John W. Lott | |
|---|---|
 John Lott in Oberwolfach 2010. | |
| Born | January 12, 1959 |
| Alma mater | University of California, Berkeley |
| Scientific career | |
| Fields | Mathematics |
| Institutions | University of California, Berkeley University of Michigan |
| Doctoral advisor | Isadore Singer |
John William Lott (born January 12, 1959) [1] is a professor of Mathematics at the University of California, Berkeley. He is known for contributions to differential geometry.
Lott received his B.S. from the Massachusetts Institute of Technology in 1978 and M.A. degrees in mathematics and physics from University of California, Berkeley. In 1983, he received a Ph.D. in mathematics under the supervision of Isadore Singer. After postdoctoral positions at Harvard University and the Institut des Hautes Études Scientifiques, he joined the faculty at the University of Michigan. In 2009, he moved to University of California, Berkeley.
Among his awards and honors:
A 1985 article of Dominique Bakry and Michel Émery introduced a generalized Ricci curvature, in which one adds to the usual Ricci curvature the hessian of a function. [2] In 2003, Lott showed that much of the standard comparison geometry results for the Ricci tensor extend to the Bakry-Émery setting. For instance, if M is a closed and connected Riemannian manifold with positive Bakry-Émery Ricci tensor, then the fundamental group of M must be finite; if instead the Bakry-Émery Ricci tensor is negative, then the isometry group of the Riemannian manifold must be finite. The comparison geometry of the Bakry-Émery Ricci tensor was taken further in an influential article of Guofang Wei and William Wylie. [3] Additionally, Lott showed that if a Riemannian manifold with smooth density arises as a collapsed limit of Riemannian manifolds with a uniform upper bound on diameter and sectional curvature and a uniform lower bound on Ricci curvature, then the lower bound on Ricci curvature is preserved in the limit as a lower bound on Bakry-Émery's Ricci curvature. In this sense, the Bakry-Émery Ricci tensor is shown to be natural in the context of Riemannian convergence theory.
In 2002 and 2003, Grigori Perelman posted two papers to the arXiv which claimed to provide a proof for William Thurston's geometrization conjecture, using Richard Hamilton's theory of Ricci flow. [4] [5] Perelman's papers attracted immediate attention for their bold claims and the fact that some of their results were quickly verified. However, due to Perelman's abbreviated style of presentation of highly technical material, many mathematicians were unable to understand much of his work, especially in his second paper. Beginning in 2003, Lott and Bruce Kleiner posted a series of annotations of Perelman's work to their websites, which was finalized in a 2008 publication. [6] Their article was most recently updated for corrections in 2013. In 2015, Kleiner and Lott were awarded the Award for Scientific Reviewing from the National Academy of Sciences of the United States for their work. Other well-known expositions of Perelman's work are due to Huai-Dong Cao and Xi-Ping Zhu, and to John Morgan and Gang Tian. [7] [8]
In 2005, Max-K. von Renesse and Karl-Theodor Sturm showed that the a lower bound of the Ricci curvature on a Riemannian manifold could be characterized by optimal transportation, in particular by the convexity of a certain "entropy" functional along geodesics of the associated Wasserstein metric space. [9] In 2009, Lott and Cédric Villani capitalized upon this equivalence to define a notion of "lower bound for Ricci curvature" for a general class of metric spaces equipped with Borel measures. Similar work was done at the same time by Sturm, with the accumulated results typically referred to as "Lott-Sturm-Villani theory". [10] [11] The papers of Lott-Villani and Sturm have initiated a very large amount of research in the mathematical literature, much of which is centered around extending classical work on Riemannian geometry to the setting of metric measure spaces. [12] [13] [14] An essentially analogous program for sectional curvature bounds (from either below or above) was initiated in the 1990s by an article of Yuri Burago, Mikhail Gromov, and Grigori Perelman, following foundations laid in the 1950s by Aleksandr Aleksandrov. [15]
In the mathematical field of geometric topology, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
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 three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.
Grigori Yakovlevich Perelman is a Russian mathematician who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. In 2005, Perelman abruptly quit his research job at the Steklov Institute of Mathematics, and in 2006 stated that he had quit professional mathematics, due to feeling disappointed over the ethical standards in the field. He lives in seclusion in Saint Petersburg, and has not accepted offers for interviews since 2006.
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.
In the mathematical field of Riemannian geometry, the scalar curvature is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor.
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow, sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation.
Shing-Tung Yau is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and Professor Emeritus at Harvard University. Until 2022 he was the William Caspar Graustein Professor of Mathematics at Harvard, at which point he moved to Tsinghua.
Richard Streit Hamilton is an American mathematician who serves as the Davies Professor of Mathematics at Columbia University. He is known for contributions to geometric analysis and partial differential equations. Hamilton is best known for foundational contributions to the theory of the Ricci flow and the development of a corresponding program of techniques and ideas for resolving the Poincaré conjecture and geometrization conjecture in the field of geometric topology. Grigori Perelman built upon Hamilton's results to prove the conjectures, and was awarded a Millennium Prize for his work. However, Perelman declined the award, regarding Hamilton's contribution as being equal to his own.
Mikhael Leonidovich Gromov is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of Institut des Hautes Études Scientifiques in France and a professor of mathematics at New York University.
Tian Gang 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 geometry, Gromov-Witten theory, and geometric analysis.
In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds, which are Kähler and Ricci-flat.
John Willard Morgan is an American mathematician known for his contributions to topology and geometry. He is a Professor Emeritus at Columbia University and a member of the Simons Center for Geometry and Physics at Stony Brook University.
Bruce Alan Kleiner is an American mathematician, working in differential geometry and topology and geometric group theory.
Huai-Dong Cao is a Chinese–American mathematician. He is the A. Everett Pitcher Professor of Mathematics at Lehigh University. He is known for his research contributions to the Ricci flow, a topic in the field of geometric analysis.
The Geometry Festival is an annual mathematics conference held in the United States.
Frank Morgan is an American mathematician and the Webster Atwell '21 Professor of Mathematics, Emeritus, at Williams College. He is known for contributions to geometric measure theory, minimal surfaces, and differential geometry, including the resolution of the double bubble conjecture. He was vice-president of the American Mathematical Society and the Mathematical Association of America.
Guofang Wei is a mathematician in the field of differential geometry. She is a professor at the University of California, Santa Barbara.
Karl-Theodor "Theo" Sturm is a German mathematician working in stochastic analysis.
Gerhard Huisken is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huisken's monotonicity formula, which is named after him. With Tom Ilmanen, he proved a version of the Riemannian Penrose inequality, which is a special case of the more general Penrose conjecture in general relativity.
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