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Free subgroups of 3-manifold groups

Expositor: Misha Belolipetsky - IMPA
Ter 23 Out 2018, 15:30 - Room 236Geometria Diferencial

Resumo: We show that any cocompact Kleinian group $\Gamma$ has an exhaustive filtration by normal subgroups $\Gamma_i$ such that any subgroup of $\Gamma_i$ generated by $k_i$ elements is free, where $k_i \ge [\Gamma:\Gamma_i]^C$ and $C = C(\Gamma) > 0$. Together with this result we prove that $\log k_i \ge C_1 \mathrm{sys}_1(M_i)$, where $\mathrm{sys}_1(M_i)$ denotes the systole of $M_i$, thus providing a large set of new examples for a conjecture of Gromov. In the second theorem $C_1> 0$ is an absolute constant. We also consider a generalization of these results to non-compact finite volume hyperbolic $3$-manifolds.

In the talk, I am going to discuss the proofs of these theorems and some related open problems.

This is a joint work with Cayo Dória.

The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations

Expositor: José Manuel Jiménez - Universidad Nacional de Colombia, Sede Medellin
Ter 23 Out 2018, 15:30 - Room 232Análise / Equações Diferenciais Parciais

Resumo: In this talk we consider a class of initial value problems (IVP) associated to the family of fractional two-dimensional Benjamin-Ono (BO) equations

$$u_t+D_x^{\alpha} u_x +\mathcal Hu_{yy}+uu_x =0,\hskip30pt (x,y)\in\mathbb R^2,\; t\geq 0,\hskip15pt {\rm (1)}\\u(x,y,0)=u_0(x,y), \hskip35pt {\;}$$where $\;u(x,y,t)\in\mathbb R$, $\;0<\alpha\leq1$, $D_x^{\alpha}$ denotes the operator defined through the Fourier transform by\begin{align}(D_x^{\alpha}f)\widehat{\;}(\xi,\eta):=|\xi|^{\alpha}\widehat{f}(\xi,\eta)\,,\notag\end{align}and $\mathcal H$ denotes the Hilbert transform with respect to the variable $x$.

When $\alpha=1$ the equation in (1), called Shrira equation, is a bidimensional generalization of the BO equation$$u_t+\mathcal H u_{xx} +u \partial_x u =0,\qquad\qquad x\in\mathbb R,\; t\geq 0,$$and was deduced by Pelinovsky and Shrira in connection with the propagation of long-wave weakly nonlinear two-dimensional perturbations in parallel boundary-layer type shear flows.

Inspired by recent works of Linares, Pilod and Saut, we study the relation between the amount of dispersion and the size of the Sobolev space in order to have local well-posedness of the family of IVPs (1).

Automorphism group of Fano models of blow-up of $mathbb{P}^{n}$

Expositor: Inder Kaur - IMPA
Qua 24 Out 2018, 15:30 - Room 228Álgebra

Resumo: Let $S$ be the blow-up of $\mathbb{P}^{n}$ in $n+3$ points, where $n$ is odd. When $n \geq 3$, this is no longer a Fano variety. However, it admits a Fano model which is isomorphic to the moduli space of rank $2$ stable parabolic bundles on $\mathbb{P}^1$ with weights $(0,\frac{1}{2})$. We use the modular interpretation to compute the automorphism group of the Fano model. This is jont work with Carolina Araujo, Thiago Fassarella and Alex Massarenti.