RESEARCH STATEMENT SER-WEI FU

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1 RESEARCH STATEMENT SER-WEI FU 1. Introduction My research interests centers around hyperbolic geometry, geometric group theory, low-dimensional topology, and dynamical systems. These topics are tied together in my work through the study of closed curves on a surface. The utility of this approach to geometry and dynamics is famously illustrated by Thurston s work on hyperbolic geometry, Teichmüller theory, and mapping class groups which is based heavily on his analysis of simple closed curves, see [10, 26, 28]. The classical work of Fricke and Klein proves that a hyperbolic metric is determined (up to isotopy) by the lengths of a finite set of simple closed curves. In my thesis, the motivating question was the extent to which the Fricke-Klein result holds for certain family of flat metrics. A flat metric is the metric determined by a holomorphic quadratic differential. Duchin-Leininger-Rafi in [5] proved that there is no such finite set of curves for the space of all flat metrics. I extended the result to strata of flat metrics, in a natural stratification, with sufficiently large dimension. On the other hand, I proved that the obstruction to finding such a finite set is a global one. Specifically, I proved that every point in a stratum has a neighborhood and a finite set of curves so that the lengths determine any metrics within the neighborhood, see Section 2. In Section 3, I will describe a family of deformations of quadratic differentials using a new construction I call flat grafting. The goal of that project is to analyze, by way of analogy, the flat grafting rays comparing them with grafting and earthquake rays in Teichmüller space. On the global scale, the deformation can be used to construct canonical paths between any two points in the space. In the local sense, I study the changes in the strata structure under flat grafting. Describing a neighborhood of a quadratic differential that intersects multiple strata is in fact a configuration problem in disguise, which is the main motivation for developing new tools. In other directions, I worked with Jayadev Athreya studying the behavior of the systole function with respect to the earthquake flow over the moduli space, see Section 4. A generic earthquake trajectory is dense in the moduli space and hence the limit infimum of the systole function will be zero. On the once-punctured torus, my work quantifies the rate that the limit infimum of the systole function goes to zero. In particular, each dense trajectory encodes a continued fraction expansion of an irrational number. The golden ratio in this case comes from an earthquake trajectory that goes arbitrarily deep into the thin part of the moduli space while taking the maximal amount of time. I am also working on a project with Matthew Stover observing the relation between graphs embedded on surfaces with moduli spaces of polygons, see Section 5. We are interested in embedded graphs that partition the surface into polygons. 1

2 2 SER-WEI FU Through this setup, we can map metrics on the surface to polygons with sidegluings. Our main question is regarding sufficient conditions on the graph and the surface so that the preimage of equilateral polygons is the whole space of metrics. 2. Length spectral rigidity For a closed surface S, let C(S) denote the set of (homotopy classes of) closed curves on S. Given a metric m on S, let l C(S) (m) denote the point of R C(S) that records the minimal length of every homotopy class. The length spectral rigidity question for a family of (isotopy classes of) metrics G asks whether the length spectrum l C(S) thought of as a function G R C(S) is injective. In 1990, Otal [24] gave a positive answer for the class of negatively curved metrics using geodesic currents introduced by Bonahon [2]. This has since been extended and refined by a number of authors [3, 4, 7, 8, 14]. Motivated by the results of [5], I am interested in length spectral rigidity results either restricting to simple closed curves or a finite set of closed curves when considering the space of flat metrics Flat(S). We say that Σ C(S) is length spectrally rigid over G Flat(S) if l Σ is an injective map over G. The set Σ is said to be locally length spectrally rigid at ρ over G if there exists a neighborhood N ρ G such that l Σ is injective over N ρ. In my case, G is a stratum of the space of flat metrics and I will use the notation Flat(S, α) to denote the stratum determined by α which encodes the cone angles and the holonomy. To describe my results, we recall that Thurston described a topology on the set of isotopy classes of simple closed curves, and proved that the closure in this topology is naturally his space of projective measured foliations PMF(S). Duchin- Leininger-Rafi [5] proved that for a surface S g,n of genus g with n punctures in which 3g + n 3 2 has the following property. A set of simple closed curves Σ is spectrally rigid for Flat(S) if and only if Σ = PMF(S). In particular, for any non-dense set of simple closed curves, they constructed a family of flat metrics on which l Σ : Flat(S) R Σ is constant. My first result in [9] extends and refines this result of [5]. Theorem 1. If (3g + n 3) 2 and dim(flat(s, α)) > (6g + 2n 6), then a set of simple closed curves Σ is length spectrally rigid over Flat(S, α) if and only if Σ = PMF(S). The main ingredient of the proof is a construction of positive dimensional families of flat metrics on which l Σ is constant. We note that our construction is considerably more flexible than that of [5], producing families in all of Flat(S) of dimension roughly three times that of [5]. Theorem 1 is at first somewhat surprising as Flat(S, α) is finite dimensional, which suggests that one should be able to find a finite set of curves which is spectrally rigid. In both [5] and in my construction, the families of metrics constructed can potentially come from very far away, suggesting that the difficulty is a global one, rather than a local one. I show in [9] that this is indeed the case by proving that at least locally, one can find a finite rigid set. Theorem 2. For any ρ Flat(S, α), there exists a finite set of closed curves Σ C(S) such that Σ is locally spectrally rigid at ρ over Flat(S, α).

