IMPA - O Instituto de Matemática Pura e Aplicada

Almgren proved that the only closed embedded minimal surface of genus zero in the round three-sphere is the totally geodesic equator, and Brendle proved that the only embedded minimal tous is the Clifford torus. In genus two, the Lawson surface $\xi_{2,1}$ is still the only known example, and whether it is the only one is an open problem. Its isometry group has order 48, and every uniqueness theorem available so far assumes a large part of it: Kapouleas and Wiygul assume the full isometry group, and Kusner, Lü and Wang certain subgroups of index two.

In this talk, I will explain how the symmetry hypothesis can be weakened to a subgroup of index six: the dihedral group $D_4$ of order eight, generated by two reflections across orthogonal totally geodesic two-spheres together with a half-turn about a great circle. This is a joint work with Joaquín Pérez.

We study the Cauchy problem for the isentropic hypo-viscous compressible Navier-Stokes equations under general pressure laws. By using convex integration method we provides the first non-uniqueness result of weak solutions to viscous compressible fluid. Due to the difficulties caused by the relative rigidity of the pressure and by the viscosity, our proof features new constructions of building blocks for both the density and momentum, which are, particularly, designed to respect the compressible structure. We provide first applications of a temporally intermittent but spatially homogeneous building block scheme and a mild density perturbation. It also applies to the compressible Euler equations and the hypo-viscous incompressible Navier-Stokes equations. This is a joint work with Peng Qu, Zirong Zeng, and Deng Zhang.

For a compact symplectic manifold, the quantum volume element is the summation of quantum products of elements in dual bases of the cohomology space. This is a deformation of the Euler class of the manifold. In this talk I will describe application of this element to Virasoro conjecture for Gromov-Witten invariants and complexity of quantum cohomology.

We show that for the standard map family, for all parameter values except one, the principal fixed point of the mapping has a transverse homoclinic point.