A page of mathematical autobiography

As my natural taste has always been to look forward rather than backward this is a task which I did not care to undertake. Now, however, I feel most grateful to my friend Mauricio Peixoto for having coaxed me into accepting it. For it has provided me with my first opportunity to cast an objective glance at my early mathematical work, my algebro-geometric phase. As I see it at last it was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry. But I must not push the metaphor too far. The time which I mean to cover runs from 1911 to 1924, from my doctorate to my research on fixed points. At the time I was on the faculties of the Universities of Nebraska (two years) and Kansas (eleven years). As was the case for almost all our scientists of that day my mathematical isolation was complete. This circumstance was most valuable in that it enabled me to develop my ideas in complete mathematical calm. Thus I made use most uncritically of early topology à la Poincaré, and even of my own later developments. Fortunately someone at the Académie des Sciences (I always suspected Emile Picard) seems to have discerned "the harpoon for the whale" with pleasant enough consequences for me. To close personal recollections, let me tell you what made me turn with all possible vigor to topology. From the p0 formula of Picard, applied to a hyperelliptic surface $ (topologically the product of 4 circles) I had come to believe that the second Betti number R2($) = 5, whereas clearly i?2($) = 6. What was wrong? After considerable time it dawned upon me that Picard only dealt with finite 2-cycles, the only useful cycles for calculating periods of certain double integrals. Missing link? The cycle at infinity, that is the plane section of the surface at infinity. This drew my attention to cycles carried by an algebraic curve, that is to algebraic cycles, and • • • the harpoon was in! My general plan is to present the first concepts of algebraic geometry, then follow up with the early algebraic topology of Poincaré plus some of my own results on intersections of cycles. I will then discuss the topology of an algebraic surface. The next step will be a