Ian Agol

Last updated
Ian Agol
Ian Agol, Aarhus 2012.jpg
Ian Agol at Aarhus University, August 2012
Born (1970-05-13) May 13, 1970 (age 54)
Alma mater California Institute of Technology
University of California, San Diego
Known for Virtually Haken conjecture
Freedman–He–Wang conjecture
Wise's conjecture
Marden tameness conjecture
Awards Breakthrough Prize in Mathematics (2016) [1]
Veblen Prize in Geometry (2013)
Senior Berwick Prize (2012)
Clay Research Award (2009)
Scientific career
Fields Mathematics
Institutions University of California, Berkeley
Doctoral advisor Michael Freedman

Ian Agol[ needs IPA ] (born May 13, 1970) is an American mathematician who deals primarily with the topology of three-dimensional manifolds. [2]

Contents

Education and career

Agol graduated with B.S. in mathematics from the California Institute of Technology in 1992 and obtained his Ph.D. in 1998 from the University of California, San Diego. At UCSD, his advisor was Michael Freedman and his thesis was Topology of Hyperbolic 3-Manifolds. [3] He is a professor at the University of California, Berkeley [4] and a former professor at the University of Illinois at Chicago. [5]

Contributions

In 2004, Agol proved the Marden tameness conjecture, a conjecture of Albert Marden. [6] It states that a hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold. The conjecture was also independently proven by Danny Calegari and David Gabai, and implies the Ahlfors measure conjecture. [6]

In 2012, he announced a proof of the virtually Haken conjecture, which was published a year later. [7] The conjecture (now theorem) states that every aspherical 3-manifold is finitely covered by a Haken manifold.

In 2022, he posted on the ArXiv a proof of Cameron Gordon's 1981 conjecture on knot theory saying that ribbon concordance forms a partial ordering on the set of knots. [8] [9]

Awards and honors

Agol, Calegari, and Gabai received the 2009 Clay Research Award for their proof of the Marden tameness conjecture. [6]

In 2005, Agol was a Guggenheim Fellow. [10] In 2012 he became a fellow of the American Mathematical Society. [11]

In 2013, Agol was awarded the Oswald Veblen Prize in Geometry, along with Daniel Wise. [12]

In 2015, he was awarded the 2016 Breakthrough Prize in Mathematics, "for spectacular contributions to low dimensional topology and geometric group theory, including work on the solutions of the tameness, virtually Haken and virtual fibering conjectures." [13]

In 2016, he was elected to the National Academy of Sciences. [14]

Personal

His identical twin brother, Eric Agol, [15] [16] [17] is an astronomy professor at the University of Washington in Seattle. [18]

Related Research Articles

<span class="mw-page-title-main">William Thurston</span> American mathematician (1946–2012)

William Paul Thurston was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.

<span class="mw-page-title-main">Figure-eight knot (mathematics)</span> Unique knot with a crossing number of four

In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure-eight knot is a prime knot.

In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries.

In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface.

<span class="mw-page-title-main">3-manifold</span> Mathematical space

In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

In mathematics, more precisely in topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to −1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries.

In the mathematical subfield of 3-manifolds, the virtually fibered conjecture, formulated by American mathematician William Thurston, states that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.

In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions.

<span class="mw-page-title-main">Surface subgroup conjecture</span>

In mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere. This problem is listed as Problem 3.75 in Robion Kirby's problem list.

In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold.

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.

David Gabai is an American mathematician and the Hughes-Rogers Professor of Mathematics at Princeton University. His research focuses on low-dimensional topology and hyperbolic geometry.

The Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society for notable research in geometry or topology. It was funded in 1961 in memory of Oswald Veblen and first issued in 1964. The Veblen Prize is now worth US$5000, and is awarded every three years.

<span class="mw-page-title-main">Hyperbolic volume</span> Normalized hyperbolic volume of the complement of a hyperbolic knot

In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological invariant of the link. As a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture.

In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space by an arithmetic Kleinian group.

<span class="mw-page-title-main">Tomasz Mrowka</span> American mathematician

Tomasz Mrowka is an American mathematician specializing in differential geometry and gauge theory. He is the Singer Professor of Mathematics and former head of the Department of Mathematics at the Massachusetts Institute of Technology.

<span class="mw-page-title-main">Danny Calegari</span> American mathematician

Danny Matthew Cornelius Calegari is a mathematician and, as of 2023, a professor of mathematics at the University of Chicago. His research interests include geometry, dynamical systems, low-dimensional topology, and geometric group theory.

In mathematics, the Ahlfors conjecture, now a theorem, states that the limit set of a finitely-generated Kleinian group is either the whole Riemann sphere, or has measure 0.

<span class="mw-page-title-main">Daniel Wise (mathematician)</span> American mathematician (born 1971)

Daniel T. Wise is an American mathematician who specializes in geometric group theory and 3-manifolds. He is a professor of mathematics at McGill University.

Albert Marden is an American mathematician, specializing in complex analysis and hyperbolic geometry.

References

  1. Lamb, Evelyn (8 November 2015), "By Solving the Mysteries of Shape-Shifting Spaces, Mathematician Wins $3-Million Prize", Scientific American
  2. Mackenzie, Dana; Cipra, Barry (December 20, 2006). What's happening in the mathematical sciences. American Mathematical Society. pp. 15–16. ISBN   978-0-8218-3585-2.
  3. Ian Agol at the Mathematics Genealogy Project.
  4. "Ian Agol". University of California, Berkeley Department of Mathematics. Retrieved June 25, 2011.
  5. "Ian Agol". University of Illinois at Chicago. Archived from the original on June 16, 2011. Retrieved June 25, 2011.
  6. 1 2 3 "Clay Research Award". Clay Mathematics Institute. Archived from the original on June 26, 2011. Retrieved June 25, 2011.
  7. Agol, Ian (2013). "The virtual Haken conjecture. With an appendix by Agol, Daniel Groves, and Jason Manning" (PDF). Documenta Mathematica . 18: 1045–1087. doi:10.4171/dm/421. MR   3104553. S2CID   255586740. Archived from the original (PDF) on 2023-03-26. Retrieved 2019-08-21.
  8. Sloman, Leila (2022-05-18). "How Complex Is a Knot? New Proof Reveals Ranking System That Works". Quanta Magazine . Retrieved 2022-05-20.
  9. Agol, Ian (2022-01-10). "Ribbon concordance of knots is a partial order". arXiv: 2201.03626 [math].
  10. "Ian Agol – Guggenheim Fellows Finder". John Simon Guggenheim Memorial Foundation. Archived from the original on September 21, 2012. Retrieved June 25, 2011.
  11. List of Fellows of the American Mathematical Society, retrieved 2012-11-03.
  12. Joint Mathematics Meetings Prize Booklet: January 2013 Prizes and Awards: Oswald Veblen Prize in Geometry, pp. 14–18
  13. "Breakthrough Prizes Give Top Scientists the Rock Star Treatment". New York Times . Nov 8, 2015.
  14. National Academy of Sciences Members and Foreign Associates Elected, News from the National Academy of Sciences, National Academy of Sciences, May 3, 2016, archived from the original on May 6, 2016, retrieved 2016-05-14.
  15. "Interview with Ian Agol" (PDF). Notices of the American Mathematical Society . 63 (1): 24. January 2016.
  16. "Obituaries – Alan Agol". Visalia Times-Delta . October 4, 2005. p. C2. Archived from the original on November 7, 2012.
  17. "Alan Agol". Marin Independent Journal . October 5, 2005.
  18. "Eric Agol". University of Washington Department of Astronomy. Retrieved June 25, 2011.