IMPA - O Instituto de Matemática Pura e Aplicada

The min-max theory of the length/area functional in the space of hypersurfaces of a compact Riemannian manifold defines several numbers, called min-max widths, which are critical values of the functional. We will discuss situations where the knowledge of these numbers allow to recover, in full or in part, geometric properties of the manifold itself. This is joint work with F. Marques and A. Neves.

An old question steaming from Klein's Erlangen Program can be phrased
in modern terms as: Is a given geometric object uniquely determined by
its group of symmetries? The first part of this talk consists of an
introduction to the problem with some selected examples from outside
algebraic geometry.

In the second part of the talk we come to the setting of algebraic
geometry, where we show that, in general, the answer to the above
question is negative. After restricting the class of varieties we will
show an instance where the answer is affirmative. Indeed, we show that
complex affine toric surfaces are determined by the abstract group
structure of their regular automorphism groups in the category of
complex normal affine surfaces using properties of the Cremona group.

Let M be a properly embedded, connected, complete surface in R^3 with boundary a convex planar curve C, satisfying an elliptic equation H=f(H^2-K), where H and K are the mean and the Gauss curvature respectively – which we will refer to as Weingarten equation. In this talk, we discuss how the symmetries of C may induce symmetries of the whole M. When M is contained in one of the two halfspaces determined by C, we give sufficient conditions for M to inherit the symmetries of C. In particular, when M is vertically cylindrically bounded, we get that M is rotational if C is a circle. In the case in which the Weingarten equation is linear, we give a sufficient condition for such a surface to be contained in a halfspace. Both results are generalizations of results of Rosenberg and Sa Earp, for constant mean curvature surfaces, to the Weingarten setting. In particular, our results also recover and generalize the constant mean curvature case.

In this talk, we speak about our current results from [1] on the rank of Jacobian varieties of the family of algebraic curves of genus ≥ 1 defined by the affine equations $y^s=ax^r+b$  over a number field k, where r ≥ 2 and s ≥ 2 are fixed integers. Assuming the strong version of Lang’s conjecture on varieties of general type, we show that the Mordell-Weil rank of the Jacobian varieties of these curves is uniformly bounded. The proof proceeds by constructing a parameter space for curves in the family with a given number of rational points and analyzing the geometry of its fibers, which are shown to be complete intersection curves of increasing genus.

[1]: Sajad Salami, On the rank of Jacobian of the curves $y^s=ax^r+b$, accepted in J. Number Theory