Topics in geometric representation theory and enumerative geometry
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.
The first half, on
QUANTUM GROUPS AND STABLE ENVELOPES,
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.
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.
After defining the invariants, we introduce cohomological Hall algebras (algebraic gadgets capturing the information in DT invariants), explain the "topological vertex" method to compute DT invariants and its relation to quantum groups, will explain how DT invariants change as the stability condition varies and its relation to cluster algebras, and finally explain the relation to Gromov--Witten invariants due to Maulik--Okounkov--Pandharipande--Thomas.
The first half, on
QUANTUM GROUPS AND STABLE ENVELOPES,
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.
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.
After defining the invariants, we introduce cohomological Hall algebras (algebraic gadgets capturing the information in DT invariants), explain the "topological vertex" method to compute DT invariants and its relation to quantum groups, will explain how DT invariants change as the stability condition varies and its relation to cluster algebras, and finally explain the relation to Gromov--Witten invariants due to Maulik--Okounkov--Pandharipande--Thomas.
Lecturer
Date
1st April ~ 12th June, 2026
Website
Prerequisite
Comfort with basic algebraic geometry, cohomology and category theory would be helpful but not essential. No physics knowledge assumed.
Syllabus
[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.
[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.
Reference
n.b. the PANEL time chooser is broken for me. I would like to choose:
3 April 13:00-16:00
8 April - 10 June 13:00 - 16:00 (every Wednesday)
8 May 13:00 - 16:00
3 April 13:00-16:00
8 April - 10 June 13:00 - 16:00 (every Wednesday)
8 May 13:00 - 16:00
Audience
Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
No
Notes Public
Yes
Language
English
Lecturer Intro
Alexei Latyntsev is a mathematician specializing in algebraic geometry, geometric representation theory, and enumerative invariants. He completed a first-class BA and a distinction MA in Mathematics at Trinity College, Cambridge, followed by a PhD at the University of Oxford. He has extensive teaching and mentoring experience, having served as a lecturer, teaching assistant, and tutor at the University of Oxford and SDU, covering a wide range of advanced mathematics courses. He is also an active organizer of research seminars, including the Oxford Physics and Geometry Seminar and the online Factorisation Algebras seminar.