3:45PM: Mustafin VarietiesBernd SturmfelsA Mustafin variety is a degeneration of projective space induced by a point configuration in a BruhatTits building. The special fiber is reduced and CohenMacaulay, and its irreducible components form interesting combinatorial patterns. For configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision. This connects our study to tropical and toric geometry. For general configurations, the irreducible components of the special fiber are rational varieties, and any blowup of projective space along a linear subspace arrangement can arise. A detailed study of Mustafin varieties is undertaken for configurations in the BruhatTits tree of $ PGL(2)$ and in the twodimensional building of $PGL(3)$. The latter yields the classification of Mustafin triangles into $38$ combinatorial types.

5:00PM: Finiteness conjectures about the projective coordinate rings of X^n/G as n variesAndrew SnowdenLet G be a reductive group (such as GL(n) or a finite group) acting on a finitely generated ring R. The ring of invariants R^G is then wellbehaved (in the sense that it is also finitely generated). What happens when we look at the sequence of invariant rings R^G, (R tensor R)^G, (R tensor R tensor R)^G, etc? Are the terms wellbehaved, in a uniform sense? (The motivation for considering this sequence comes from geometry, where it corresponds to looking at the diagonal action on products of a given space.) I conjecture that these rings are wellbehaved, in a uniform sense. I will explain the precise sense in which I mean this. I will also explain two features of the conjecture: first, that for any G and R it can be thought of as a very explicit problem in graph theory (I will give several examples of G's and R's and the corresponding graph theory problems); and second, that it can be rephrased into a more abstract and structural statement about certain algebraic objects, which may indicate that it fits into a broader theory.
