Topics in Geometric Representation Theory
and Enumerative Geometry

When: Starting April 3, 2027, running through early June 2027

Where: BIMSA/Tsinghua, Shuangqing, and online

Instructor: Alyosha Latyntsev (alyoshalatyntsev@gmail.com)

This is a course overviewing one of mathematics' main attempts in the past two decades at translating results from quantum field theory and string theory into the language of algebraic geometry and representation theory, and what open problems remain.

Course Overview

Part I: Quantum Groups and Stable Envelopes

The first half will explain the beautiful story of what an (affine) quantum group is, and how to make them using differential equations or geometry.

We explain their construction using the Knizhnik–Zamolodchikov (q-)differential equations due to Drinfeld and Kohno, how you can build them from the (singular/K/elliptic) cohomology of certain schemes using the stable envelope construction of Maulik and Okounkov, and solve the KZ equations using quasimaps.

Part II: Enumerative Invariants and Wall-Crossing

The second half is about enumerative invariants and wall-crossing, where we explain Gromov–Witten invariants (counting curves) and Donaldson–Thomas invariants (counting stable objects in Calabi-Yau manifolds) and their algebraic structure.

Course Structure

Part I: Quantum Groups and Stable Envelopes

A1: The quantum group U_q(g)

B1: Nakajima quiver varieties

A2: Knizhnik–Zamolodchikov equations and the Drinfeld–Kohno Theorem

B2: Stable envelopes and Yangians

A3: qKZ equations and elliptic KZ equations

B3: K-theoretic and elliptic stable envelopes. Vertex functions and the construction of solutions to qKZ by quasimaps

Part II: Enumerative Geometry and Wall-Crossing

C1: Virtual classes and torus localisation

D1: (Cohomological) Hall algebras and Donaldson–Thomas invariants

C2: The topological vertex

D2: Kontsevich–Soibelman wall-crossing and Bridgeland scattering invariants

C3: Statement of MNOP's GW = DT Theorem

D3: Scattering diagrams and tropical geometry

Course Materials

References

[ES] Etingof, P., Schiffmann, O. Lectures on Quantum Groups. International Press.
[MO] Maulik, D., Okounkov, A. Quantum groups and quantum cohomology. Astérisque.
[Na] Nakajima, H. Lectures on Hilbert schemes of points on surfaces. AMS.
[Ok] Okounkov, A. Lectures on K-theoretic computations in enumerative geometry.
[MNOP] Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R. Gromov–Witten theory and Donaldson–Thomas theory, I. Compositio Math.
[KS] Kontsevich, M., Soibelman, Y. Holomorphic Floer theory I.
[KS2] Kontsevich, M., Soibelman, Y. Stability structures, motivic Donaldson-Thomas invariants and cluster transformations.
[Br] Bridgeland, T. Stability conditions on triangulated categories. Ann. of Math.
[HK] Hacking, P., Keel, S. Mirror symmetry and cluster algebras.
[AKMV] Aganagic, M., Klemm, A., Mariño, M., Vafa, C. The topological vertex. Comm. Math. Phys.
[GPS] Gross, M., Pandharipande, R., Siebert, B. The tropical vertex. Duke Math. J.
[BPSS] Bridgeland, T., Pandharipande, R., Stoppa, J., Smith, I. Scattering diagrams and Hall algebras.
[BZN] Ben-Zvi, D., Nadler, D. Loop spaces and connections. J. Topology.
[Ko] Kohno, T. Conformal field theory and topology. AMS.
[Dr] Drinfeld, V.G. On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q̄/Q). Leningrad Math. J.

Topics in Geometric Representation Theoryand Enumerative Geometry

Course Overview

Part I: Quantum Groups and Stable Envelopes

Part II: Enumerative Invariants and Wall-Crossing

Course Structure

Part I: Quantum Groups and Stable Envelopes

Part II: Enumerative Geometry and Wall-Crossing

Course Materials

References

Topics in Geometric Representation Theory
and Enumerative Geometry