Enumerative geometry seminar

Coorganized with Yang Zhou

Wednesdays at 2pm

Yinbang Lin, Tongji University
Special time: Moved to 3:30 pm!, 04/23, SIMIS room 1210

Riemann-Roch and Brill-Noether problems over surfaces
Motivated by the Verlinde/Segre correspondence over surfaces, we try to bound the dimension of the global sections of semistable sheaves in terms of the rank and the first Chern class. This improves the explicit Le Potier-Simpson bound when the first Chern class is small compared to the rank. In some cases, we also obtain the asymptotic bound as the second Chern class goes to infinity, using Bridgeland stability conditions. Understanding these bounds is the foundation of the Brill-Noether problem. Over K3 surfaces of Picard number one, we show examples of Brill-Noether loci which are nonempty and irreducible of expected dimensions. This is work in progress jointly with Thomas Goller and Zhixian Zhu.

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.