[1]
E. S. Mahmoodian and M. Mirzakhani :
“Decomposition of complete tripartite graphs into 5-cycles ,”
pp. 235–241
in
Combinatorics advances: Papers presented at the 25th annual Iranian mathematics conference
(Tehran, 28–31 March 1994 ).
Edited by C. J. Colbourn and E. S. Mahmoodian .
Mathematics and its Applications 329 .
Kluwer Academic Publishers (Dordrecht ),
1995 .
MR
1366852
Zbl
0837.05088
incollection
Abstract
People
BibTeX
A result of Sotteau on the necessary and sufficient conditions for decomposing the complete bipartite graphs into even cycles has been shown in many occasions, that it is a very important tool in the theory of graph decomposition into even cycles. In order to have similar tools in the case of odd cycle decomposition, obviously bipartite graphs are not suitable to be considered. Searching for such tools, we have considered decomposition of complete tripartite graphs, \( K_{r,s,t} \) , into 5-cycles. There are some necessary conditions that we have shown their sufficiency in the case of \( r = t \) , and some other cases. Our conjecture is that these conditions are always sufficient.
@incollection {key1366852m,
AUTHOR = {Mahmoodian, E. S. and Mirzakhani, Maryam},
TITLE = {Decomposition of complete tripartite
graphs into 5-cycles},
BOOKTITLE = {Combinatorics advances: {P}apers presented
at the 25th annual {I}ranian mathematics
conference},
EDITOR = {Colbourn, Charles J. and Mahmoodian,
Ebadollah S.},
SERIES = {Mathematics and its Applications},
NUMBER = {329},
PUBLISHER = {Kluwer Academic Publishers},
ADDRESS = {Dordrecht},
YEAR = {1995},
PAGES = {235--241},
DOI = {10.1007/978-1-4613-3554-2_15},
NOTE = {(Tehran, 28--31 March 1994). MR:1366852.
Zbl:0837.05088.},
ISSN = {2066-5997},
ISBN = {9781461335566},
}
[2]
M. Mirzakhani :
“A small non-4-choosable planar graph ,”
Bull. Inst. Combin. Appl.
17
(1996 ),
pp. 15–18 .
MR
1386951
Zbl
0860.05029
article
BibTeX
@article {key1386951m,
AUTHOR = {Mirzakhani, Maryam},
TITLE = {A small non-4-choosable planar graph},
JOURNAL = {Bull. Inst. Combin. Appl.},
FJOURNAL = {Bulletin of the Institute of Combinatorics
and its Applications},
VOLUME = {17},
YEAR = {1996},
PAGES = {15--18},
NOTE = {MR:1386951. Zbl:0860.05029.},
ISSN = {1183-1278},
}
[3]
M. Mirzakhani :
“A simple proof of a theorem of Schur ,”
Am. Math. Mon.
105 : 3
(March 1998 ),
pp. 260–262 .
MR
1615548
Zbl
0916.15004
article
Abstract
BibTeX
In [1905], I. Schur proved that the maximum number of mutually commuting linearly indepenedent complex matrices of order \( n \) is \( \lfloor n^2/4\rfloor+1 \) . Forty years later, Jacobson [1944] gave a simpler derivation of Schur’s Theorem and extended it from algebrically closed fields. We present a simpler proof of this theorem.
@article {key1615548m,
AUTHOR = {Mirzakhani, M.},
TITLE = {A simple proof of a theorem of {S}chur},
JOURNAL = {Am. Math. Mon.},
FJOURNAL = {The American Mathematical Monthly},
VOLUME = {105},
NUMBER = {3},
MONTH = {March},
YEAR = {1998},
PAGES = {260--262},
DOI = {10.2307/2589084},
NOTE = {MR:1615548. Zbl:0916.15004.},
ISSN = {0002-9890},
}
[4]
M. Mirzakhani :
Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves .
Ph.D. thesis ,
Harvard University ,
2004 .
Advised by C. T. McMullen .
