The most current versions of my papers will always be available from my website; it is possible that the arXiv version may be slightly outdated.
- Noninjectivity of the monodromy of certain equicritical strata, with Peter Huxford (preprint).
Abstract:
An equicritical stratum is the locus of univariate monic squarefree complex polynomials where the critical points have prescribed multiplicities.
Tracking the positions of both roots and critical points, there is a natural "monodromy map" taking the fundamental group into a braid group.
We show here that when there are exactly two critical points, this monodromy map is noninjective.
- Topological monodromy kernels for fundamental groups of discriminant complements (submitted).
Abstract:
A linear system on a smooth complex algebraic surface gives rise to a family of smooth curves in the surface. Such a family has a
topological monodromy representation valued in the mapping class group of a fiber. Extending arguments of Kuno, we show that if the image of this
representation is of finite index, then the kernel is infinite. This applies in particular to linear systems on smooth toric surfaces and on smooth
complete intersections. In the case of plane curves, we extend the techniques of Carlson-Toledo to show that the kernel is quite rich (e.g. it contains
a nonabelian free group).
- Siegel–Veech Constants for Cyclic Covers of Generic Translation Surfaces, with David Aulicino, Aaron Calderon, Carlos Matheus, and Martin Schmoll (submitted).
Abstract:
We compute the asymptotic number of cylinders, weighted by their area to any non-negative power, on any cyclic branched cover of any generic
translation surface in any stratum. Our formulas depend only on topological invariants of the cover and number-theoretic properties of the degree:
in particular, the ratio of the related Siegel–Veech constants for the locus of covers and for the base stratum component is independent of the number
of branch values. One surprising corollary is that this ratio for area3 Siegel–Veech constants is always equal to the reciprocal of the the degree of the
cover. A key ingredient is a classification of the connected components of certain loci of cyclic branched covers.
- Monodromy and vanishing cycles for complete intersection curves, with Ishan Banerjee (submitted).
Abstract:
We compute the topological monodromy of every family of complete intersection curves.
Like in the case of plane curves previously treated by the second-named author, we find the answer
is given by the r-spin mapping class group associated to the maximal root of the adjoint line bundle.
Our main innovation is a suite of tools for studying the monodromy of sections of a tensor product of
very ample line bundles in terms of the monodromy of sections of the factors, allowing for an
induction on (multi-)degree.
- Monodromy of stratified braid groups, II, Res. Math. Sci., to appear.
Abstract:
The space of monic squarefree polynomials has a stratification according to the multiplicities of the critical points, called the equicritical
stratification. Tracking the positions of roots and critical points, there is a map from the fundamental group of a stratum into a braid group.
We give a complete determination of this map. It turns out to be characterized by the geometry of the translation surface structure on \(\mathbb{CP}^1\)
induced by the logarithmic derivative \(df/f\) of a polynomial in the stratum.
- Families of elliptic curves over the four-pointed configuration space and exceptional sequences for the braid group on four strands, with Will Chen (submitted).
Abstract:
We show that the configuration space of four unordered points in \(\mathbb{C}\) with barycenter 0 is isomorphic to the space of triples \((E,Q,\omega)\),
where \(E\) is an elliptic curve, \(Q\in E^\circ\) a nonzero point, and \(\omega\) a nonzero holomorphic differential on \(E\). At the level of
fundamental groups, our construction unifies two classical exceptional exact sequences involving the braid group \(B_4\): namely, the sequence
\(1\rightarrow F_2\rightarrow B_4\rightarrow B_3\rightarrow 1\), where \(F_2\) is a free group of rank 2, related to Ferrari's solution of the quartic,
and the sequence \(1\rightarrow \mathbb{Z} \rightarrow B_4\rightarrow\operatorname{Aut}^+(F_2)\rightarrow 1\) of Dyer-Formanek-Grossman.
- Connected components of the topological surgery graph of a unicellular collection, with Abdoul Karim Sane (submitted).
