Curve counting, moduli, and logarithmic geometry

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An effective method to study the geometry of complicated spaces is to examine how other spaces are able to sit inside them. For instance, there exist surfaces in three-dimensional space that contain precisely 27 straight lines. We might therefore conclude that such surfaces must have a fundamentally different geometry than familiar flat 2-dimensional space, where one can draw a line between any two points. Calculations such as these captured the imagination of mathematicians for centuries, but in the 1990s, theoretical physics gave birth to a powerful new form of this idea. The physicists recognised that these simple minded "curve counting" questions were relevant and computable invariants of certain physical models in string theory. In the decades since, the invariants have had impacts on countless faraway corners of the pure mathematics world. This proposal seeks to understand the modern avatars of these curve counting invariants. The spaces in question will be solution sets to systems of polynomial equations, known as algebraic varieties. The methods of the proposal lie at the nexus of two young subjects known as logarithmic and tropical geometry. The process of solving a system of polynomial equations can be broken up into two steps. One can first find solutions that have the right order of magnitude, or precisely, the set of possible sizes of solutions. As an analogy, rather than calculating the product of 212 and 330 exactly, one can eyeball that the answer is about 60000. While this is the wrong answer, it gives a good enough estimate for many purposes. Tropical geometry seeks to apply this logic to geometry itself, by finding geometric structures that are simple, but reflect a useful approximation of a true geometry. Logarithmic geometry is the technical bridge that allows one to return to the subtle world of polynomial systems. Tropical geometry itself has roots in optimisation theory and theoretical physics, and applications reaching as far as statistics and auction theory. The fundamental goal of this research proposal is to understand how these tropical geometric structures control curve counting invariants, and seeks to build and exploit a bridge between these two directions of mathematical inquiry. Concrete objectives will be to address several long standing questions concerning the structure of curve counting invariants, and to use tropical methods to make complete and effective calculations in algebraic geometry, that go beyond what has been achieved without tropical input.