MR
2705986
phdthesis
People
BibTeX
@phdthesis {key2705986m,
AUTHOR = {Mirzakhani, Maryam},
TITLE = {Simple geodesics on hyperbolic surfaces
and the volume of the moduli space of
curves},
SCHOOL = {Harvard University},
YEAR = {2004},
PAGES = {130},
URL = {https://search.proquest.com/docview/305191605},
NOTE = {Advised by C. T. McMullen.
MR:2705986.},
ISBN = {9780496791750},
}
[5]
M. Mirzakhani :
“Weil–Petersson volumes and intersection theory on the moduli space of curves ,”
J. Am. Math. Soc.
20 : 1
(2007 ),
pp. 1–23 .
MR
2257394
Zbl
1120.32008
article
Abstract
BibTeX
In this paper, we establish a relationship between the Weil–Petersson volume \( V_{g,n}(b) \) of the moduli space \( \mathcal{M}_{g,n}(b) \) of hyperbolic Riemann surfaces with geodesic boundary components of lengths \( b_1,\dots \) , \( b_n \) , and the intersection numbers of tautological classes on the moduli space \( \overline{\mathcal{M}}_{g,n} \) of stable curves. As a result, by using the recursive formula for \( V_{g,n}(b) \) obtained in the author’s “Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces”, preprint, 2003, we derive a new proof of the Virasoro constraints for a point. This result is equivalent to the Witten–Kontsevich formula.
@article {key2257394m,
AUTHOR = {Mirzakhani, Maryam},
TITLE = {Weil--{P}etersson volumes and intersection
theory on the moduli space of curves},
JOURNAL = {J. Am. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical
Society},
VOLUME = {20},
NUMBER = {1},
YEAR = {2007},
PAGES = {1--23},
DOI = {10.1090/S0894-0347-06-00526-1},
NOTE = {MR:2257394. Zbl:1120.32008.},
ISSN = {0894-0347},
}
[6]
M. Mirzakhani :
“Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces ,”
Invent. Math.
167 : 1
(2007 ),
pp. 179–222 .
MR
2264808
Zbl
1125.30039
article
Abstract
BibTeX
In this paper we investigate the Weil–Petersson volume of the moduli space of curves with marked points. We develop a method for integrating geometric functions over the moduli space of curves, and obtain an effective recursive formula for the volume \( V_{g,n}(L_1,\dots \) , \( L_n) \) of the moduli space \( \mathcal{M}_{g,n}(L_1,\dots \) , \( L_n) \) of hyperbolic Riemann surfaces of genus \( g \) with \( n \) geodesic boundary components. We show that \( V_{g,n}(L) \) is a polynomial whose coefficients are rational multiples of powers of \( \pi \) . The constant term of the polynomial \( V_{g,n}(L) \) is the Weil–Petersson volume of the moduli spacemof closed surfaces of genus \( g \) with \( n \) marked points.
@article {key2264808m,
AUTHOR = {Mirzakhani, Maryam},
TITLE = {Simple geodesics and {W}eil--{P}etersson
volumes of moduli spaces of bordered
{R}iemann surfaces},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {167},
NUMBER = {1},
YEAR = {2007},
PAGES = {179--222},
DOI = {10.1007/s00222-006-0013-2},
NOTE = {MR:2264808. Zbl:1125.30039.},
ISSN = {0020-9910},
}
[7]
M. Mirzakhani :
“Random hyperbolic surfaces and measured laminations ,”
pp. 179–198
in
In the tradition of Ahlfors–Bers, IV
(Ann Arbor, MI, 19–22 May 2005 ).
Edited by D. Canary, J. Gilman, J. Heinonen, and H. Masur .
Contemporary Mathematics 432 .
American Mathematical Society (Providence, RI ),
2007 .
Available open access
here .
