Lesley Sibner
Encyclopedia
Leslie Sibner is a mathematician
and professor
of mathematics
at Polytechnic Institute of New York University. She earned her Bachelors at City College
CUNY in Mathematics. She completed her doctorate at Courant Institute NYU in 1964 under the joint supervision of Lipman Bers
and Cathleen Morawetz. Her thesis concerned partial differential equations of mixed-type.
for a year. She was a Fulbright Scholar at the Institut Henri Poincaré in Paris the following year. At this time, in addition to solo work on the Tricomi equation and compressible flow
s, she began working with her husband Robert Sibner on a problem suggested by Lipman Bers
: do there exists compressible flows on a Riemann Surface
? As part of her work in this direction, she studied Differential Geometry and Hodge Theory
eventually proving a nonlinear Hodge DeRham Theorem with Robert Sibner based on a physical interpretation of one dimensional harmonic forms on closed manifolds. The techniques are related to her prior work on compressible flows. They kept working together on related problems and applications of this important work for many years.
In 1967 she joined the faculty at Polytechnic University in Brooklyn, New York. In 1969 she proved the Morse Index Theorem for Degenerate Elliptic Operators by extending classical Sturm-Liouville theory
.
In 1971-1972 she spent a year at the Institute for Advanced Study
where she met Michael Atiyah
and Raoul Bott
. She realized she could use her knowledge of analysis to solve geometric problems related to the Atiyah–Bott fixed-point theorem
. In 1974,
Lesley and Robert Sibner produced a constructive proof of the Riemann-Roch Theorem in 1974.
Karen Uhlenbeck
suggested that Leslie Sibner work on Yang-Mills Equations. In 1979-1980 she visited Harvard University where she learned Gauge Field Theory from Clifford Taubes
. This lead results about point singularities in the Yang-Mills equation and the Yang-Mills-Higgs equations. Her interest in singularities soon brought her deeper into geometry, leading to a classification of singular connections and to a condition for removing two-dimensional singularities in work with Robert Sibner.
Realizing that instantons could under certain circumstances be viewed as monopoles, the Sibners and Uhlenbeck constructed non-minimal unstable critical points of the Yang-Mills functional over the four-sphere in 1989. She was invited to present this work at the Geometry Festival
. She was a Bunting Scholar at Radcliff Institute in 1991. For the subsequent decades, Leslie Sibner focussed on gauge theory
and gravitational instantons. Although the research sounds very physical, in fact throughout her career, Leslie Sibner applied physical intuition to prove important geometric and topological theorems.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and professor
Professor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
at Polytechnic Institute of New York University. She earned her Bachelors at City College
City College of New York
The City College of the City University of New York is a senior college of the City University of New York , in New York City. It is also the oldest of the City University's twenty-three institutions of higher learning...
CUNY in Mathematics. She completed her doctorate at Courant Institute NYU in 1964 under the joint supervision of Lipman Bers
Lipman Bers
Lipman Bers was an American mathematician born in Riga who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups.-Biography:...
and Cathleen Morawetz. Her thesis concerned partial differential equations of mixed-type.
Research career
In 1964, Lesley Sibner became an instructor at Stanford UniversityStanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...
for a year. She was a Fulbright Scholar at the Institut Henri Poincaré in Paris the following year. At this time, in addition to solo work on the Tricomi equation and compressible flow
Compressible flow
Compressible flow is the area of fluid mechanics that deals with fluids in which the fluid density varies significantly in response to a change in pressure. Compressibility effects are typically considered significant if the Mach number of the flow exceeds 0.3, or if the fluid undergoes very large...
s, she began working with her husband Robert Sibner on a problem suggested by Lipman Bers
Lipman Bers
Lipman Bers was an American mathematician born in Riga who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups.-Biography:...
: do there exists compressible flows on a Riemann Surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
? As part of her work in this direction, she studied Differential Geometry and Hodge Theory
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...
eventually proving a nonlinear Hodge DeRham Theorem with Robert Sibner based on a physical interpretation of one dimensional harmonic forms on closed manifolds. The techniques are related to her prior work on compressible flows. They kept working together on related problems and applications of this important work for many years.
In 1967 she joined the faculty at Polytechnic University in Brooklyn, New York. In 1969 she proved the Morse Index Theorem for Degenerate Elliptic Operators by extending classical Sturm-Liouville theory
Sturm-Liouville theory
In mathematics and its applications, a classical Sturm–Liouville equation, named after Jacques Charles François Sturm and Joseph Liouville , is a real second-order linear differential equation of the form...
.
In 1971-1972 she spent a year at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
where she met Michael Atiyah
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...
and Raoul Bott
Raoul Bott
Raoul Bott, FRS was a Hungarian mathematician known for numerous basic contributions to geometry in its broad sense...
. She realized she could use her knowledge of analysis to solve geometric problems related to the Atiyah–Bott fixed-point theorem
Atiyah–Bott fixed-point theorem
In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M , which uses an elliptic complex on M...
. In 1974,
Lesley and Robert Sibner produced a constructive proof of the Riemann-Roch Theorem in 1974.
Karen Uhlenbeck
Karen Uhlenbeck
Karen Keskulla Uhlenbeck is a professor and Sid W. Richardson Regents Chairholder in the Department of Mathematics at The University of Texas in Austin. In 1998 she was selected to be a Noether Lecturer. In 2000, she became a recipient of the National Medal of Science...
suggested that Leslie Sibner work on Yang-Mills Equations. In 1979-1980 she visited Harvard University where she learned Gauge Field Theory from Clifford Taubes
Clifford Taubes
Clifford Henry Taubes is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology.-Early career:Taubes received his Ph.D...
. This lead results about point singularities in the Yang-Mills equation and the Yang-Mills-Higgs equations. Her interest in singularities soon brought her deeper into geometry, leading to a classification of singular connections and to a condition for removing two-dimensional singularities in work with Robert Sibner.
Realizing that instantons could under certain circumstances be viewed as monopoles, the Sibners and Uhlenbeck constructed non-minimal unstable critical points of the Yang-Mills functional over the four-sphere in 1989. She was invited to present this work at the Geometry Festival
Geometry Festival
-1985 at Penn:* Marcel Berger* Pat Eberlein* Jost Eschenburg* Friedrich Hirzebruch* Blaine Lawson* Leon Simon* Scott Wolpert* Deane Yang-1986 at Maryland:* Uwe Abresch, Explicit constant mean curvature tori...
. She was a Bunting Scholar at Radcliff Institute in 1991. For the subsequent decades, Leslie Sibner focussed on gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
and gravitational instantons. Although the research sounds very physical, in fact throughout her career, Leslie Sibner applied physical intuition to prove important geometric and topological theorems.
External links
- Noether Brochure
- Notable women in mathematics: a biographical dictionary Edited by Charlene Morrow, Teri Perl, Greenwood Press, Westport CT 1998. http://books.google.com/books?id=u7nqH3RzUusC&pg=PA229&lpg=PA229&dq=lesley+sibnersource=bl&ots=GmsWwk_WJg&sig=Q7WNdZMK-4AAFcMG3yqvSWva_j0&hl=enei=PRJGS8KrC8KylAfHlrEP&sa=X&oi=book_result&ct=result&resnum=6&ved=0CBMQ6AEwBTgK#v=onepageq=lesley%20sibner&f=false