Tropical and Non-Archimedean Analytic Methods in Algebraic Geometry

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Algebraic geometry studies solution sets of systems of polynomial equations (algebraic varieties) and has many applications in areas within mathematics and beyond, including physics, computer science, and engineering. The research supported by this award will center on using modern degeneration techniques, especially those from the field of tropical geometry, to study classical spaces from algebraic geometry. The main goal of tropical geometry is transforming questions about algebraic varieties into questions about polyhedral complexes. A process called tropicalization attaches a polyhedral complex to an algebraic variety. The polyhedral complex, a combinatorial object, encodes some of the geometry of the original algebraic variety. The research develops further tools for the study of algebraic varieties in terms of their so called "nonarchimedean analytification". Overall this project will refine, abstract, and generalize these new methods and explore deeper applications to open problems in algebraic geometry. In conjunction with this research program, the PI will continue to direct the SUMRY program for undergraduate research in mathematics at Yale, providing not only research opportunities for the undergraduate participants, but also mentorship opportunities for the graduate students and postdocs leading small group projects. The PI will extend the tropical independence methods developed in proofs of the Gieseker-Petri theorem and the maximal rank conjecture for quadrics, developing new notions of tropicalization of linear series and combining piecewise linear methods with reduction of rational functions to pursue further progress toward the maximal rank conjecture and strong maximal rank conjecture. He will continue his work using the combinatorial topology of moduli spaces of stable tropical curves to extract information about the top weight cohomology of moduli spaces of curves, looking for additional structures such as hidden filtrations whose graded pieces are representation stable with respect to the action induced by permutation of the marked points. The PI also aims to develop foundations for a tropical theory of vector bundles, by generalizing the theory of convergence Newton polygons for vector bundles with connection on algebraic curves to vector bundles with integrable connections on higher dimensional varieties, using toric vector bundles as a test case.