We investigate the transversality of holomorphic mappings between CR submanifolds of complex spaces. In equidimension case, we show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target, provided that the source manifold is of finite type and the map is of generic full rank. In different dimensions, the situation is more delicate. We will show that under certain restrictions on the dimensions and the rank of Levi forms, the mappings for which the set points where the map has degenerate rank is of codimension at least 2, is transversal to the target. In addition, we show that under more restrictive conditions on the manifolds, finite holomorphic mappings are transversal.

We show that any $k$-uniform hypergraph with $n$ edges contains two
edge
disjoint subgraphs of size $\tilde{\Omega}(n^{2/(k+1)})$ for $k=4,5$
and
$6$. This result is best possible up to a logarithmic factor due to a
upper bound construction of Erd\H{o}s, Pach, and Pyber who show there
exist $k$-uniform hypergraphs with $n$ edges and with no two edge
disjoint isomorphic subgraphs with size larger than
$\tilde{O}(n^{2/(k+1)})$. Furthermore, this result extends results
Erd\H{o}s, Pach and Pyber who also established the lower bound for $k=2$
(eg. for graphs), and of Gould and R\"odl who established the result for

I will introduce a new geometric flow on complex, non-Kahler
manifolds. I will exhibit Perelman-type functionals for this flow, and some regularity results. Finally I will present an optimal regularity conjecture and discuss its relationship to the long open problem of the classification of Class VII surfaces. This is joint work with G.Tian.