Mirror Symmetry, Hodge Theory and Differential Equations

The following is the report on the Oberwolfach workshop Mirror Symmetry, Hodge Theory and Differential Equations (April 2015), which brought together researchers from various areas such as quantum cohomology, complex algebraic geometry, Hodge theory representation theory etc. Mathematics Subject Classification (2010): 14J33, 32S40, 14D07, 34Mxx, 53D45. Introduction by the Organisers The workshop Mirror Symmetry, Hodge Theory and Differential Equations, which took place from April 20 to 24, 2015 aimed at reporting on recent developments on research topics related to various areas of pure mathematics such as Hodge theory, linear differential equations, quantum cohomology, category theory and representation theory, to name only a few. The workshop had 25 participants, with a wide geographical horizon. The list of participants included several young postdocs, the workshop was an excellent opportunity for them to present their results to a larger audience. The meeting has highlighted the intense activity in these mathematical domains, as well as the strong interaction with other mathematical topics, such as Langlands correspondence, representation theory and irregular differential equations. The workshop consisted of 19 talks and many informal discussions, supplemented with some evening discussion sessions. Some talks were meant to give an overview on a particular field relevant for the main subject of the workshop, others reported on precise new results and and a few ones mainly contained new ideas or work in progress. For example, Duco van Straten talked on joint work with 1202 Oberwolfach Report 22/2015 A. Mellit and V. Golyshev on geometric Langlands correspondence and congruence differential equations, motivated by the classification of Fano manifolds by using quantum cohomology and tools from Langlands correspondence. Two talks, by Clélia Pech and Konstanze Rietsch (partly on joint work with R. Marsh and L. Williams) were on mirror symmetry statements for non-toric varieties such as homogeneous spaces and more specifically Grassmannians (including equivariant aspects). A whole day was concerned with talks related to various kinds of hypergeometric equations. Uli Walther (based on joint work with L. Matusevich and E. Miller as well as M. Schulze) introduced us to the techniques of Euler-Koszul homology, Thomas Reichelt used these to describe how the formalism of mixed Hodge modules can be applied to the study of Gelfand-Kapranov-Zelevinsky differential systems. Takuro Mochizuki explained his recent work on twistor modules and GKZ-systems, and Alberto Castaño Domı́nguez explained results on hypergeometric description of the cohomology of the Dwork family. Hiroshi Iritani reported on his recent work on mirror statements involving the big quantum cohomology rings, whereas Alessandro Chiodo overviewed his results (with Y. Ruan and H. Iritani) on the Landau-Ginzburg/Calabi-Yau correspondence. The talk of Lev Borisov showed a surprising application of mirror symmetry: From some specific examples of so-called double mirror Calabi-Yau families (i.e., families having the same mirror) one can derive that the class of the affine line is a zero divisor in the Grothendieck ring. Categorical aspects of quantum cohomology and mirror symmetry have been discussed in several talks: Dmytro Shklyarov explained how to (re)construct GaußManin cohomology from categories (like matrix factorizations), and Etienne Mann talked about his joint work with M. Robalo on categorification of Gromov-Witten invariants. Todor Milanov reported on a recent construction which builds on the oscillating integrals as well as the period integrals in singularity theory and Landau-Ginzburg models: It establishes a vertex operator algebra structure on a certain Fock space associated to these data. Alexey Basalaev explained how to endow cohomological field theories with an action of SL(2,C) and how to show modularity properties of certain potentials. In a somewhat different spirit, Emmanuel Scheidegger talked on joint work with M. Alim, H. Movasati and S.T. Yau on the construction of a certain Lie algebra of vector fields on the moduli space of Calabi-Yau threefolds and its relation to the holomorphic anomaly equations. Ana Ros Camacho introduced a new equivalence relation for homogeneous polynomials, called orbifold equivalence, by imposing the existence of a matrix factorization with nonzero quantum dimension for the difference of the two polynomials, and considers the case of ADE singularities. Helge Ruddat reported on joint work with B. Siebert on the construction of canonical coordinates for a degenerating family of Calabi-Yau varieties. Mirror Symmetry, Hodge Theory and Differential Equations 1203 In an overview talk, Philip Boalch described examples of wild character varieties as finite dimensional multiplicative symplectic quotients. He related these examples to a 1764 paper of Euler, and showed that Euler’s continuant polynomials are group valued moment maps. The last talk of the conference by Marco Hien was concerned with a topological approach to the recent work of d’Agnolo-Kashiwara on Riemann-Hilbert correspondence for arbitrary differential systems. Summarizing, we feel that we had an extremely interesting meeting with many beautiful talks covering a large variety of subjects. The discussions that took place between the talks (as well as during the traditional hike to St. Roman on Wednesday afternoon) were quite stimulating. The enthusiasm of the participants as well as the great atmosphere at MFO largely contributed to the success of the workshop. The meeting showed that the subject of mirror symmetry, in all its ramifications, is as vibrant as ever and many open questions are still ahead of us. Acknowledgements: The MFO and the workshop organizers would like to thank the Simons Foundation for supporting Takuro Mochizuki in the “Simons Visiting Professors” program at the MFO. The organizers also acknowledge the financial support of the ANR-DFG program SISYPH (ANR-13-IS01-0001-01/02, DFG No HE 2287/4-1 & SE 1114/5-1). Mirror Symmetry, Hodge Theory and Differential Equations 1205 Workshop: Mirror Symmetry, Hodge Theory and Differential Equations