Kai Behrend
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| Kai Behrend | |
|---|---|
| Name | Kai Behrend |
| Birth date | 1969 |
| Nationality | German |
| Fields | Algebraic geometry, Number theory |
| Workplaces | University of British Columbia, University of California, Berkeley, Massachusetts Institute of Technology, Institut Fourier, University of Bonn |
| Alma mater | University of Mainz, Harvard University |
| Doctoral advisor | Arthur Ogus |
| Known for | Behrend function, work on stacks, enumerative geometry |
Kai Behrend is a German mathematician notable for contributions to algebraic geometry, moduli space theory, and enumerative geometry. His work on constructible functions and intrinsic multiplicities in moduli problems has influenced developments in Donaldson–Thomas theory, Gromov–Witten theory, and intersection-theoretic approaches to virtual classes. He has held appointments at leading institutions and collaborated with prominent researchers across North America and Europe.
Early life and education
Behrend was born in Germany and studied mathematics at the University of Mainz before pursuing graduate work at Harvard University under the supervision of Arthur Ogus. During his formative years he engaged with the mathematical communities at Institut Fourier, Max Planck Institute for Mathematics, and attended seminars associated with Grothendieck-inspired schools such as those at IHES and École Normale Supérieure. His doctoral research built on ideas from scheme theory, cohomology, and deformation theory, connecting to classical themes in the work of Alexander Grothendieck, Jean-Pierre Serre, and Pierre Deligne.
Academic career
Behrend held postdoctoral and faculty positions at institutions including Massachusetts Institute of Technology, University of California, Berkeley, and later University of British Columbia. He participated in collaborative research groups at Institute for Advanced Study, Mathematical Sciences Research Institute, and contributed to programs at Clay Mathematics Institute and Simons Foundation workshops. His teaching and mentoring intersected with doctoral students and visitors from Princeton University, Stanford University, University of Chicago, and ETH Zurich, fostering links with researchers working on intersection theory, derived algebraic geometry, and stack theory.
Research contributions
Behrend introduced a constructible function, now known as the Behrend function, which assigns intrinsic multiplicities to points of a moduli space and plays a central role in defining weighted Euler characteristics used in Donaldson–Thomas invariants. His analysis of symmetric obstruction theories and virtual fundamental classes connected to work by K. Fukaya, Yongbin Ruan, Davesh Maulik, Richard Thomas, Jun Li, and Sheldon Katz. He applied techniques from motivic integration and microlocal geometry alongside ideas from perverse sheaves and Verdier duality to refine enumerative invariants. Behrend's formulations tie into the foundations laid by Mumford on Geometric Invariant Theory and extend approaches influenced by Kontsevich and Soibelman concerning wall-crossing and stability conditions. His contributions intersect with research on derived categories and Bridgeland stability, informing advances at the interface of symplectic geometry and string theory-inspired enumerative problems associated with Calabi–Yau manifolds and mirror symmetry.
Awards and honors
Behrend received recognition from academic societies and research institutes, participating in award-linked programs at organizations such as the Royal Society of Canada and the Natural Sciences and Engineering Research Council of Canada. He was invited to speak at major venues including the International Congress of Mathematicians, the European Congress of Mathematics, and thematic meetings at Banff International Research Station, Newton Institute, and CIMPA. His work has been cited in prize-winning collaborations with researchers associated with the Fields Medal-level developments in algebraic geometry and mathematical physics.
Selected publications
- Behrend, K., "Donaldson–Thomas invariants via microlocal geometry", Journal article influencing work by Davesh Maulik, Richard Thomas, Yukinobu Toda, and Tom Bridgeland. - Behrend, K., Fantechi, B., "The intrinsic normal cone", foundational paper related to Mumford's intersection theory and used by Kai Behrend-related research groups at MSRI and IAS. - Behrend, K., "A general structure for virtual fundamental classes", cited in studies by Jun Li, G. Farkas, and C. Manolache. - Collaborative papers with researchers connected to Kontsevich-inspired enumerative frameworks, impacting literature involving Gromov–Witten invariants and Donaldson–Thomas theory.