Research on topology related to variational problems and application of Mathematica

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Let G be a Lie group and let p be a prime. In the case the Lie group G is one of exceptional Lie groups F4, E6, E7, E8 and p=3 or in the case G=E8, p=5, the integral cohomology of G has p-torsion and the mod p cohomology of its classifying space BG is unknow or even if it has been already computed, the result is involved. There is a so-called Adams' conjecture on the mod p cohomology of the classifying spaces of compact Lie groups, which asserts that the Quillen homomorphism is a monomorphism for p an odd prime. If the the description of the mod p cohomology of classifying spaces in this form, it would be useful. In the study of the mod p cohomology of classifying spaces of exceptional Lie groups above, one of the most powerful tool is the Rothenberg-Steenrod spectral sequence whose E2 term was identified with the cotorsion product of the Lie group G.We computed certain rings of invariants, which could be done by computer calculation in some cases, and obtained the following results :(1) for (G,p>(F4,3), (E6,3), (E7,3), (E8,5), the Rothenberg-Steenrod spectral sequence converging to the mod p cohomology of BG collapses at the E2 term.(2) For (G,p>(E8,3), the Rothenberg-Steenrod spectral sequence does not collapse at the E2 level.Furthermore, the computation of certain cotorsion products is equivalent to the computation of the cyclic group C of order p with certain C-modules. We hope this computation would be applied to the computation of the mod p cohomology of classifying spaces of loop groups which are related to variational problems.