Logarithmic and Non-Archimedean Gromov-Witten invariants

Since the early 1990s, curve-counting invariants have played an increasingly important role in algebraic geometry. Typically, these invariants give meaningful answers to questions such as "How many
curves of a fixed degree and genus are contained in a given algebraic variety?" There has been recently a great deal of development in logarithmic Gromov-Witten theory, which allows the imposition of tangency conditions with divisors. This is allows a good theory of Gromov-Witten invariants for degenerations of algebraic varieties.

More recently, Tony Yue Yu has developed a theory of non-Archimedean Gromov-Witten invariants, which also provides a good theory of invariants for degenerations of algebraic varieties. This leads to the first fundamental question: what is the relationship between these two kinds of invariants. Johnston will work towards a comparison result for these invariants. Initially, he will also explore the recent work of Keel-Yu on constructions of mirror pairs using non-Archimedean algebraic geometry, where a direct comparison with the Gross-Siebert approach is likely to be more readily achieved. Once this is done, he will prove a general comparison result. This will allow the two theories to be used interchangeably where appropriate.