While the general surface of degree at least 4 in projective 3-space contains no lines, the maximum possible number of lines on any surface of degree at least 4 over a field k is a classical question dating back to at least Clebsch's work in 1861. When the degree is less than the (positive) characteristic and always in characteristic 0, the number of lines has an upper bound which is quadratic in the degree when the degree is at least 4. In contrast, once the degree is at least one more than the characteristic, it has long been known that there are surfaces with vastly more lines. In this talk, we answer this classical question over an arbitrary field. In particular, we prove that the maximum number of lines on any smooth surface of degree d over any field k is $d^4-3d^3+3d^2$ and show that, up to projective equivalence, a unique surface obtains this sharp upper bound in the infinitely many degrees and characteristics where it is obtained.

Poroelasticity equations arise from many applications in geophysics and biomechanics, so numerical simulations of poroelasticity equations are of great interest. In this talk I discuss advanced finite element methods for poroelasticity and related problems. In the first part, I introduce parameter-robust discretization of poroelasticity and explain that efficient preconditioners can be obtained by the operator preconditioning approach. In the second part, I present hybridizable discontinuous Galerkin (HDG) methods for the problems that Stokes/Navier-Stokes equations and porous/poroelastic equations are coupled with interfaces. The talk is based on joint works with K.-A. Mardal (University of Oslo), M. E. Rognes (Simula Research Laboratory), A. Cesmelioglu (Oakland University) S. Rhebergen (University of Waterloo), and other collaborators.