This will be a discussion of the paper https://arxiv.org/abs/1604.02618

This will be a quick introduction to tropical algebra and the main results from the paper https://arxiv.org/pdf/1604.00113.pdf

Classifying line arrangements on the plane is a problem that has been around for a long time. There has been a lot of work from the perspective of incidence geometry, but after a paper of Hirzebruch in in 80's, it has also attracted the attention of algebraic geometers for the applications that it has on classifying complex algebraic surfaces of general type. In this talk I will present various results around this problem, I will show some specific questions that are still open, and I will explain how it relates to complex surfaces of general type. 
 

We consider the one parameter mirror families W of the Calabi-Yau 3-folds with Picard-Fuchs  equations of hypergeometric type. By mirror symmetry the  even D-brane masses of orginial Calabi-Yau manifolds M can be identified with four periods with respect to an integral symplectic basis of $H_3(W,\mathbb{Z})$ at the point of maximal unipotent monodromy. We establish that the masses of the D4 and D2 branes at the conifold are given by the two algebraically independent values of the L-function of the weight four holomorphic Hecke eigenform with eigenvalue one of $\Gamma_0(N)$. For the quintic in  $\mathbb{P}^4$ it this Hecke eigenform of $\Gamma_0(25)$ was as found by Chad Schoen.  It was discovered  by de la Ossa, Candelas and Villegas that  its  coefficients $a_p$ count the number of  solutions of  the mirror quinitic at the conifold over the finite number field $\mathbb{F}_p$ . Using the theory of periods and quasi-periods of $\Gamma_0(N)$ and the special geometry pairing on Calabi-Yau 3 folds we can fix further values in the connection matrix between the maximal unipotent monodromy point and the conifold point.  

We discuss mathematical studies on the Vafa-Witten theory, one of topological twists of N=4 super Yang-Mills theory in four dimensions, from the viewpoints of both differential and algebraic geometry. After mentioning backgrounds and motivation, we describe some issues to construct mathematical theory of this Vafa-Witten one, and explain possible ways to sort them out by analytic and algebro-geometric methods, the latter is joint work with Richard Thomas.

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by n colours with at least n+1 edges of each colour there must be a matching that uses each colour exactly once (such a matching is called rainbow). This conjecture recently have been proved asymptotically by Pokrovskiy. In this talk, I will consider the same question without the bipartiteness assumption. It turns out that in any multigraph with bounded edge multiplicities, that is properly edge-coloured by n colours with at least n+o(n) edges of each colour, there must be a matching of size n-O(1) that uses each colour at most once. This is joint work with Peter Keevash.