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Colin Adams (mathematician)
Encyclopedia
Colin Conrad Adams is a mathematician
primarily working in the areas of hyperbolic 3-manifold
s and knot theory
. His book, The Knot Book, has been praised for its accessible approach to advanced topics in knot theory
. He is currently Francis Christopher Oakley Third Century Professor of Mathematics at Williams College
, where he has been since 1985. He writes "Mathematically Bent", a column of math humor for the Mathematical Intelligencer
.
in mathematics
from the University of Wisconsin–Madison
in 1983. His dissertation was entitled "Hyperbolic Structures on Link Complements" and supervised by James Cannon.
is the unique cusped hyperbolic 3-manifold
of smallest volume. The proof utilizes horoball
-packing arguments. Adams is known for his clever use of such arguments utilizing horoball patterns and his work would be used in the later proof by Cao and Meyerhoff that the smallest cusped orientable hyperbolic 3-manifolds are precisely the figure-eight knot
complement
and its sibling manifold.
Adams has investigated and defined a variety of geometric invariants of hyperbolic link
s and hyperbolic 3-manifolds in general. He developed techniques for working with volumes of special classes of hyperbolic links. He proved augmented alternating links, which he defined, were hyperbolic. In addition, he has defined almost alternating and toroidally alternating links. He has often collaborated and published this research with students from SMALL, an undergraduate summer research program at Williams.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
primarily working in the areas of hyperbolic 3-manifold
Hyperbolic 3-manifold
A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
s and knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
. His book, The Knot Book, has been praised for its accessible approach to advanced topics in knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
. He is currently Francis Christopher Oakley Third Century Professor of Mathematics at Williams College
Williams College
Williams College is a private liberal arts college located in Williamstown, Massachusetts, United States. It was established in 1793 with funds from the estate of Ephraim Williams. Originally a men's college, Williams became co-educational in 1970. Fraternities were also phased out during this...
, where he has been since 1985. He writes "Mathematically Bent", a column of math humor for the Mathematical Intelligencer
Mathematical Intelligencer
The Mathematical Intelligencer is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common amongst such journals.-Mathematical Conversations:...
.
Academic career
Adams received a B.Sc. from MIT in 1978 and a Ph.D.Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
from the University of Wisconsin–Madison
University of Wisconsin–Madison
The University of Wisconsin–Madison is a public research university located in Madison, Wisconsin, United States. Founded in 1848, UW–Madison is the flagship campus of the University of Wisconsin System. It became a land-grant institution in 1866...
in 1983. His dissertation was entitled "Hyperbolic Structures on Link Complements" and supervised by James Cannon.
Work
Among his earliest contributions is his theorem that the Gieseking manifoldGieseking manifold
In mathematics, the Gieseking manifold, is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161...
is the unique cusped hyperbolic 3-manifold
Hyperbolic 3-manifold
A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
of smallest volume. The proof utilizes horoball
Horoball
In hyperbolic geometry, a horoball is an object in hyperbolic n-space: the limit of a sequence of increasing balls sharing a tangent hyperplane and its point of tangency. Its boundary is called a horosphere. For n = 2 a horosphere is called a horocycle.This terminology is due to William...
-packing arguments. Adams is known for his clever use of such arguments utilizing horoball patterns and his work would be used in the later proof by Cao and Meyerhoff that the smallest cusped orientable hyperbolic 3-manifolds are precisely the figure-eight knot
Figure-eight knot (mathematics)
In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot...
complement
Knot complement
In mathematics, the knot complement of a tame knot K is the complement of the interior of the embedding of a solid torus into the 3-sphere. To make this precise, suppose that K is a knot in a three-manifold M. Let N be a thickened neighborhood of K; so N is a solid torus...
and its sibling manifold.
Adams has investigated and defined a variety of geometric invariants of hyperbolic link
Hyperbolic link
In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry...
s and hyperbolic 3-manifolds in general. He developed techniques for working with volumes of special classes of hyperbolic links. He proved augmented alternating links, which he defined, were hyperbolic. In addition, he has defined almost alternating and toroidally alternating links. He has often collaborated and published this research with students from SMALL, an undergraduate summer research program at Williams.
Books
- C. Adams, The Knot Book: An elementary introduction to the mathematical theory of knots. Revised reprint of the 1994 original. American Mathematical Society, Providence, RI, 2004. xiv+307 pp. ISBN 0-8218-3678-1
- C. Adams, J. Hass, A. Thompson, How to Ace Calculus: the Streetwise Guide. W. H. Freeman and Company, 1998. ISBN 0-7167-3160-6
- C. Adams, J. Hass, A. Thompson, How to Ace the Rest of Calculus: the Streetwise Guide. W. H. Freeman and Company, 2001. ISBN 0-7167-4174-9
- C. Adams, Why Knot?: An Introduction to the Mathematical Theory of Knots. Key College, 2004. ISBN 1-931914-22-2
- C.Adams, R. Franzosa, "Introduction to Topology: Pure and Applied." Prentice Hall, 2007. ISBN 0-1318-4869-0
- C. Adams, "Riot at the Calc Exam and Other Mathematically Bent Stories." American Mathematical Society, 2009. ISBN 0-8218-4817-8
Selected publications
- C. Adams, Thrice-punctured spheres in hyperbolic $3$-manifolds. Trans. Amer. Math. Soc. 287 (1985), no. 2, 645—656.
- C. Adams, Augmented alternating link complements are hyperbolic. Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), 115—130, London Math. Soc. Lecture Note Ser., 112, Cambridge Univ. Press, Cambridge, 1986.
- C. Adams, The noncompact hyperbolic $3$-manifold of minimal volume. Proc. Amer. Math. Soc. 100 (1987), no. 4, 601—606.
- C. Adams and A. Reid, Systoles of hyperbolic $3$-manifolds. Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 1, 103—110.
- C. Adams; A. Colestock; J. Fowler; W. Gillam; E. Katerman. Cusp size bounds from singular surfaces in hyperbolic 3-manifolds. Trans. Amer. Math. Soc. 358 (2006), no. 2, 727—741