L3

The effective cone of the Hilbert scheme of points in $P^2$ has

finitely many chambers corresponding to finitely many birational models.

In this talk, I will identify these models with moduli of

Bridgeland-stable two-term complexes in the derived category of

coherent sheaves on $P^2$ and describe a

map from (a slice of) the stability manifold of $P^2$

to the effective cone of the Hilbert scheme that would explain the

correspondence. This is joint work with Daniele Arcara and Izzet Coskun.

It is known that generic and universal structures and Ramsey classes are related. We explain this connection and complement it by some new examples. Particularly we disscuss universal and Ramsey classes defined by existence and non-existence of homomorphisms.

The 2-dimensional assignment problem (minimum cost matching) is solvable in polynomial time, and it is known that a random instance of size n, with entries chosen independently and uniformly at random from [0,1], has expected cost tending to π^2/6.  In dimensions 3 and higher, the "planar" assignment problem is NP-complete, but what is the expected cost for a random instance, and how well can a heuristic do?  In d dimensions, the expected cost is of order at least n^{2-d} and at most ln n times larger, but the upper bound is non-constructive.  For 3 dimensions, we show a heuristic capable of producing a solution within a factor n^ε of the lower bound, for any constant ε, in time of order roughly n^{1/ε}.  In dimensions 4 and higher, the question is wide open: we don't know any reasonable average-case assignment heuristic.