In the first forty or so years of the twentieth century there were two principal strands in algebraic geometry; one was the geometric approach of the Italian school, and the other was the transcendental approach as represented by Lefschetz and Hodge and continued by Kodaira and Spencer. In the last half-century a third strand, the algebraic approach of Weil, Zariski, and Grothendieck was added, and all three strands have now become intertwined. There is no better illustration of this than the Weil conjectures. These purely arithmetical statements were formulated by Weil, who also understood that they could be proved if a ''suitable'' cohomology theory could be developed for varieties defined over a field of finite characteristic.
Such a cohomology theory was introduced by Grothendieck, and as is well known, the Weil program was completed by Deligne, who in effect used an inductive procedure reminiscent of the Lefschetz pencil method to prove an arithmetic analogue of the Hard Lefschetz Theorem.
When we look at algebraic geometry today, we not only see the intertwining of the historical strands within the field, but equally, we see algebraic geometry intertwined with the rest of mathematics and central to the ongoing developments of the field. One illustration of this is the use of algebraic geometry to generate solutions of special differential equations, both ordinary and partial. Here I mention the work originating from the Russian school that. To some extent it may be said that we have come full circle to the historical roots of algebraic geometry in the study of special transcendental functions arising from abelian integrals, abelian sums, and periods as explained above.
In all of these developments, the topological properties of algebraic varieties, as part of the infrastructure of algebraic geometry, play a central role. I would like to mention a very beautiful recent development that exemplifies both a style and subject that are direct descendents of Lefschetz. This influence is manifest today in his theorems, some of which were stated above, in the intertwining of topology and algebraic geometry, and in his overall approach to mathematics.
Much of Lefschetz' work in topology is concerned with the notion of "fixed point. The first important result of fixed-point theory was proved by L. Brouwer in It asserts that, if E is a closed n -cell, then every mapping of E into itself has at least one fixed point. This result becomes false if E is replaced by a space with a more complicated topological structure.
On the other hand, any mapping of S into itself has a well-defined degree d f [intuitively, d f is the number of times that f maps S around itself; a more precise definition is given in the next paragraph]. If t q is the trace of this matrix, the Lefschetz number of f is the alternating sum L f of the integers t q. The homomorphisms f q depend only on the homotopy class of the map f and therefore the same is true for the Lefschetz number.
In his first proof of the fixed-point theorem in , 1 , Lefschetz made the additional assumption that X is an orientable closed n -manifold. One can approximate the map f by a map g that has only a finite number of fixed points and that is well-behaved near each fixed point x in the sense that g maps some neighborhood of x homomorphically upon another neighborhood.
The Cartesian square of X is an orientable 2n -manifold Y, and the diagonal D and the graph G of g can be regarded as n -cycles in Y. Their intersection consists of all points x,x such that x is a fixed point of g ; it is a zero-cycle y of Y whose Kronecker index I y is easily seen to equal the sum of the indices of the fixed points of g. While this proof is attractive, it suffers from the disadvantage that it fails to include the Brouwer theorem as a special case. It was to remedy this situation that Lefschetz invented relative homology.
Like many other results of the time, the Lefschetz Duality Theorem was awkward to state because the correct concepts had not yet been developed. Expressed in modern language, it asserts that, if X is a compact, oriented n -manifold with regular boundary A, then the relative ho-. For the purpose of proving the fixed-point theorem, it suffices to know that the homology groups H q X,A and H n-q X are dual vector spaces. This was sufficient to modify the proof above to cover the case of manifolds with regular boundary, and this was done by Lefschetz in , 2.
The importance of the Lefschetz duality theorem was not limited to this application. On the other hand, if A is a subcomplex of a triangulation of the n -sphere S , and if U is a regular neighborhood of A, then S - U is a manifold with regular boundary b U , to which we may apply the Lefschetz duality theorem to conclude that Hq [ S - U,b U ] and H n-q S-U are isomorphic.
Using standard by now! Thus, the Lefschetz duality theorem appears as a unifying factor, connecting two important but apparently unrelated results. Not content with this version of the fixed-point theorem, Lefschetz continued to seek generalizations. In Hopf had proved the theorem for arbitrary compacy polyhedra, but with some restriction on the map f. By Lefschetz had succeeded in removing the latter restrictions ,2 , and by he was able to remove the hypothesis of triangulability of X , requiring instead that X be a compactum which is homologically locally connected in a suitable sense ,4.
