![Page 1: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/1.jpg)
admcycles
A Sage-package for computations in the cohomology ring of the modulispace of stable curves
Johannes Schmitt
July 2019
Johannes Schmitt admcycles July 2019 1 / 15
![Page 2: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/2.jpg)
Table of Contents
1 Moduli spaces of curves and their cohomology
2 The admcycles-package
Johannes Schmitt admcycles July 2019 2 / 15
![Page 3: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/3.jpg)
Table of Contents
1 Moduli spaces of curves and their cohomology
2 The admcycles-package
Johannes Schmitt admcycles July 2019 3 / 15
![Page 4: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/4.jpg)
The moduli space of stable curves
Definition (Deligne-Mumford 1969)
Let g , n ≥ 0 be integers (with 2g − 2 + n > 0).
Mg ,n =
(C , p1, . . . , pn) :
C compact complex algebraiccurve of arithmetic genus gwith at worst nodal singularitiesp1, . . . , pn ∈ C distinct smooth pointsAut(C , p1, . . . , pn) finite
/iso
Johannes Schmitt admcycles July 2019 4 / 15
![Page 5: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/5.jpg)
The moduli space of stable curves
Definition (Deligne-Mumford 1969)
Let g , n ≥ 0 be integers (with 2g − 2 + n > 0).
Mg ,n =
(C , p1, . . . , pn) :
C compact complex algebraiccurve of arithmetic genus gwith at worst nodal singularities
p1, . . . , pn ∈ C distinct smooth pointsAut(C , p1, . . . , pn) finite
/iso
Johannes Schmitt admcycles July 2019 4 / 15
![Page 6: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/6.jpg)
The moduli space of stable curves
Definition (Deligne-Mumford 1969)
Let g , n ≥ 0 be integers (with 2g − 2 + n > 0).
Mg ,n =
(C , p1, . . . , pn) :
C compact complex algebraiccurve of arithmetic genus gwith at worst nodal singularitiesp1, . . . , pn ∈ C distinct smooth points
Aut(C , p1, . . . , pn) finite
/iso
Johannes Schmitt admcycles July 2019 4 / 15
![Page 7: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/7.jpg)
The moduli space of stable curves
Definition (Deligne-Mumford 1969)
Let g , n ≥ 0 be integers (with 2g − 2 + n > 0).
Mg ,n =
(C , p1, . . . , pn) :
C compact complex algebraiccurve of arithmetic genus gwith at worst nodal singularitiesp1, . . . , pn ∈ C distinct smooth pointsAut(C , p1, . . . , pn) finite
/iso
Johannes Schmitt admcycles July 2019 4 / 15
![Page 8: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/8.jpg)
The moduli space of stable curves
Fact
Mg ,n is smooth, compact, connected space of C-dimension 3g − 3 + n.
Johannes Schmitt admcycles July 2019 5 / 15
![Page 9: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/9.jpg)
The moduli space of stable curves
Fact
Mg ,n is smooth, compact, connected space of C-dimension 3g − 3 + n.
Johannes Schmitt admcycles July 2019 5 / 15
![Page 10: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/10.jpg)
The moduli space of stable curves
Fact
Mg ,n is smooth, compact, connected space of C-dimension 3g − 3 + n.
Johannes Schmitt admcycles July 2019 5 / 15
![Page 11: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/11.jpg)
The moduli space of stable curves
Fact
Mg ,n is smooth, compact, connected space of C-dimension 3g − 3 + n.
Johannes Schmitt admcycles July 2019 5 / 15
![Page 12: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/12.jpg)
The moduli space of stable curves
Fact
Mg ,n is smooth, compact, connected space of C-dimension 3g − 3 + n.
Johannes Schmitt admcycles July 2019 5 / 15
![Page 13: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/13.jpg)
Tautological classes in the cohomology ring of Mg ,n
Facts
Inside the cohomology H∗(Mg ,n)there are natural tautological classes[Γ, α], indexed by certain decoratedgraphs.