MR
2342816
Zbl
1129.32011
incollection
Abstract
People
BibTeX
@incollection {key2342816m,
AUTHOR = {Mirzakhani, Maryam},
TITLE = {Random hyperbolic surfaces and measured
laminations},
BOOKTITLE = {In the tradition of {A}hlfors--{B}ers,
{IV}},
EDITOR = {Canary, Dick and Gilman, Jane and Heinonen,
Juha and Masur, Howard},
SERIES = {Contemporary Mathematics},
NUMBER = {432},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2007},
PAGES = {179--198},
DOI = {10.1090/conm/432/08308},
URL = {http://www.ams.org/books/conm/432/8308/conm432-8308.pdf},
NOTE = {(Ann Arbor, MI, 19--22 May 2005). Available
open access at http://math.stanford.edu/~mmirzakh/Papers/ABC.pdf.
MR:2342816. Zbl:1129.32011.},
ISSN = {0271-4132},
ISBN = {9780821842270},
}
[8]
M. Mirzakhani :
“Growth of the number of simple closed geodesics on hyperbolic surfaces ,”
Ann. Math. (2)
168 : 1
(2008 ),
pp. 97–125 .
MR
2415399
Zbl
1177.37036
article
Abstract
BibTeX
In this paper, we study the growth of \( s_X(L) \) , the number of simple closed geodesics of length \( \leq L \) on a complete hyperbolic surface \( X \) of finite area. We also study the frequencies of different types of simple closed geodesics on \( X \) and their relationship with the Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces.
@article {key2415399m,
AUTHOR = {Mirzakhani, Maryam},
TITLE = {Growth of the number of simple closed
geodesics on hyperbolic surfaces},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {168},
NUMBER = {1},
YEAR = {2008},
PAGES = {97--125},
DOI = {10.4007/annals.2008.168.97},
NOTE = {MR:2415399. Zbl:1177.37036.},
ISSN = {0003-486X},
}
[9]
M. Mirzakhani :
“Ergodic theory of the earthquake flow ,”
Int. Math. Res. Not.
2008
(2008 ).
MR
2416997
Zbl
1189.30087
article
Abstract
BibTeX
In this paper we investigate the dynamics of the earthquake flow defined by Thurston on the bundle \( \mathcal{PM}_g \) of geodesic measured laminations. This flow is a natural generalization of twisting along simple closed geodesics. We discuss the relationship between the Teichmüller horocycle flow on the bundle \( \mathcal{QM}_g \) of holomorphc quadratic differentials, and the earthquake flow. In fact, the basic ergodic properties of the Teichmüller horocycle flow are better understood [17]. In this paper, we show:
The earthquake flow and the Teichmüller horocycle flow are measurably isomorphic.
@article {key2416997m,
AUTHOR = {Mirzakhani, Maryam},
TITLE = {Ergodic theory of the earthquake flow},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
VOLUME = {2008},
YEAR = {2008},
DOI = {10.1093/imrn/rnm116},
NOTE = {MR:2416997. Zbl:1189.30087.},
ISSN = {1073-7928},
}
[10]
E. Lindenstrauss and M. Mirzakhani :
“Ergodic theory of the space of measured laminations ,”
Int. Math. Res. Not.
2008
(2008 ).
MR
2424174
Zbl
1160.37006
article
Abstract
People
BibTeX
@article {key2424174m,
AUTHOR = {Lindenstrauss, Elon and Mirzakhani,
Maryam},
TITLE = {Ergodic theory of the space of measured
laminations},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
VOLUME = {2008},
YEAR = {2008},
DOI = {10.1093/imrn/rnm126},
NOTE = {MR:2424174. Zbl:1160.37006.},
ISSN = {1073-7928},
}
[11]
M. Mirzakhani :
“On Weil–Petersson volumes and geometry of random hyperbolic surfaces ,”
pp. 1126–1145
in
Proceedings of the International Congress of Mathematicians
(Hyderabad, India, 19–27 August 2010 ),
vol. 2 .
Edited by R. Bhatia, A. Pal, G. Rangarajan, V. Srinivas, and M. Vanninathan .
Hindustan Book Agency (New Delhi ),
2010 .
MR
2827834
Zbl
1239.32013
incollection
Abstract
People
BibTeX
This paper investigates the geometric properties of random hyperbolic surfaces with respect to the Weil–Petersson measure. We describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces. First, we study the asymptotic behavior of Weil–Petersson volumes of the moduli spaces of hyperbolic surfaces of genus \( g \) as \( g\to\infty \) . Then we apply these asymptotic estimates to study the geometric properties of random hyperbolic surfaces, such as the length of the shortest simple closed geodesic of a given combinatorial type.
