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The main subjects of my PhD project are the Thurston norm and taut foliations on 3-manifolds.
The Thurston norm is a seminorm on the second real homology group of a compact orientable 3-manifold M. The unit ball of this norm is a convex polyhedron, whose shape data (e.g. number of vertices, regularity) measures the complexity of the surfaces sitting in the ambient 3-manifold. Some of the principal general problems I address are:
Which polyhedra can be realised as the unit ball of the Thurston norm on some 3-manifold?
Given a 3-manifold, compute the unit ball of its Thurston norm.
A foliation is a partition F of M into injectively immersed surfaces - called leaves - locally looking like the subdivision of the space into horizontal planes. A foliation F is taut if every leaf intersects a closed transversal to the foliation. Taut foliations were introduced by Novikov and brought to prominence by Sullivan and Thurston. By Thurston and Roussarie, every taut foliation of M offers new insight about the topology of surfaces inside M. For this reason, classifying taut foliations on 3-manifolds has been an important problem for topologists in the last seven decades.
A first way to study taut foliations is through their tangent distribution. The plane field TF determines a second cohomology class e(F), called the Euler class of F. Thurston showed that the Euler class of a taut foliation must lie inside the dual Thurston ball B* of M. Thurston asked whether every integral class in B* is the Euler class of some taut foliation. Kronheimer and Mrowka further showed that there are only finitely many homotopy classes of plane fields of taut foliations. My research addresses the problem of classifying taut foliations in multiple ways: first via their Euler classes, then up to homotopy of plane fields, and finally up to homotopy of plane fields of taut foliations.
Publications (click on titles for access to papers)
Publications (click on titles for access to papers)
Journal of Knot Theory and Its Ramifications Vol. 34, No. 12, 2550059 (2025)
We show that, when M is the complement of a 2-bridge link L, the Thurston ball of M has at most 8 faces. The proof of this result strongly relies on a description of essential surfaces in 2-bridge link complements given by Floyd and Hatcher. Then, we exhibit norm-minimizing representatives for every integral homology class and use them to compare the complexity of the Thurston ball with the complexities of L and of M.
As an example, we show that all the vertices of the Thurston ball lie on the bisectors if and only if M fibers over the circle with fiber a surface with boundary equal to a longitude of one component and some meridians of the other. Finally, we use 2-bridge links in satellite constructions to find 2-component links whose complements in the 3-sphere have Thurston balls with arbitrarily many vertices.
Journal of Topology and Analysis Vol. 18, No. 5, pp. 1449--1475 (2026)
The Thurston norm of a closed oriented graph manifold is a sum of absolute values of linear functionals, and either each or none of the top-dimensional faces of its unit ball are fibered. We show that, conversely, every norm that can be written as a sum of absolute values of linear functionals with rational coefficients is the nonvanishing Thurston norm of some graph manifold, with respect to a rational basis on its second real homology. Moreover, we can choose such graph manifold either to fiber over the circle or not.
In particular, every symmetric polygon with rational vertices is the unit polygon of the nonvanishing Thurston norm of a graph manifold fibering over the circle. In dimension 3 or more, many symmetric polyhedra with rational vertices are not realizable as nonvanishing Thurston norm ball of any graph manifold. However, given such a polyhedron, we show that there is always a graph manifold whose nonvanishing Thurston norm ball induces a finer partition into cones over the faces.
Given a cooriented branched surface B fully carrying a foliation F, we use the dual graph of B to define a simplicial 1-cycle representing the Poincaré dual of the Euler class of F. This construction provides an effective way to track how the Euler class of a foliation carried by a hierarchy changes when the orientation of a decomposing disk is reversed. We also show how our formula generalises previous results of Lackenby and Dunfield.
By the classical work of Thurston and Gabai, the Thurston norm of any compact oriented irreducible 3-manifold with toroidal boundary is determined by finitely many taut sutured manifold hierarchies. We give an explicit procedure to extract this information from such hierarchies via the maw dual graph construction. As an application, we compute the Thurston norm of the exterior of all alternating and some non-alternating pretzel links with three components. Using these computations, we give a negative answer to a question of Baker--Taylor. Namely, we exhibit an infinite family of manifolds, and an infinite family of knots in each manifold, such that the wrapping number associated with these knots is not a seminorm on the respective second real homology group.
Next, we use Gabai's work to show that if a taut surface in a link exterior is disjoint from a boundary torus and does not represent a corner of the Thurston norm ball, then the corresponding link component is algebraically split from the rest of the link. Moreover, the surface remains norm-minimising under every non-longitudinal Dehn filling on that boundary component and, under an assumption of atoroidality, is a leaf of a taut foliation transverse to the filling core.
Other
Other
As Master thesis I studied a proof of Thurston's Theorem about the hyperbolization of the 3-manifolds that fiber over the circle. Unfortunately, the thesis is in Italian, but perhaps it will be in English one day too!
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