Jump to content

Pedro Ontaneda: Difference between revisions

Add links
From Wikipedia, the free encyclopedia
Browse history interactively
← Previous editNext edit →
Content deleted Content added
VisualWikitext
WL RW (talk | contribs)
17 edits
m Added a reference to the page Hyperbolic groups
WL RW (talk | contribs)
17 edits
m removed the stub label
(3 intermediate revisions by 2 users not shown)
Line 14: Line 14:
| doctoral_advisor = [[Lowell Jones]]
| doctoral_advisor = [[Lowell Jones]]
| doctoral_students =
| doctoral_students =
| known_for = [[Riemannian Hyperbolization]]
| known_for = [[Riemannian hyperbolization]]
| author_abbrev_bot =
| author_abbrev_bot =
| author_abbrev_zoo =
| author_abbrev_zoo =
Line 37: Line 37:
One of Ontaneda's most celebrated contributions is the construction of manifolds that admit negatively curved Riemannian metrics but do not admit locally symmetric ones.
One of Ontaneda's most celebrated contributions is the construction of manifolds that admit negatively curved Riemannian metrics but do not admit locally symmetric ones.
More precisely, he showed that for any <math>n \geq 4</math> and for any <math>\varepsilon > 0</math> there exists a closed Riemannian <math>n</math>-manifold <math>N</math> satisfying the following two properties:<ref>{{Cite journal
More precisely, he showed that for any <math>n \geq 4</math> and for any <math>\varepsilon > 0</math> there exists a closed Riemannian <math>n</math>-manifold <math>N</math> satisfying the following two properties:<ref>{{Cite journal
| last1 = Ontaneda
| author=Pedro Ontaneda
| first1 = Pedro
| title = Riemannian hyperbolization
| title = Riemannian hyperbolization
| journal = Publ. Math. Inst. Hautes Études Sci.
| journal = Publ. Math. Inst. Hautes Études Sci.
Line 50: Line 49:
# All the [[sectional curvature]]s of <math>N</math> are in <math>[-1-\varepsilon, -1]</math>.
# All the [[sectional curvature]]s of <math>N</math> are in <math>[-1-\varepsilon, -1]</math>.
# <math>N</math> is not homeomorphic to a locally symmetric space.
# <math>N</math> is not homeomorphic to a locally symmetric space.
In particular, the fundamental group of <math>N</math> is [[Hyperbolic group|Gromov hyperbolic]] but not isomorphic to a lattice in a Lie group of rank 1.
In particular, the fundamental group of <math>N</math> is [[Hyperbolic group|Gromov hyperbolic]] but not isomorphic to a uniform lattice in a Lie group of rank 1.


These manifolds are obtained via the so-called '''Riemannian hyperbolization procedure''', developed by Ontaneda, which is a smooth version of the strict hyperbolization procedure introduced by [[Ruth Charney]] and [[Michael W. Davis]].<ref>{{cite journal|url=https://dx.doi.org/10.1016/0040-9383%2894%2900027-I|title=Strict hyperbolization|year=1995 |doi=10.1016/0040-9383(94)00027-I |last1=Charney |first1=Ruth M.|author1-link=Ruth Charney |last2=Davis |first2=Michael W. |journal=[[Topology (journal)|Topology]] |volume=34 |issue=2 |pages=329–350 }}</ref>
These manifolds are obtained via the '''Riemannian hyperbolization procedure''' developed by Ontaneda in a series of papers, which is a smooth version of the strict hyperbolization procedure introduced by [[Ruth Charney]] and [[Michael W. Davis]].<ref>{{cite journal|url=https://dx.doi.org/10.1016/0040-9383%2894%2900027-I|title=Strict hyperbolization|year=1995 |doi=10.1016/0040-9383(94)00027-I |author1=Ruth Charney |author1-link=Ruth Charney |author2=Michael W. Davis |author2-link=Michael W. Davis |journal=[[Topology (journal)|Topology]] |volume=34 |issue=2 |pages=329–350 }}</ref>
The obstruction to being locally symmetric comes from the fact that Ontaneda's manifolds have non trivial rational [[Pontryagin classes]].
The obstruction to being locally symmetric comes from the fact that Ontaneda's manifolds have nontrivial rational [[Pontryagin classes]].
Note that the restriction on the dimension is necessary. Indeed, if a surface admits a negatively curved metric, then it admits one that is locally isometric to the real hyperbolic plane, as a consequence of the [[uniformization theorem]]. A similar statement holds for <math>3</math>-manifolds thanks to the [[hyperbolization theorem]].
The restriction to dimension <math>n \geq 4</math> is necessary. Indeed, if a surface admits a negatively curved metric, then it admits one that is locally isometric to the real [[hyperbolic plane]], as a consequence of the [[uniformization theorem]]. A similar statement holds for <math>3</math>-manifolds thanks to the [[hyperbolization theorem]].


