Seminários

Subspace stabilisers in hyperbolic lattices

Resumo: In a joint work with Nikolay Bogachev, Alexander Kolpakov, and Leone Slavich we discovered an interesting connection between totally geodesic subspaces of a hyperbolic manifold or orbifold and finite subgroups of the commensurator of its fundamental group. We call the totally geodesic subspaces associated to the finite subgroups by fc-subspaces. It appears that these subspaces have some remarkable properties. We show that in an arithmetic orbifold all totally geodesic subspaces of sufficiently small codimension are fc and there are infinitely many of them, while in non-arithmetic cases there are only finitely many fc-subspaces and their number is bounded in terms of volume. In particular, all totally geodesic subspaces of an arithmetic hyperbolic 3-orbifold are fc-subspaces. In the talk, I will begin with an introduction to the topic and then discuss some results and their proofs.

Genuine deformations of Euclidean hypersurfaces in higher codimensions

Resumo: In this presentation, we extend the classification given by Sbrana and Cartan for Riemannian manifolds $M^n$ with at least two non-congruent immersions in $\mathbb{R}^{n+1}$. We give a characterization of some families of Euclidean hypersurfaces which possess (non-trivial) isometric immersions in higher codimensions. We generalize concepts introduced by Sbrana and Cartan, such as a conjugate submanifold and species. Finally, we will see some examples of such manifolds.

Flags of AG codes and curves with many points

Resumo: Let ${\mathbb F}_q$ be the finite field with $q$ elements. A flag of linear codes $$C_0 \subset C_1 \subset \cdots \subset C_s$$ is said to satisfy the {\it isometry-dual property} if there exists a vector ${\bf x}\in ({\mathbb F}_q^*)^n$ such that $C_i={\bf x} \cdot C_{s-i}^\perp$, where $C_i^\perp$ denotes the dual code of the code $C_i$. Consider ${\mathcal F/\mathbb{ F}_q}$ a function field and let $P$ and $Q_1,\ldots, Q_t$ be rational places in ${\mathcal F}$. Let the divisor $D$ be the sum of pairwise different places of ${\mathcal F}$ such that $P, Q_1,\ldots, Q_t$ are not in $\mbox{supp}(D)$.

We study flags of $(t+1)$-algebraic geometry point codes $$C_{\mathcal L}(D, a_0P+\sum_{i=1}^t\beta_iQ_i)\subset C_{\mathcal L}(D, a_1P+\sum_{i=1}^t\beta_iQ_i))\subset \dots \subset C_{\mathcal L}(D, a_sP+\sum_{i=1}^t\beta_iQ_i)$$ for any tuple of integers $\beta_1,\dots,\beta_t$ and for an increasing sequence of integers $a_0,\dots,a_s$. We apply the obtained results to the broad class of Kummer extensions defined by affine equations of the form $y^m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $\mbox{gcd}(r, m)=1$. In particular, depending on the place $P$ and for $D$ an $Aut({\mathbb F}_q(x, y)/{\mathbb F}_q(x))$-invariant sum of rational places of ${\mathcal F}$ such that $P, Q_i \notin \mbox{supp} D$, we obtain necessary and sufficient conditions on $m$ and $\beta_i$'s such that the flag has the isometry-dual property. If we have time, we are also going to show the importance of having explicitly algebraic curves with many points in order to obtain {\it good } codes and how they can be constructed.