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Revision as of 08:01, 10 August 2024

Aaron C. Naber (born November 16, 1982) is an American mathematician and mathematical physicist. [1]

Education and career

His father, Gregory L. Naber, became professor emeritus of mathematics at California State University, Chico.[2][3] Aaron Naber graduated in 2005 with a B.S. in mathematics from Pennsylvania State University. He received in 2009 his Ph.D. in mathematics from Princeton University.[4] His Ph.D. thesis (Ricci solitons and collapsed spaces) was supervised by Gang Tian.[5] At Massachusetts Institute of Technology (MIT), Naber was from 2009 to 2012 a Moore Instructor and from 2012 to 2013 an assistant professor. At Northwestern University he was from 2013 to 2015 an associate professor and was appointed in 2015 Kenneth F. Burgess Professor for Mathematics.[4] In 2024 he was appointed a permanent faculty member in the School of Mathematics of the Institute for Advanced Study.[6]

Research

Naber does research on nonlinear harmonic maps, minimal varifolds, general elliptic partial differential equations, geometric analysis, the calculus of variations, and differential geometry with applications in mathematical physics to Yang-Mills theories and Einstein manifolds.[7] His research on the development of Riemannian manifolds under Ricci flow and mean curvature flow and related regularity questions is particularly noteworthy. A major problem in the proof of the Poincaré conjecture by Grigori Perelman consists of the singularities of the Ricci flow. In his doctoral dissertation, Naber extended the investigation from the three dimensions investigated by Perelman to manifolds having four or more dimensions (with bounded non-negative curvature) and investigated shrinking soliton solutions.[8] With Gang Tian, he investigated the geometric structure of collapsing n-dimensional Riemannian manifolds with uniformly bounded sectional curvature and in particular that in four and fewer dimensions a smooth orbifold structure results outside a finite number of points. In 2015, he and Robert Haslhofer succeeded in finding new estimates and a definition of weak solutions for the Ricci flow, even for the non-continuous case, by studying the embedding of the Ricci flow in an infinite-dimensional stochastic analytical structure.[4]

Awards and honors

In 2014 Naber was awarded a two-year Sloan Research Fellowship and was an invited speaker with talk The structure and meaning of Ricci curvature at the International Congress of Mathematicians in Seoul.[4] In 2018 he received the New Horizon in Mathematics Prize[9] and was elected a Fellow of the American Mathematical Society.[10] In 2023 the Institut de Mathématiques de Toulouse awarded him the Fermat Prize.[11] In 2024 Naber was elected a Member of the National Academy of Sciences.[12]

Selected publications

  • with Gang Tian: Geometric structure of collapsing Riemannian manifolds, Part 1, Arxiv 2008, Part 2, Arxiv 2009 (N*-bundles and Almost Ricci Flat Spaces)
  • with Jeff Cheeger: Lower Bounds on Ricci Curvature and Quantitative Behavior of Singular Sets, Inventiones Math., vol. 191, 2013, pp. 321–339. Arxiv 2011
  • Characterizations of Bounded Ricci Curvature on Smooth and NonSmooth Spaces, Arxiv 2013.
  • with Jeff Cheeger: Regularity of Einstein Manifolds and the Codimension 4 Conjecture, Annals of Mathematics, vol. 182, 2014, pp. 1093–1165, Arxiv
  • with Tobias Colding: Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications, Annals of Mathematics, vol. 176, 2012, pp. 1173–1229. Arxiv 2011
  • with Daniele Valtorta: Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps, Annals of Mathematics, vol. 185, 2017, pp. 131–227.
  • with Robert Haslhofer: Ricci Curvature and Bochner Formulas for Martingales, Arxiv 2016
  • with Wenshuai Jiang: L2 Curvature Bounds on Manifolds with Bounded Ricci Curvature, Arxiv 2016
  • with Daniele Valtorta: Energy identity for stationary Yang-Mills, Arxiv 2016
  • with Jeff Cheeger and Wenshuai Jiang: Rectifiability of singular sets in noncollapsed spaces with Ricci curvature bounded below, Arxiv 2018
  • Sectional Sampler. Analysis of nonlinear geometric equations, March 2019, Notices of the AMS, p. 408
  • with Elia Bruè and Daniele Semola, Fundamental Groups and the Milnor Conjecture, Arxiv 2023
  • with Elia Bruè and Daniele Semola, Six dimensional counterexample to the Milnor Conjecture, Arxiv 2023
  • with Nicholas Edelen and Daniele Valtorta: Rectifiable Reifenberg and uniform positivity under almost calibrations, Arxiv 2024

References

  1. ^ "Aaron Naber - Scholars | Institute for Advanced Study". 17 April 2024.
  2. ^ "Aaron Naber: 2018 New Horizons Prize laureate". YouTube. Breakthrough.
  3. ^ Naber, Gregory L. (2013). Topology, Geometry, and Gauge Fields: Interactions. Springer.
  4. ^ a b c d "C.V. for Aaron Naber" (PDF). Department of Mathematics, Northwestern University.
  5. ^ Aaron Naber at the Mathematics Genealogy Project
  6. ^ "Three World-Leading Mathematicians Join IAS Faculty - Press Release | Institute for Advanced Study". July 2024.
  7. ^ "Homepage of Aaaron Naber". Mathematics Department, Northwestern University.
  8. ^ He published partial results before his 2009 doctoral dissertation, Noncompact Shrinking 4-Solitons with Nonnegative Curvature, Arxiv 2007
  9. ^ "Aaron Naber | 2018 New Horizons in Mathematics Prize". breakthroughprize.org.
  10. ^ "List of Fellows (sorted by last name)". American Mathematical Society.
  11. ^ Fermat Prix 2023
  12. ^ "Nine mathematicians elected to National Academy of Sciences". News from the AMS, American Mathematical Society (ams.oeg). April 30, 2024.
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