I work in geometric
topology. My main research interest concerns the construction and
study of 3manifolds and
4manifolds via
combinatorial and geometric tools like spines, polyhedra, and Dehn
surgery. I am also interested in hyperbolic
geometry and quantum topology.
Complexity of 3manifolds: As defined by Matveev, the complexity of a 3manifold is the minimum number of vertices of a simple spine, and equals the minimum number of tetrahedra in a triangulation in the most interesting cases. With various collaborators and one computer, we have produced tables of:
Complexity of 4manifolds: I would like to define a reasonable complexity on closed orientable 4manifolds. I have tried essentially two approaches:
Dehn surgery: A Dehn surgery is exceptional when the resulting 3manifold is not hyperbolic. A lot of effort has been devoted in the last 30 years to understanding exceptional surgeries: see this recent survey of Cameron Gordon. An extremely ambituous question would be the following: can we understand all exceptional surgeries on all knots/links in the 3sphere? There are infinitely many exceptional fillings on a multicomponent link, but it is possible to describe all of them with a finite amount of data. Such data form sometimes a surprisingly short list, which can be prouduced via computer in a reasonably short amunt of time. Consider for instance the following chain links These are conjecturally the links with i = 1, ..., 5 components having smallest hyperbolic volume. Using a python code on SnapPy it is possible to classify all exceptional surgeries on such links in few minutes, and the amount of data necessary to describe them is roughly constant for i = 1, 2, ..., 5: on each link, less than 10 exceptional surgeries are enough to "generate" (via symmetries) all the exceptional surgeries. See [22], with Petronio and Roukema. The python code is available on this page and can be used on any link. See the detailed instructions there. Normal surfaces: Normal and (octagonal) almost normal surfaces generalize to knormal surfaces, belonging the cases k=0 and 1 respectively. With Evgeny Fominykh we gave a short proof that a minimal triangulation of an irreducible 3manifold does not contain any knormal sphere (with few exceptions) in [15].
Kirby moves: The short paper [19] answers a nice question on Mathoverflow about Kirby calculus. I show that there is a finite collection of local moves that connected any two surgery presentations of the same 3manifold via framed links in the threesphere. Quantum invariants: I have more recently been interested in quantum invariants. In [23] with Costantino we use these mysterious objects to construct an analytic family of representations of the mapping class group defined on the unit disc, that includes the finite representations at the roots of unity. It would be nice to get similar analytic families for other 3dimensional objects. In [25] with Carrega we study the relation between quantum invariants, shadows, and ribbon surfaces. We have extended a theorem of Eisermann that connects quantum invariant and ribbon surfaces in the 3spheres. Hyperbolic 4manifolds: Various hyperbolic manifolds can be constructed by assembling hyperbolic regular polytopes. In [21] with Kolpakov we use the ideal hyperbolic 24cell to build hyperbolic fourmanifolds with an arbitrary number of cusps, and in [24] with Kolpakov and Tschantz we use the 120cell to build some hyperbolic fourmanifolds with connected geodesic boundary of controlled volume. In [27] I prove that every finitevolume hyperbolic 3manifold that decomposes into rightangled regular polyhedra is geodesically contained in a hyperbolic 4manifold having controlled volume. The paper [29] is a survey on hyperbolic fourmanifolds. In [30] with Riolo we define a deformation relating two noncommensurable hyperbolic fourmanifolds through cone manifolds with cone singularity an immersed surface. This family may be interpreted as a hyperbolic Dehn filling. Tropical geometry: I have been interested in tropical geometry for some time. In [26] with Golla we study the topological notion of decomposing a fourmanifold into pairofpants that arises naturally from this area. Spines of minimal area In [28], with Novaga, Pluda, and Riolo, we raise the question whether every closed riemannian manifold has a spine of minimal area (that is, codimension one Hausdorff measure). We answer it affirmatively in dimension 2 and study the spines of minimal lengths on constant curvature surfaces. We introduce the spine systole, a proper function on moduli spaces. 
some paintings of my
mother
