Mikhail Kapranov in nLab

Mikhail Kapranov is a professor of mathematics at Kavli IPMU in Tokyo.

• old Yale website; old Toronto website

• A list of publications html

• lecture at msri 2000 Noncommutative neighborhoods and noncommutative Fourier transform: link

Selected writings

Introducing Serre functors:

• Alexei I. Bondal, Mikhail M. Kapranov, Representable functors, Serre functors, and mutations, Mathematics of the USSR-Izvestiya 35 3 (1990) 519-541 [doi:10.1070/IM1990v035n03ABEH000716]

On algebraic K-theory and Morse theory:

• Mikhail Kapranov, Masahico Saito: Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions, in: Higher homotopy structure in topology and mathematical physics (Poughkeepsie, N.Y. 1996), Contemporary Mathematics 227, AMS (1999) 191–225 [ams:CONM/227, pdf]

On Rozansky-Witten invariants:

• Mikhail Kapranov, Rozansky–Witten invariants via Atiyah classes, Compositio Math. 115 (1999), no. 1, 71–113 (MR2000h:57056, doi, alg-geom/9704009)

On 2-representations:

• Nora Ganter, Mikhail Kapranov: Representation and character theory in 2-categories, Adv. Math. 217 5 (2008) 2268-2300 [math.AT/0602510, doi:10.1016/j.aim.2007.10.004]

• Nora Ganter, Mikhail Kapranov: Symmetric and exterior powers of categories, Journal of Transformation Groups 19 1 (2014) 57-103 [arxiv/1110.4753, doi:10.1007/s00031-014-9255-z]

Discussion of superalgebra as induced from free groupal symmetric monoidal categories (abelian 2-groups) and hence ultimately from the sphere spectrum (cf. super 2-algebra and spectral superscheme):

• Mikhail Kapranov, Categorification of supersymmetry and stable homotopy groups of spheres, talk at Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky (1953-2013) (April 2013) [pdf, video:YT]

  Abstract:. The “minimal sign skeleton” necessary to formulate the Koszul sign rule is a certain Picard category, a symmetric monoidal category with all objects and morphisms invertible. It can be seen as the free Picard category generated by one object and corresponds, by Grothendieck‘s dictionary, to the truncation of the spherical spectrum SS in degrees 0 and 1, so that {±1}\{\pm 1\} appears as the first stable homotopy group of spheres π n+1(S n)\pi_{n+1}(S^n). This suggest a “higher” or categorified versions of super-mathematics which utilize deeper structure of S S . The first concept on this path is that of a supersymmetric monoidal category which is categorified version of the concept of a supercommutative algebra.

• Mikhail Kapranov, Supergeometry in mathematics and physics, in Gabriel Catren, Mathieu Anel, (eds.) New Spaces for Mathematics and Physics (arXiv:1512.07042)

• Mikhail Kapranov, Super-geometry, talk at New Spaces for Mathematics & Physics, IHP Paris, Oct-Sept 2015 (video recording)

Towards a noncommutative integral transform:

• Mikhail Kapranov, Noncommutative geometry and path integrals, In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics 270 (2009) doi arXiv:math.QA/0612411
• Mikhail Kapranov, Free Lie algebroids and the space of paths, Sel. math., New ser. 13, 277 (2007) doi arXiv:math:AG/0702584
• Mikhail Kapranov, Membranes and higher groupoids, arXiv:1502.06166

On higher Segal spaces

• Tobias Dyckerhoff, Mikhail Kapranov, Higher Segal spaces I, arxiv:1212.3563; now part of the book T. Dyckerhoff, M. Kapranov, Higher Segal spaces, Springer LNM 2244 (2019) doi

• Mikhail Kapranov, Higher Segal spaces, talk at IHES (2012) (video)

• Kapranov's noncommutative geometry

• Kapranov-Voevodsky 2-vector spaces

• superalgebra

• higher parallel transport

• Higher Kac-Moody algebras