A^1-homotopy theory, also known as motivic homotopy theory, is an up-and-coming area of mathematics that unites the two worlds of algebraic geometry and algebraic topology by applying tools from algebraic topology, in particular homotopy theory, to algebraic varieties.
This yields new tools to attack classical problems in algebraic geometry and related areas.
I started my research studying examples of algebraic varieties that are A^1-contractible, i.e., algebraic varieties that look like a point in this A^1-homotopy category.
Right now my research is focused on the application of A^1-homotopy theory to enumerative geometry resulting in the new and fast growing area A^1-enumerative geometry which allows to study questions in enumerative geometry over an arbitrary base field. Currently, I am particularly interested in applying methods from tropical geometry in A^1-enumerative geometry.
- Quadratic Counts of Twisted Cubics, with Marc Levine, accepted for publication in Trends in Mathematics, book in memory of Alberto Collino, arXiv
- Bézoutians and the A^1-degree, with Thomas Brazelton and Stephen McKean, accepted for publication in Algebra & Number Theory arXiv
- Quadratic types and the dynamic Euler number of lines on a quintic threefold, published in 2022 in Advances in Mathematics. https://doi.org/10.1016/j.aim.2022.108508 arXiv
- Computing A^1-Euler numbers with Macaulay2, published in Res. Math. Sci., 2023, arXiv
- Applications to A^1-enumerative geometry of the A^1-degree, with Kirsten Wickelgren,
Res Math Sci 8, 24 (2021). https://doi.org/10.1007/s40687-021-00255-6, arXiv
- A^1-contractibility of affine modifications, with Adrien Dubouloz and Paul Arne Østvær, published 2019 in International Journal of Mathematics. https://doi.org/10.1142/S0129167X19500691, arXiv
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