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 algebraic variety, smooth, complex affine variety of complex dimension n or, more generally, if V is any Stein manifold of dimension n, then V admits a ...

and the Frankel conjecture. 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 university, public Land-grant university, land-grant research university in Berkeley, California, United States. Founded in 1868 and named after t ...

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 in communications material, formerly and colloquially UCSD) is a public university, public Land-grant university, land-grant research university in San Diego, California, United States. Es ...

, Frankel was a longtime member of the

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

Princeton, New Jersey The Municipality of Princeton is a Borough (New Jersey), borough in Mercer County, New Jersey, United States. It was established on January 1, 2013, through the consolidation of the Borough of Princeton, New Jersey, Borough of Princeton and Pri ...

. He is known for his work in global differential geometry,

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 physics theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phe ...

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

Stanford University Leland Stanford Junior University, commonly referred to as Stanford University, is a Private university, private research university in Stanford, California, United States. It was founded in 1885 by railroad magnate Leland Stanford (the eighth ...

and

Brown University Brown University is a Private university, private Ivy League research university in Providence, Rhode Island, United States. It is the List of colonial colleges, seventh-oldest institution of higher education in the US, founded in 1764 as the ' ...

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 is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

on a

closed manifold In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The onl ...

, 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 provid ...

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 Arnol ...

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 alg ...

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, the ...

using

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

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

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 form ...

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 t ...

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 differentiabl ...

. 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 pseudo-Riemannian manifold that preserves the metric tensor. Killing vector fields are the infinitesimal generators of isom ...

for which the corresponding one-parameter group of isometries acts by

holomorphic mapping In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas of charts to the open unit disc in the complex coordinate space \mathbb^n, such that the transition maps are holomorp ...

s, Frankel used the

Cartan formula The Cartan formula in mathematics may refer to two different formulae in differential geometry or algebraic topology. Cartan formula in differential geometry The Cartan formula in differential geometry states: : \mathcal L_X = \mathrm d \, \iota_ ...

to show that the

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

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, more specifically in differential geometry, the de Rham theorem says that the ring homomorphism from the de Rham cohomology to the singular cohomology given by integration is an isomorphism. The Poincaré lemma implies that the de R ...

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 foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott function ...

'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 the ...

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 O ...

, 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