The seminar consists of 4 intensive lectures and 1 research talk. The goal of the lecture is to provide an introduction to theta functions, and understand why cluster variables are theta functions.

## Speaker

Man Wai Cheung (Kavli IPMU)

## Schedule

*Lectures*

- 2022/12/7 (Wed) 15:30~16:30, 17:00~18:00
- 2022/12/8 (Thu) 11:00~12:00, 13:30~14:30

*Research talk*

- 2022/12/8 (Thu) 15:00~16:00

## Venue

Hybrid style at I-site Namba, Osaka Metropolitan University

## Lectures

*Title*: Lectures on Theta functions and Cluster variables

*Abstract*:

Gross-Hacking-Keel-Konstevish introduced theta bases as bases for cluster algebras. In the lectures, we will study the construction of theta functions, and understand why cluster variables are theta functions. We will also look at the multiplication rule for theta functions. Then we can see how the collections of theta functions give bases to cluster algebras. Afterward, we will go back to review the whole construction again for cluster algebras with principal coefficients. With this setting, we will see how to obtain the A-, X- theta functions, i.e. the x, y cluster variables in the sense of Fomin-Zelevinsky. Then we will understand how to visualize c and g vectors on scattering diagrams from the point of view in Cluster algebra IV by Fomin-Zelevinsky and tropical duality proved by Nakanishi. At the end, we will apply the relation between cluster algebras, quiver representations, and cluster categories to this theta bases construction.

## Research Talk

*Title*:
Counting tropical curves by quiver representation

*Abstract*:

Mikhalkin established the correspondence between holomorphic curves and tropical curves on toric surfaces. The result is generalized by Nishinou and Siebert and many others. Tropical curve (disk) counts are then seen as the algebro-geometric analogue holomorphic disk countings in mirror symmetry. Together with Travis Mandel, we have developed a description of the counting in terms of quiver representations.

## Registration

Online (Zoom):

https://omu-ac-jp.zoom.us/meeting/register/tJAqc-6qrDsqHd1liynMBStdgUfcxX2RkWam

In-person:

Please e-mail to the following address: preprojective2022@gmail.com

## Support

Osaka Central Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics) JPMXP0619217849.