Abstract: We describe how simplicial and relative categories form a model of $(\infty ,1)$-categories.

Abstract: I will give an introduction to tropical geometry. The main philosophy is that the limit at infinity of algebraic objects (i.e. things defined by polynomials) are piecewise linear objects. Roughly speaking, the "tropical variety" of an algebraic variety is a polyhedral complex (i.e. a union of convex polyhedrons) which encodes the behavior at infinity of the variety. The fundamental theorem of tropical geometry states that different ways to define the tropical variety, using valuation map, initial ideals and (min, +) algebra are the same. Tropical geometry is intimately related to the Grobner basis theory as well as toric geometry. Finally, I will explain new developments to extend tropical geometry to subvarieties in matrix groups (e.g. GL(n)). Singular values of matrices and Smith normal forms make an appearance. For the most part the talk should be understandable for anybody with basic knowledge of algebra and geometry e.g. knowing the definition of a polynomial.