Ailana Margaret Fraser | |
---|---|
Born | Toronto, Ontario, Canada |
Occupation | Professor of Mathematics |
Academic background | |
Alma mater | Stanford University |
Academic work | |
Discipline | Mathematics,Geometric Analysis |
Institutions | University of British Columbia |
Ailana Margaret Fraser is a Canadian mathematician and professor of mathematics at the University of British Columbia. [1] She is known for her work in geometric analysis and the theory of minimal surfaces. [2] Her research is particularly focused on extremal eigenvalue problems and sharp eigenvalue estimates for surfaces,min-max minimal surface theory,free boundary minimal surfaces,and positive isotropic curvature. [3]
Fraser was born in Toronto,Ontario. [2] She received her Ph.D. from Stanford University in 1998 under the supervision of Richard Schoen. [2] [4] After postdoctoral studies at the Courant Institute of New York University,she taught at Brown University before moving to UBC. [2]
Fraser is well-known for her 2011 work with Schoen on the first "Steklov eigenvalue" of a compact Riemannian manifold-with-boundary. This is defined as the minimal nonzero eigenvalue of the "Dirichlet to Neumann" operator which sends a function on the boundary to the normal derivative of its harmonic extension into the interior. In the two-dimensional case,Fraser and Schoen were able to adapt Paul Yang and Shing-Tung Yau's use of the Hersch trick in order to approximate the product of the first Steklov eigenvalue with the length of the boundary from above,by topological data. [5] [6]
Under an ansatz of rotational symmetry,Fraser and Schoen carefully analyzed the case of an annulus,showing that the metric optimizing the above eigenvalue-length product is obtained as the intrinsic geometry of a geometrically meaningful part of the catenoid. By use of the uniformization theorem for surfaces with boundary,they were able to remove the condition of rotational symmetry,replacing it by certain weaker conditions;however,they conjectured that their result should be unconditional.
In general dimensions,Fraser and Schoen developed a "boundary" version of Peter Li and Yau's "conformal volume." [7] By building upon some of Li and Yau's arguments,they gave lower bounds for the first Steklov eigenvalue in terms of conformal volumes,in addition to isoperimetric inequalities for certain minimal surfaces of the unit ball.
Fraser won the Krieger–Nelson Prize of the Canadian Mathematical Society in 2012 [2] and became a fellow of the American Mathematical Society in 2013. [8] In 2018 the Canadian Mathematical Society listed her in their inaugural class of fellows [9] and in 2021 awarded her,along with Marco Gualtieri,the Cathleen Synge Morawetz Prize. [10] In 2022 she was awarded a Simons Fellowship. [11]
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.
Eugenio Calabi was an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, specializing in differential geometry, partial differential equations and their applications.
In the mathematical field of differential geometry, there are various splitting theorems on when a pseudo-Riemannian manifold can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian manifolds, although there has also been research into splitting of Lorentzian manifolds.
Cathleen Synge Morawetz was a Canadian mathematician who spent much of her career in the United States. Morawetz's research was mainly in the study of the partial differential equations governing fluid flow, particularly those of mixed type occurring in transonic flow. She was professor emerita at the Courant Institute of Mathematical Sciences at the New York University, where she had also served as director from 1984 to 1988. She was president of the American Mathematical Society from 1995 to 1996. She was awarded the National Medal of Science in 1998.
Richard Melvin Schoen is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984.
Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from University of California at Berkeley. Cheng then spent some years as a post-doctoral fellow and assistant professor at Princeton University and the State University of New York at Stony Brook. Then he became a full professor at University of California at Los Angeles. Cheng chaired the Mathematics departments of both the Chinese University of Hong Kong and the Hong Kong University of Science and Technology in the 1990s. In 2004, he became the Dean of Science at HKUST. In 2012, he became a fellow of the American Mathematical Society.
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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 mapping also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy. As such, the theory of harmonic maps contains both the theory of unit-speed geodesics in Riemannian geometry and the theory of harmonic functions.
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.
The Krieger–Nelson Prize is presented by the Canadian Mathematical Society in recognition of an outstanding woman in mathematics. It was first awarded in 1995. The award is named after Cecilia Krieger and Evelyn Nelson, both known for their contributions to mathematics in Canada.
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Robert Bartnik was an Australian mathematician based at Monash University. He was known for his contributions to the rigorous mathematical study of general relativity. He received his bachelor's and master's degrees from Melbourne University and a PhD in mathematics from Princeton University in 1983, where his advisor was Shing-Tung Yau. In 2004 he was elected to the Australian Academy of Science, with citation:
Professor Bartnik is renowned internationally for the application of geometric analysis to mathematical problems arising in Einstein's theory of general relativity. His work is characterised by his ability to uncover new and anticipated phenomena in space-time geometry, often employing sophisticated tools from linear and nonlinear partial differential equations as well as elaborate numerical computations. He has contributed greatly to our understanding of the properties of the Einstein equations and gravitation.
The Geometry Festival is an annual mathematics conference held in the United States.
Herbert Blaine Lawson, Jr. is a mathematician best known for his work in minimal surfaces, calibrated geometry, and algebraic cycles. He is currently a Distinguished Professor of Mathematics at Stony Brook University. He received his PhD from Stanford University in 1969 for work carried out under the supervision of Robert Osserman.
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.
In differential geometry, Yau's conjecture is a mathematical conjecture which states that any closed Riemannian 3-manifold has an infinite number of smooth closed immersed minimal surfaces. It is named after Shing-Tung Yau, who posed it as the 88th entry in his 1982 list of open problems in differential geometry.
William Hamilton Meeks III is an American mathematician, specializing in differential geometry and minimal surfaces.
David Allen Hoffman is an American mathematician whose research concerns differential geometry. He is an adjunct professor at Stanford University. In 1985, together with William Meeks, he proved that Costa's surface was embedded. He is a fellow of the American Mathematical Society since 2018, for "contributions to differential geometry, particularly minimal surface theory, and for pioneering the use of computer graphics as an aid to research." He was awarded the Chauvenet Prize in 1990 for his expository article "The Computer-Aided Discovery of New Embedded Minimal Surfaces". He obtained his Ph.D. from Stanford University in 1971 under the supervision of Robert Osserman.
Peter Wai-Kwong Li is an American mathematician whose research interests include differential geometry and partial differential equations, in particular geometric analysis. After undergraduate work at California State University, Fresno, he received his Ph.D. at University of California, Berkeley under Shiing-Shen Chern in 1979. Presently he is Professor Emeritus at University of California, Irvine, where he has been located since 1991.
Joseph Harold Sampson Jr. was an American mathematician known for his work in mathematical analysis, geometry and topology, especially his work about harmonic maps in collaboration with James Eells. He obtained his Ph.D. in mathematics from Princeton University in 1951 under the supervision of Salomon Bochner.