Speaker(s):
Brett Parker (Australian National University)
Organizer(s):
River Chiang (National Cheng Kung University)
Kaoru Ono (RIMS, Kyoto University)
1. Outline & Descriptions
Holomorphic curves play a central role in symplectic topology. They can be regarded as 2-dimensional analogues of a geodesics within a symplectic manifold, or as trajectories traced out by interacting strings in string theory, and provide a rich geometric framework for understanding symplectic topology. In many situations, holomorphic curves can be studied using 1-dimensional piecewise-linear objects called tropical curves. In the first lecture, I will explain the geometry behind the appearance of tropical curves, and explain why it is useful to employ a category blending tropical geometry with usual differential or algebraic geometry. In the remaining lectures, I will introduce the category of exploded manifolds, and explain how using such a category provides a guiding framework for proving gluing formulae and understanding holomorphic curves under a wide class of degenerations including normal crossing degenerations. Importantly, the transversality and intersection theory required for gluing theorems takes place within the category of exploded manifolds, so I will spend some time on transversality, intersection theory, and the implicit function theorem within the category of exploded manifolds.
Brett Parker (Australian National University) November 24, 2023 10:00-12:00 台灣大學次震宇宙館509研討室
Abstract:
What do J-holomorphic curves look like? I’ll give an introduction to some geometric tools for understanding how J-holomorphic curves behave, working towards understanding how holomorphic curves can look like piecewise linear graphs called tropical curves. I hope to give geometric intuition about why working in a category with two different length scales is helpful.
Brett Parker (Australian National University) November 30, 2023 16:00-18:00 台灣大學次震宇宙館509研討室
Abstract:
I will give some basic examples of exploded manifolds, explain the link to tropical algebra and logarithmic geometry, and give geometric intuition for these spaces.
Brett Parker (Australian National University) December 1, 2023 10:00-12:00 台灣大學次震宇宙館509研討室
Abstract:
I will discuss tangent spaces, the implicit function theorem, transversality and fibre products, and cohomology theories for exploded manifolds, concentrating on the important differences from usual smooth geometry. (Such differences are forced by the geometry of holomorphic curves in normal crossing degenerations, so are relevant to other methods of studying these moduli spaces.)
Brett Parker (Australian National University) December 1, 2023 14:00-16:00 台灣大學次震宇宙館509研討室
Abstract:
I will talk about J-holomorphic curves in exploded manifolds, and a version of the implicit function theorem for the dbar equation in families. This gives a different perspective on gluing analysis.
Brett Parker (Australian National University) December 7, 2023 16:00-18:00 台灣大學次震宇宙館509研討室
Abstract:
I will give a gentle introduction to the moduli stack of curves in exploded manifolds, explain how Kuranishi charts naturally embed in this moduli stack, and sketch how Gromov—Witten invariants are defined using this machinery. I hope to emphasis how using this moduli stack makes the construction of Gromov—Witten invariants very natural.
Brett Parker (Australian National University) December 8, 2023 10:00-12:00 台灣大學次震宇宙館509研討室
Abstract:
I will briefly sketch some example calculations of Gromov—Witten invariants in the Calabi—Yau 3-fold setting. In this setting each tropical curve contributes to a count of Gromov—Witten invariants in all genus, in a kind of quantum deformation of the usual tropical counts of curves.
