Counting curves on Calabi-Yau 3-folds


This project investigates problems in enumerative geometry, which is one of the most fundamental and classical subjects in mathematics. For example, one can ask a question like 'How many lines pass through two given points in the plane?'. Sometimes the set of all curves satisfying certain geometric properties is finite, but sometimes not. A modern approach to the enumerative geometry is to assign numbers, called `virtual invariants' to the spaces parametrising all curves we want to count, whether or not they are finite sets. There are several different virtual invariants, and some of them are introduced by physicists in string theory.

This project aims to solve enumerative problems for a particular kind of spaces called Calabi-Yau (CY) 3-folds. Due to their rich symmetry, CY 3-folds appear in various research areas such as algebraic geometry, symplectic geometry, representation theory, category theory, mirror symmetry, and string theory. In the last three decades, CY 3-folds are the central research objects in these fields and a great deal of new mathematics has arisen from studying curves inside CY 3-folds and their numerical invariants.

Specifically, we plan to understand the relationship between the so-called Gromov-Witten (GW) theory and Gopakumar-Vafa (GV) theory. The exciting point of this kind of question is that by bridging two different theories, one can sometimes solve a very difficult conjecture in one theory via relatively easy facts in the other theory. In our case, we will investigate the brand new GV theory and use it to solve long-standing problems in GW theory.

Another intriguing point of this project is that we have various approach to tackle the problem. We will use both classical and modern methods in various fields such as (derived) algebraic geometry, topology, category theory, representation theory, combinatorics, mirror symmetry, and string theory.

We expect that the candidate is interested in or familiar with one of the above areas. Through the project, he or she will be excited to see how these areas are connected and combinations of different techniques provide powerful tools to solve problems.


Start Date: 1st October 2023

Further Details:

This PhD project is funded by The Faculty of Science & Engineering at The University of Liverpool and will start on 1st October 2023.

Successful candidates who meet the University of Liverpool eligibility criteria will be awarded a Faculty of Science & Engineering studentship for 3.5 years, covering UK tuition fees and an annual tax-free stipend (e.g. £17,688 p.a. for 2022-23).

Faculty of Science & Engineering students benefit from bespoke graduate training and £5,000 for training, travel and conferences. 

The Faculty of Science & Engineering is committed to equality, diversity, widening participation and inclusion. Academic qualifications are considered alongside non-academic experience. Our recruitment process considers potential with the same weighting as past experience. Students must complete a personal statement profoma and ensure this is included in their online application.

How to Apply:

All applicants must complete the personal statement proforma. This is instead of a normal personal/supporting statement/cover letter. The proforma is designed to standardise this part of the application to minimise the difference between those who are given support and those who are not. The proforma can be found here: More information on the application process can be found here: When applying online, students should ensure they include the department name in the ‘Programme Applied For’ section of the online form, as well as the Faculty of Science & Engineering as the ‘studentship type’ in the finance section.

Application Web Address: 


Open to UK applicants

Funding information

Funded studentship

UK students are only eligible for a fully-funded  Faculty of Science & Engineering studentship; overseas students are eligible to apply if they can financially cover the difference in UK and Overseas tuition fees, cover the costs of their student visa, NHS health surcharge, travel insurance and transport to the UK, as these are excluded from the funding.