Title：Enumerative geometry of curves: classical and modern approaches

Speaker：陈旭佳  （纽约州立大学石溪分校）

Time & Room：

   

时  间

1月2日

16:00-17:30

1月3日

16:00-17:30

1月4日

16:00-17:30

教  室

五教 5107

二教 2204

管理科研楼 1218

Abstract: The problem of enumerating plane curves of various types goes back at least to the 19th century. Kontsevich's recursion, motivated by string theory and proved by Ruan-Tian in the early 1990s, determines counts of rational curves in many complex surfaces. Welschinger defined invariant signed counts of real rational curves in real surfaces (complex surfaces with a conjugation) in 2003. Solomon interpreted Welschinger's invariants as holomorphic disk counts in 2006 and proposed Kontsevich-type recursions for them in 2007, along with an outline for adapting Ruan-Tian's homotopy style argument to the real setting. For many symplectic fourfolds, these recursions determine all invariants from basic inputs. 

The first talk will overview the classical perspectives on counting low-degree rational curves in the plane, which involve only basic linear algebra and topology. The last part of the first talk and the second talk will introduce the modern perspective of moduli spaces of pseudo-holomorphic maps in symplectic topology which underpins Ruan-Tian's proof of Kontsevich's recursion. The last talk will present my proof of Solomon's recursions which combines the moduli space perspective with new topological insights for pulling back cobordisms by continuous maps that may not be relatively orientable.