Abstract:
A unicellular collection on a surface is a collection of curves whose complement is a single disk. There is a natural surgery operation on unicellular
collections, endowing the set of such with a graph structure where the edge relation is given by surgery. Here we determine the connected components of
this graph, showing that they are enumerated by a certain homological "surgery invariant". Our approach is group-theoretic and proceeds by understanding
the action of the mapping class group on unicellular collections. In the course of our arguments, we determine simple generating sets for the stabilizer
in the mapping class group of a mod-2 homology class, which may be of independent interest.
- Generating the homology of covers of surfaces, with Marco Boggi and Andrew Putman, Bull. Lond. Math. Soc., to appear.
Abstract:
Putman and Wieland conjectured that if \(\tilde{\Sigma} \rightarrow \Sigma\) is a finite branched cover between closed
oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of \(H_1(\tilde{\Sigma};\mathbb{Q})\) under the action of lifts
to \(\tilde{\Sigma}\) of mapping classes on \(\Sigma\) are infinite. We prove that this holds if \(H_1(\tilde{\Sigma};\mathbb{Q})\) is
generated by the homology classes of lifts of simple closed curves on \(\Sigma\). We also prove that the subspace
of \(H_1(\tilde{\Sigma};\mathbb{Q})\) spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie
on subsurfaces homeomorphic to 2-holed spheres, and we prove that \(H_1(\tilde{\Sigma};\mathbb{Q})\) is generated
by the homology classes of lifts of loops on \(\Sigma\) lying on subsurfaces homeomorphic to 3-holed
spheres.
- Stratified braid groups: monodromy (submitted).
Abstract: The space of monic squarefree complex polynomials has a stratification according to the multiplicities of the critical points.
We introduce a method to study these strata by way of the infinite-area translation surface associated to the logarithmic derivative \(df/f\) of the
polynomial. We determine the monodromy of these strata in the braid group, thus describing which braidings of the roots are possible if the orders of the
critical points are required to stay fixed. Mirroring the story for holomorphic differentials on higher-genus surfaces, we find the answer is governed by
the framing of the punctured disk induced by the horizontal foliation on the translation surface.
- Holomorphic maps between configuration spaces of Riemann surfaces, with Lei Chen (submitted).
Abstract: We prove a suite of results classifying holomorphic maps between configuration spaces of
Riemann surfaces; we consider both the ordered and unordered setting as well as the cases of
genus zero, one, and at least two. We give a complete classification of all holomorphic maps
\(\operatorname{Conf}_n(\mathbb{C})\to \operatorname{Conf}_m(\mathbb{C})\) provided that \(n\ge 5\)
and \(m\le 2n\) extending the Tameness Theorem of Lin, which is the case \(m = n\). We also give a
complete classification of holomorphic maps between ordered configuration spaces of Riemann surfaces
of genus at most one (answering a question of Farb), and show that the higher genus setting is closely linked to the still-mysterious
``effective de Franchis problem''. The main technical theme of the paper is that holomorphicity allows
one to promote group-theoretic rigidity results to the space level.
- Totally symmetric sets in the general linear group, with Noah Caplinger, Michigan Math. J., to appear.
Abstract: A totally symmetric set is a finite subset of a group for which any permutation of the elements can be realized by conjugation in the ambient group.
Such sets are rigid under homomorphisms, and so exert a great deal of control over the algebraic structure. In this paper we introduce a more
general perspective on total symmetry, and formulate a notion of "irreducibility" for totally symmetric sets in the general linear group. We classify
irreducible totally symmetric sets, as well as those of maximal cardinality.
- Simple closed curves in stable covers of surfaces, Trans. Amer. Math. Soc. 376 (2023), no. 9, 6447–6473.
Abstract:Let \(f: X \to Y\) be a regular covering of a surface Y of finite type with nonempty boundary, with finitely-generated
(possibly infinite) deck group G. We give necessary and sufficient conditions for an integral homology class on X to admit
a representative as a connected component of the preimage of a nonseparating simple closed curve on Y, possibly after passing
to a "stabilization", i.e. a G-equivariant embedding of covering spaces \(X \hookrightarrow X^+\).
- Surface bundles and the section conjecture, with Wanlin Li, Daniel Litt, and Padmavathi Srinivasan, Math. Ann. 386 (2023), no. 1-2, 877--942.
