Transversal study on Ultradiscretizing phenomena in algebraic geometry, learning theory and biological mathematics

(1)对于14个单峰例外超曲面奇点，我们在K3曲面的派生范畴中构造了代数圈，并证明了可以从它们的范畴交数恢复动态图。(2)通过构造具体的对应，我们证明了著名的95个加权K3超曲面实际上是75个族。(3)我们首次引入了排序问题的几何观点。证明了临界路径的变化发生在热带超曲面上，发展了一种全新的路径转换可视化方法。(4)对热带椭圆曲线进行了拓扑分类。(5)讨论了神经网络能力获取的代数几何方法。

(1) For the fourteen unimodal exceptional hypersurface singularities, we have constructed algebraic cycles in the derived category of K3 surfaces, and shown that we can recover the Dynkin diagrams from their categorical intersection numbers.(2) We have shown that the famous ninety-five families of weighted K3 hypersurfaces are, in fact, essentially seventy-five families by constructing concrete correspondences.(3) We introduced a geometric point of view to the scheduling problems for the first time, showed that the change of critical paths occurred at a tropical hypersurface and developed totally new method to visualize the transition of paths.(4) We did a topological classification of tropical elliptic curves.(5) We discussed about algebro-geometric method for acquiring abilities by neural networks.