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MACI SIGNED

Moduli, Algebraic Cycles, and Invariants

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EC-Contrib. €

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Project "MACI" data sheet

The following table provides information about the project.

Coordinator
EIDGENOESSISCHE TECHNISCHE HOCHSCHULE ZUERICH 

Organization address
address: Raemistrasse 101
city: ZUERICH
postcode: 8092
website: https://www.ethz.ch/de.html

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country Switzerland [CH]
 Total cost 2˙496˙055 €
 EC max contribution 2˙496˙055 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2017-ADG
 Funding Scheme ERC-ADG
 Starting year 2018
 Duration (year-month-day) from 2018-09-01   to  2023-08-31

 Partnership

Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    EIDGENOESSISCHE TECHNISCHE HOCHSCHULE ZUERICH CH (ZUERICH) coordinator 2˙496˙055.00

Map

 Project objective

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 basic links between the algebraic geometry of moduli spaces and path integrals in quantum field theory were made. The virtual fundamental class plays an essential role in these connections. I propose to study the algebraic cycle theory of basic moduli spaces. The guiding questions are: What are the most important cycles? What is the structure of the algebra of cycles? How can the classes of geometric loci be expressed? The virtual fundamental class and the associated invariants often control the answers. A combination of virtual localization, degeneration, and R-matrix methods together with new ideas from log geometry will be used in the study.

Most of the basic moduli spaces in algebraic geometry related to varieties of dimension at most 3 -- including the moduli of curves, the moduli of maps, the moduli of surfaces, and the moduli of sheaves on 3-folds -- will be considered. The current state of the study of the algebraic cycle theory in these cases varies from rather advanced (for the moduli of curves) to much less so (for the moduli of surfaces). There is a range of rich open questions which I will attack: Pixton's conjectures for the moduli of curves, the structure of the ring of Noether-Lefschetz loci for the moduli of K3 surfaces, the holomorphic anomaly equation in Gromov-Witten theory, and conjectures governing descendents for the moduli of sheaves. The dimension 3 restriction is often necessary for a good deformation theory and the existence of a virtual fundamental class.

 Publications

year authors and title journal last update
List of publications.
2019 Hyenho Lho, Rahul Pandharipande
Crepant resolution and the holomorphic anomaly equation for [C3/Z3]
published pages: 781-813, ISSN: 0024-6115, DOI: 10.1112/plms.12248
Proceedings of the London Mathematical Society 119/3 2020-04-24
2019 R. Pandharipande, D. Zvonkine, D. Petersen
Cohomological field theories with non-tautological classes
published pages: 191-213, ISSN: 0004-2080, DOI: 10.4310/arkiv.2019.v57.n1.a10
Arkiv för Matematik 57/1 2020-04-24
2019 RAHUL PANDHARIPANDE, HSIAN-HUA TSENG
HIGHER GENUS GROMOV–WITTEN THEORY OF AND ASSOCIATED TO LOCAL CURVES
published pages: , ISSN: 2050-5086, DOI: 10.1017/fmp.2019.4
Forum of Mathematics, Pi 7 2020-04-24
2019 R. Pandharipande, A. Pixton, D. Zvonkine
Tautological relations via $r$-spin structures
published pages: 439-496, ISSN: 1056-3911, DOI: 10.1090/jag/736
Journal of Algebraic Geometry 28/3 2020-04-24
2020 H. Fan, L. Wu, F. You
Structures in genus‐zero relative Gromov–Witten theory
published pages: 269-307, ISSN: 1753-8416, DOI: 10.1112/topo.12131
Journal of Topology 13/1 2020-04-24
2020 A. Oblomkov, A. Okounkov, R. Pandharipande
GW/PT Descendent Correspondence via Vertex Operators
published pages: 1321-1359, ISSN: 0010-3616, DOI: 10.1007/s00220-020-03686-4
Communications in Mathematical Physics 374/3 2020-04-24
2019 Dörfler, Julian; Roth, Marc; Schmitt, Johannes; Wellnitz, Philip
\"Counting induced subgraphs: An algebraic approach to #W[1]-hardness\"
published pages: , ISSN: , DOI: 10.4230/LIPIcs.MFCS.2019.26
Leibniz International Proceedings in Informatics, 138 2 2020-04-24
2019 Hyenho Lho, Rahul Pandharipande
Holomorphic Anomaly Equations for the Formal Quintic
published pages: 1-40, ISSN: 2096-6075, DOI: 10.1007/s42543-018-0008-0
Peking Mathematical Journal 2/1 2020-04-24
2019 Schmitt, Johannes
Geometrically defined cycles on moduli spaces of curves
published pages: , ISSN: , DOI: 10.3929/ethz-b-000358887
4 2020-04-24
2019 Rahul Pandharipande, Hsian-Hua Tseng
The Hilb/Sym correspondence for $${mathbb {C}}^2$$ C 2 : descendents and Fourier–Mukai
published pages: 509-540, ISSN: 0025-5831, DOI: 10.1007/s00208-019-01891-8
Mathematische Annalen 375/1-2 2020-04-24

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