Tomasz Mrowka

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Tomasz Mrowka, Aarhus 2011.

Tomasz Mrowka (born September 9, 1961 in State College , Pennsylvania ) is an American mathematician who deals with differential geometry and three- and four-dimensional topology .

Mrowka studied at the Massachusetts Institute of Technology (MIT) (Bachelor's degree 1983) and received his PhD in 1989 from the University of California, Berkeley with Clifford Taubes and Robion Kirby (A Local Mayer-Vietoris Principle for Yang-Mills Moduli Spaces). He then worked as a post-doc at the Mathematical Sciences Research Institute (MSRI) in Berkeley in 1988/89 , from 1989 to 1991 at Stanford University and from 1991 at Caltech , where he was professor from 1992 to 1996. In 1995 he was visiting professor at Harvard University and MIT. From 1996 he was professor at MIT, since 2007 as Simons Professor of Mathematics .

Mrowka worked, often with Peter Kronheimer , on the topology of 4-manifolds following the fundamental work of Simon Donaldson . Together, Mrowka and Kronheimer proved a structure theorem for the Donaldson invariants . In 1994 he and Kronheimer proved the Thom conjecture that algebraic curves among the connected curves with the same homology class that are smoothly embedded in the complex projective plane are distinguished by the fact that they have minimal gender (gender, a topological invariant, is in turn with the algebraic curves determined by their degree). In doing so, they used the Seiberg-Witten theory that had just been developed at the time . 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 P. M. N. Feehan and T. G. Leness on the Witten conjecture about Donaldson and Seiberg-Witten invariants, adhesion theorems for Donaldsonine 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 and Kronheimer proved that the Khovanov homology distinguishes trivial knots (that is, recognizes unknot ).

Larry Guth and Ruan Yongbin are among his PhD students .

He was a Sloan Research Fellow from 1993 to 1995 and a Clay Mathematics Visiting Professor in 1995. In 2007 he and Kronheimer received the Oswald Veblen Prize and in 2011 both received the Joseph L. Doob Prize for their book Monopoles and Three-Manifolds . He is a member of the American Academy of Arts and Sciences (2007) and the National Academy of Sciences (2015). In 1994 he was invited speaker at the International Congress of Mathematicians (Embedded surfaces and the structure of Donaldson's polynomial invariants). In 2018 he is plenary speaker at the ICM in Rio with Kronheimer ( Knots, three-manifolds and instantons ).

Fonts

  • with Gompf: Irreducible 4-manifolds need not be complex. Ann. of Math. (2) 138 (1993) no. 1, 61-111.
  • with Kronheimer: Gauge theory for embedded surfaces. I. Topology 32 (1993) no. 4, 773-826. II. Topology, 34 (1995), no. 1, 37-97.
  • with Kronheimer: Embedded surfaces and the structure of Donaldson's polynomial invariants. Journal of Differential Geometry, Vol. 41, 1995, 573-734.
  • with Kronheimer: The genus of embedded surfaces in the projective plane. Mathematical Research Letters, Vol. 1, 1994, 797-808.
  • with Kronheimer: Monopoles and contact structures. Invent. Math. 130 (1997) no. 2, 209-255.
  • with Ozsváth, Yu Seiberg-Witten monopoles on Seifert fibered spaces. Comm. Anal. Geom. 5 (1997), no. 4, 685-791.
  • with Kronheimer: Witten's conjecture and property P. Geometry and Topology, Vol. 8, 2004, 295-310. ArXiv
  • with Kronheimer, Ozsváth, Szabó: Monopoles and lens space surgeries. Ann. of Math. (2) 165 (2007), no. 2, 457-546.
  • with Kronheimer: Khovanov homology is an unknot-detector. Publ. Math. Inst. Hautes Études Sci. No. 113 (2011), 97-208.
  • with Kronheimer "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 Mrowka, Tomasz
BRIEF DESCRIPTION American mathematician
DATE OF BIRTH September 9, 1961
PLACE OF BIRTH State College , Pennsylvania