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Topology of moduli spaces and operads
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. , vii, 45 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 13:02
National Category
Geometry
Identifiers
URN: urn:nbn:se:kth:diva-122389ISBN: 978-91-7501-759-4 (print)OAI: oai:DiVA.org:kth-122389DiVA: diva2:622090
Public defence
2013-05-31, Sal E2, Lindstedtsvägen 3, KTH, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

QC 20130520

Available from: 2013-05-20 Created: 2013-05-20 Last updated: 2013-05-20Bibliographically approved
List of papers
1. The structure of the tautological ring in genus one
Open this publication in new window or tab >>The structure of the tautological ring in genus one
2014 (English)In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 163, no 4, 777-793 p.Article in journal (Refereed) Published
Abstract [en]

We prove Getzler's claims about the cohomology of the moduli space of stable curves of genus one, that is, that the even cohomology ring is spanned by the strata classes and that all relations between these classes follow from the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) relation and Getzler's relation. In particular, the even cohomology ring is isomorphic to the tautological ring.

Keyword
Mixed Hodge Structure, Configuration-Spaces, Intersection Theory, Moduli Spaces, Cohomology, Curves
National Category
Geometry
Identifiers
urn:nbn:se:kth:diva-122385 (URN)10.1215/00127094-2429916 (DOI)000332751000004 ()2-s2.0-84897805523 (Scopus ID)
Note

QC 20140414. Updated from manuscript to article in journal.

Available from: 2013-05-20 Created: 2013-05-20 Last updated: 2017-12-06Bibliographically approved
2. The Gorenstein conjecture fails for the tautological ring of M̅ 2,n
Open this publication in new window or tab >>The Gorenstein conjecture fails for the tautological ring of M̅ 2,n
2014 (English)In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 196, no 1, 139-161 p.Article in journal (Refereed) Published
Abstract [en]

We prove that for N equal to at least one of the integers 8, 12, 16, 20 the tautological ring is not Gorenstein. In fact, our N equals the smallest integer such that there is a non-tautological cohomology class of even degree on . By work of Graber and Pandharipande, such a class exists on , and we present some evidence indicating that N is in fact 20.

Keyword
Moduli Spaces, Local Systems, Eisenstein Cohomology, Configuration-Spaces, Abelian Surfaces, Curves
National Category
Geometry
Identifiers
urn:nbn:se:kth:diva-122386 (URN)10.1007/s00222-013-0466-z (DOI)000333160600003 ()2-s2.0-84896400395 (Scopus ID)
Note

QC 20140428

Available from: 2013-05-20 Created: 2013-05-20 Last updated: 2017-12-06Bibliographically approved
3. Cohomology of local systems on loci of d-elliptic Abelian surfaces
Open this publication in new window or tab >>Cohomology of local systems on loci of d-elliptic Abelian surfaces
2013 (English)In: The Michigan mathematical journal, ISSN 0026-2285, E-ISSN 1945-2365, Vol. 62, no 4, 705-720 p.Article in journal (Refereed) Published
Abstract [en]

We consider the loci of d-elliptic curves in $M_2$, and corresponding loci of d-elliptic surfaces in $A_2$. We show how a description of these loci as quotients of a product of modular curves can be used to calculate cohomology of natural local systems on them, both as mixed Hodge structures and $\ell$-adic Galois representations. We study in particular the case d=2, and compute the Euler characteristic of the moduli space of n-pointed bi-elliptic genus 2 curves in the Grothendieck group of Hodge structures.

National Category
Algebra and Logic
Identifiers
urn:nbn:se:kth:diva-49336 (URN)10.1307/mmj/1387226161 (DOI)000330420800003 ()2-s2.0-84892168133 (Scopus ID)
Note

QC 20140227. Updated from manuscript to article in journal.

Available from: 2011-11-25 Created: 2011-11-25 Last updated: 2017-12-08Bibliographically approved
4. On the operad structure of admissible G-covers
Open this publication in new window or tab >>On the operad structure of admissible G-covers
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We describe the modular operad structure on the moduli spaces of pointed stable curves equipped with an admissible G-cover. To do this we are forced to introduce the notion of an operad colored not by a set but by the objects of a groupoid. This construction interpolates in some sense between “framed” and “colored” versions of operads; we hope that it will be of independent interest. An algebra over this operad is the same thing as a G-equivariant CohFT. Our main theorem is an extension of the symmetric function formalism for modular operads to this setting; we prove an analogue of the formula of Getzler and Kapranov describing the effect of the “free modular operad” functor on the level of symmetric functions.

National Category
Algebra and Logic
Identifiers
urn:nbn:se:kth:diva-48902 (URN)
Note

QS 2011

Available from: 2011-11-24 Created: 2011-11-24 Last updated: 2013-05-20Bibliographically approved
5. Minimal models, GT-action and formality of the little disk operad
Open this publication in new window or tab >>Minimal models, GT-action and formality of the little disk operad
2014 (English)In: Selecta Mathematica, New Series, ISSN 1022-1824, E-ISSN 1420-9020, Vol. 20, no 3, 817-822 p.Article in journal (Refereed) Published
Abstract [en]

We give a new proof of formality of the operad of little disks. The proof makes use of an operadic version of a simple formality criterion for commutative differential graded algebras due to Sullivan. We see that formality is a direct consequence of the fact that the Grothendieck-Teichmuller group operates on the chain operad of little disks.

Keyword
Formal operad, Grothendieck-Teichmuller group, Drinfel'd associator
National Category
Geometry
Identifiers
urn:nbn:se:kth:diva-122382 (URN)10.1007/s00029-013-0135-5 (DOI)000337756600004 ()2-s2.0-84902369451 (Scopus ID)
Note

QC 20140805. Updated from manuscript to article in journal.

Available from: 2013-05-20 Created: 2013-05-20 Last updated: 2017-12-06Bibliographically approved

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