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

In 1989-90 the Mathematical Sciences study Institute carried out a application on Algebraic Topology and its functions. the most components of focus have been homotopy idea, K-theory, and functions to geometric topology, gauge idea, and moduli areas. Workshops have been carried out in those 3 parts. This quantity involves invited, expository articles at the issues studied in this software. They describe fresh advances and element to attainable new instructions. they need to end up to be important references for researchers in Algebraic Topology and comparable fields, in addition to to graduate scholars.

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12. Since Qt is invariant and bounded in W, one finds that Qt = S(t)Qt (for t > r) is bounded in V. 6 Robustness of Attractors. It is hard to overstate the importance of the role played by the Stability Lemma in the study of the longtime dynamics of infinite dimensional systems. , upper semicontinuous dependence on parameters) of attractors. Let Qto be an attractor for a given II:-contracting semiflow So(t) . o(t) = So(t). (t) is robust at Qto, or upper semicontinuous with respect to A at A = AO, provided that, for every to> 0, there is a neighborhood 0 = O(to) of AO in A such that for each A EO, the semi flow S>.

Now if Al consists of a finite number of points, then there is a U E Al with the property that U E Bn for all n > r. On the other hand, if Al is an infinite set, then there is a point of accumulation U E A 2 . Since the sets Bn are closed and monotone, one has U E B n , for all n > r. Thus in either case, one has U E B. For Item (6) we let tn and Un be sequences that satisfy tn --* 00, as n --* 00, and Un E Bt n • Define An as An = Clw { Urn: m def ~ n} . Then An is bounded and closed with An J AnH and ",(An) :S ",(Bt n ) --* 0.

Other examples will be developed in the text. 2. Compact and ",-Contracting Semiftows. Many of the applications of the theory of semiflows to partial differential equations, or to differential equations with time delays, occur in an important setting wherein the given semiflow has some smoothing property. In this section we will examine two of these properties: compactness and ",-contracting. , ClwS(t)B is compact. The number r(B) is referred to as a compactification time for S(t)B . If r(B) = to can be chosen independent of B, then we will say that the semiflow 0' is compact for t > to .