Topics in Geometric Representation Theory and Enumerative Geometry

Spring semester 2026

When: 13:00 - 16:00 Wednesdays and 9:30 - 12:30 Tuesdays, ending 10th June 2026

Where: Tsinghua, Shuangqing, Room 715. Online possible but strongly discouraged; zoom available at https://bimsa.net/activity/Topingeoreptheandenugeo/

Instructor: Alyosha Latyntsev, BIMSA (alyoshalatyntsev@gmail.com)

Feedback form: https://forms.gle/ThqV2XQnxfNWQeEF8

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 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)

A2: Knizhnik–Zamolodchikov (KZ) equations and the Drinfeld–Kohno Theorem.

B1: Nakajima quiver varieties

B2: Stable envelopes and Yangians

B3: Examples

A3: qKZ equations and construction of solutions using quasimaps

Each point takes ~2-4 hours. Main references for this part: [Dr], [ES], [MO].

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 and discussion

D3: Scattering diagrams and tropical geometry

Exact material in Part II determined by available time and interest from the participants.

Course Materials

Lecture Notes (PDF)

References

[AKMV] Aganagic, M., Klemm, A., Mariño, M., Vafa, C. The topological vertex. Comm. Math. Phys.
[BPSS] Bridgeland, T., Pandharipande, R., Stoppa, J., Smith, I. Scattering diagrams and Hall algebras.
[Br] Bridgeland, T. Stability conditions on triangulated categories. Ann. of Math.
[BZN] Ben-Zvi, D., Nadler, D. Loop spaces and connections. J. Topology.
[Dr] Drinfeld, V.G. On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q̄/Q). Leningrad Math. J.
[ES] Etingof, P., Schiffmann, O. Lectures on Quantum Groups. International Press.
[GPS] Gross, M., Pandharipande, R., Siebert, B. The tropical vertex. Duke Math. J.
[HK] Hacking, P., Keel, S. Mirror symmetry and cluster algebras.
[Ko] Kohno, T. Conformal field theory and topology. AMS.
[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.