A Tribute to the Life and Work of Professor Bumsig Kim:

Categorical and Enumerative Aspects of Mirror Symmetry




September 19-23, 2022                                      KIAS, Room 1503

Program Home > Program




◆ Mon (19 Sep)


Yuan-Pin Lee (Utah, Academia Sinica)

- Title: QK = GV, two integral invariants on Calabi--Yau threefolds

- Abstract: On Calabi--Yau threefolds there are two types of integral invariants, quantum K-invariants and Gopakumar--Vafa invariants. In this talk, I will explain a joint project with You-Cheng Chou which aims to show that the quantum K-invariants and Gopakumar invariants are equivalent. At genus zero, this is a conjecture by Jockers--Mayr and Garoufalidis--Scheidegger, and a proof of the JMGS conjecture will be presented.


Mark Shoemaker (Colorado State)

- Title: A Kleiman criterion for GIT stack quotients

- Abstract: Kleiman’s criterion states that, for X a projective scheme, a divisor D is ample if and only if it pairs positively with every element of the closure of the cone of curves.  In other words, the cone of ample divisors in N^1(X) is the interior of the nef cone.  In this talk I will present an analogous statement for a GIT stack quotient [X // G].  In this new context, the ample cone of X is replaced by a cell in the vGIT decomposition of the G-ample cone C^G(X), and curves in X are replaced by quasimaps to [X // G].


Jeremy Guere (Grenoble)

- Title: Congruences on K-theoretic Gromov-Witten invariants

- Abstract: In this talk, I will present how enlarging the scope of the localization formula to other groups than a torus action could provide new computational results on the Gromov-Witten theory of algebraic varieties. In particular, I will focus on finite symmetries of the famous quintic threefold and show explicit computations of its K-theoretic Gromov-Witten invariants modulo 41. This approach works similarly in other enumerative theories, such as the recent GLSM theory developed among others by Favero-Kim.



◆ Tue (20 Sep)


Duiliu Emanuel Diaconescu (Rutgers, online)

- Title: Character varieties and Gromov-Witten invariants 

- Abstract: A conjectural relation will be presented, between counting rational points on wild character varieties and counting curves in Calabi-Yau threefolds. Based on work with W.-y. Chuang, R. Donagi, S. Nawata and Tony Pantev. 


Yong-Geun Oh (IBS-CGP, Postech)

- Title: Contact instantons and entanglement of Legendrian links

- Abstract: We introduce a conformally invariant nonlinear sigma model on the bulk of contact manifolds with boundary condition on the Legendrian links in any odd dimension. We call any finite energy solution a contact instanton. We also explain its Hamiltonian-perturbed equation and establish the Gromov-Floer-Hofer type convergence result for (Hamiltonian-perturbed) contact instantons of finite energy and construct its compactification of the moduli space. Then we explain how we can apply this analytical machinery in the study of contact topology and contact Hamiltonian dynamics. As an example, we explain our proof of a conjecture of Sandon and Shelukhin on the translated points of contactomorphisms on compact contact manifold. This  is the  contact analog to the Arnold's conjecture type in symplectic geometry.


Michel van Garrel (Birmingham)

- Title: Log Mirror Symmetry

- Abstract: Consider the log K3 surface (S,E) given by the pair of the projective plane and an elliptic curve. In 2001, N. Takahashi built the mirror family M to (S,E) and predicted how to extract from period integral of M the log Gromov-Witten invariants of maximal tangency of (S,E). In joint work with Ruddat and Siebert, we prove this conjecture. In fact, we work more generally with the mirror constructions of the Gross-Siebert programme. We form the periods on the Gross-Siebert mirror family and prove that they computes the log Gromov-Witten invariants.


Hyenho Lho (CNU)

- Title : Equivariant quasimap invariants of Grassmanian

- Abstract : I will review the joint result with Prof. Bumsig Kim about the calculation of elliptic quasimap invariants for quintic threefold. The methods used in this study can be easily generalized to other GIT targets. I will generalize this method to calculate the equivariant quasimap invariants of Calabi-Yau complete intersections in Gr(2,4) and Gr(2,5). And I will further discuss how to calculate the general Grassmannian cases.



◆ Wed (21 Sep)


Aaron Bertram (Utah)

- Title: Instability of Syzygies

- Abstract: A stability condition on the derived category of coherent sheaves on a complex projective manifold $X$ determines stable objects of a fixed Chern class invariant and also filtrations of unstable objects. In families, this partitions the base of the family into locally closed subschemes (the ``Shatz stratification''). As the stability condition varies, one obtains refinements of these partitions, which are actually not stratifications, but whose ultimate refinement may be. Our main example is the betti table stratification of graded Gorenstein quotient rings of the polynomial ring in three variables, which we explain with stability conditions. This is joint work with Brooke Ullery. 


Helge Ruddat (Mainz)

- Title: The proper Landau-Ginzburg potential is the open mirror map

- Abstract: Zaslow, Gräfnitz and I compute the Landau-Ginzburg superpotential of the mirror of P^2 relative a smooth elliptic curve. This infinite power series is a generating function for 2-contact point invariants of (P^2,E) and equals the open mirror map for outer Aganagic-Vafa branes in the canonical bundle K_X. The proof of this fact makes use of the log version of the degeneration formula in Gromov-Witten theory which was proved jointly with Bumsig Kim.



