Morgan is the founder of SMALL, one of the largest and best known summer undergraduate Mathematics research programs. In 2012 he became a fellow of the American Mathematical Society.[4]
Frank Morgan is also an avid dancer. He gained fame for his work "Dancing the Parkway".[5]
Mathematical work
Double bubble
He is known for proving, in collaboration with Michael Hutchings, Manuel Ritoré, and Antonio Ros, the Double Bubble conjecture, which states that the minimum-surface-area enclosure of two given volumes is formed by three spherical patches meeting at 120-degree angles at a common circle.
He has also made contributions to the study of manifolds with density, which are Riemannian manifolds together with a measure of volume which is deformed from the standard Riemannian volume form. Such deformed volume measures suggest modifications of the Ricci curvature of the Riemannian manifold, as introduced by Dominique Bakry and Michel Émery.[6] Morgan showed how to modify the classical Heintze-Karcher inequality, which controls the volume of certain cylindrical regions in the space by the Ricci curvature in the region and the mean curvature of the region's cross-section, to hold in the setting of manifolds with density. As a corollary, he was also able to put the Levy-Gromov isoperimetric inequality into this setting. Much of his current work deals with various aspects of isoperimetric inequalities and manifolds with density.
Publications
Textbooks
Calculus Lite. Third edition. A K Peters/CRC Press, Natick, MA, 2001. ISBN1-56881-157-8
Geometric measure theory. A beginner's guide. Fifth edition. Illustrated by James F. Bredt. Elsevier/Academic Press, Amsterdam, 2016. viii+263 pp. ISBN978-0-12-804489-6
The math chat book. MAA Spectrum. Mathematical Association of America, Washington, DC, 2000. xiv+113 pp. ISBN0-88385-530-5
Real analysis. American Mathematical Society, Providence, RI, 2005. viii+151 pp. ISBN0-8218-3670-6
Real analysis and applications. Including Fourier series and the calculus of variations. American Mathematical Society, Providence, RI, 2005. x+197 pp. ISBN0-8218-3841-5
Riemannian geometry. A beginner's guide. Second edition. A K Peters, Ltd., Wellesley, MA, 1998. x+156 pp. ISBN1-56881-073-3
Notable articles
Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros. Proof of the double bubble conjecture. Ann. of Math. (2) 155 (2002), no. 2, 459–489. doi:10.2307/3062123
↑ D. Bakry and Michel Émery. Diffusions hypercontractives. Séminaire de probabilités, XIX, 1983/84, 177–206. Lecture Notes in Math., 1123, Springer, Berlin, 1985.
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.
Grigori Yakovlevich Perelman is a Russian mathematician and geometer who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. In 2005, Perelman resigned from his research post in Steklov Institute of Mathematics and in 2006 stated that he had quit professional mathematics, owing to feeling disappointed over the ethical standards in the field. He lives in seclusion in Saint Petersburg and has declined requests for interviews since 2006.
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, Yau was the William Caspar Graustein Professor of Mathematics at Harvard, at which point he moved to Tsinghua.
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In -dimensional space the inequality lower bounds the surface area or perimeter of a set by its volume ,
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.
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.
Thierry Aubin was a French mathematician who worked at the Centre de Mathématiques de Jussieu, and was a leading expert on Riemannian geometry and non-linear partial differential equations. His fundamental contributions to the theory of the Yamabe equation led, in conjunction with results of Trudinger and Schoen, to a proof of the Yamabe Conjecture: every compact Riemannian manifold can be conformally rescaled to produce a manifold of constant scalar curvature. Along with Yau, he also showed that Kähler manifolds with negative first Chern classes always admit Kähler–Einstein metrics, a result closely related to the Calabi conjecture. The latter result, established by Yau, provides the largest class of known examples of compact Einstein manifolds. Aubin was the first mathematician to propose the Cartan–Hadamard conjecture.
Leon Melvyn Simon, born in 1945, is a Leroy P. Steele Prize and Bôcher Prize-winning mathematician, known for deep contributions to the fields of geometric analysis, geometric measure theory, and partial differential equations. He is currently Professor Emeritus in the Mathematics Department at Stanford University.
John William Lott is a professor of Mathematics at the University of California, Berkeley. He is known for contributions to differential geometry.
Guofang Wei is a mathematician in the field of differential geometry. She is a professor at the University of California, Santa Barbara.
In the mathematical theory of minimal surfaces, the double bubble theorem states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a standard double bubble: three spherical surfaces meeting at angles of 120° on a common circle. The double bubble theorem was formulated and thought to be true in the 19th century, and became a "serious focus of research" by 1989, but was not proven until 2002.
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.
Sylvestre F. L. Gallot is a French mathematician, specializing in differential geometry. He is an emeritus professor at the Institut Fourier of the Université Grenoble Alpes, in the Geometry and Topology section.
In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, known as Cartan–Hadamard manifolds. The conjecture, which is named after French mathematicians Élie Cartan and Jacques Hadamard, may be traced back to work of André Weil in 1926.
Emanuel Milman : is a professor at the Technion – Israel Institute of Technology’s Faculty of Mathematics, where he holds the prestigious Yitzhak Modai Academic Chair. His primary areas of research are Analysis and Geometry.
This page is based on this Wikipedia article Text is available under the CC BY-SA 4.0 license; additional terms may apply. Images, videos and audio are available under their respective licenses.
