A Three-Day School and Research Workshop
Preparatory Meeting for the 2027 Summer School on the Theory of Atoms
Simons Center for Geometry and Physics (SCGP)
October 7–9, 2026
Overview
Birational geometry has recently acquired a new and unexpected source of invariants. Rather than being extracted from the geometry of a variety directly, these invariants are read off from its quantum cohomology and from the Hodge-theoretic structures attached to it. The workshop is devoted to these developments, to the techniques that make them computable, and to the questions they open. The meeting will take place at the Simons Center for Geometry and Physics (SCGP), Stony Brook, New York, from October 7 to October 9, 2026. Two mini-courses, of two lectures each, run from Wednesday morning through Thursday morning and build the necessary background from the ground up. The research talks begin on Thursday at 11:00 and continue through Friday. The organizing theme is the theory of atoms, introduced by Katzarkov, Kontsevich, Pantev, and Yu [11], in which a canonical decomposition of the quantum cohomology of a smooth projective variety produces new birational invariants. The construction draws on classical and non-commutative Hodge theory [10] together with genus-zero Gromov– Witten theory, and relies in an essential way on the decomposition theorem for the quantum D-module of a blowup due to Iritani [9] and on the decomposition and framing theory for F-bundles developed in [8]. Equivariant extensions of the formalism, obtained by combining atoms with the modular symbols of Kontsevich, Pestun, and Tschinkel, are developed in [2]. Applications include obstructions to rationality and to linearizability of group actions that are invisible to older techniques.
Each mini-course lecture lasts one hour and forty minutes; each research talk lasts fifty minutes. All times indicated below are local times at the Simons Center (Eastern Time, New York).
Relation to the 2027 Summer School
This workshop is conceived as a preparatory meeting for the Summer School on the Theory of Atoms to be held in 2027. Its purpose is twofold: to assemble, in a compact and self-contained form, the prerequisites that participants of the summer school might be assumed to have; and to allow the organizers and lecturers to test and calibrate the structure of the extended program. Accordingly, the two mini-courses of October 7–8 are designed as a first pass through the material that will be developed at greater length and in greater generality in 2027. Participants who intend to take part in the 2027 summer school are, therefore , particularly encouraged to attend. More details on the summer school, including dates, venue, and the full list of lecturers, will be announced in due course.
Aims
Among the main objectives of the school are:
• to explain what the new quantum-cohomological invariants of birational geometry are, and what they can and cannot detect;
• to introduce the basic geometric and enumerative structures entering the theory of atoms, starting from genus-zero Gromov–Witten theory;
• to explain how Gromov–Witten theory gives rise to Frobenius-type structures and to the A-model F-bundle;
• to introduce the non-Archimedean framework used in the study of convergence and analytic continuation of the relevant structures;
• to explain decomposition phenomena for F-bundles and their role in the definition of Hodge atoms;
• to present explicit examples, with particular emphasis on flag varieties and related homogeneous spaces;
• to make explicit the dictionary with the corresponding structures in two-dimensional field theory and topological string theory;
• to discuss current applications, ongoing research, and open problems; and
• to prepare the ground for the 2027 summer school.
Schedule
Each mini-course lecture lasts one hour and forty minutes; each research talk lasts fifty minutes. All times indicated below are local times at the Simons Center (Eastern Time, New York).
The period from 15:45 to 16:20 is kept free of sessions on all three days.
Day 1: Wednesday, October 7, 2026
Time Activity
10:00–11:40 Mini-course I, Lecture 1 — P. A. Muniz Martins Lunch break
11:40–13:30 Lunch Break
13:30–15:10 Mini-course I, Lecture 2 — P. A. Muniz Martins Break
15:10–16:20 Break
16:20–18:00 Mini-course II, Lecture 1 — L. F. Cavenaghi
Mini-course I lecturer: Pedro Antonio Muniz Martins (Unicamp)
10:00–11:40 Lecture I: Introduction to Gromov–Witten Invariants and Localization
The first lecture introduces the basic ideas of Gromov–Witten theory, with an emphasis on the genus-zero theory relevant to quantum cohomology. We discuss moduli spaces of stable maps, evaluation morphisms, curve classes, and the geometric meaning of Gromov–Witten invariants. The second part of the lecture introduces equivariant cohomology and the localization principle. Particular attention will be paid to torus actions and to the localization techniques that make explicit Gromov–Witten computations possible. The general formalism will then be specialized to flag varieties, where the geometry of torus-fixed points and invariant curves provides a natural combinatorial framework for calculations. The lecture concludes with a computational demonstration illustrating how Gromov–Witten invariants of complete intersections in flag varieties can be approached algorithmically.
