Monroe H. Martin Lectures

The first Monroe H. Martin lecture series (this series of lectures is in honor of Monroe H. Martin ), will be given by Caucher Birkar.

Zoom Meeting ID: 920 4917 8642

Lectures 1: Feb. 16 (Baltimore time: 11 am-12 pm)

Lectures 2: Feb. 18 (Baltimore time: 11 am-12 pm)

Lectures 3: Feb. 19 (Baltimore time: 12 pm-1 pm)

*The first lecture is for the general audience. But the other two are for people with an algebraic geometry background.

Title: Classification theory of algebraic varieties.

Abstract: The classification of algebraic varieties is at the heart of algebraic geometry. With roots in the ancient world the theory saw great advances in dimensions one and two in the 19th century and the first half of 20th century. It was only in the 1970-80's that a general framework was formulated, and by the early 1990's a satisfactory theory was developed in dimension 3. The last 30 years has seen great progress in all dimensions.

In the first lecture I will try to give a historical perspective and discuss the theory in general terms. I will explain how the theory is based on birational transformations and moduli considerations.

In the second lecture I will discuss log Calabi-Yau fibrations. This is a class of spaces which includes Fano and Calabi-Yau varieties and their local counterparts. They are of great importance in the classification theory and well beyond.

In the third lecture I will talk about generalised pairs. This is a recently developed notion generalising the notions of varieties and pairs. It has found many applications and fits well into the classification theory.

The second Monroe H. Martin lecture series (this series of lectures is in honor of Monroe H. Martin ), will be given by Chenyang Xu. (postponed due to COVID-19)

Lecture 1: 4:00PM - 5:00PM, March 23, Maryland 110

Lecture 2: 4:00PM - 5:00PM, March 24, Shaffer 303

Lecture 3: 4:00PM - 5:00PM, March 25, Krieger 205

Title: Algebraic K-stability theory of Fano varieties

Abstract: The notion of K-stability has been first defined by differential geometers to characterise the existence of canonical metrics on a polarised variety. In the case of Fano varieties, where K-stability corresponds to the existence of a Kahler-Einstein metric, the purely algebraic study of K-stability now has become an independent subject in higher dimensional algebraic geometry, which attracts lots of recent interests. In our lectures, we will explain abundant recent progress that people have made by using deep machinery such as the minimal model program. We will also discuss some remaining challenging questions.

Lecture 1: In our first lecture, we will discuss some equivalent characterisations of K-stability of Fano varieties, which form the foundation of the algebraic K-stability theory.

Lecture 2-3: We will discuss some main topics in algebraic geometers’ research of K-stability of Fano varieties. One featured application is to give a construction of moduli spaces of K-stable Fano varieties, called K-moduli. This has essentially solved the longstanding question of constructing well behaved moduli spaces for general Fano varieties, though some key property is still only conjectural. If time permits, we will also discuss some other progress, such as establishing K-stability for explicit Fano varieties; a local K-stability theory for singularities, etc.