# Seminars

## TODAY

Algebraic curves and foliations

**Speaker:**

*Hossein Movasati*- IMPA

**Holomorphic Foliations**

**Abstract: **Consider a field $k$ of characteristic $0$, not necessarily algebraically closed, and a fixed algebraic curve $f=0$ defined by a tame polynomial $f\in k[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic foliations in the plane ${\mathbb A}^2_k$ having $f=0$ as a fixed invariant curve is generated as $k[x,y]$-module by at most four elements, three of them are the trivial foliations $fdx,fdy$ and $df$. Our proof is algorithmic and constructs the fourth foliation explicitly. Using Serre's GAGA and Quillen-Suslin theorem,
we show that for a suitable field extension $K$ of $k$ such a module over $K[x,y]$ is actually generated by two elements, and therefore, such curves are free divisors in the
sense of K. Saito. After performing Groebner basis for this module, we observe that in many well-known examples $K=k$. This is a joint work with C. Camacho and with an appendix by C. Hertling, https://arxiv.org/abs/2101.08627

## UPCOMING

Global well-posedness and scattering for the inhomogeneous nonlinear Schrödinger equation

**Speaker:**

*Luiz Gustavo Farah*- UFMG

**Analysis and Partial Differential Equations**

**Abstract: **We consider the inhomogeneous nonlinear Schrödinger (INLS) equation
\begin{equation*}%\label{INLS}
i u_t +\Delta u+|x|^{-b}|u|^\alpha u = 0, \,\,\, x\in \mathbb{R}^N,
\end{equation*}
in the intercritical regime $\frac{4-2b}{N}<\alpha<\frac{4-2b}{N-2}$, with $0<b<\min\{N/2,2\}$.

In this talk we discuss global well-posedness and scattering results for the INLS equation in the radial and non-radial settings. These results were obtained in collaboration with Mykael Cardoso (UFPI), Carlos Guzmán (UFF) and Jason Murphy (Missouri S&T).

The lecture link is meet.google.com/okd-onwg-naz

Minimal Surfaces in Hyperbolic 3-manifolds

**Speaker:**

*Baris Coskunuzer*- UT Dallas

**Differential Geometry**

**Abstract: **In this talk, we will show the existence of smoothly embedded closed minimal surfaces in infinite volume hyperbolic 3-manifolds. The talk will be non-technical, and accessible to graduate students.

*Remark: the link to access the talk will be made available half an hour before the talk, at http://w3.impa.br/~l.ambrozio/seminario.html. Alternatively, please contact the organiser Lucas Ambrozio at l.ambrozio@impa.br in order to receive the link by email.*