Celebratio Mathematica

P. F. Baum and R. Bott: “On the zeroes of meromorphic vector-fields,” pp. 29–47 in Essays on topology and related topics: Mémoires dédiés à Georges de Rham [Essays on topology and related topics: Memoirs dedicated to Georges de Rham] (Geneva, 26–28 March 1969). Edited by A. Haefliger and R. Narasimhan. Springer (Berlin), 1970. MR 261635 Zbl 0193.52201 incollection

Let \( M \) be a compact complex analytic manifold and let \( x \) be a holomorphic vector-field on \( M \). In an earlier paper by one of us (see [Bott 1967]) it was shown that the behavior of \( x \) near its zeroes determined all the Chern numbers of \( M \) and the nature of this determination was explicitly given where \( x \) had only nondegenerate zeroes. The primary purpose of this note is to extend this result to meromorphic fields, or equivalently to sections \( s \) of \( T\otimes L \) where \( T \) is the holomorphic tangent bundle to \( M \) and \( L \) is a holomorphic line bundle. We will also drop the non-degeneracy assumption of the zeroes of \( s \), but we treat only the case where \( s \) vanishes at isolated points \( \{p\} \).

Georges de Rham	Related
Raghavan Narasimhan	Related
Raoul H. Bott	Related	Volume
André Haefliger	Related

P. Baum and R. Bott: “Singularities of holomorphic foliations,” J. Diff. Geom. 7 : 3–4 (1972), pp. 279–342. To S. S. Chern and D. C. Spencer on their 60th birthdays. MR 377923 Zbl 0268.57011 article

Raoul H. Bott	Related	Volume
Donald Clayton Spencer	Related
S. S. Chern	Related	Volume

P. Baum: “Working with Bott (rocking and rolling with Raoul),” pp. xxii–xxiii in Raoul Bott: Collected papers, vol. 3: Foliations. Edited by R. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston), 1995. MR 1321887 incollection

Raoul H. Bott	Related	Volume
Robert Duncan MacPherson	Related

P. Baum: “Dirac operator and \( K \)-theory for discrete groups,” pp. 97–107 in A celebration of the mathematical legacy of Raoul Bott (Montreal, 9–13 June 2008). Edited by P. R. Kotiuga. CRM Proceedings & Lecture Notes 50. American Mathematical Society (Providence, RI), 2010. MR 2648889 Zbl 1201.19001 incollection

Let \( G \) be a locally compact, Hausdorff, and second countable (i.e. the topology has a countable base) topological group. Examples are Lie groups, \( p \)-adic groups, adelic groups, and discrete groups. In 1980, P. Baum and A. Connes proposed an answer to the problem of calculating the \( K \)-theory of the reduced \( C^* \) algebra of \( G \). When true, the conjecture has several corollaries. Among these are exhaustion of the discrete series by Dirac induction, Novikov conjecture on homotopy invariance of higher signatures, amd Gromov–Lawson–Rosenberg conjecture on Riemannian metrics of positive scalar curvature. Since 1980, the conjecture has been proved for many examples. Although it seems quite possible that eventually there will be a discrete group which is a counter-example, at the present moment there is no really good candidate for a counter-example. This talk will explain the conjecture, concentrating on the case of discrete groups from the point of view of the index of elliptic operators. The talk is intended for non-specialists. All the basic definitions (\( C^* \) algebra, \( K \)-theory etc) will be carefully stated.

Raoul H. Bott	Related	Volume
Peter Robert Kotiuga	Related

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