◆ Thur (22 Sep)


Yukinobu Toda (IPMU)

- Title: Categorical and K-theoretic Donaldson-Thomas theory of C^3

- Abstract: The generating series of Donaldson-Thomas invariants associated with Hilbert schemes of points on the three-dimensional affine space is known to form a MacMahon function, whose coefficients are numbers of plane partitions. In this talk, I will give a categorical and K-theoretic analogue of the above formula. I will consider the triangulated categories of matrix factorizations of super-potentials whose critical loci are Hilbert schemes of points, and show the existence of their semi-orthogonal decompositions which are regarded as categorification of the above MacMahon formula. In fact there exist explicitly constructed objects in each semi-orthogonal summand, whose cardinality is the number of plane partitions, which generate the torus localized K-group of the above category of matrix factorizations. This is a joint work with Tudor Padurariu.


Young-Hoon Kiem (SNU)

- Title: Derived cosection localization

- Abstract: A Deligne-Mumford stack X equipped with a perfect obstruction theory whose obstruction sheaf admits a cosection has its virtual fundamental class localized to the zero locus Y of the cosection. Since Y often turns out to be an interesting space on its own, one may wonder if there is a natural induced perfect obstruction theory whose virtual fundamental class is the cosection localized virtual fundamental class. From the viewpoint of derived algebraic geometry, the cosection should be thought of as a (-1)-shifted 1-form and the Lagrangian intersection of its graph with the zero section in the dual obstruction cone has a natural (-2)-shifted symplectic structure if the 1-form is closed. By a recent work of Jeongseok Oh and Richard Thomas, we then have a virtual fundamental class of the zero locus Y of the cosection. In this talk, I will discuss the above mentioned constructions and show that the cosection localized virtual fundamental class of X coincides with the virtual fundamental class of Y. Based on a joint work with Hyeonjun Park.


Bhamidi Sreedhar (IBS-CGP)

- Title : Hochschild homology of matrix factorization categories of Deligne-Mumford Stacks

- Abstract:   This talk is based on a joint work with Dongwook Choa and Professor Bumsig Kim. In this talk we will discuss a Hirzebruch-Riemann-Roch (HRR) type theorem for matrix factorization categories of Deligne-Mumford stacks. We will first discuss a proof of a Hochschild-Kostant-Rosenberg type isomorphism and show how it can be used to define a Chern character formula which allows us to prove the HRR type theorem.


Jeongseok Oh (Imperial)

- Title: Localized Chern characters for 2-periodic complexes

- Abstract: We have started this project Spring 2017 that was my last term of PhD. It studies a comparison of the quasimap theory of a complete intersection and that of the corresponding gauged linear sigma model. This joint work with Prof. Bumsig Kim was very special to me since I could absorb a part of his knowledge of intersection theory during this one year joint work which eventually became my main field of research after PhD. In this talk, I would like to share the thing I have learned from him and introduce our joint work. I hope I can deliver properly how much he is a great and kind PhD advisor.



◆ Fri (23 Sep)


Andrei Caldararu (Wisconsin)

- Title: Higher structures on Hochschild chains of categories of matrix factorizations

- Abstract: In this talk I shall present an overview of structures that exist on the Hochschild chains of any cyclic A-infinity category, with an emphasis on the special case of categories of matrix factorizations. Professor Bumsig Kim made important contributions to the study of the Hochschild and negative cyclic homology theories of such matrix factorizations. In my talk I shall present some of his works, and explain how these (conjecturally) fit with works of Kyoji Saito in singularity theory. I will conclude with a discussion of categorical enumerative invariants associated to categories of matrix factorizations, joint work in progress with Si Li and Junwu Tu.


Alexander Polishchuk (Oregon)

- Title: Spin G-Cohomological Field Theories

- Abstract: This is a report on a joint ongoing work with He, Shen and Vaintrob. Let G be a finite group. G-Cohomological Field Theories were introduced by Jarvis-Kaufmann-Kimura as a simultaneous extension of orbifold Gromov-Witten invariants of quotient stacks X/G and of the corresponding Fantechi-Gottsche rings. Using an analogy between the moduli spaces of admissible covers and those of generalized spin structures, we define an analog of this picture in the Landau-Ginzburg setting.


Taejung Kim (KNUE)

- Title: Standard conjecture D for local stacky matrix factorizations

- Abstract: After introducing Grothendieck's standard conjecture D and M. Brown and M. Walker’s work on the non-commutative generalization of the standard conjecture for the differential graded category of matrix factorizations associated to an isolated hypersurface singularity, we will explain its extension to the case of local stacky matrix factorizations for a finite group.


David Favero (Alberta, Minnesota)

- Title: CoFTs for GLSMs

- Abstract: I will discuss the construction of enumerative invariants for GLSMs based on joint work with Professor Bumsig Kim (see arXiv:2006.12182) where we proved that these invariants form a Cohomological Field Theory.  These invariants should generalize both FJRW theory and Gromov-Witten theory (for GIT quotients of affine space).   They are obtained by forming the analogue of a virtual fundamental class which lives in the twisted Hodge complex over a certain "moduli space of maps to the GLSM".  This virtual fundamental class roughly comes as the Atiyah class of a "virtual matrix factorization" associated to the GLSM data, for which I will outline the construction.