3 RESEARCH STATEMENT 3 3. Flat grafting The space of quadratic differentials QD(S), seen as the cotangent bundle over the Teichmüller space T (S), is a very fascinating space. Studying the properties of the earthquake flow and the grafting ray in [6,16,19] motivated me to construct deformations of QD(S) that I call flat graftings. The constructions are related to Masur and Zorich s hole transports in [23], and Wright s cylinder deformations in [29]. A flat grafting deformation is defined by an initial quadratic differential q, a simple closed curve γ, and a vector v R 2, the height of the flat cylinder that will be glued in along the curve. We define the flat grafting ray to be the one-parameter family obtained by changing the length of the vector v. The straight line foliation of q parallel to v is a measured foliation that is invariant on the flat grafting ray. Every flat grafting deformation can be written as a horizontal flat grafting conjugated by a rotation. Theorem-in-progress 1. Let q QD(S) and γ be a simple closed curve. The projective class of the vertical foliation of the horizontal grafting ray H t (q, γ) converges to γ in PMF(S). As a corollary, combining flat grafting deformations with scaling and the Teichmüller flow, we can take q arbitrarily close to any desired quadratic differential. The space QD(S) is naturally stratified by the cone angles of the singularities of the induced singular flat metric. Flat grafting deformations describe all possible configurations of the splitting of singularities, i.e., a cone point splitting into a finite number of cone points of smaller cone angles. Theorem-in-progress 2. Let q be a quadratic differential in the closure of a stratum QD(S, α). There exists a simple closed curve γ such that the horizontal grafting ray of q with respect to γ lie inside QD(S, α) for all t Cusp excursions for earthquake flows Mumford s compactness criterion tells us that to tend to infinity in the moduli space, the length of some curve must tend to zero. The length of the shortest curve in a hyperbolic metric, or the systole function, is thus a natural function. In particular, the behavior of this function along various flows provides useful geometrical and dynamical information. The non-divergence property of horocyclic flows and earthquake flows proved by Minsky-Weiss [22] shows that the orbits of these flows spend only a small amount of time in the thin-part (i.e. the place where the systole is small). However, the ergodicity of the two flows (see [17, 20]) implies that the systole goes to zero (along a subsequence) for almost every trajectory. These contrasting properties motivate further investigations. We consider the quantity C f ([X], λ) := lim sup t log l sys (E t (X, λ)), f(t) where [X] is a point in the moduli space M(S), λ ML(S) is a measured lamination on S, E t describes the time t earthquake flow on M(S) ML(S), and l sys : M(S) R + is the systole with respect to the hyperbolic metric X. See [16, 27] for more details on the earthquake map. We observe that ergodicity of the earthquake flow implies that C f (X, λ) is constant almost everywhere.

4 4 SER-WEI FU In [1], Athreya proved that C log t is a constant almost everywhere for the horocyclic flow. The typical value of C log t is also studied in [18, 25], where the result is called a logarithmic law on flows. The close relation between the horocyclic flow and the earthquake flow shown in [20] motivates us to develop an earthquake flow analogue. However, hyperbolic geometry proved otherwise for the once-punctured torus. Theorem-in-progress 3. For almost every ([X], λ) M(S 1,1 ) ML(S 1,1 ) with respect to a natural measure that is finite over the unit-bundle, we have log l C sys (E t (X, λ)) t ([X], λ) = lim sup =. t t The computation used in the proof shows that C t is closely related to the continued fraction expansion of the slope of λ with respect to X. The classical work on continued fractions allows us to conclude C t = a.e. on S 1,1 as well as finding examples where C t is nonzero and finite. In particular, the smaller the exponents of the continued fraction expansion is, the limit infimum of the systole function goes to zero slower, hence the golden ratio corresponds to the minimum of C t for irrational λ. 5. Graphs on surfaces and moduli spaces of polygons Graphs on surfaces have been extensively studied from the combinatorial and the topological viewpoints, see [12, 21]. From a metric geometry point of view, we are interested in geodesic embeddings of graphs on surfaces. The well-known theorem of Tutte states that every 3-connected planar graph has a straight-line embedding with convex polygon faces. A similar question can be asked for any graph that can be 2-cell-embedded (the complement of the graph is a collection of disks) on a closed surface equipped with a locally Euclidean metric. In particular, we take the natural metric on graphs and ask the following question. Question. If G is a graph that has a 2-cell-embedding on a torus, what properties of G will guarantee the existence of a straight-line embedding where all edges are of the same length on some flat torus? Moreover, is there any graph with degree at least 3 at each vertex that has such an embedding for every flat torus? The complement of an edge-isometric (edges of equal length) 2-cell embedded graph G on a flat torus is a collection of equilateral polygons equipped with edgegluings. The moduli spaces of equilateral polygons were studied in [15] and the embeddings we are looking for form natural subsets of the moduli spaces. The 2-rose and the 3-rose embedded on the torus correspond to subsets of the moduli space of equilateral quadrilaterals and equilateral hexagons. For both graphs, almost every flat torus does not admit edge-isometric embeddings. The graphs K 4 and K 3,3, however, can be edge-isometric embedded on a large open set of flat tori. The project combines different fields of study and we expect interesting topological descriptions for the moduli space of the embeddings. References [1] Jayadev S. Athreya, Cusp excursions on parameter spaces, J. Lond. Math. Soc. (2) 87 (2013), no. 3, [2] Francis Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), no. 1,

