Georgia Institute of Technology College of Sciences

The dual complex of a semistable degeneration records the combinatorics of the intersections in the special fiber. More generally, one can associate a polyhedral dual complex to any toroidal degeneration. It is natural to ask for connections between the geometry of an algebraic variety and the combinatorial properties of its dual complex. In this talk, I will explain one such result: The dual complex of an n-dimensional uniruled variety has the homotopy type of an (n-1)-dimensional simplicial complex. The key technical tool is a specialization map to dual complexes and a balancing condition for these specialization.

Tropicalization involves passing to an ordered group M, usually taken to be (R, +) or (Q, +), and viewed as a semifield. Although there is a rich theory arising from this viewpoint, idempotent semirings possess a restricted algebraic structure theory, and also do not reflect important valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques. Our research in tropical algebra has focused on coping with the fact that the max-plus’ algebra lacks negation, which is used throughout the classical structure theory of modules. At the outset one is confronted with the obstacle that different cosets need not be disjoint, which plays havoc with the traditional structure theory. Building on an idea of Gaubert and his group (including work of Akian and Guterman), we introduce a general way of artificially providing negation, in a manner similar to the construction of Z from N but with one crucial difference necessitated by the fact that the max-plus algebra is not additively cancelative! This leads to the possibility of defining many auxiliary tropical structures, such as Lie algebras and exterior algebras, and also providing a key ingredient for a module theory that could enable one to use standard tools such as homology.

Let det_n be the homogeneous polynomial obtained by taking the determinant of an n x n matrix of indeterminates. In this presentation linear maps called Young flattenings will be defined and will be used to show new lower bounds on the symmetric border rank of det_n.

Given a non-isotrivial elliptic curve E over K=Fq(t), there is always a finite extension L of K which is itself a rational function field such that E(L) has large rank. The situation is completely different over complex function fields: For "most" E over K=C(t), the rank E(L) is zero for any rational function field L=C(u). The yoga that suggests this theorem leads to other remarkable statements about rational curves on surfaces generalizing a conjecture of Lang.