Gökova Geometry Topology Institute

This two-week research session will bring together experts in algebraic geometry, symplectic geometry, homological algebra and mathematical physics.

The Theory of Atoms is a new framework introduced by Ludmil Katzarkov, Maxim Kontsevich, Tony Pantev and Tony Yue Yu. It combines classical Hodge theory with ideas coming from Gromov–Witten theory, quantum multiplication and symplectic geometry. The resulting objects, called Hodge atoms, are designed to separate birationally meaningful information from the additional contributions created by blow-ups. Thus, the theory provides a promising new approach to some of the central problems of modern birational geometry.

The main goal of this research session is to consolidate the foundations of the Theory of Atoms and develop its applications. One of our principal objectives will be to establish, clarify and refine the blow-up formulas underlying the theory. We plan to study these formulas in several complementary settings and to understand how atomic decompositions behave under birational transformations. We will then apply this structure to the construction of new invariants and new obstructions to rationality and stable rationality.

A substantial part of the program will be devoted to equivariant birational geometry. We plan to develop equivariant and orbifold versions of the Theory of Atoms and to compare them with modular-symbol, motivic and Burnside-type invariants. These comparisons may provide new tools for studying group actions on algebraic varieties, including questions of equivariant rationality, stable linearizability and birational conjugacy. We will test the resulting methods on explicit finite-group actions on rationally connected varieties.

Another important direction concerns derived categories. We aim to investigate the relationship between Hodge atoms and atomic or semiorthogonal decompositions of derived categories. In particular, we plan to compare the information carried by atoms with categorical birational invariants, categorical measures and motivic decompositions. We also hope to clarify the relation between the Theory of Atoms and the noncommutative Minimal Model Program.

The interaction with the classical Minimal Model Program will be one of the central themes of the session. We plan to study how atoms change under divisorial contractions, flips and flops, and how they behave for Mori fibre spaces. A natural question is whether atomic decompositions can refine the birational classification supplied by the Minimal Model Program, or distinguish varieties and group actions that cannot be separated by presently known invariants.

The symplectic side of the theory will also play an essential role. We plan to develop and apply techniques from quantum cohomology, Floer theory, quantum connections and Givental formalism. These methods should make it possible to compute atoms in concrete examples and to relate the algebraic-geometric theory to the symplectic topology of the corresponding varieties. Special attention will be paid to examples in which the quantum product, the associated differential equations and the relevant birational transformations can be described explicitly.

During our stay at the Gokova Geometry Topology Institute, we plan to combine foundational work with detailed computations and case studies. Our aim is to obtain new blow-up formulas, construct equivariant and categorical versions of atomic invariants, investigate their behavior under the Minimal Model Program, and produce explicit applications to rationality problems. We also hope that the session will establish a common language connecting birational geometry, derived categories, motivic invariants and symplectic techniques.

For further questions about this event please contact : Ivan Cheltsov