5 RESEARCH STATEMENT 5 [3] Christopher B. Croke, Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv. 65 (1990), no. 1, [4] C. Croke, A. Fathi, and J. Feldman, The marked length-spectrum of a surface of nonpositive curvature, Topology 31 (1992), no. 4, [5] Moon Duchin, Christopher J. Leininger, and Kasra Rafi, Length spectra and degeneration of flat metrics, Invent. Math. 182 (2010), no. 2, [6] David Dumas and Michael Wolf, Projective structures, grafting and measured laminations, Geom. Topol. 12 (2008), no. 1, [7] Albert Fathi, Le spectre marqué des longueurs des surfaces sans points conjugués, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 9, (French, with English summary). [8] Jeffrey Frazier, Length spectral rigidity of non-positively curved surfaces, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.) University of Maryland, College Park. [9] Ser-Wei Fu, Length spectra and strata of flat metrics, Geom. Dedicata 173 (2014), [10] A. Fathi, F. Laudenbach, and V. Poénaru, Travaux de Thurston sur les surfaces, Société Mathématique de France, Paris, 1991 (French). Séminaire Orsay; Reprint of Travaux de Thurston sur les surfaces, Soc. Math. France, Paris, 1979 [ MR (82m:57003)]; Astérisque No (1991). [11] Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, [12] Jonathan L. Gross and Thomas W. Tucker, Topological graph theory, Dover Publications, Inc., Mineola, NY, Reprint of the 1987 original [Wiley, New York; MR (88h:05034)] with a new preface and supplementary bibliography. [13] John Hubbard and Howard Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), no. 3-4, [14] Sa ar Hersonsky and Frédéric Paulin, On the rigidity of discrete isometry groups of negatively curved spaces, Comment. Math. Helv. 72 (1997), no. 3, [15] Michael Kapovich and John Millson, On the moduli space of polygons in the Euclidean plane, J. Differential Geom. 42 (1995), no. 2, [16] Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, [17] Howard Masur, Ergodic actions of the mapping class group, Proc. Amer. Math. Soc. 94 (1985), no. 3, [18], Logarithmic law for geodesics in moduli space, (Göttingen, 1991/Seattle, WA, 1991), Contemp. Math., vol. 150, Amer. Math. Soc., Providence, RI, 1993, pp [19] Curtis T. McMullen, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998), no. 2, [20] Maryam Mirzakhani, Ergodic theory of the earthquake flow, Int. Math. Res. Not. IMRN 3 (2008), Art. ID rnm116, 39. [21] Bojan Mohar and Carsten Thomassen, Graphs on surfaces, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, [22] Yair Minsky and Barak Weiss, Nondivergence of horocyclic flows on moduli space, J. Reine Angew. Math. 552 (2002), [23] Howard Masur and Anton Zorich, Multiple saddle connections on flat surfaces and the principal boundary of the moduli spaces of quadratic differentials, Geom. Funct. Anal. 18 (2008), no. 3, [24] Jean-Pierre Otal, Le spectre marqué des longueurs des surfaces à courbure négative, Ann. of Math. (2) 131 (1990), no. 1, (French). [25] Dennis Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math. 149 (1982), no. 3-4, [26] William P. Thurston, The Geometry and Topology of Three-manifolds, Princeton Lecture Notes (1980). [27], Earthquakes in two-dimensional hyperbolic geometry, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp [28], On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, [29] Alex Wright, Cylinder deformations in orbit closures of translation surfaces, Geom. Topol. 19 (2015), no. 1,