory K(n). By regarding these classifying spaces as the homotopy types of certain differentiable stacks, their construction can be viewed as a stack version of SpanierWhitehead type construction. From this point of view, we will extend their results and prove a K(n)version of Poincare duality for DeligneMumford stacks. A few examples of stacks defined by finite groups and moduli stack of Riemann surfaces will be discussed at the end
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
Equivariant algebraic ktheory of products of motivic circles by Tracy Leah Nance(
)
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
Fix an algebraically closed field k of characteristic zero. We describe a method which produces deloopings of K'(k) in the Gm(k)direction via a homotopy limit over pth power maps, and examine the outcome of analogous constructions in an equivariant setting. These constructions provide a technique for studying actions of finite groups on motivic spheres which cannot be described by a usual smash product of Gm's with group action
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
Fix an algebraically closed field k of characteristic zero. We describe a method which produces deloopings of K'(k) in the Gm(k)direction via a homotopy limit over pth power maps, and examine the outcome of analogous constructions in an equivariant setting. These constructions provide a technique for studying actions of finite groups on motivic spheres which cannot be described by a usual smash product of Gm's with group action
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])
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
Characteristic classes of surface bundles(
Book
)
1 edition published in 2004 in English and held by 1 WorldCat member library worldwide
1 edition published in 2004 in English and held by 1 WorldCat member library worldwide
Some results on Ktheory, topological Hochschild homology, and parameterized spectra by Jonathan Alfred Campbell(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
This thesis comprises three separate papers. The first proves a duality result relating the topological Hochschild homology of a ring spectrum to the SpanierWhitehead dual of the topological Hochschild homology of the Koszul dual ring spectrum. The second, proves an additivity result for the transfer Atheory of infinity topoi, as introduced by Barwick. The third presents partial results of joint work with John Lind. In the full joint paper we create a machine which has as input 2Vector bundles and as output parameterized kumodules
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
This thesis comprises three separate papers. The first proves a duality result relating the topological Hochschild homology of a ring spectrum to the SpanierWhitehead dual of the topological Hochschild homology of the Koszul dual ring spectrum. The second, proves an additivity result for the transfer Atheory of infinity topoi, as introduced by Barwick. The third presents partial results of joint work with John Lind. In the full joint paper we create a machine which has as input 2Vector bundles and as output parameterized kumodules
Filtered floer and symplectic homology via GromovWitten theory by Lus 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
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
The moduli space of real curves and a Z/2equivariant MadsenWeiss theorem by Nisan Alexander Stiennon(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
Galatius, Madsen, Tillmann, and Weiss proved that the classifying space of the category of 2cobordisms is equivalent to the loopspace of a particular Thom spectrum. We show that this is in fact a Z/2equivariant equivalence, where we equip all spaces with a Z/2action which is motivated by complex conjugation of complex curves. In order to do this, we prove an equivariant delooping theorem which shows that grouplike topological monoids with Z/2action are Z/2equivalent to loopspaces. Furthermore, we motivate our choice of Z/2action by showing that it determines a Z/2space BDiff_g whose fixed points classify real curves
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
Galatius, Madsen, Tillmann, and Weiss proved that the classifying space of the category of 2cobordisms is equivalent to the loopspace of a particular Thom spectrum. We show that this is in fact a Z/2equivariant equivalence, where we equip all spaces with a Z/2action which is motivated by complex conjugation of complex curves. In order to do this, we prove an equivariant delooping theorem which shows that grouplike topological monoids with Z/2action are Z/2equivalent to loopspaces. Furthermore, we motivate our choice of Z/2action by showing that it determines a Z/2space BDiff_g whose fixed points classify real 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
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Related Identities
 Tillmann, U. L. (Ulrike Luise) 1962 Editor
 Sinha, Dev (Dev Prakash) 1971 Editor
 Stanford University Department of Mathematics
 Cohen, Ralph L. 1952 Thesis advisor
 Madsen, Ib
 Ionel, Eleny Thesis advisor
 Aarhus Universitet Matematisk Institut
 Carlsson, G. (Gunnar) 1952 Thesis advisor
 Carlsson, Gunnar Thesis advisor
 Eliashberg, Y. 1946 Thesis advisor
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