Enumerative seminar

Coorganized with Yang Zhou

Wednesdays at 2pm

Hyeonjun Park, KIAS
Thursday 06/12, SIMIS room TBA

Lagrangian classes, Donaldson-Thomas theory, and gauged linear sigma models
In this talk, I will explain the construction of Lagrangian classes for perverse sheaves in cohomological Donaldson-Thomas theory, whose existence was conjectured by Joyce. The two key ingredients are a relative version of the DT perverse sheaves and a hyperbolic version of the dimensional reduction theorem. As a special case, we recover Borisov-Joyce/Oh-Thomas virtual classes in DT4 theory.

As applications, I will explain how to construct the following structures from the Lagrangian classes: (1) cohomological Hall algebras for 3-Calabi-Yau categories, (2) relative Donaldson-Thomas invariants for Fano 4-folds with anti-canonical divisors, (3) refined surface counting invariants for Calabi-Yau 4-folds, (4) cohomological field theories for gauged linear sigma models.

This is joint work in progress with Adeel Khan, Tasuki Kinjo, and Pavel Safronov.

Past talks

Yehao Zhou, SIMIS & Fudan University
04/16, SCMS room 102, Fudan University

Stable envelope for critical loci
In this talk, we will introduce a generalization of Maulik-Okounkov’s stable envelopes to equivariant critical cohomology. In the case of a tripled quiver variety with standard cubic potential, this reduces to MO’s stable envelope for the Nakajima variety of the corresponding doubled quiver along the dimensional reduction. We define non-abelian stable envelopes for quivers with potentials following a similar construction of Aganagic-Okounkov, and use them to relate critical COHAs to the abelian stable envelopes. Explicit computations are given in three examples: 1) Verma modules and higher spin representations of the Yangian of sl(2); 2) oscillator representations of the shifted Yangian of sl(2); 3) fundamental representation of the Yangian of sl(2|1). This talk is based on joint work in progress with Yalong Cao, Andrei Okounkov, and Zijun Zhou.

Will Donovan, Tsinghua University
On Monday as an exception! 2pm 03/31, SIMIS room 1310

Exceptional surfaces in 3-folds and derived symmetries Video
Crepant resolutions of 3-fold singularities may contain elaborate configurations of exceptional surfaces. Using toric cases as a guide, I review some known contributions of these configurations to the derived autoequivalence group of the resolution, particularly from the work of Seidel-Thomas, and discuss work in progress with Luyu Zheng.

Yingchun Zhang, Zhejiang University
03/26, SCMS room 102, Fudan University

Quantum cohomology/quantum K rings and cluster algebras
I will introduce a relation between the quantum cohomology ring/quantum K ring of a quiver variety and the cluster algebra. More explicitly, given a quiver with potential, there is an injective ring homomorphism from the cluster algebra to quantum cohomology/quantum K ring of the corresponding quiver variety. This relation has been proved for A and D type quivers.