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.
日期
2026年04月01日 至 06月12日
网站
修课要求
Comfort with basic algebraic geometry, cohomology and category theory would be helpful but not essential. No physics knowledge assumed.
课程大纲
[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.
参考资料
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
听众
Advanced Undergraduate
, Graduate
, 博士后
, Researcher
视频公开
不公开
笔记公开
公开
语言
英文
讲师介绍
阿列克谢·拉滕采夫，是一位专注于代数几何、几何表示论和枚举不变量的青年数学家。他于剑桥大学三一学院获得数学学士（一等荣誉）和硕士（优秀）学位，师从伊恩·格罗诺夫斯基。随后在牛津大学基督堂学院获得数学博士学位。他拥有丰富的教学与学术组织经验，曾在牛津大学、南丹麦大学等机构讲授多门课程，并是多个研讨会和阅读小组（如牛津物理与几何研讨会、分解代数在线研讨会等）的创始人或组织者。