@incollection {key2827834m,
AUTHOR = {Mirzakhani, Maryam},
TITLE = {On {W}eil--{P}etersson volumes and geometry
of random hyperbolic surfaces},
BOOKTITLE = {Proceedings of the {I}nternational {C}ongress
of {M}athematicians},
EDITOR = {Bhatia, Rajendra and Pal, Arup and Rangarajan,
G. and Srinivas, V. and Vanninathan,
M.},
VOLUME = {2},
PUBLISHER = {Hindustan Book Agency},
ADDRESS = {New Delhi},
YEAR = {2010},
PAGES = {1126--1145},
DOI = {10.1142/9789814324359_0089},
NOTE = {(Hyderabad, India, 19--27 August 2010).
MR:2827834. Zbl:1239.32013.},
ISBN = {9788185931083},
}
[12]
A. Eskin and M. Mirzakhani :
“Counting closed geodesics in moduli space ,”
J. Mod. Dyn.
5 : 1
(2011 ),
pp. 71–105 .
MR
2787598
Zbl
1219.37006
article
Abstract
People
BibTeX
We compute the asymptotics, as \( R \) tends to infinity, of the number \( N(R) \) of closed geodesics of length at most \( R \) in the moduli space of compact Riemann surfaces of genus \( g \) . In fact, \( N(R) \) is the number of conjugacy classes of pseudo-Anosov elements of the mapping class group of a compact surface of genus \( g \) of translation length at most \( R \) .
@article {key2787598m,
AUTHOR = {Eskin, Alex and Mirzakhani, Maryam},
TITLE = {Counting closed geodesics in moduli
space},
JOURNAL = {J. Mod. Dyn.},
FJOURNAL = {Journal of Modern Dynamics},
VOLUME = {5},
NUMBER = {1},
YEAR = {2011},
PAGES = {71--105},
DOI = {10.3934/jmd.2011.5.71},
NOTE = {MR:2787598. Zbl:1219.37006.},
ISSN = {1930-5311},
}
[13]
J. Athreya, A. Bufetov, A. Eskin, and M. Mirzakhani :
“Lattice point asymptotics and volume growth on Teichmüller space ,”
Duke Math. J.
161 : 6
(2012 ),
pp. 1055–1111 .
MR
2913101
Zbl
1246.37009
article
Abstract
People
BibTeX
We apply some of the ideas of Margulis’s Ph.D. dissertation to Teichmüller space. Let \( X \) be a point in Teichmüller space, and let \( B_R(X) \) be the ball of radius \( R \) centered at \( X \) (with distances measured in the Teichmüller metric). We obtain asymptotic formulas as \( R \) tends to infinity for the volume of \( B_R(X) \) , and also for the cardinality of the intersection of \( B_R(X) \) with an orbit of the mapping class group.
@article {key2913101m,
AUTHOR = {Athreya, Jayadev and Bufetov, Alexander
and Eskin, Alex and Mirzakhani, Maryam},
TITLE = {Lattice point asymptotics and volume
growth on {T}eichm\"uller space},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {161},
NUMBER = {6},
YEAR = {2012},
PAGES = {1055--1111},
DOI = {10.1215/00127094-1548443},
NOTE = {MR:2913101. Zbl:1246.37009.},
ISSN = {0012-7094},
}
[14]
M. Mirzakhani :
“Growth of Weil–Petersson volumes and random hyperbolic surfaces of large genus ,”
J. Diff. Geom.
94 : 2
(2013 ),
pp. 267–300 .
MR
3080483
Zbl
1270.30014
article
Abstract
BibTeX
In this paper, we investigate the geometric properties of random hyperbolic surfaces of large genus. We describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces. First, we study the asymptotic behavior of Weil-Petersson volume \( V_{g,n} \) of the moduli spaces of hyperbolic surfaces of genus \( g \) with \( n \) punctures as \( g\to\infty \) . Then we discuss basic geometric properties of a random hyperbolic surface of genus gg with respect to the Weil–Petersson measure as \( g\to\infty \) .