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
Johannes Schmitt admcycles July 2019 6 / 15
![Page 14: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/14.jpg)
Tautological classes in the cohomology ring of Mg ,n
Facts
Inside the cohomology H∗(Mg ,n)there are natural tautological classes[Γ, α], indexed by certain decoratedgraphs.
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
Johannes Schmitt admcycles July 2019 6 / 15
![Page 15: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/15.jpg)
Tautological classes in the cohomology ring of Mg ,n
Facts
Inside the cohomology H∗(Mg ,n)there are natural tautological classes[Γ, α], indexed by certain decoratedgraphs.
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
Johannes Schmitt admcycles July 2019 6 / 15
![Page 16: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/16.jpg)
Tautological classes in the cohomology ring of Mg ,n
Facts
Inside the cohomology H∗(Mg ,n)there are natural tautological classes[Γ, α], indexed by certain decoratedgraphs.
Example
[1 2
1
2
]·
[2 1
1
2
]
=
[1 1 1
1
2
]
Johannes Schmitt admcycles July 2019 6 / 15
![Page 17: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/17.jpg)
Tautological classes in the cohomology ring of Mg ,n
Facts
Inside the cohomology H∗(Mg ,n)there are natural tautological classes[Γ, α], indexed by certain decoratedgraphs.
Example
[1 2
1
2
]·
[2 1
1
2
]
=
[1 1 1
1
2
]
Johannes Schmitt admcycles July 2019 6 / 15
![Page 18: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/18.jpg)
The tautological ring RH∗(Mg ,n) ⊂ H∗(Mg ,n)
Properties
explicit, finite list of generators [Γ, α] as Q-vector space
combinatorial description of intersection product [Γ, α] · [Γ′, α′](Graber-Pandharipande, 2003)
list of many linear relations between the generators(Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012,Pandharipande-Pixton-Zvonkine 2013)
effective description of isomorphism RHdim(Mg ,n) ∼= Q(Witten 1991, Kontsevich 1992)
The admcycles-package
All of these (and more!) are now implemented and available in theSage-package admcycles.
Johannes Schmitt admcycles July 2019 7 / 15
![Page 19: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/19.jpg)
The tautological ring RH∗(Mg ,n) ⊂ H∗(Mg ,n)
Properties
explicit, finite list of generators [Γ, α] as Q-vector space
combinatorial description of intersection product [Γ, α] · [Γ′, α′](Graber-Pandharipande, 2003)
list of many linear relations between the generators(Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012,Pandharipande-Pixton-Zvonkine 2013)
effective description of isomorphism RHdim(Mg ,n) ∼= Q(Witten 1991, Kontsevich 1992)
The admcycles-package
All of these (and more!) are now implemented and available in theSage-package admcycles.
Johannes Schmitt admcycles July 2019 7 / 15
![Page 20: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/20.jpg)
Mathematical connections and applications
Example: Strata of differentials, flat surfaces
Strata of abelian, quadratic, . . . differentials naturally live inprojectivized Hodge bundle over Mg ,n [BCG+18, BCG+19]
Recursive description of their cohomology classes in terms oftautological classes [Sau19]
Description of Masur-Veech volumes for strata of abelian differentialsin terms of intersection numbers [CMSZ19]
Further examples
Mapping class group
Ribbon graphs
Tropical geometry
Integrable systems
Gromov-Witten invariants and enumerative geometry
Johannes Schmitt admcycles July 2019 8 / 15
![Page 21: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/21.jpg)
Mathematical connections and applications
Example: Strata of differentials, flat surfaces
Strata of abelian, quadratic, . . . differentials naturally live inprojectivized Hodge bundle over Mg ,n [BCG+18, BCG+19]
Recursive description of their cohomology classes in terms oftautological classes [Sau19]
Description of Masur-Veech volumes for strata of abelian differentialsin terms of intersection numbers [CMSZ19]
Further examples
Mapping class group
Ribbon graphs
Tropical geometry
Integrable systems
Gromov-Witten invariants and enumerative geometry
Johannes Schmitt admcycles July 2019 8 / 15
![Page 22: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/22.jpg)
Mathematical connections and applications
Example: Strata of differentials, flat surfaces
