Peter Kronheimer

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Peter Benedict Kronheimer (* 1963 in London ) is a British mathematician who studies differential geometry and three- and four-dimensional topology.

biography

Kronheimer attended the City of London School and studied at Merton College of Oxford University , where he graduated in 1984 with a BA degree and in 1986 Michael Atiyah received his doctorate ( ALE Gravitational Instantons ). He then spent two years at Balliol College, Oxford, and the Institute for Advanced Study for two years before returning to Merton College, Oxford as a tutor and fellow. In 1995 he went to Harvard University , where he is William Caspar Graustein Professor of Mathematics.

Kronheimer worked, often with Tomasz Mrowka from the Massachusetts Institute of Technology (MIT), on the topology of 4-manifolds following the fundamental work of Simon Donaldson , with whom Kronheimer also wrote a book. Together, Kronheimer and Mrowka proved a structure theorem for the Donaldson invariants . In 1994 he proved (using the Seiberg-Witten theory ) with Mrowka the Thom conjecture that algebraic curves among the connected curves with the same homology class embedded smoothly in the complex projective plane are distinguished by their minimal gender (gender, a topological invariant, is again determined in the case of the algebraic curves by their degree). In 2003 he and Mrowka proved the " Property-P conjecture " of the knot theory with the help of different methods of the (differential) topology of 3-dimensional manifolds (results on tight foliage by David Gabai , relationship to contact structures ), a theorem about symplectic fillings of Contact manifolds from Eliashberg , the non - vanishing theorem by Clifford Taubes for symplectic 4-manifolds , results from PMN Feehan and TG Leness on the Witten conjecture about Donaldson and Seiberg-Witten invariants, adhesion theorems for Donaldsonin variants using Instanton- Floer homology , and the theorem von Floer on exact triangles in Instanton-Floer homology. The Property P conjecture says that the 3-manifold generated by stretching surgery (with parameters p, q, where q is not equal to zero) along a non-trivial node in a non-trivial fundamental group .

In 2011 he proved with Mrowka that the Khovanov homology can distinguish trivial knots (that is, recognizes unknot ).

Kronheimer was inducted into the Royal Society in 1997 . He is plenary speaker with Mrowka at the ICM 2018 ( Knots, three-manifolds and instantons ).

His PhD students include Ian Dowker , Jacob Rasmussen , and Ciprian Manolescu .

He is married and has two sons.

Prizes and awards

Fonts

  • The construction of ALE spaces as hyper-Kähler quotients. J. Differential Geom. 29 (1989) no. 3, 665-683.
  • with Mrowka: Gauge theory for embedded surfaces. I. Topology 32 (1993) no. 4, 773-826. II. Topology, 34 (1995), no. 1, 37-97.
  • mit Mrowka: Embedded surfaces and the structure of Donaldson's polynomial invariants. Journal of Differential Geometry, Vol. 41, 1995, 573-734.
  • with Mrowka: The genus of embedded surfaces in the projective plane. Mathematical Research Letters, Vol. 1, 1994, 797-808.
  • with Mrowka: Monopoles and contact structures. Invent. Math. 130 (1997) no. 2, 209-255.
  • Minimal gender in S 1 × M 3 . Invent. Math. 135 (1999), no. 1, 45-61.
  • with Mrowka: Witten's conjecture and property P. Geometry and Topology, Vol. 8, 2004, 295-310. ArXiv
  • with Mrowka, Ozsváth, Szabó: Monopoles and lens space surgeries. Ann. of Math. (2) 165 (2007), no. 2, 457-546.
  • mit Mrowka: Khovanov homology is an unknot-detector. Publ. Math. Inst. Hautes Études Sci. No. 113 (2011), 97-208.
  • with Donaldson: The geometry of 4-manifolds , Oxford University Press 1990, 1997
  • with Mrowka: Monopoles and 3-Manifolds , Cambridge University Press 2007

Web links

References

  1. ↑ also independently proven by John Morgan , Zoltán Szabó , Clifford Taubes
  2. ↑ Kronheimer, Mrowka: Khovanov homology is an unknot-detector, Publ. Math., Inst. Hautes Étud. Sci., Vol. 113, 2011, pp. 97-208
personal data
SURNAME Kronheimer, Peter
ALTERNATIVE NAMES Kronheimer, Peter Benedict (full name)
BRIEF DESCRIPTION British mathematician
DATE OF BIRTH 1963
PLACE OF BIRTH London