What is...?

Tropical enumerative geometry is a branch of combinatorial algebraic geometry that aims to count algebraic objects (usually curves on some surface passing through a number of points) by turning them into combinatorial objects, called tropical curves...

CAT(0) cube complexes are cell complexes whose cells are cubes, whose naturally defined metric is non-positively curved in some precise sense. 

They can be equivalently defined in a variety of ways, which a priori looks very different. They naturally...

Open book decompositions provide a topological decomposition of a given manifold. We focus on dimension three. While the definition seems to be purely topological, it encodes information about fibered knots, surface dynamics, contact structures of...

In the 1970s, Viro's method paved an important path in the study of the topology of real algebraic varieties and became a precursor to tropical geometry. This method involves subdividing an integer polytope and using the information from each of its...

Persistence modules offer a way to analyze how features, such as connected components or holes, evolve as a space is gradually changed. One can think of a persistence module as a sequence of vector spaces, each corresponding to a particular stage of...