**Winter School on Algebraic Curves, Riemann Surfaces and Moduli Spaces**

March 4-8, 2019

Morningside Center of Mathematics, CAS

**Speakers:**

Maksym Fedorchuk (Boston College)

Carlo Gasbarri (University of Strasbourg)

Samuel Grushevsky (Stonybrook University)

Eduard Looijenga (Tsinghua University)

Ngaiming Mok (Hong Kong University)

Athanase Papadopoulos (University of Strasbourg)

Alexander Schmitt (Free University of Berlin)

**Organizers: **

Baohua Fu

Lizhen Ji

Juergen Jost

Athanase Papadopoulos

Xiaotao Sun

Shing-Tung Yau

**Abstract:**

Riemann surfaces, algebraic curves, vector bundles on algebraic curves and their moduli spaces are fundamental objects in the contemporary mathematics. They are connected to a major portion of the mainstream of mathematics, and a proper understanding of their rich structures will be both satisfying and beneficial. This winter school will offer courses on a broad range of topics related to the above topics.

**Conference Schedule:**

March 4, 2019 | |||

07:50-08:25 |
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08:25-08:30 |
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08:30-09:15 | Eduard Looijenga (Tsinghua University) | Moduli spaces of curves and the representations of the mapping class group (I) | |

09:25-10:10 | Alexander Schmitt (Free University of Berlin) | Introduction to holomorphic vector bundles on compact Riemann surfaces (I) | |

10:10-10:30 | Tea Break | ||

10:30-11:15 | Maksym Fedorchuk (Boston College) | The existence and construction of different moduli spaces of curves (I) | |

11:25-12:10 | Athanase Papadopoulos (University of Strasbourg) | The metric theory of Teichmuller spaces of compact Riemann surfaces (I) | |

12:10-13:30 | Lunch, Fourth floor of Wuke Restaurant (物科餐厅四楼) | ||

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13:30-14:15 | Alexander Schmitt (Free University of Berlin) | Introduction to holomorphic vector bundles on compact Riemann surfaces (II) | |

14:25-15:10 | Carlo Gasbarri (University of Strasbourg) | Introduction to Diophantine geometry over function fields (I) | |

15:10-15:30 | Tea Break | ||

15:30-16:15 | Eduard Looijenga (Tsinghua University) | Moduli spaces of curves and the representations of the mapping class group (II) | |

16:25-17:10 | Athanase Papadopoulos (University of Strasbourg) | The metric theory of Teichmuller spaces of compact Riemann surfaces (II) | |

March 5, 2019 | |||

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08:30-09:15 | Ngaiming Mok (Hong Kong University) | Geometric structures and substructures on uniruled projective manifolds (I) | |

09:25-10:10 | Eduard Looijenga (Tsinghua University) | Moduli spaces of curves and the representations of the mapping class group (III) | |

10:10-10:30 | Tea Break | ||

10:30-11:15 | Samuel Grushevsky (Stonybrook University) | Moduli of curves with a differential, and compactifications (I) | |

11:25-12:10 | Maksym Fedorchuk (Boston College) | The existence and construction of different moduli spaces of curves (II) | |

12:10-13:30 | Lunch, Fourth floor of Wuke Restaurant (物科餐厅四楼) | ||

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13:30-14:15 | Athanase Papadopoulos (University of Strasbourg) | The metric theory of Teichmuller spaces of compact Riemann surfaces (III) | |

14:25-15:10 | Carlo Gasbarri (University of Strasbourg) | Introduction to Diophantine geometry over function fields (II) | |

15:10-15:30 | Tea Break | ||

15:30-17:30 | Ngaiming Mok (Hong Kong University) | Geometric structures and substructures on uniruled projective manifolds (II, III) | |

17:30-19:30 |
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March 6, 2019 | |||

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08:30-09:15 | Alexander Schmitt (Free University of Berlin) | Introduction to holomorphic vector bundles on compact Riemann surfaces (III) | |

09:25-10:10 | Carlo Gasbarri (University of Strasbourg) | Introduction to Diophantine geometry over function fields (III) | |

10:10-10:30 | Tea Break | ||

10:30-11:15 | Samuel Grushevsky (Stonybrook University) | Moduli of curves with a differential, and compactifications (II) | |

11:25-12:25 | Ngaiming Mok (Hong Kong University) | Geometric structures and substructures on uniruled projective manifolds (IV) | |

12:25-13:30 | Lunch, Fourth floor of Wuke Restaurant (物科餐厅四楼) | ||

13:30-17:10 | Free Discussion | ||

March 7, 2019 | |||

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08:30-09:15 | Carlo Gasbarri (University of Strasbourg) | Introduction to Diophantine geometry over function fields (IV) | |

09:25-10:10 | Maksym Fedorchuk (Boston College) | The existence and construction of different moduli spaces of curves (III) | |

10:10-10:30 | Tea Break | ||

10:30-11:15 | Eduard Looijenga (Tsinghua University) | Moduli spaces of curves and the representations of the mapping class group (IV) | |

