396 SECTIONAL TRANSACTIONS —A*. 



Prof. W. V. D. Hodge, F.R.S. — Some applications of harmonic integrals 

 (12.0). 



On an analytic variety which is an absolute orientable manifold a ^-fold 

 integral which is exact has the property that its value taken over a bounding 

 ^-cycle is zero, but it may have a non-zero value when taken over a p-cycle 

 which is not homologous to zero ; this we call the period of the integral 

 on the cycle. It is known that if R^ is the ^th Betti number of the manifold 

 there exist exact integrals which have arbitrarily assigned periods on R^ 

 independent ^-cycles. If the manifold carries a Riemannian metric 



gijdx^dxJ 



we can associate with a ^-fold integral 



^ij Pi, . .. i,dxi^ . . . dxif (i) 



an (« — p)-io\d integral 



j)!(w-j))! jV^-^*' • • ■ ''^■' • • • >''^''*' . . . g'"'^ Pk, . . . hdxh . . . dxJ'f (2) 



If the integrals (i) and (2) are both exact we say that (i) is a harmonic 

 integral. It follows that (2) is also harmonic. It is known that there exists 

 exactly one p-iold harmonic integral having assigned periods on Rp in- 

 dependent ^-cycles of the manifold. 



In a mathematical theory in which an analytic manifold appears it is 

 often possible to assign a Riemannian metric to the manifold in such a way 

 that the harmonic integrals prove useful weapons in the development of 

 the theory. 



Dr. B. Kaufmann. — Topological methods in the theory of conformal repre- 

 sentation (12.45). 



In the theory of conformal representation there are two main groups of 

 problems : the ' inner ' problems concerning mappings of plane regions, 

 and the problems on boundary relations. The most important result in 

 the first group is Riemann's fundamental theorem, and in the second 

 Fatou's theorem. The development of the theory in these two directions 

 has led to two well-known topological conceptions : the Riemann surface, 

 and ideal elements (prime ends). Through Hilbert's work at the beginning 

 of this century it became possible to extend the inner mapping theorems 

 to regions of arbitrary connectivity (Hilbert, Kobe). But the boundary 

 problems in the general case remained unsolved. However, the general 

 theory of ideal elements makes it possible to approach these problems. 

 The existence as well as the nature and the structure of the ideal elements 

 is revealed by a close study of a certain o-dimensional group of limit cycles 

 in an w-dimensional region by methods of an appropriate homology 

 theory. This group can be turned into a certain abstract space in which 

 the ideal elements can be seen and described. With the help of a method 

 of canonical dissections of regions of infinite connectivity (which might 

 be called Souslin dissections) some first results on boundary relations are 

 obtained. These ultimate results can be understood without any knowledge 

 of the theory of ideal elements. 



Afternoon. 

 Visit to works of British Tabulating Machine Co., Letchworth. 



