Calabi-Yau metrics can degenerate in a 1-parameter family by varying the complex structure, and a basic invariant is the dimension of the essential skeleton, which is an integer between 0 and n. The case of zero is the context of noncollapsed degeneration of Donaldson-Sun theory, while the case of n is the context of the SYZ conjecture.

The rank versus genus conjecture, proposed by Waldhausen in 1978, asks whether the rank of the fundamental group of a 3-manifold equals its Heegaard genus. While there are known counterexamples to this conjecture, it still holds for link exteriors in the 3-sphere. In this talk, we discuss this problem and we describe families of links for which this conjecture is true.

Given a free group F and a pseudovariety V of finite groups, i.e. a class of finite groups closed under taking subgroups, quotients and finitary direct products, we endow F with its pro-V topology. Given an arbitrary finitely generated subgroup H of F, some classical topological decidability problems are: Is it decidable whether H is dense? Is it decidable whether H is closed?  

In this talk, we will study a gauge-theoretic construction of (branched) complex projective structures on a closed Riemann surface X of genus g>1. This construction underlies the celebrated conformal limit of Gaiotto and provides a preliminary understanding of its geometry.  More concretely, the non-abelian Hodge correspondence maps a polystabl SL(2,R)-Higgs bundle on X to a connection which, in some cases, is the holonomy of a branched hyperbolic structure.

Mirror symmetry is a somewhat mysterious phenomenon that relates the geometry of two distinct Calabi-Yau manifolds. In the realm of trying to understand this relationship an equation for a connection on a line bundle in a Kahler manifold appeared. This is commonly called the deformed Hermitian-Yang-Mills equation and I will explain what it is and the current joint work with Benoit Charbonneau and Rosa Sena-Dias which explicitly solves this equation on a specific setting. This helps in understanding the problem of the existence of solutions and ruling out possible stability conditions.

In this talk, I will discuss recent joint work with Alexeev-Engel-Schaffler that provides explicit, combinatorial descriptions of stable degenerations of numerically polarized Enriques surfaces of degree 2. We show that this can be identified with a semitoroidal compactification of the Hodge-theoretic period domain, leveraging Alexeev-Engel’s seminal work on compactifications of moduli spaces of K3 surfaces via integral-affine geometry.

I will explain how to express the space of almost complex structures J(M) on a six-manifold M as the quotient of a space of sections of a sphere bundle over M by an S^1 action. This leads to a computation of the rational homotopy type of J(M) in most cases, as well as a simple description of the homotopy type in others, such as when M is the six-sphere. In dimension 6, two orthogonal, almost complex structures generically intersect at a finite number of points inside twistor space. I will give a formula for the intersection number and discuss some applications.

A twisted rational map over an algebraically closed and complete non-archimedean field K of characteristic 0 is a composition of a rational map over K and certain continuous automorphism of K. We explore the dynamical properties of tame twisted rational maps.

We consider germs of holomorphic vector fields at the origin in dimension 3 with non-isolated singularities that are tangent to a holomorphic 

We consider germs of holomorphic vector fields at the origin in dimension 3 with non-isolated singularities that are tangent to a holomorphic foliation of codimension one, referred to as a 2-flag of foliations. Our focus lies on cases where this geometric structure originates from second-orde