Strata of abelian, quadratic, . . . differentials naturally live inprojectivized Hodge bundle over Mg ,n [BCG+18, BCG+19]
Recursive description of their cohomology classes in terms oftautological classes [Sau19]
Description of Masur-Veech volumes for strata of abelian differentialsin terms of intersection numbers [CMSZ19]
Further examples
Mapping class group
Ribbon graphs
Tropical geometry
Integrable systems
Gromov-Witten invariants and enumerative geometry
Johannes Schmitt admcycles July 2019 8 / 15
![Page 23: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/23.jpg)
Table of Contents
1 Moduli spaces of curves and their cohomology
2 The admcycles-package
Johannes Schmitt admcycles July 2019 9 / 15
![Page 24: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/24.jpg)
History and authors
Based on earlier implementation by Aaron Pixton
Interface and extension by Johannes Schmitt and Jason van Zelm forstudy of admissible cover cycles ([Sv18]) – hence the name
Conversion to proper Sage-package and lots of improvements byVincent Delecroix
Available on https://gitlab.com/jo314schmitt/admcycles
Johannes Schmitt admcycles July 2019 10 / 15
![Page 25: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/25.jpg)
A first computation
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
+
[1 0 2
1 2
]
Johannes Schmitt admcycles July 2019 11 / 15
![Page 26: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/26.jpg)
A first computation
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
+
[1 0 2
1 2
]
Johannes Schmitt admcycles July 2019 11 / 15
![Page 27: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/27.jpg)
A first computation
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
+
[1 0 2
1 2
]
Johannes Schmitt admcycles July 2019 11 / 15
![Page 28: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/28.jpg)
A first computation
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
+
[1 0 2
1 2
]
Johannes Schmitt admcycles July 2019 11 / 15
![Page 29: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/29.jpg)
A first computation
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
+
[1 0 2
1 2
]
Johannes Schmitt admcycles July 2019 11 / 15
![Page 30: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/30.jpg)
A first computation
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
+
[1 0 2
1 2
]
Johannes Schmitt admcycles July 2019 11 / 15
![Page 31: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/31.jpg)
A first computation
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
+
[1 0 2
1 2
]
Johannes Schmitt admcycles July 2019 11 / 15
![Page 32: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/32.jpg)
A first computation
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
+
[1 0 2
1 2
]
Johannes Schmitt admcycles July 2019 11 / 15
![Page 33: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/33.jpg)
A first computation
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
+
[1 0 2
1 2
]
Johannes Schmitt admcycles July 2019 11 / 15
![Page 34: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/34.jpg)
A first computation
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
+
[1 0 2
1 2
]
Johannes Schmitt admcycles July 2019 11 / 15
![Page 35: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/35.jpg)
A first computation
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
+
[1 0 2
1 2
]
Johannes Schmitt admcycles July 2019 11 / 15
![Page 36: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/36.jpg)
A first computation
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
+
[1 0 2
1 2
]
Johannes Schmitt admcycles July 2019 11 / 15
![Page 37: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/37.jpg)
A first computation
Example
[1 2
1
2
]
·
[2 1
1
2
]
=
[1 1 1
1
2
]
+
[1 0 2
1 2
]
Johannes Schmitt admcycles July 2019 11 / 15
![Page 38: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/38.jpg)
Useful resources and further reading
This presentation (see also appendix)
https://gitlab.com/jo314schmitt/admcycles
README.rst – detailed installation instructionsexamples.sage – example computations with comments
https://people.math.ethz.ch/~schmittj/manual.pdf
User manual with explanations of basic functions and samplecomputations
http://www-personal.umich.edu/~janda/taut.pdf
