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Cohomology of moduli spaces

English title Cohomology of moduli spaces
Applicant Pandharipande Rahul
Number 162928
Funding scheme Project funding (Div. I-III)
Research institution Departement Mathematik ETH Zürich
Institution of higher education ETH Zurich - ETHZ
Main discipline Mathematics
Start/End 01.10.2015 - 30.09.2018
Approved amount 699'402.00
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Keywords (5)

curves; Algebra; Geometry; moduli; sheaves

Lay Summary (German)

Lead
Algebraische Geometrie ist das Studium von Varietäten - den Nullstellenmengen von polynomialen Gleichungen in mehreren Variablen. Dieses Fachgebiet hat eine zentrale Rolle in der Mathematik, mit Verbindungen zur Zahlentheorie, Darstellungstheorie und Topologie. Modulprobleme in der algebraischen Geometrie beschäftigen sich mit dem Verhalten von Varietäten unter der Variation der Koeffizienten der definierenden Polynome. Am Ende des 20. Jahrhunderts wurden einige fundamentale Zusammenhänge zwischen der algebraischen Geometrie von Modulräumen und Integralen in Quantenfeldtheorien hergestellt. Ich schlage hier vor, die volle Geometrie von einigen Modulräumen mit Verbindungen zur mathematischen Physik zu untersuchen. Dieses Studium umfasst die Integrale in Verbindung mit Quantenfeldtheorien und vieles mehr. Das Ergebnis des Projekts wird ein sehr viel besseres Verständnis von Modulräumen, den assoziierten algebraischen Strukturen (Ringen) und den zugehörigen Integralen sein.
Lay summary
Ich werde mich vor allem auf Modulräume von Kurven, Abbildungen und Garben und auf Modulräume von K3-Flächen konzentrieren. Dies sind alles grundlegende Objekte in der algebraischen Geometrie, die im gesamten Fachgebiet vorkommen. In den vergangenen Jahren gab es hier beachtliche Fortschritte: Pixtons Vermutungen für den Modulraum von Kurven, den Beweis der Gromov-Witten-Stabile-Paare-Ko
rrespondenz auf vollständigen Schnitten für den Modulraum von Garben und die Berechnung der Picardgruppen für den Modulraum von K3-Flächen. Das Fachgebiet hat ein Stadium erreicht, in dem schneller und signifikanter Fortschritt möglich ist.

Die von mir vorgeschlagene Herangehensweise ist eine Mischung aus neuen Geometrien und neuen Techniken. Letztere basieren auf drei grundlegenden Ideen: Deformationen von algebraischen Objekten, Degenerationen von algebraischen Objekten und Symmetrien der zugehörigen Theorien (Gruppenoperationen oder höhere Strukturen, wie Kohomologische Feldtheorien). Die neuen Geometrien werden verwendet werden, um neue Zusammenhänge zwischen den bekannten Modulräumen zu sehen.

Direct link to Lay Summary Last update: 29.09.2015

Responsible applicant and co-applicants

Employees

Publications

Publication
Stable quotients and the holomorphic anomaly equation
Lho Hyenho, Pandharipande Rahul (2018), Stable quotients and the holomorphic anomaly equation, in Advances in Mathematics, 332, 349-402.
A correspondence of good G-sets under partial geometric quotients
Schmitt Johannes (2018), A correspondence of good G-sets under partial geometric quotients, in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 59(2), 343-360.
THE MODULI SPACE OF TWISTED CANONICAL DIVISORS
Farkas Gavril, Pandharipande Rahul (2018), THE MODULI SPACE OF TWISTED CANONICAL DIVISORS, in Journal of the Institute of Mathematics of Jussieu, 17(03), 615-672.
A calculus for the moduli space of curves
PandharipandeRahul (2018), A calculus for the moduli space of curves, in Proc. Sympos. Pure Math., 97.1, 459-487.
A compactification of the moduli space of self-maps of $\mathbb {CP}^1$ via stable maps
Schmitt Johannes (2017), A compactification of the moduli space of self-maps of $\mathbb {CP}^1$ via stable maps, in Conformal Geometry and Dynamics of the American Mathematical Society, 21(11), 273-318.
Double ramification cycles on the moduli spaces of curves
Janda F., Pandharipande R., Pixton A., Zvonkine D. (2017), Double ramification cycles on the moduli spaces of curves, in Publications mathématiques de l'IHÉS, 125(1), 221-266.
Gromov-Witten/Pairs correspondence for the quintic 3-fold
Pandharipande R., Pixton A. (2017), Gromov-Witten/Pairs correspondence for the quintic 3-fold, in Journal of the American Mathematical Society, 30(2), 389-449.
Feynman amplitudes and limits of heights
Amini Omid, Amini Omid, Блох Спенсер, Bloch Spencer J, Burgos Gil José Ignacio, Burgos Gil José Ignacio, Fresán Javier, Fresán Javier (2016), Feynman amplitudes and limits of heights, in Известия Российской академии наук. Серия математическая, 80(5), 5-40.
CohFTs with non-tautological classes
PandharipandeRahul, ZvonkineDimitri, CohFTs with non-tautological classes, in Arkiv for Matematik.
Dimension theory of the moduli space of twisted k-differentials
SchmittJohannes, Dimension theory of the moduli space of twisted k-differentials, in Documenta Mathematica.
Multiple zeta values: from numbers to motives
Burgos GilJ. I., JavierFresan, Multiple zeta values: from numbers to motives, in Clay Mathematics Proceedings.
Relations in the tautological ring of the moduli space of K3 surfaces
PandharipandeRahul, YinQizheng, Relations in the tautological ring of the moduli space of K3 surfaces, in Jour. EMS.
The combinatorics of Lehn's conjecture
MarianAlina, OpreaDragos, PandharipandeRahul, The combinatorics of Lehn's conjecture, in Jour. of the Math. Soc. of Japan.

