History of Algebraic Geometry

Cohomology Theories and Algebraic Geometry: from Poincare to Grothendieck


One trend of current mathematics is the unity between seemingly different subjects, and (co-)homology theories account for one important thread which runs through large part of today's mathematics. The interaction between cohomology theories and algebraic geometry has been a particularly fruitful and active one. One purpose of this topic is to understand how cohomological methods have changed algebraic geometry from Poincare to Grothendieck, and indeed all through the twentieth century, often via interaction with other mathematical disciplines, esp. topology. 

More specifically, we are inviting experts from both algebraic geometry and related subjects and historians of mathematics to gain a systematic understanding of the following topics: 

(1). Homology in the works of Poincaré and Picard

(2). From Lefschetz to the variation of Hodge structures

(3). The apparent absence of cohomolgy in Italian Algebraic Geometry and its rewriting by van der Waerden, Zariski, Weil

(4). Sheaf cohomology and applications: works of Serre, Cartan, Kodaira and others

(5). Hirzebruch-Riemann-Roch and generalizations, Chern characters

(6). Grothendieck, schemes and homological algebra

(7). Chow rings

(8). So-called Weil cohonologies

(9). Generalized cohomology theories, such as K-theory, motives, motivic cohomology

It might be helpful to quote from an article in the Bulletin of the AMS by Zariski in 1956:

"The cohomological methods, in conjunction with the powerful tool of harmonic integrals, were remarkably effective in the solution of global complex-analytic problems in general, and of problems of classical algebraic geometry in particular (Chern, Hirzebruch, Kodaira-Spencer, Serre, and others). It is natural to ask whether the cohomological methods can be equally effective in abstract algebraic geometry where the method of harmonic integrals is no longer available.”


One specific goal of this conference is to understand the development of cohomology methods after this article of Zariski, in particular Grothendieck's Tohoku paper

"Sur quelques points d'algèbre homologique." Tohoku Math. J. 9 (1957) 119--221,

and his vision set out  in his ICM 1958 talk:

"The cohomology theory of abstract algebraic varieties". 1960 Proc. Internat. Congress Math. in Edinburgh, 1958, pp. 103--118.

Bringing together experts from algebraic geometry and from the history of mathematics we hope to achieve a global understanding of the kind of transformation that the cohomological point of view has brought about. The goal will be to tell the story of a collection of invariants of geometrical objects which have emancipated themselves to become an autonomous realm of objects. 

We also hope to publish a book covering the above topics based on talks of this conference.

We believe that bringing together experts in algebraic geometry with some common interest in the historical development on cohomological methods in algebraic geometry will be both fruitful and enjoyable.