Time: from August 11 to August 30
Venue：E4-233 & ZOOM
ZOOM ID: 921 6725 8223
Passcode: 064984
1.
Time：12:30-14:00，August 11, 16, 23, 30
Title：Diophantine approximation over adelic curves
Speaker：Paolo Dolce
Abstract：
Part I. This is a very introductory talk about diophantine approximation over the rationals. I would like to explain the meaning of "good approximants" and the naive links with diophantine equations. I want to explain in all details the easy proofs of the well known theorems of Dirichlet and Liouville. Moreover I would like to state the main properties of the irrationality measure. I will introduce Roth's theorem (without proof) and show how it implies the finiteness of the S-unit equation.
Part II. Assuming that we all know what is an adelic curve, I want to explain, with details, my recent proof of Roth's theorem for adelic curves. Such a proof is quite simple but with some messy calculations. At the end of the talk we can discuss some open problems regarding the quantitative Roth theorem for adelic curves.
Part III. I want explain Vojta's paper "A generalisation of Theorem of Faltings and Thue-Siegel-Roth-Wirsing". Here Vojta proves the one-dimensional instance of his famous conjecture, which generalises in a single statement all the theorems mentioned in the title of the paper. Vojta says that his theorem can be proved in a quantitative way but I haven't found any follow up in the literature, so I believe that this is an open problem that we could discuss.
2.
Time：14:00-15:30，August 11, 16, 23, 30
Title：Arakelov geometry over adelic curves
Speaker：Prof. Huayi Chen
Abstract：
I will explain a recent joint work with Atsushi Moriwaki on Arakelov geometry over a structure called adelic curve. An adelic curve is a countable field equipped with a family of absolute values parametrized by a measure space, which satisfies a natural integrability condition. Note that any global field corresponds canonically to an adelic curve and it turns out that any countable field admits a non-trivial adelic curve structure (there are many adelic curve structures when the field is transcendental over its prime field). Hence our theory generalizes largely the classic Arakelov geometry. I will also explain some applications to the study of positivity properties of adelic line bundles.
3.
Time：12:30-14:00, August 14
Title：On theta invariants and volume function on arithmetic varieties
Speaker：Mounir Hajli
Abstract：
The arithmetic volume is an important invariant in Arakelov geometry which was introduced by Atsushi Moriwaki. In this talk, I will introduce a new invariant for hermitian line bundles on arithmetic varieties. I will describe the interplay between this invariant and the arithmetic volume.
