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

We first investigate the interconnection of invariants of certain group actions and time-reversibility of a class of two-dimensional polynomial systems with 1:-1 resonant singularity at the origin. 

The time-reversibility is related to the Sibirsky subvariety of the center (integrability) variety and it is known that every time-reversible system has a local analytic first integral at the origin. 

We propose a new  algorithm to obtain a generating set for the Sibirsky ideal of such polynomial  systems and investigate  some algebraic properties of this ideal.

Likelihood Geometry is the geometric study of the maximum likelihood estimation problem to an algebraic variety serving as a statistical model. In this talk, basic notions of algebraic statistics and likelihood geometry will be introduced before discussing the maximum likelihood degree of a toric variety (also known as a log-linear model in the statistics context) in more detail.

Any Poisson bracket on a differentiable manifold can be written locally as the sum of two commuting Poisson brackets: a Poisson bracket of
a symplectic form and a Poisson bracket that vanishes at a point. This is the so-called Weinstein splitting. The result is strictly local; in
general, it is impossible to globalize the two local Poisson structures.  It turns out that for holomorphic Poisson structures on compact Kähler
manifolds admitting a simply connected compact symplectic leaf, the local Weinstein splitting globalizes and produces a global splitting of

The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors

separated by a narrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction

is modeled by a family of differential equations on 2-torus depending on 3 parameters. In this talk we are going to study these parameters.