In a perforated domain, the asymptotic behavior of the fluid motion depends on the rate (inter-hole distance)/(size of the holes). We will present the standard framework and explain how to find the critical rate where "strange terms" appear for the Laplace and Navier-Stokes equations. Next, we will study Euler equations where the critical rate is totally different than for parabolic equations. These works are in collaboration with V.Bonnaillie-Noel, M.Hillairet, N.Masmoudi, C.Wang and D.Wu.

The planarity testing problem is the algorithmic problem of testing whether
a given input graph is planar, that is, whether it can be drawn in the
plane without edge crossings. Clustered planarity (c-planarity, for short)
was introduced in 1995 by Feng, Cohen, and Eades as a
generalization of graph planarity, in which the vertex set of the
input graph is endowed with a
hierarchical clustering and we seek an embedding (edge crossing-free
drawing) of the graph in the
plane that respects the clustering in a certain natural sense.

A seemingly unrelated problem of thickenability for simplicial complexes
emerged in the topology of manifolds in the 1960s. A 2-dimensional
simplicial complex is thickenable if it embeds in some orientable
3-dimensional manifold.

We study the atomic embeddability testing problem, which is a common
generalization of clustered planarity and thickenability
testing, and present a polynomial time algorithm for this problem,
thereby giving the first polynomial time algorithm for c-planarity.

Until now, it has been an open problem whether c-planarity can be
tested efficiently, despite relentless efforts. Recently, Carmesin announced
that thickenability can be tested in polynomial time.
Our algorithm for atomic embeddability combines ideas from Carmesin's
work with algorithmic tools previously developed for so-called weak
embeddability testing.

Joint work with Csaba Toth.