Abstract:
We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph \(\Gamma\) of genus \(g>2\), and every
field k, the generic curve with reduction type \(\Gamma\) over k satisfies the section conjecture. We prove many cases of this conjecture.
In so doing we produce many examples of curves satisfying the section conjecture over fields of geometric interest, and then over p-adic fields
and number fields via a Chebotarev argument.
We construct two Galois cohomology classes \(o_1\) and \(\widetilde{o_2}\), which obstruct the existence of \(\pi_1\)-sections and hence
of rational points. The first is an abelian obstruction, closely related to the period of a curve and to a cohomology class on the moduli
space of curves \(\mathscr{M}_g\) studied by Morita. The second is a 2-nilpotent obstruction and appears to be new. We study the
degeneration of these classes via topological techniques, and we produce examples of surface bundles over surfaces where these classes
obstruct sections. We then use these constructions to produce curves over p-adic fields and number fields where each class obstructs
\(\pi_1\)-sections and hence rational points.
Among our geometric results are a new proof of the section conjecture for the generic curve of genus \(g\geq 3\), and a proof of the
section conjecture for the generic curve of even genus with a rational divisor class of degree one (where the obstruction to the
existence of a section is genuinely non-abelian).
- Vanishing cycles, plane curve singularities, and framed mapping class groups, with Pablo Portilla Cuadrado, Geom. Topol. 25 (2021), no. 6, 3179–3228.
Abstract:
Let f be an isolated plane curve singularity with Milnor fiber of genus at least 5. For all such f, we give (a) an intrinsic description of
the geometric monodromy group that does not invoke the notion of the versal deformation space, and (b) an easy criterion to decide if a given
simple closed curve in the Milnor fiber is a vanishing cycle or not. With the lone exception of singularities of type A and D, we find that
both are determined completely by a canonical framing of the Milnor fiber induced by the Hamiltonian vector field associated to f. As a corollary
we answer a question of Sullivan concerning the injectivity of monodromy groups for all singularities having Milnor fiber of genus at least 7.
- Global fixed points of mapping class group actions and a theorem of Markovic, with Lei Chen, J. Topol. 15 (2022), no. 3, 1311–1324.
Abstract:
We give a short and elementary proof of the non-realizability of the mapping class group via homeomorphisms.
This was originally established by Markovic, resolving a conjecture of Thurston. With the tools established in this paper,
we also obtain some rigidity results for actions of the mapping class group on Euclidean spaces.
- Framed mapping class groups and the monodromy of strata of Abelian differentials, with Aaron Calderon, J. Eur. Math. Soc., 25(2023), no.12, 4719–4790.
Abstract:
This paper investigates the relationship between strata of abelian differentials and various mapping class groups afforded
by means of the topological monodromy representation. Building off of prior work of the authors, we show that the fundamental
group of a stratum surjects onto the subgroup of the mapping class group which preserves a fixed framing of the underlying
Riemann surface, thereby giving a complete characterization of the monodromy group. In the course of our proof we also show
that these "framed mapping class groups" are finitely generated (even though they are of infinite index) and give explicit
generating sets.
- Relative homological representations of framed mapping class groups, with Aaron Calderon, Bull. Lond. Math. Soc., 53 (2021) no. 1 204–219.
Abstract:
Let (Σ,Z) be a surface endowed with a nonempty finite set of marked points. When Σ\Z is equipped
with a preferred framing φ, the "framed mapping class group" PMod(Σ, Z)[φ] is defined as the subgroup of
the (pure) mapping class group PMod(Σ,Z) consisting of elements that preserve φ up to isotopy. Such groups
arise naturally in the study of families of translation surfaces. In this note, we determine the action of
PMod(Σ,Z)[φ] on the relative homology H1(Σ,Z;Z), describing the image as the kernel of a certain
crossed homomorphism related to classical spin structures. Applying recent work of the authors, we use this to describe the
monodromy action of the orbifold fundamental group of a stratum of abelian differentials on the relative periods.
