Studies on Diophantine Geometry and Arakelov geometry

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Chunhui Liu obtained fruitful results on counting rational points by the determinant method. During the participation of the thematic activity "Reinventing rational points" in IHP during May and June 2019, he had lots of effective communications with some experts on rational points, and finally he had a significant improvement on his understand to the density of rational points in arithmetic varieties. In his preprint "Determinant method and the pseudo-effective threshold" (arxiv: arxiv:1910.00306), he explicated a connection between the positivity of certain line bundles and the density of rational points, which seems to have a large potential application in the future. For example, we have known a lot on the pseudo-effective threshold on certain line bundles on some particular varieties, and these results can be applied to study their density of rational points. He was also writing another paper on the similar area, which will be available quite soon. Besides these, he also realized that the study of certain heights of points on Hilbert schemes will be useful to the application of determinant. Some auxiliar results were also obtained during the research process, and they deserve to be published as several small papers. I believe that once he accomplishes these subjects, we will have a novel understand to the quantitative arithmetic and rational points. Besides the work on rational points, he went on studying the Diophantine approximation over arithmetic function fields. I believe he will produce an excellent result in this area.