Collaboration

Group / person Country
Types of collaboration
Dimitri Zvonkine / Jussieu France (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Carel Faber / Utrecht Netherlands (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Davesh Maulik / Columbia Univ. United States of America (North America)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Mark Gross / Cambridge Univ. Great Britain and Northern Ireland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Aaron Pixton / Harvard (Clay) United States of America (North America)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Albrecht Klemm / Bonn Germany (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Richard Thomas / Imperial College Great Britain and Northern Ireland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Andrei Okounkov / Columbia Univ. United States of America (North America)
- in-depth/constructive exchanges on approaches, methods or results
- Publication

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
ICM 2018 (Rio de Janeiro) Talk given at a conference Geometry of the moduli space of curves - Plenary Lecture 01.08.2018 Rio de Janeiro, Brazil Pandharipande Rahul;


Associated projects

Number Title Start Funding scheme
143274 Moduli spaces of curves, sheaves, and K3 surfaces 01.10.2012 Project funding (Div. I-III)
182181 Cohomological field theories, algebraic cycles, and moduli spaces 01.01.2019 Project funding (Div. I-III)
141869 NCCR SwissMAP: The Mathematics of Physics (phase I) 01.07.2014 National Centres of Competence in Research (NCCRs)
143274 Moduli spaces of curves, sheaves, and K3 surfaces 01.10.2012 Project funding (Div. I-III)

Abstract

Algebraic geometry is the study of varieties -- the zero sets of polynomial equations in several variables. The subject has a central role in mathematics with connections to number theory, representation theory, and topology. Moduli questions in algebraic geometry concern the behavior of varieties as the coefficients of the defining polynomials vary. At the end of the 20th century, several fundamental links between the algebraic geometry of moduli spaces and path integrals in quantum field theory were made. The integrals were, in several cases, interpreted as top intersection pairings on moduli spaces. I propose here to study the full cohomology and cycle theory of several moduli spaces in algebraic geometry with connections to mathematical physics, not only the top intersection pairings. The outcome will be a much better understanding of the geometry. Moreover, the cohomology rings will likely be important and beautiful new mathematical structures.My main focus will be on the moduli spaces of curves, maps, sheaves, and $K3$ surfaces. Striking progress has been made in the past three years: Pixton's conjectures governing tautological classes on the moduli space of curves, the proof of the Gromov-Witten/Pairs correspondence for complete intersections, and the calculation of the Picard groups of the moduli of K3 surfaces have all advanced the subject. The area has entered a stage where rapid and significant progress is possible. My proposed approach to these questions uses a mix of new geometries and new techniques. The virtual fundamental class (based on the deformation-obstruction theory of the moduli spaces) plays an basic underlying role. For the moduli space of curves, the introduction of r-spin geometries and the Gromov-Witten theory of orbifolds have been crucial in recent progress. The Givental-Teleman classification of semisimple Cohomological Field Theories (CohFTs) is new and powerful tool which will be used here systematically in the study of tautological rings. The degeneration formula for Gromov-Witten and stable pairs theories will be applied for the first time to the Virasoro constraints in higher dimensions. The cohomology of the moduli space of Riemann surfaces with boundary and its interaction with the moduli space of stable curves is a new and promising area. The recent breakthroughs related to the Picard groups of the moduli of K3 surfaces open the door to a study of the associated Schubert calculus.
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