@article {key3080483m,
AUTHOR = {Mirzakhani, Maryam},
TITLE = {Growth of {W}eil--{P}etersson volumes
and random hyperbolic surfaces of large
genus},
JOURNAL = {J. Diff. Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {94},
NUMBER = {2},
YEAR = {2013},
PAGES = {267--300},
DOI = {10.4310/jdg/1367438650},
NOTE = {MR:3080483. Zbl:1270.30014.},
ISSN = {0022-040X},
}
[15]
M. Mirzakhani and P. Zograf :
“Towards large genus asymptotics of intersection numbers on moduli spaces of curves ,”
Geom. Funct. Anal.
25 : 4
(2015 ),
pp. 1258–1289 .
MR
3385633
Zbl
1327.14136
article
Abstract
People
BibTeX
We explicitly compute the diverging factor in the large genus asymptotics of the Weil–Petersson volumes of the moduli spaces of \( n \) -pointed complex algebraic curves. Modulo a universal multiplicative constant we prove the existence of a complete asymptotic expansion of the Weil–Petersson volumes in the inverse powers of the genus with coefficients that are polynomials in \( n \) . This is done by analyzing various recursions for the more general intersection numbers of tautological classes, whose large genus asymptotic behavior is also extensively studied.
@article {key3385633m,
AUTHOR = {Mirzakhani, Maryam and Zograf, Peter},
TITLE = {Towards large genus asymptotics of intersection
numbers on moduli spaces of curves},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {25},
NUMBER = {4},
YEAR = {2015},
PAGES = {1258--1289},
DOI = {10.1007/s00039-015-0336-5},
NOTE = {MR:3385633. Zbl:1327.14136.},
ISSN = {1016-443X},
}
[16]
A. Eskin, M. Mirzakhani, and A. Mohammadi :
“Isolation, equidistribution, and orbit closures for the \( \mathrm{SL}(2,\mathbb{R}) \) action on moduli space ,”
Ann. Math. (2)
182 : 2
(2015 ),
pp. 673–721 .
MR
3418528
Zbl
1357.37040
article
Abstract
People
BibTeX
We prove results about orbit closures and equidistribution for the \( \mathrm{SL}(2,\mathbb{R}) \) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of the first two authors and a certain isolation property of closed \( \mathrm{SL}(2,\mathbb{R}) \) invariant manifolds developed in this paper.
@article {key3418528m,
AUTHOR = {Eskin, Alex and Mirzakhani, Maryam and
Mohammadi, Amir},
TITLE = {Isolation, equidistribution, and orbit
closures for the \$\mathrm{SL}(2,\mathbb{R})\$
action on moduli space},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {182},
NUMBER = {2},
YEAR = {2015},
PAGES = {673--721},
DOI = {10.4007/annals.2015.182.2.7},
NOTE = {MR:3418528. Zbl:1357.37040.},
ISSN = {0003-486X},
}
[17]
M. Mirzakhani and J. Vondrák :
“Sperner’s colorings, hypergraph labeling problems and fair division ,”
pp. 873–886
in
Proceedings of the twenty-sixth annual ACM-SIAM symposium on discrete algorithms
(San Diego, CA, 4–6 January 2015 ).
Edited by P. Indyk .
SIAM (Philadelphia ),
2015 .
MR
3451084
Zbl
1371.05094
incollection
Abstract
People
BibTeX
We prove three results about colorings of the simplex reminiscent of Sperner’s Lemma, with applications in hardness of approximation and fair division.