- Higher spin mapping class groups and strata of Abelian differentials over Teichmüller space, with Aaron Calderon, Adv. Math. 389 (2021), Paper No. 107926.
Abstract:
For g ≥ 5, we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space,
establishing an analogue of a theorem of Kontsevich and Zorich in the setting of marked translation surfaces. Building off of
work of the first author, we find that the non-hyperelliptic components are classified by an
invariant known as an r--spin structure. This is accomplished by computing a certain monodromy group valued in the mapping class
group. To do this, we determine explicit finite generating sets for all r--spin stabilizer subgroups of the mapping class group,
completing a project begun by the second author in
a recent paper.
Some corollaries in flat geometry and toric geometry are obtained from these results.
- Linear-central filtrations and the image of the Burau representation, Geom. Dedicata 211, 145–163 (2021).
Abstract:
The Burau representation is a fundamental bridge between the braid group and diverse other topics in mathematics. A 1974 question of
Birman asks for a description of the image; in this paper we give an approximate answer. Since a 1984 paper of Squier it has been
known that the Burau representation preserves a certain Hermitian form. We show that the Burau image is dense in this unitary group
relative to a topology induced by a naturally-occurring filtration. We expect that the methods of the paper should extend to many
other representations of the braid group.
- Section problems for configurations of points on the Riemann sphere, with Lei Chen, Alg. Geom. Topol., 20-6 (2020), 3047--3082.
Abstract:
This paper contains a suite of results concerning the problem of adding m distinct new points to a configuration of n
distinct points on the Riemann sphere, such that the new points depend continuously on the old. Altogether, the results of
the paper provide a complete answer to the following question: given n ≥ 5, for which m can one continuously add m
points to a configuration of n points? For n ≥ 6, we find that m must be divisible by n(n-1)(n-2), and we provide
a construction based on the idea of cabling of braids. For n = 3,4, we give some exceptional constructions based on the
theory of elliptic curves.
- Arithmeticity of the monodromy of some Kodaira fibrations, with Bena Tshishiku, Compositio, 155(1), 114-157.
Abstract:
A question of Griffiths-Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic.
We resolve this in the affirmative for the class of algebraic surfaces known as Atiyah-Kodaira manifolds, which have base
and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the ``geometric"
monodromy, valued in the mapping class group of the fiber.
- The Birman exact sequence does not virtually split, with Lei Chen, Math. Res. Lett. 28 (2021), no. 2, 383–413.
Abstract:
This paper answers a basic question about the Birman exact sequence in the theory of mapping class groups.
We prove that the Birman exact sequence does not admit a section over any subgroup Γ contained in the
Torelli group with finite index. A fortiori this proves that there is no section of the Birman exact
sequence for any finite-index subgroup of the full mapping class group. This theorem was announced in a 1990
preprint of G. Mess, but an error was uncovered and described in a recent paper of the first author.
- Monodromy and vanishing cycles in toric surfaces, Inventiones, 216(1), 153-213.
Abstract:
Given an ample line bundle on a toric surface, a question of Donaldson asks which simple closed curves can be vanishing
cycles for nodal degenerations of smooth curves in the complete linear system. This paper provides a complete answer.
This is accomplished by reformulating the problem in terms of the mapping class group-valued monodromy of the linear
system, and giving a precise determination of this monodromy group.
- On the monodromy group of the family of smooth plane curves, Glasgow Math. J., to appear.
Abstract:
We consider the space of smooth complex projective plane curves of degree d. Defined over this is the tautological family
of plane curves, and hence there is a monodromy representation into the mapping class group of the fiber. We show two results
concerning this monodromy group. First, we show that the presence of an invariant known as an ``n-spin structure'' constrains
the image in ways not predicted by previous work of Beauville. Second, we show that for degree d = 5, our invariant is the only
obstruction for a mapping class to be contained in the image. This requires combining the algebro-geometric work of Lönne
with Johnson's theory of the Torelli subgroup of the mapping class group.
- Cup products in surface bundles, higher Johnson invariants, and MMM classes, Math Z., 288(3), 1377-1394.
