Theodore Frankel (June 17, 1929 – August 5, 2017) was a mathematician who introduced the

Andreotti–Frankel theorem In mathematics, the Andreotti–Frankel theorem, introduced by , states that if V is a smooth, complex affine variety of complex dimension n or, more generally, if V is any Stein manifold of dimension n, then V admits a Morse function with cr ...

and the

Frankel conjecture In the mathematical fields of differential geometry and algebraic geometry, the Frankel conjecture was a problem posed by Theodore Frankel in 1961. It was resolved in 1979 by Shigefumi Mori, and by Yum-Tong Siu and Shing-Tung Yau. In its diff ...

. Frankel received his Ph.D. from the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

in 1955. His doctoral advisor was

Harley Flanders Harley M. Flanders (September 13, 1925 – July 26, 2013) was an American mathematician, known for several textbooks and contributions to his fields: algebra and algebraic number theory, linear algebra, electrical networks, scientific computing. ...

. A Professor Emeritus of Mathematics at

University of California, San Diego The University of California, San Diego (UC San Diego or colloquially, UCSD) is a public university, public Land-grant university, land-grant research university in San Diego, California. Established in 1960 near the pre-existing Scripps Insti ...

, Frankel was a longtime member of the

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

Princeton, New Jersey Princeton is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of whi ...

. He is known for his work in

global differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

Morse theory In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differenti ...

, and

relativity theory The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...

. He joined the UC San Diego mathematics department in 1965, after serving on the faculties at Stanford University and Brown University.

Research

In the 1930s, John Synge established what is now known as

Synge's theorem In mathematics, specifically Riemannian geometry, Synge's theorem is a classical result relating the curvature of a Riemannian manifold to its topology. It is named for John Lighton Synge, who proved it in 1936. Theorem and sketch of proof Let ...

, by applying the second variation formula for arclength to a minimal loop. Frankel adapted Synge's method to higher-dimensional objects. As a consequence, he was able to prove that, when given a positively curved

Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...

on a

closed manifold In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example ...

, any two

totally geodesic This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or provi ...

compact submanifolds must intersect if their dimensions are large enough. The idea is to apply Synge's method to a minimizing geodesic between the two submanifolds. By the same approach, Frankel proved that complex submanifolds of positively curved

Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Ar ...

s must intersect if their dimensions are sufficiently large. These results were later extended by Samuel Goldberg and

Shoshichi Kobayashi was a Japanese mathematician. He was the eldest brother of electrical engineer and computer scientist Hisashi Kobayashi. His research interests were in Riemannian and complex manifolds, transformation groups of geometric structures, and Lie al ...

to allow positivity of the holomorphic bisectional curvature. Inspired by work of

René Thom René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became ...

, Frankel and

Aldo Andreotti Aldo Andreotti (15 March 1924 – 21 February 1980) was an Italian mathematician who worked on algebraic geometry, on the theory of functions of several complex variables and on partial differential operators. Notably he proved the Andreotti–F ...

gave a new proof of the

Lefschetz hyperplane theorem In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, th ...

using

. The crux of the argument is the algebraic fact that the

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...

s of the

real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

of a complex

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...

must occur in pairs of the form . This becomes relevant in the context of Lefschetz's theorem, by considering a Morse function given by the distance to a fixed point. The second-order analysis at critical points is immediately aided by the above algebraic analysis, and the homology vanishing phenomena follows via the

Morse inequalities In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...

. Given a

Killing vector field In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal gen ...

for which the corresponding one-parameter group of isometries acts by holomorphic mappings, Frankel used the Cartan formula to show that the

interior product In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of ...

of the vector field with the Kähler form is closed. Assuming that the first

Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...

is zero, the

de Rham theorem In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapt ...

applies to construct a function whose critical points coincide with the zeros of the vector field. A second-order analysis at the critical points shows that the set of zeros of the vector field is a nondegenerate critical manifold for the function. Following

Raoul Bott Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian- American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions whi ...

's development of Morse theory for critical manifolds, Frankel was able to establish that the Betti numbers of the manifold are fully encoded by the Betti numbers of the critical manifolds, together with the index of his Morse function along these manifolds. These ideas of Frankel were later important for works of

Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded t ...

and

Nigel Hitchin Nigel James Hitchin FRS (born 2 August 1946) is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics. He is a Professor Emeritus of Mathematics at the University of ...

, among others.

Major Publications

Articles * * * Textbooks * *

References

{{DEFAULTSORT:Frankel, Theodore 20th-century American mathematicians 21st-century American mathematicians University of California, San Diego faculty 1929 births 2017 deaths University of California, Berkeley alumni Institute for Advanced Study people Brown University faculty Stanford University faculty