By Gunnar Carlsson (auth.), Gunnar E. Carlsson, Ralph L. Cohen, Wu-Chung Hsiang, John D. S. Jones (eds.)

In 1989-90 the Mathematical Sciences learn Institute carried out a software on *Algebraic Topology and its **Applications*. the most parts of focus have been homotopy conception, K-theory, and functions to geometric topology, gauge thought, and moduli areas. Workshops have been performed in those 3 parts. This quantity comprises invited, expository articles at the subject matters studied in this software. They describe contemporary advances and aspect to attainable new instructions. they need to turn out to be precious references for researchers in Algebraic Topology and comparable fields, in addition to to graduate students.

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0 ... h,m(s) ) hm(s) . i,k{s)) < deg(fkk(s) for i < k. Now, if we fix the degree, d, of D, then L deg(fii (s)) = d and if we further fix degrees of the Iii themselves we obtain a finite partition of Rat:t{Gm,n) into strata. 52 RALPH L. COHEN AND R. JAMES MILGRAM Example. Suppose m = 2, n = 3, and d = 2, then there are three patterns Here RI has complex dimension 10, RII has complex dimension 9 and RIll has complex dimension 8. This gives a stratification of Rat2(G2,3) consisting of three strata.

Lond. Math. Soc. 50, 1-26. [DK] S. Donaldson and P. Kronheimer, The Geometry of Four- Manifolds, Oxford Univ. Press, 1990. [FU] D. Freed and K. Uhlenbeck, Instantons and four- manifolds, MSRI Publications, vol. 1, Springer Verlag, 1984. [F] M. K. B. Thomas, editors 150 (1990), Lond. Math. Soc. Lect. Notes. H. Gottlieb, On fibre spaces and the evaluation map, Ann. of Math. 87, 42-55. C. Hurtubise, The classification of monopoles for the classical groups, Comm. Math. Phys. 120, 613-64l. R. Hartshorne, Stable vector bundles and instantons, Comm.

HOMOTOPY OF GAUGE THEORETIC MODULI SPACES 41 A major difficulty with proving the Atiyah-Jones conjectures was the intractibility of the "reality" condition, condition (3), above in the definition of the ADHM-instantons. Attempts were made to live with it, for example in the work of Hartshorne, [Ha] , and attempts were made to remove it, [BM3], but each ran into trouble at about k = 2 or 3, though Hartshorne was able to characterize M2' An alternative approach was initiated by Donaldson [D2, A].