So John and I agreed and wrote the obituary below. His devotion to math, his disdain for formality and convention, his openness and what John and others call his naiveté struck a chord with me. I got to know him when he visited Harvard and John, Shurik (as he was known) and I ran a seminar on “Existence theorems”. Now he is a hero of mine, the person that I met most deserving of the adjective “genius”. John Tate and I were asked by Nature magazine to write an obituary for Alexander Grothendieck. Here is the beginning excerpt of Mumford’s blog post 2 explaining why he and John Tate (his coauthor for the post) needed to talk about the concept of a scheme in their post: But since Nature is bills itself as “An international journal, published weekly, with original, groundbreaking research spanning all of the scientific disciplines ” Mumford assumed the readers of Nature would be interested not only in where Grothendieck was born and died, but in what he actually accomplished in his life, and why he is admired for his mathematics. My colleague Edward Frenkel published a brief non-technical obituary about Grothendieck in the New York Times, and perhaps that is what Nature had in mind for its journal as well. The obituary Mumford was asked to write was for Alexander Grothendieck, a leading and towering figure in 20th century mathematics who built many of the foundations for modern algebraic geometry. Now I have to say that I have heard of obituaries being retracted, but never of an obituary being rejected. The Mumford rejection is all the more disturbing because it happened after he was invited by Nature to write the obituary in the first place! It therefore came as a surprise to me to read his post titled “ Can one explain schemes to biologists?” in which he describes the rejection by the journal Nature of an obituary he was asked to write. Among his many awards are the MacArthur Fellowship, the Shaw Prize, the Wolf Prize and the National Medal of Science. A lot of his work is connected to neuroscience and therefore biology. For those readers of the blog who do not follow mathematics, it is relevant to what I am about to write that David Mumford won the Fields Medal in 1974 for his work in algebraic geometry, and afterwards launched another successful career as an applied mathematician, building on Ulf Grenader’s Pattern Theory and making significant contributions to vision research. First there was a blog post by David Mumford, a professor emeritus of applied mathematics at Brown University, published on December 14th. I try not to become preoccupied with the two cultures problem, but this holiday season I have not been able to escape it. For constantly I feel that I am moving among two groups- comparable in intelligence, identical in race, not grossly different in social origin, earning about the same incomes, who have almost ceased to communicate at all, who in intellectual, moral and psychological climate have so little in common that instead of crossing the campus from Evans Hall to the Li Ka Shing building, I may as well have crossed an ocean. I think it is through living among these groups and much more, I think, through moving regularly from one to the other and back again that I have become occupied with the problem that I’ve christened to myself as the ‘two cultures’. I have had, of course, intimate friends among both biologists and mathematicians. In particular, with $J=I$, $IA \rightarrow I(A/K)=IA/K$ is an isomorphism so that $K=0$.I’m a (50%) professor of mathematics and (50%) professor of molecular & cell biology at UC Berkeley. There have been plenty of days when I have spent the working hours with biologists and then gone off at night with some mathematicians. The former sequence is exact, the second sequence is exact except maybe in $JA$, and the vertical arrows are surjective (and an isomorphism for the $A/K$ one).ĭiagram chasing shows that the second sequence is exact, but $JK \subset MIA=0$ so that $JA \cong J(A/K)$. So consider the morphism of complexes from $0 \rightarrow J \otimes K \rightarrow J \otimes A \rightarrow J \otimes (A/K) \rightarrow 0$ to $0 \rightarrow JK \rightarrow JA \rightarrow J(A/K) \rightarrow 0$ (where $J$ is any proper ideal of $R$). Now, $A/K$ is flat over $R$, and $0 \rightarrow K \rightarrow A \rightarrow A/K\rightarrow 0$ is exact, so it remains exact when tensored by anything. It follows that $B=IB +\phi(A) \subset MB+\phi(A)=M\phi(A)+\phi(A)=\phi(A)$ and $\phi$ is surjective, of kernel $K$. Let $t \in M$, then $\phi(ta) \in tb +IMB=tb$, so that $\phi(MA)=MB$. Then there exists $x \in A$ such that $\phi(a) \in b+IB$.
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