Ontaneda also contributed a "remarkable"<ref>Boris Hasselblatt, Review of "New partially hyperbolic dynamical systems I", ''MathSciNet'', {{MR|3455236}}.</ref> expansion to the classification of [[dynamical system]]s by showing that [[Anosov diffeomorphism]]s exist on many manifolds of high dimension; see his 2015 paper.
Ontaneda also contributed a "remarkable"<ref>Boris Hasselblatt, Review of "New partially hyperbolic dynamical systems I", ''MathSciNet'', {{MR|3455236}}.</ref> expansion to the classification of [[dynamical system]]s by showing that [[Anosov diffeomorphism]]s exist on many manifolds of high dimension; see his 2015 paper.
Line 81: Line 80:


{{Authority control}}
{{Authority control}}

{{US-mathematician-stub}}

Revision as of 13:04, 24 November 2024

Pedro Ontaneda
Alma materStony Brook University (Ph.D., 1994)
Known forRiemannian hyperbolization
Scientific career
FieldsTopology
Differential geometry
InstitutionsBinghamton University
Federal University of Pernambuco
Doctoral advisorLowell Jones

Pedro Ontaneda Portal is a Peruvian-American mathematician specializing in topology and differential geometry. He is a distinguished professor at Binghamton University, a unit of the State University of New York.[1]

Education and career

Ontaneda received his Ph.D. in 1994 from Stony Brook University (another unit of SUNY), advised by Lowell Jones.[2] Subsequently he taught at the Federal University of Pernambuco in Brazil. He moved to Binghamton University in 2005.

Mathematical contributions

Ontaneda's work deals with the geometry and topology of aspherical spaces, with particular attention to the relationship between exotic structures and negative or non-positive curvature on manifolds.

Classical examples of Riemannian manifolds of negative curvature are given by real hyperbolic manifolds, or more generally by locally symmetric spaces of rank 1. One of Ontaneda's most celebrated contributions is the construction of manifolds that admit negatively curved Riemannian metrics but do not admit locally symmetric ones. More precisely, he showed that for any and for any there exists a closed Riemannian -manifold satisfying the following two properties:[3]

  1. All the sectional curvatures of are in .
  2. is not homeomorphic to a locally symmetric space.

In particular, the fundamental group of is Gromov hyperbolic but not isomorphic to a uniform lattice in a Lie group of rank 1.

These manifolds are obtained via the Riemannian hyperbolization procedure developed by Ontaneda in a series of papers, which is a smooth version of the strict hyperbolization procedure introduced by Ruth Charney and Michael W. Davis.[4] The obstruction to being locally symmetric comes from the fact that Ontaneda's manifolds have nontrivial rational Pontryagin classes. The restriction to dimension is necessary. Indeed, if a surface admits a negatively curved metric, then it admits one that is locally isometric to the real hyperbolic plane, as a consequence of the uniformization theorem. A similar statement holds for -manifolds thanks to the hyperbolization theorem.

Ontaneda also contributed a "remarkable"[5] expansion to the classification of dynamical systems by showing that Anosov diffeomorphisms exist on many manifolds of high dimension; see his 2015 paper.

Selected publications

  • F. T. Farrell, L. E. Jones, and P. Ontaneda (2007), "Negative curvature and exotic topology." In Surveys in Differential Geometry, Vol. XI, pp. 329–347, International Press, Somerville, MA.
  • F. Thomas Farrell and Pedro Ontaneda (2010), "On the topology of the space of negatively curved metrics." Journal of Differential Geometry 86, no. 2, pp. 273–301.
  • Andrey Gogolev, Pedro Ontaneda, and Federico Rodriguez Hertz (2015), "New partially hyperbolic dynamical systems I." Acta Mathematica 215, no. 2, pp. 363–393.
  • Pedro Ontaneda (2020), "Riemannian hyperbolization." Publ. Math. Inst. Hautes Études Sci. 131, pp. 1–72.

References

  1. ^ "Five Binghamton faculty promoted to distinguished ranks", BingUNews, Binghamton University, May 9, 2024, retrieved 2024-05-08
  2. ^ Pedro Ontaneda at the Mathematics Genealogy Project
  3. ^ Pedro Ontaneda (2020). "Riemannian hyperbolization". Publ. Math. Inst. Hautes Études Sci. 131: 1–72. doi:10.1007/s10240-020-00113-1.
  4. ^ Ruth Charney; Michael W. Davis (1995). "Strict hyperbolization". Topology. 34 (2): 329–350. doi:10.1016/0040-9383(94)00027-I.
  5. ^ Boris Hasselblatt, Review of "New partially hyperbolic dynamical systems I", MathSciNet, MR3455236.
Retrieved from "https://en.wikipedia.org/enwiki/w/index.php?title=Pedro_Ontaneda&oldid=1259307770"