Aaron Naber (born November 16, 1982) is an American mathematician.

Education and career

Aaron Naber graduated in 2005 with a B.S. in mathematics from

Pennsylvania State University The Pennsylvania State University (Penn State or PSU) is a Public university, public Commonwealth System of Higher Education, state-related Land-grant university, land-grant research university with campuses and facilities throughout Pennsyl ...

. He received his Ph.D. in mathematics in 2009 from

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

. His Ph.D. thesis ''(Ricci solitons and collapsed spaces)'' was supervised by

Gang Tian Tian Gang (; born November 24, 1958) is a Chinese mathematician. He is a professor of mathematics at Peking University and Higgins Professor Emeritus at Princeton University. He is known for contributions to the mathematical fields of Kähler g ...

. From 2009 to 2012, Naber was a Moore Instructor at

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...

(MIT) and was then an assistant professor from 2012 to 2013. From 2013 to 2015, Naber was at

Northwestern University Northwestern University (NU) is a Private university, private research university in Evanston, Illinois, United States. Established in 1851 to serve the historic Northwest Territory, it is the oldest University charter, chartered university in ...

as an associate professor and in 2015 was appointed Kenneth F. Burgess Professor for Mathematics. In 2024, he was appointed a permanent faculty member in the School of Mathematics of the

Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

Research

Naber does research on nonlinear

harmonic map In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a ...

s, minimal

varifold In mathematics, a varifold is, loosely speaking, a measure-theoretic generalization of the concept of a differentiable manifold, by replacing differentiability requirements with those provided by rectifiable sets, while maintaining the general alg ...

s, general

elliptic partial differential equation In mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDEs are frequently used to model steady states, unlike parabolic PDE and hyperbolic PDE which gene ...

geometric analysis Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of ...

, the

calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

, and

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

with applications in mathematical physics to Yang-Mills theories and

Einstein manifold In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is ...

s. 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. With Gang Tian, he investigated the geometric structure of collapsing n-dimensional

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

s with uniformly bounded

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a po ...

and in particular that in four and fewer dimensions a smooth

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space. D ...

structure results outside a finite number of points. As a postdoctoral student Naber and Tobias Colding solved the constant dimension conjecture for lower Ricci curvature, which shows limits of manifolds with lower Ricci curvature have a well defined dimension. As a postdoc and later assistant professor at MIT, Naber and Jeff Cheeger introduced the notion of quantitative stratification to Lower Ricci curvature. The estimates and techniques caught on in a wide variety of nonlinear equations, including nonlinear harmonic maps, minimal surfaces, mean curvature flow, and Yang Mills. During his time at Northwestern, Naber and Cheeger proved the codimension four conjecture, showing in particular that Einstein manifolds have controlled singular sets. This work was extended with Wenshuai Jiang in order to prove sharp rectifiability of the singular sets. During this time Naber gave a characterization of Einstein manifolds, or more generally spaces with bounded Ricci curvature, through the analysis of path space of the manifold. This work was generalized with Robert Haslhofer to give a full generation of the Bakry-Emery-Ledoux estimates for martingales on path space. Near the end of his time at Northwestern, Elia Brue, Naber and Daniele Semola gave a counterexample to the Milnor conjecture, showing the existence of spaces with nonnegative Ricci curvature and infinitely generated fundamental group. Naber and Daniele Valtorta have also done a series of works on nonlinear harmonic maps. Together they developed a stratification theory for nonlinear harmonic maps, which broadly extended the results of Schoen/Uhlenbeck from Hausdorff dimension estimates to finite measure and rectifiable structure for singular sets. The techniques were general and generalized by many others, applying to many situations in which the dimension reduction ideas of Federer had worked, including minimal surfaces, Yang-Mills, Q-valued harmonic maps. Valtorta and Naber have also resolved the Energy Identity conjecture, first for Yang-Mills and later for nonlinear harmonic maps using very different sets of ideas.

Awards and honors

In 2014 Naber was awarded a two-year

Sloan Research Fellowship The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This program is one of the oldest of its kind in the United States. ...

and was an invited speaker with talk ''The structure and meaning of Ricci curvature'' at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

Seoul Seoul, officially Seoul Special Metropolitan City, is the capital city, capital and largest city of South Korea. The broader Seoul Metropolitan Area, encompassing Seoul, Gyeonggi Province and Incheon, emerged as the world's List of cities b ...

. In 2018 he received the New Horizon in Mathematics Prize and was elected a Fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

. In 2023 Naber was awarded a Simons Investigator award. In 2023 the ''

Institut de Mathématiques de Toulouse ''Institut de Mathématiques de Toulouse'' (Toulouse Mathematics Institute; IMT) is a research laboratory of the mathematics community of the Toulouse area in France. It is partially supported by the French public research agency CNRS as unit UMR ...

'' awarded him the

Fermat Prize The Fermat prize of mathematics, mathematical research biennially rewards research works in fields where the contributions of Pierre de Fermat have been decisive: * Statements of variational principles * Foundations of probability and analytic geo ...

. In 2024 Naber was elected a Member of the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...

Publications

* with Gang Tian: Geometric structure of collapsing Riemannian manifolds, Part 1
Arxiv 2008Part 2, Arxiv 2009
(N*-bundles and Almost Ricci Flat Spaces) * with

Jeff Cheeger Jeff Cheeger (born December 1, 1943) is an American mathematician and Silver Professor at the Courant Institute of Mathematical Sciences of New York University. His main interest is differential geometry and its connections with topology and an ...

: 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 Tobias Holck Colding (born 1963) is a Danish mathematician working on geometric analysis, and low-dimensional topology. He is the great grandchild of Ludwig August Colding. Biography He was born in Copenhagen, Denmark, to Torben Holck 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, Comm. in Pure and Applied Math, Vol 71 Iss 6
Arxiv 2016
* with Wenshuai Jiang: L² Curvature Bounds on Manifolds with Bounded Ricci Curvature, Annals of Mathematics, vol. 193-1
Arxiv 2016
* with Daniele Valtorta: Energy identity for stationary Yang-Mills, Inventiones, vol. 216
Arxiv 2016
* with Jeff Cheeger and Wenshuai Jiang: Rectifiability of singular sets in noncollapsed spaces with Ricci curvature bounded below, Annals of Mathematics, vol. 193-2
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, Annals of Mathematics, to appear
Arxiv 2023
(See Milnor conjecture (Ricci curvature).) * with Elia Bruè and Daniele Semola, Six dimensional counterexample to the Milnor Conjecture
Arxiv 2023
* with Daniele Valtorta: Energy Identity for Stationary Harmonic Maps
Arxiv 2023
* with Nicholas Edelen and Daniele Valtorta: Rectifiable Reifenberg and uniform positivity under almost calibrations
Arxiv 2024

References

External links

* * {{DEFAULTSORT:Naber, Aaron 1982 births Living people 21st-century American mathematicians Differential geometers Partial differential equation theorists Pennsylvania State University alumni Princeton University alumni Northwestern University faculty Institute for Advanced Study faculty Sloan Research Fellows Members of the United States National Academy of Sciences