(1) [第16期北京大学特别数学讲座]Rationally Connected Varieties

Speaker: Jason Starr, Stony Brook University
Date: From 2013-07-01 To 2013-07-12
Venue: Room 29 at Quan Zhai, BICMR

The class of rationally connected varieties provides a birational class of interesting varieties, which includes Fano varieties, unirational varieties etc. It has been an active and fertile subject attracting algebraic geometers from different viewpoints. This series of lectures targets to give a comprehensive study of its most important geometry and arithmetic properties.

Speaker: Jason Starr

Date: 1-12 July, 2013 Every Tuesday and Thursday, 10:00-12:00

Venue: Room 29 at Quan Zhai, BICMR

Lecture 1: Rationally Connected Varieties

Abstract: There is a fruitful algebro-geometric analogue of topology that replaces continuous maps from the closed interval by algebraic maps from the projective line. This leads to an analogue of "path connected spaces", namely "rationally connected varieties": a beautiful class of varieties studied by Kollár-Miyaoka-Mori and Campana that includes all rational varieties, all Fano manifolds, and many moduli spaces. This analogy leads to a conjecture of Kollár-Miyaoka-Mori that was proved by Graber, Harris and myself: every algebraic fibration over a curve whose general fiber is rationally connected admits a section.

Lecture 2: Rationally Simply Connected Varieties

Abstract: Continuing the analogy from last time, Barry Mazur suggested a notion of "rationally simply connected varieties". After initial work of Harris and myself, this was developed by de Jong, Xuhua He and myself. We prove that a sufficiently "transverse" fibration over a surface that has vanishing "elementary obstruction" and whose general fiber is rationally simply connected admits a rational section. I will also explain an important generalization due to Yi Zhu.

Lecture 3: The Weak Approximation Conjecture of Hassett-Tschinkel

Abstract: For a rationally connected fibration over a curve, the topological analogy suggest a stronger result than the Graber Harris-Starr theorem, namely the Weak Approximation Conjecture of Hassett-Tschinkel: every formal local section of a rationally connected fibration can be approximated to arbitrary order by an algebraic section. I will explain work towards this conjecture by Hassett-Tschinkel at places of good reduction, Hassett-Tschinkel, Knecht and Chenyang Xu for log del Pezzo surfaces, Hassett and de Jong-Starr for rationally simply connected fibrations, Roth-Starr, and recent exciting work of Zhiyu Tian and Runpu Zong for places of potentially good reduction (in a strong sense), and work of Zhiyu Tian unconditionally proving weak approximation for cubic surfaces.

Lecture 4: Rationally connected varieties over finite fields, PAC fields, and global function

Abstract: I will explain a beautiful counterpart due to Hélène Esnault about rationally connected varieties over finite fields, as well as extensions by Esnault and Esnault-Xu. I will also explain a counterpart due to János Kollár over "PAC fields", settling Ax's conjecture in characteristic 0, as well as important extensions by Hogadi-Xu. I will finish by explaining joint work with Xu combining these results with the de Jong-He-Starr theorem to prove existence of rational points of rationally simply connected varieties over global function fields and over function fields over PAC fields.

(2) [第16期北京大学特别数学讲座]Introduction to the Min-max theory for minimal surfaces

Speaker: Xin Zhou, MSRI
Date: From 2013-07-05 To 2013-07-25
Venue: Room 09 at Quan Zhai, BICMR

Peking University sixteenth special lectures in mathematics, the time : 2013.7.5-7.25, on Monday, Wednesday, Friday 10:00-12:00 ,place : Mathematics center Quan Zhai 9 classroom. Xin Zhou obtained his master degree at Peking University under the supervision of Prof. Gang Tian; obtained his PhD at Stanford University; and will do his Postdoc research at MSRI and MIT from this coming fall. His research topic is minimal surfaces and general relativity.

Title: Introduction to the Min-max theory for minimal surfaces

Speaker: Xin Zhou

Date&Time: 2013.7.5-7.25, on Monday, Wednesday, Friday 10:00-12:00

Venue: Room 09 at Quan Zhai, BICMR

Abstract:
As a powerful method to construct minimal surfaces, Almgren and Pitts developed the min-max theory in the 1970's. There are a fare mount of works after that. Recently, the proof of the Willmore conjecture by Marques and Neves essentially used the min-max theory. Here we will give an introduction to this theory. Our final goal it to introduce the simplified theory by Colding and De Lellis. This simplified work reviewed all the main ideas of the Almgren-Pitts theory, but used limited knowledge of the geometric measure theory, which makes it easier to understand.We will start from the basic minimal surfaces theory. We plan to cover the basic variational formulae, the classical Plateau problem, the curvature estimates for stable minimal hypersurfaces, and some basic knowledge of varifold theory. Then we will begin introducing Colding-De Lellis'paper.Time permitted, we might talk about some related topics, like the index bounded problem, genus bounds, the Sack-Uhlenbeck's method, or even a brief introduction to the Almgren-Pitts'setting.