First, we prove a coloring lemma conjectured by Ene and Vondrák [2014]: Let
\begin{align*} V_{k,q} &= \bigl\{ \mathbf{v}\in \mathbb{Z}_+^k: \textstyle\sum_{i=1}^k v_i = q \bigr\} \quad\text{and}\\ E_{k,q} &= \bigl\{ \{\mathbf{a} + \mathbf{e}_1,\mathbf{a} + \mathbf{e}_2,\dots, \mathbf{a} + \mathbf{e}_k\}\,:\,\mathbf{a}\in \mathbb{Z}_+^k, \textstyle\sum_{i=1}^k a_i = q-1 \bigr\} . \end{align*}
Then for every Sperner-admissible labeling (\( \ell: V_{k,q} \to [k] \) such that \( v_{\ell(\mathbf{v})} > 0 \) for each \( \mathbf{v}\in V_{k,q} \) ), there are at least \( {q+k-3\choose k-2} \) non-monochromatic hyperedges in \( E_{k,q} \) . This implies an optimal Unique-Games hardness of \( (k{-}1{-}\epsilon) \) -approximation for the Hypergraph Labeling with Color Lists problem [Chekuri and Ene 2011]: Given a \( k \) -uniform hypergraph \( H = (V,E) \) with color lists \( L(v)\subseteq [k] \) , \( \forall v\in V \) , find a labeling \( \ell(v)\in L(v) \) that minimizes the number of non-monochromatic hyperedges. We also show that a \( (k{-}1) \) -approximation can be achieved.
Second, we show that in contrast to Sperner’s Lemma, there is a Sperner-admissible labeling of \( V_{k,q} \) such that every hyperedge in \( E_{k,q} \) contains at most 4 colors. We present an interpretation of this statement in the context of fair division: There is a preference function on
\[ \Delta_{k,q} = \bigl\{ \mathbf{x}\in\mathbb{R}_+^k: \textstyle\sum_{i=}^k x_i = q \bigr\} \]
such that for any division of \( q \) units of a resource, \( (x_1,x_2,\dots \) , \( x_k)\in\Delta_{k,q} \) such that
\[ \textstyle\sum_{i=1}^k \lfloor x_i \rfloor = q-1 ,\]
at most 4 players out of \( k \) are satisfied.
Third, we prove that there are subdivisions of the simplex with a fractional labeling (analogous to a fractional solution for Min-CSP problems) such that every hyperedge in the subdivision uses only labelings with 1 or 2 colors. This means that a natural LP cannot distinguish instances of Hypergraph Labeling with Color Lists that can be labeled so that every hyperedge uses at most 2 colors, and instances that must have a rainbow hyperedge. We prove that this problem is indeed NP-hard for \( k = 3 \) .
@incollection {key3451084m,
AUTHOR = {Mirzakhani, Maryam and Vondr\'ak, Jan},
TITLE = {Sperner's colorings, hypergraph labeling
problems and fair division},
BOOKTITLE = {Proceedings of the twenty-sixth annual
{ACM}-{SIAM} symposium on discrete algorithms},
EDITOR = {Indyk, Piotr},
PUBLISHER = {SIAM},
ADDRESS = {Philadelphia},
YEAR = {2015},
PAGES = {873--886},
DOI = {10.1137/1.9781611973730.60},
NOTE = {(San Diego, CA, 4--6 January 2015).
MR:3451084. Zbl:1371.05094.},
ISBN = {9781611973747},
}
[18]
M. Mirzakhani and A. Wright :
“The boundary of an affine invariant submanifold ,”
Invent. Math.
209 : 3
(2017 ),
pp. 927–984 .
MR
3681397
Zbl
06786974
article
Abstract
People
BibTeX
We study the boundary of an affine invariant submanifold of a stratum of translation surfaces in a partial compactification consisting of all finite area Abelian differentials over nodal Riemann surfaces, modulo zero area components. The main result is a formula for the tangent space to the boundary. We also prove finiteness results concerning cylinders, a partial converse to the Cylinder Deformation Theorem, and a result generalizing part of the Veech dichotomy.
@article {key3681397m,
AUTHOR = {Mirzakhani, Maryam and Wright, Alex},
TITLE = {The boundary of an affine invariant
submanifold},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {209},
NUMBER = {3},
YEAR = {2017},
PAGES = {927--984},
DOI = {10.1007/s00222-017-0722-8},
NOTE = {MR:3681397. Zbl:06786974.},
ISSN = {0020-9910},
}