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In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers (algebraic), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory (for example, "a knot invariant is a rule that assigns to any knot K a quantity φ(K) such that if K and K' are equivalent then φ(K) = φ(K')."). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification, both in "enumeration" and "duplication removal".
A knot invariant is a quantity defined on the set of all knots, which takes the same value for any two equivalent knots. For example, a knot group is a knot invariant.
Typically a knot invariant is a combinatorial quantity defined on knot diagrams. Thus if two knot diagrams differ with respect to some knot invariant, they must represent different knots. However, as is generally the case with topological invariants, if two knot diagrams share the same values with respect to a [single] knot invariant, then we still cannot conclude that the knots are the same.
Richard Streit Hamilton was an American mathematician who served as the Davies Professor of Mathematics at Columbia University.
Mikhael Leonidovich Gromov is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of Institut des Hautes Études Scientifiques in France and a professor of mathematics at New York University.
Friedrich Ernst Peter Hirzebruch ForMemRS was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as "the most important mathematician in Germany of the postwar period."
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In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover that is a Haken manifold.
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The Maryam Mirzakhani Prize in Mathematics is awarded by the U.S. National Academy of Sciences "for excellence of research in the mathematical sciences published within the past ten years." Named after the Iranian mathematician Maryam Mirzakhani, the prize has been awarded every four years since 1988.
Otto Haupt was a German mathematician.
Gerhard Huisken is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huisken's monotonicity formula, which is named after him. With Tom Ilmanen, he proved a version of the Riemannian Penrose inequality, which is a special case of the more general Penrose conjecture in general relativity.
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References
- ↑ "The previous award winners". Nov 2024. Retrieved December 8, 2024.
- ↑ O'Connor, J.J.; Robertson, E.F. (Nov 2006). "Hans Grauert". MacTutor. Retrieved December 8, 2024.
- ↑ "Von Staudt-Preis für Stefan Hildebrandt". DMV notices. 2 (4): 4. April 1994. Retrieved Dec 9, 2024.
- ↑ "Don Zagier" . Retrieved December 8, 2024.
- ↑ "Verleihung des Karl Georg Christian von Staudt-Preises an Gunter Harder und Friedhelm Waldhausen" . Retrieved 2021-08-31.
- ↑ "2013 Staudt Prize awarded to Prof. Rapoport". Friedrich-Alexander-Universität Erlangen-Nürnberg. 2013-02-04. Retrieved 2020-12-08.
- ↑ "Mathematician Burkhard Wilking receives Staudt Prize 2022" . Retrieved December 8, 2024.
- ↑ "Wolfgang Lück receives the von Staudt Prize". 2024-11-06. Retrieved 2024-12-08.
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