The extensions of the fixed point theorem to more general spaces are not simply generalizations for their own sake. Indeed, fixed point theorems often appear in analysis as tools for proving existence theorems. To mention a very simple example, consider the ordinary first-order boundary problem. The correspondence that associates to each function y the function defined by the right-hand side of the latter equation may be regarded as a mapping f of a suitable function space into itself.
And a solution of the equation is nothing but a fixed point of f.
To be sure, the function spaces appearing here and in other places in analysis are far from being compact, and so the Lefschetz theorem does not apply directly. Nevertheless, this point of view has been a very powerful one in modern analysis. The importance of Lefschetz' work, however, is not limited to the study of fixed-point theorems. The notion of a fixed point of a map of a space into itself can be thought of as a special case of that of a coincidence. This suggested to Lefschetz the idea of defining intersections in an orientable manifold M.
This he succeeded in doing with the result that the graded. All this took place before the discovery of cohomology. While Lefschetz did not define cohomology groups, he introduced pseudo-cycles in ,4. They were not defined intrinsically but, rather, were defined in terms of an embedding of the space in question in a sphere, and were used only as a tool for the proof of one of the versions of his fixed-point theorem. It was not until the late s that the modern treatment of cohomology and cup products was given.
Other of Lefschetz' ideas that by now have thoroughly permeated the subject include singular homology theory and relative homology. While Lefschetz was not the first to use singular chains, his Colloquium Lectures ,1 gave the first formal treatment of the theory. His theory had some mild defects the chain groups turned out not to be free , but these were corrected by Eilenberg in , and the resulting theory has been of the greatest importance. As for relative homology groups, they are principal ingredients in the axiomatic treatment of the homology theory by Eilenberg and Steenrod, which has been so influential in the development of the subject in the last thirty or so years.
Lefschetz was nearly sixty years old when he turned to differential equations, and he devoted the last twenty-five years of his life to the subject. He wrote over forty papers, articles, and books in this field and formed around him a vigorous and distinguished school, guiding and encouraging students and young mathematicians to work on problems of significance.
In fact, he rekindled interest in a subject that had been nearly totally neglected in the United. States, and he recognized its mathematical importance and practical implications. Although Lefschetz' own contributions to differential equations, control theory, and dynamical systems are not comparable to his great work in algebraic geometry and topology, he nevertheless wrote noteworthy original papers in these areas.
His main interests centered around the theory of dissipative as distinct from conservative dynamical systems, including structural stability, and the resolution of singularities of critical points and bifurcating periodic orbits. Dissipative dynamical systems are important in engineering problems where friction and resistance are essential ingredients. Such dynamical systems can be represented as vector fields on the phase-space manifold. Let S be the set of all C 1 vector fields on a compact differentiable manifold M without boundary and assign to S the C 1 -topology.
Two systems V 1 and V 2 of S are said to be qualitatively equivalent if there is a homomorphism of M mapping the unparameterized solution curves of V 1 onto those of V 2. Structurally stable differential systems are important in applied problems where the parameters of the physical process are known only approximately. Lefschetz stimulated and guided work on these qualitative problems of global analysis. Peixoto proved that the structurally stable systems on a compact surface form an open dense subset of S. Markus proved that, on arbitrary n -dimensional manifolds M, a structurally stable system must necessarily have isolated and elementary critical points and periodic orbits.
Lefschetz was the first person from outside the former Soviet Union to recognize the importance of Liapunov's. He opened up the field of the mathematical theory of control, and in , one of his students, Donald Bushaw, gave the first complete solution of a nontrivial problem in optimal control. Among his other original contributions was his work on the behavior of solutions of analytic differential equations near an isolated singular point. He gave a complete characterization and a constructive procedure for obtaining all the solution curves of a two-dimensional system near an isolated critical point that pass through this critical point ,1.
For a two-dimensional analytic system for which the coefficient matrix of the linear variational equation of an isolated critical point has both roots zero but is not identically zero, he proved that there can be at most a single nested oval of orbits ,1. He gave one of the best treatments of the method of determining the stability of an isolated equilibrium point of an n -dimensional system for which the linear variational equation has some zero roots ,1.
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