Short mathematical introduction of the tautological ring
Johannes Schmitt admcycles July 2019 12 / 15
![Page 39: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/39.jpg)
Applications of admcycles (and Pixton’s program)
Verifying theorems in special cases
New recursion formulas for intersection numbers [GHW19]Bielliptic Hodge integrals [PT17]
Falsifying theorems in special cases
Computing formulas for interesting cycle classes
Admissible cover cycles [Sv18]
Exploring conjectures
Recursion for fundamental class of strata of differentials [FP18, Sch18]Intersections of strata of differentials [HPS17]Recursion for hyperelliptic cycles (work in progress with RenzoCavalieri)
Johannes Schmitt admcycles July 2019 13 / 15
![Page 40: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/40.jpg)
Project proposals
Improving cache function
Problem: currently, working memory (not time!) is a bottleneck formany computations due to handwritten cache function
Skills: basic Sage programming; Time horizon: 3-7 hours
Possible Impact: access to new computations (e.g. hyperelliptic cyclein M7)
Improving computation of Double Ramification Cycle
Problem: currently compute set of flows (in Z/rZ) on graph bybrute-force trial-and-error, limiting number of accessible cases
Skills: graph algorithms; Time horizon: 1-3 days
Possible Impact: active research field (Buryak-Rossi; Wu;Pandharipande, . . .) with interest in doing concrete computations
Johannes Schmitt admcycles July 2019 14 / 15
![Page 41: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/41.jpg)
Project proposals
Improving cache function
Problem: currently, working memory (not time!) is a bottleneck formany computations due to handwritten cache function
Skills: basic Sage programming; Time horizon: 3-7 hours
Possible Impact: access to new computations (e.g. hyperelliptic cyclein M7)
Improving computation of Double Ramification Cycle
Problem: currently compute set of flows (in Z/rZ) on graph bybrute-force trial-and-error, limiting number of accessible cases
Skills: graph algorithms; Time horizon: 1-3 days
Possible Impact: active research field (Buryak-Rossi; Wu;Pandharipande, . . .) with interest in doing concrete computations
Johannes Schmitt admcycles July 2019 14 / 15
![Page 42: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/42.jpg)
Project proposals
Improving cache function
Problem: currently, working memory (not time!) is a bottleneck formany computations due to handwritten cache function
Skills: basic Sage programming; Time horizon: 3-7 hours
Possible Impact: access to new computations (e.g. hyperelliptic cyclein M7)
Improving computation of Double Ramification Cycle
Problem: currently compute set of flows (in Z/rZ) on graph bybrute-force trial-and-error, limiting number of accessible cases
Skills: graph algorithms; Time horizon: 1-3 days
Possible Impact: active research field (Buryak-Rossi; Wu;Pandharipande, . . .) with interest in doing concrete computations
Johannes Schmitt admcycles July 2019 14 / 15
![Page 43: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/43.jpg)
Project proposals
Implement conjectural recursion for fundamental class of strata ofdifferentials
Problem: the papers [FP18, Sch18] propose an algorithm forrecursively computing classes of strata of differentials, which has yetto be implemented
Skills: basic familiarity with moduli of curves, operations intautological ring as implemented in admcycles
Time horizon: 1-3 days
Possible Impact: useful in ongoing research concerning strata ofdifferentials, possible use in implementation by Jonathan Zachhuber
Thank you for your attention!
Johannes Schmitt admcycles July 2019 15 / 15
![Page 44: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/44.jpg)
Project proposals
Implement conjectural recursion for fundamental class of strata ofdifferentials
Problem: the papers [FP18, Sch18] propose an algorithm forrecursively computing classes of strata of differentials, which has yetto be implemented
Skills: basic familiarity with moduli of curves, operations intautological ring as implemented in admcycles
Time horizon: 1-3 days
Possible Impact: useful in ongoing research concerning strata ofdifferentials, possible use in implementation by Jonathan Zachhuber
Thank you for your attention!
