I will present existence and uniqueness results for theCauchy problem as in the title.
Each problem to be solved at the study group will be discussed.
The Cox ring of a variety is an analogue of the homogeneous coordinate ring of projective space. Cox rings are not defined for every variety and even when they are defined, they need not be finitely generated. Varieties for which the Cox ring is finitely generated are called Mori dream spaces and, as the name suggests, they are particularly well-suited for the Minimal Model Program. Such varieties include toric varieties and del Pezzo surfaces.
I will report on joint work with T. Várilly and M. Velasco where we introduce a class of smooth projective surfaces having finitely generated Cox ring. This class of surfaces contains toric surfaces and (log) del Pezzo surfaces.
Standard quantum logic, as intitiated by Birkhoff and von Neumann, suffers from severe problems which relate quite directly to interpretational issues in the foundations of quantum theory. In this talk, I will present some aspects of the so-called topos approach to quantum theory, as initiated by Isham and Butterfield, which aims at a mathematical reformulation of quantum theory and provides a new, well-behaved form of quantum logic that is based upon the internal logic of a certain (pre)sheaf topos.
Fix a positive integer $k$, and consider the class of all graphs which do not have $k+1$ vertex-disjoint cycles. A classical result of Erdos and P\'{o}sa says that each such graph $G$ contains a blocker of size at most $f(k)$. Here a {\em blocker} is a set $B$ of vertices such that $G-B$ has no cycles.
We give a minor extension of this result, and deduce that almost all such labelled graphs on vertex set $1,\ldots,n$ have a blocker of size $k$. This yields an asymptotic counting formula for such graphs; and allows us to deduce further properties of a graph $R_n$ taken uniformly at random from the class: we see for example that the probability that $R_n$ is connected tends to a specified limit as $n \to \infty$.
There are corresponding results when we consider unlabelled graphs with few disjoint cycles. We consider also variants of the problem involving for example disjoint long cycles.
This is joint work with Valentas Kurauskas and Mihyun Kang.