13:30 – 15:10 Lecture II: Frobenius Manifolds, Euler Vector Fields, and Materialization
The afternoon lecture introduces Frobenius manifolds and the geometric structures naturally associated with genus-zero Gromov–Witten theory. Beginning with the WDVV equations and the quantum product, we discuss the compatibility between multiplication, metric, and flat structures that underlies the definition of a Frobenius manifold. Particular attention will be given to Euler vector fields and the additional grading information they encode. We will examine multiplication by the Euler vector field, its spectral behavior, and the role played by the corresponding eigenvalue data in the geometry of the theory. The lecture then turns to the materialization procedure and explains how the formal structures arising from quantum cohomology can be related to analytic geometric objects. This provides one of the bridges between Gromov–Witten theory, Frobenius geometry, and the structures that later enter the definition of atoms.
Mini-course II lecturer: Leonardo F. Cavenaghi (International Center for Mathematical Sciences – Sofia, ICMS-Sofia)
16:20–18:00 Lecture 1: Berkovich Geometry and the A-Model F-Bundle
This lecture introduces the non-Archimedean geometric framework used in the construc- tion of the A-model objects appearing in the theory of atoms. We begin with the basic notions of non-Archimedean analytic geometry and Berkovich analytic spaces, empha- sizing those features that are relevant to families arising from quantum cohomology. We then discuss how this analytic framework can be used to formulate and study con- vergence properties of Gromov–Witten potentials. Particular attention will be given to the transition from formal quantum-cohomological data to analytic families. The second part of the lecture introduces F-manifolds and F-bundles. After reviewing the integrability condition for the multiplication on the tangent sheaf, we construct the A-model F -bundle and describe the geometric information encoded by its multiplication, connection, and spectral data. The relation with the genus-zero Gromov–Witten theory developed earlier in the day will be made explicit throughout the discussion.
Day 2: Thursday, October 8, 2026
Time Activity
09:30–11:10 Mini-course II, Lecture 2 — L. F. Cavenaghi Coffee break
11:20–12:10 Tony Pantev (University of Pennsylvania)**
12:10— Lunch break; afternoon free for informal discussion
Speaker: Leonardo F. Cavenaghi (ICMS-Sofia)
9:30 – 11:10 Lecture II: Decomposition Theorems for F-Bundles and Hodge Atoms
The second lecture develops the structural results for F-bundles that lead to the notion of atoms. We first discuss local and global decomposition phenomena and the role of spectral data in separating the different components of the corresponding geometric structure. The blowup decomposition of the quantum D-module will be presented as the main source of birational control. We then introduce the Hodge-theoretic locus and explain the passage from the spectral decomposition of the A-model F-bundle to the definition of Hodge atoms.The discussion will emphasize both the geometric motivation for the construction and the information retained by the resulting components. Examples will be used to illustrate how atoms can encode geometric features that are not immediately visible at the level of ordinary cohomological invariants.
Research Talks
The research part of the workshop opens on Thursday late morning and continues through Friday. The talks are devoted to recent developments in the theory of atoms, mirror symmetry, Hodge-theoretic structures, and related aspects of algebraic and complex geometry. They are intended to connect the foundational material of the mini-courses with current research questions and applications, and to outline the program of the 2027 summer school.
09:30–10:20 Maxim Kontsevich (IHÉS)**
10:30–11:20 Andrew Harder (Lehigh University)
11:20–13:00 Lunch Break
13:00–13:50 Daniel Pomerleano (University of Massachusetts Boston)
13:55–14:45 Leonardo F. Cavenaghi (ICMS-Sofia)
14:50–15:40 Jiaji Cai (Stony Brook University)
15:40–16:20 Coffee Break
16:20–17:10 Ludmil Katzarkov (SCGP)
**Remote Talk