11:25-12:25 | Ngaiming Mok (Hong Kong University) | Geometric structures and substructures on uniruled projective manifolds (V) | |

12:25-13:30 | Lunch, Fourth floor of Wuke Restaurant (物科餐厅四楼) | ||

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13:30-14:15 | Alexander Schmitt (Free University of Berlin) | Introduction to holomorphic vector bundles on compact Riemann surfaces (IV) | |

14:25-15:10 | Maksym Fedorchuk (Boston College) | The existence and construction of different moduli spaces of curves (IV) | |

15:10-15:30 | Tea Break | ||

15:30-16:15 | Samuel Grushevsky (Stonybrook University) | Moduli of curves with a differential, and compactifications (III) | |

16:25-17:10 | Eduard Looijenga (Tsinghua University) | Moduli spaces of curves and the representations of the mapping class group (V) | |

March 8, 2019 | |||

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08:30-09:15 | Samuel Grushevsky (Stonybrook University) | Moduli of curves with a differential, and compactifications (IV) | |

09:25-10:10 | Athanase Papadopoulos (University of Strasbourg) | The metric theory of Teichmuller spaces of compact Riemann surfaces (IV) | |

10:10-10:30 | Tea Break | ||

10:30-11:15 | Carlo Gasbarri (University of Strasbourg) | Introduction to Diophantine geometry over function fields (V) | |

11:25-12:10 | Samuel Grushevsky (Stonybrook University) | Moduli of curves with a differential, and compactifications (V) | |

12:10-13:30 | Lunch, Fourth floor of Wuke Restaurant (物科餐厅四楼) | ||

| |||

13:30-14:15 | Athanase Papadopoulos (University of Strasbourg) | The metric theory of Teichmuller spaces of compact Riemann surfaces (V) | |

14:25-15:10 | Alexander Schmitt (Free University of Berlin) | Introduction to holomorphic vector bundles on compact Riemann surfaces (V) | |

15:10-15:30 | Tea Break | ||

15:30-16:15 | Maksym Fedorchuk (Boston College) | The existence and construction of different moduli spaces of curves (V) | |

**Minicourses:**

*Maksym Fedorchuk*

A lecture series on the existence and construction of different moduli spaces of curves, with a special emphasis on the cases when the stacks in question are only algebraic and not Deligne-Mumford.

*Carlo Gasbarri*

(Download)

*Introduction to Diophantine geometry over function fields*

Prerequisites: A standard course of algebraic geometry: let's say at the level of Harshorne book.

- Analogy between number fields and functions fields.

-rational points, integral points/ curves over varieties

-heights over function fields, relationship with intersection theory and analogue to Northcott theorem.

-curves of genus zero over function fields always have rational points

-curves of genus 1: Mordell Weil theorem

-curves of higher genus: Mordell conjecture (over function fields)

- some open problems.

*Samuel Grushevsky*

*Moduli of curves with a differential, and compactifications*

We will discuss the geometry of the moduli spaces of Riemann surfaces together with a meromorphic differential. There is a natural action of SL(2,R) action on this space, which leads to an interesting dynamical system, studying which is the field of Teichmuller dynamics. On the other hand, it's closely related to the moduli spaces of Riemann surfaces together with a meromorphic function, which is natural from the point of view of symplectic field theory and integrable hierarchies. We will discuss these motivations and some fundamental questions, and then describe some compactifications of these moduli spaces.

**Eduard Looijenga**

*Moduli spaces of curves and the representations of the mapping class group*

After recalling some basic material such as the notions appearing in the title, we focus

on (possibly virtual) representations of the mapping class group that appear in ``mathematical nature''

and the way algebraic geometry can sometimes help us to gain more insight in such (in principle purely topological) objects.

**Ngaiming Mok**

*Geometric structures and substructures on uniruled projective manifolds*

*Athanase Papadopoulos*

*Alexander Schmitt*

**Useful Information:**

**Transportation**

You can take the following 3 vehicles and then go to hotel or conference building by walk according to the local maps below.

l Airport Shuttle

Taking airport shuttle line 5 to Zhongguancun (中关村) Station (the last station) which costs about 30 Yuan.

http://en.bcia.com.cn/traffic/airbus/index.shtml

l Subway

Taking subway line 10 to Zhichunlu (知春路) or Zhichunli (知春里) Station which costs about 30 Yuan from airport or 5 Yuan from train station.

l Taxi

Taking taxi to Baofusiqiao (保福寺桥) which costs about 100 Yuan from airport or 50 Yuan from train station. You can print the following message and show it to taxi driver.

**您好，请送我到辽宁大厦（保福寺桥）。**

Please take me to the Liaoning International Hotel.

**Hotel**

All speakers will stay in Liaoning International Hotel. http://www.liaoninginternationalhotel.com

**Local Maps**

**Photo:**

No. 55, Zhongguancun East Road, Haidian District, Beijing 100190

Phone: 86-10-82541165 Email: mcmadmin@math.ac.cn Technical Support：Qingyun Software