Fall '25 AG Postdoc Seminar

Location: Math 507, Time: 1:45-3:00 on Friday, Organizers: James Hotchkiss and Alekos Robotis

The topic of the seminar this semester is the recent paper by Katzarkov, Kontsevich, Pantev, and Yu [KKPY] . Some other relevant references and sources are collected below.

• Hodge theoretic aspects of mirror symmetry by Katzarkov, Kontsevich, Pantev [KKP]
• Decomposition and framing of F-bundles and applications to quantum cohomology by Hinault, Yu, Zhang, and Zhang [HYZZ]
• Quantum cohomology of blowups by Hiroshi Iritani [I]
• Introduction to non-Archimedean geometry by Piotr Achinger [A]
• Notes on stable maps and quantum cohomology by William Fulton and Rahul Pandharipande [FP]

Below is the list of talks, speakers, and topics:

• September 12: Introduction/overview - Alekos Robotis. A brief overview of the ideas and concepts introduced in [KKPY].
• September 19: Gromov-Witten Invariants - Peter Moody. This talk covers the main definitions and first results about genus 0 Gromov-Witten invariants. Reference: [FP]
• September 26: Quantum Cohomology - This talk introduces the structures of quantum cohomology constructed using Gromov-Witten theory and their relations to the A-model F-bundle introduced in [KKPY].
• October 3: F-bundles - This talk covers 1) the basic language of non-Archimedean geometry needed to formulate the results on F-bundles and 2) the definitions of first properties of F-bundles from section 2 of [HYZZ]. References: [A] and [HYZZ]
• October 10: Decomposition Theorems for F-manifolds - This talk covers 1) the Frobenius theorem in non-Archimedean and formal geometry and 2) decomposition theorems for F-manifolds. References: [HYZZ] sections 3.1 and 3.2.
• October 17: Decompositions of Maximal F-bundles - This talk covers the spectral decomposition theorem for maximal F-bundles.
• October 24: Iritani's Blowup Formula - This talk gives an overview of the recent work of Iritani on the blowup formula in the language of quantum D-modules. Reference: [I]
• October 31: KKPY Blowup Formula - This talk explains the proof of the blowup formula of [KKPY] (i.e. Theorem 4.5) and mentions the similar result for projective bundles. Time permitting, section 5.4 of [HYZZ] could also be discussed.
• November 7: Hodge Atoms - This talk introduces the notion of Hodge Atoms, used in the construction of new birational invariants using the theory of F-bundles. Reference: [KKPY] section 5.
• November 14: Cubic Fourfolds - This talk explains the applications of the theory of Hodge Atoms to proving irrationality of very general cubic fourfolds. Reference: [KKPY] sections 6.1 and 6.2.
• November 21: More examples - This talk discusses further examples of Hodge Atoms of projective varieties. Reference: [KKPY] section 6.
• December 5: Bonus Talk - TBD!