Tropical algebraic geometry is the geometry of the tropical semiring (R, min, +). Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete description of the families of quadrics through four points in the tropical projective plane and a counterexample to the incidence version of Pappus' Theorem.

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Abstract. In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semidefinite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems, show how the degree principle for the symmetric group can be computationally exploited and also propose some methods to efficiently compute in the geometric quotient.

Abstract. We propose a new hierarchical approach to understand the complexity of the open problem of computing a Nash equilibrium in a bimatrix game. Specifically, we investigate a hierarchy of bimatrix games (A, B) which results from restricting the rank of the matrix A + B to be of fixed rank at most k. For every fixed k, this class strictly generalizes the class of zero-sum games, but is a very special case of general bimatrix games. We show that even for k = 1 the set of Nash equilibria of these games can consist of an arbitrarily large number of connected components. While the question of exact polynomial time algorithms to find a Nash equilibrium remains open for games of fixed rank, we can provide polynomial time algorithms for finding an ε-approximation.

Abstract. We answer a question of David Larman, by proving the following result. Any four unit balls in three-dimensional space, whose centers are not collinear, have at most twelve common tangent lines. This bound is tight. IntroductionThe screen of a computer monitor consists of small pixels. Suppose that we are given a three-dimensional scene consisting of several objects and a viewpoint. Generating an image of this scene ("rendering" the scene) is a basic task in computer graphics and in computational geometry, which amounts to determining the visible object(s) at each pixel. The methods developed for the solution of such problems have an extensive literature under the labels "ray tracing" and "hidden surface removal" (see, e.g., [7] and [20]). This field has served as a rich source of problems on geometric, combinatorial, and algebraic properties of systems of lines in their interaction with geometric objects.For instance, we can assume that all of our objects are unit balls in a region A, and we want to determine which balls are not visible from any viewpoint outside of A (see [29]