Johannes Schmitt admcycles July 2019 15 / 15
![Page 45: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/45.jpg)
Table of Contents
3 Appendix: Definition of the tautological ring
4 Bibliography
Johannes Schmitt admcycles July 2019 1 / 7
![Page 46: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/46.jpg)
Recursive boundary structure
To (C , p1, . . . , pn) ∈Mg ,n we can associate a stable graph Γ(C ,p1,...,pn)
Conversely, given a stable graph Γ we have a gluing map
ξΓ :∏
v∈V (Γ)
Mg(v),n(v) =M1,3 ×M2,1 →M3,2
Johannes Schmitt admcycles July 2019 2 / 7
![Page 47: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/47.jpg)
Recursive boundary structure
Proposition
The map ξΓ is finite with image equal to
{(C , p1, . . . , pn) : Γ(C ,p1,...,pn) = Γ}.
Johannes Schmitt admcycles July 2019 3 / 7
![Page 48: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/48.jpg)
The tautological ring
Definition: the tautological ring
The tautological ring RH∗(Mg ,n) ⊂ H∗(Mg ,n) is spanned as a Q-vectorsubspace by elements
[Γ, α] = (ξΓ)∗
product of κ, ψ-classes︸ ︷︷ ︸α
on∏
v∈V (Γ)
Mg(v),n(v)
Example [ κ1
1 2
1
2
]= (ξΓ)∗ (κ1 ⊗ ψh) ∈ RH∗(M3,2),
for ξΓ :M1,3 ×M2,1 →M3,2 and α = κ1 ⊗ ψh ∈ H∗(M1,3 ×M2,1)
Johannes Schmitt admcycles July 2019 4 / 7
![Page 49: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/49.jpg)
Table of Contents
3 Appendix: Definition of the tautological ring
4 Bibliography
Johannes Schmitt admcycles July 2019 5 / 7
![Page 50: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/50.jpg)
References I
Matt Bainbridge, Dawei Chen, Quentin Gendron, Samuel Grushevsky,and Martin Moller, Compactification of strata of Abelian differentials,Duke Math. J. 167 (2018), no. 12, 2347–2416. MR 3848392
, Strata of k-differentials, Algebr. Geom. 6 (2019), no. 2,196–233. MR 3914751
Dawei Chen, Martin Moller, Adrien Sauvaget, and Don Zagier,Masur-Veech volumes and intersection theory on moduli spaces ofabelian differentials, arXiv e-prints (2019), arXiv:1901.01785.
Gavril Farkas and Rahul Pandharipande, The moduli space of twistedcanonical divisors, J. Inst. Math. Jussieu 17 (2018), no. 3, 615–672.MR 3789183
Harald Grosse, Alexander Hock, and Raimar Wulkenhaar, A Laplacianto compute intersection numbers on Mg ,n and correlation functions inNCQFT, arXiv e-prints (2019), arXiv:1903.12526.
Johannes Schmitt admcycles July 2019 6 / 7
![Page 51: admcycles - A Sage-package for computations in the cohomology … · 2019. 7. 23. · (Faber-Zagier 2000, Pandharipande-Pixton 2010, Pixton 2012, Pandharipande-Pixton-Zvonkine 2013)](https://reader035.vdocuments.mx/reader035/viewer/2022071604/613f333fa7a58608c268c50e/html5/thumbnails/51.jpg)
References II
D. Holmes, A. Pixton, and J. Schmitt, Multiplicativity of the doubleramification cycle, ArXiv e-prints (2017), accepted by Doc. Math.
Rahul Pandharipande and Hsian-Hua Tseng, Higher genusGromov-Witten theory of the Hilbert scheme of points of the planeand CohFTs associated to local curves, arXiv e-prints (2017),arXiv:1707.01406.
Adrien Sauvaget, Cohomology classes of strata of differentials, Geom.Topol. 23 (2019), no. 3, 1085–1171. MR 3956890
Johannes Schmitt, Dimension theory of the moduli space of twistedk-differentials, Doc. Math. 23 (2018), 871–894. MR 3861042
Johannes Schmitt and Jason van Zelm, Intersections of loci ofadmissible covers with tautological classes, arXiv e-prints (2018),arXiv:1808.05817.
Johannes Schmitt admcycles July 2019 7 / 7