Foundations of Gromov-Witten theory

Lecturer: Yannik Schüler-Hammer
Time and Location: Thu 16:15-18:00, HG G 26.3

Content

This seminar introduces Gromov-Witten theory, starting from basics in intersection theory and discussing the geometry of moduli spaces of stable curves and maps. The goal is to define Gromov-Witten invariants, develop methods for their computation. For instance, we will show that there are 2’875 lines on a general quintic threefold. The learning objectives are:

for students to gain understanding about the foundations and methods in Gromov-Witten theory,
to develop their skills in reading research-level mathematical texts,
to enhance their presentation and communication skills.

For each session, reading material will be assigned along with a session leader. The session leader will present the assigned material and answer questions from the audience, with the goal of making each session a collaborative and active mathematical conversation rather than a passive lecture. Assessment will be based on participation in the seminar, with particular emphasis on the session led by each student.

Course organisation

You can find the slides from our first meeting here. After the first meeting, please fill in this form indicating up to four preferred topics in order of preference until Fri 18 Sept 20:00. The assignment of talks will be announced by email and on this website Tue 22 Sept.

You can find more detailed information and references for each talk in this note.

#	date	topic	presenter
0	17.09.	Introduction and assignment of talks	Yannik Schuler
1	24.09.	Riemann surfaces and Hurwitz numbers	Roland Hafner
2	01.10.	Intersection theory	Rivaldo Cifuentes Monroy
3	08.10.	Intersection theory on the moduli space of curves
4	15.10.	Moduli space of genus zero stable maps
5	22.10.	Moduli space of higher genus stable maps
6	29.10.	Equivariant intersection theory and localisation
7	05.11.	Localisation for stable maps to projective space
8	12.11.	Relative stable maps and the ELSV formula
9	19.11.	Hurwitz numbers and Fock space
10	26.11.	The stationary Gromov-Witten theory of \( \mathbb{P}^1 \)
11	03.12.	The Gromov-Witten/Hurwitz correspondence
12	10.12.
13	17.12.

Giving a seminar talk

A good seminar talk is structured, engaging, and clear. When preparing your presentation, start by identifying the key ideas: What are the main concepts? Why are they important? How do they fit into the broader context of quiver representations? Before diving into technical proofs, take some time to provide intuition and motivation. Here are a few practical tips:

Organisation: Plan your talk with a clear structure — begin with an overview, introduce necessary definitions, state the main results, and then explain proofs or computations step by step.
Examples: Illustrate abstract ideas with concrete examples. Quiver representations generalise concepts you have already encountered in linear algebra, so well-chosen examples can greatly aid understanding.
Notation and clarity: Be mindful of notation and avoid overwhelming the audience with too many symbols at once. Whenever possible, explain formulas in words.
Time management: Practice beforehand to ensure your talk fits within the allotted time. If a proof is too long, highlight only the key steps and refer to the book for details.
Engagement: Encourage questions and interaction. If a concept is tricky, take a moment to check if everyone is following before moving on.

For additional tips on giving a seminar talk, see the advice provided by Johannes Schmitt. [This guideline was taken with slight modifications from these notes, section 1.1]

Requirement for passing

In order to pass the seminar, you have to:

give a talk from the above list to the other participants
attend (most of) the talks of the other participants </ul>