Abstract:
In this paper we prove a family of results connecting the problem of computing cup products in surface bundles to various
other objects that appear in the theory of the cohomology of the mapping class group and the Torelli group. We show that
N. Kawazumi's twisted MMM class m0,k can be used to compute k-fold cup products in surface bundles, and that m0,k
provides an extension of the higher Johnson invariant τk-2 as a twisted cohomology class on the full mapping class
group. These results are used to show that the behavior of the restriction of the even MMM classes e2i to the Torelli group of a
surface with one boundary component is completely determined by the image of the higher Johnson invariants, and to give a
partial answer to a question of D. Johnson. We also use these ideas to show that all surface bundles with monodromy in the
Johnson kernel have cohomology rings isomorphic to that of a trivial bundle, implying the vanishing of all higher Johnson
invariants when restricted to the Johnson kernel.
- On the non-realizability of braid groups by diffeomorphisms, with Bena Tshishiku, Bull. Lond. Math. Soc. 48 (2016), no. 3, 457--471.
Abstract:
For every compact surface S of finite type (possibly with boundary components but without punctures), we show that when n is
sufficiently large there is no lift σ of the surface braid group Bn(S) to Diff(S,n), the group of
diffeomorphisms preserving n marked points and restricting to the identity on the boundary. Our methods are applied to give a
new proof of Morita's non-lifting theorem in the best possible range. These techniques extend to the more general setting of
spaces of codimension-2 embeddings, and we obtain corresponding results for spherical motion groups, including the
string motion group.
- Surface bundles over surfaces with arbitrarily many fiberings, Geom. Topol. 19-5 (2015), 2901--2923.
Abstract:
In this paper we give the first example of a surface bundle over a surface with at least three fiberings. In fact, for each
n ≥ 3 we construct 4-manifolds E admitting at least n distinct fiberings
pi: E → Σgi as a surface
bundle over a surface with base and fiber both closed surfaces of negative Euler characteristic. We give examples of surface
bundles admitting multiple fiberings for which the monodromy representation has image in the Torelli group, showing the
necessity of all of the assumptions made in the main theorem of our recent
paper.
Our examples show that the number of surface bundle structures that can be realized on a 4-manifold E with Euler characteristic d grows
exponentially with d.
- Cup products, the Johnson homomorphism, and surface bundles over surfaces with multiple fiberings, Alg. Geom. Topol. 15-6 (2015), 3613--3652.
Abstract:
Let Σg → E → Σh be a surface bundle over a surface with monodromy representation ρ: π1 Σh → Mod(Σg)
contained in the Torelli group. In this paper we express the integral cup product structure in H*E in terms
of the Johnson homomorphism. This is applied to the question of obtaining an upper bound on the maximal n such that p1: E → Σh1,..., pn: E → Σhn
are fibering maps realizing E as the total space of a surface bundle over a surface in n distinct ways. We prove that any
nontrivial surface bundle over a surface with monodromy contained in the Johnson kernel fibers in a unique way.
- Sandpiles and Dominos, with Laura Florescu, Daniela Morar, David Perkinson and Tianyuan Xu, The Electronic Journal of Combinatorics, Volume 22(1), 2015.
Abstract:
We consider the subgroup of the abelian sandpile group of the grid graph consisting of configurations of sand that are symmetric with respect to central vertical and horizontal axes. We show that the size of this group is (i) the number of domino tilings of a corresponding weighted rectangular checkerboard; (ii) a product of special values of Chebyshev polynomials; and (iii) a double-product whose factors are sums of squares of values of trigonometric functions. We provide a new derivation of the formula due to Kasteleyn and to Temperley and Fisher for counting the number of domino tilings of a 2m x 2n rectangular checkerboard and a new way of counting the number of domino tilings of a 2m x 2n checkerboard on a Möbius strip.
- A Note on the Critical Group of a Line Graph, with David Perkinson and Tianyuan Xu, The Electronic Journal of Combinatorics, Volume 18(1), 2011.
Abstract:
This note answers a question posed by Levine in arxiv:math.CO/0906.2809. The main result is Theorem 1 which shows that under certain circumstances a critical group of a directed graph is the quotient of a critical group of its directed line graph.