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The Gaussian phantom network theory plays a fundamental role in the study of the elastic behavior of polymer networks such as rubbers and gels [H. M. James and E. Guth (1947), R. Kubo (1947)]. It is also fundamental in the understanding of the properties of networks such as microscopic fluctuations and neutron scattering behavior. However, the elastic behavior and the scattering properties have been theoretically studied mainly for networks with no loops such as tree graphs. It was nontrivial to derive the elastic modulus for networks with loops, although the effects of loops have been investigated in recent experiments [B. D. Olsen et al. (2016, 2019)]. Furthermore, the assumption of fixed crosslinks formulated in the James-Guth theory has been fiercely criticized by several researchers, so that its main consequences on elasticity are not considered seriously or appropriately in polymer textbooks.
In the present talk we briefly review the standard elastic theory of polymer networks and argue its potential difficulties. We then reformulate and generalize the James-Guth t theory of polymer networks in terms of homology in topology, so that we derive useful exact results not only for the Gaussian networks but also for polymer networks consisting of chains with nonlinear potentials. We also derive the scattering functions for deformed phantom networks through homology. For instance, we present a systematic method for exactly calculating the nonlinear elasticity of such polymer networks that consist of finitely extensible random walk chains. Some of the derived results are found to be consistent with molecular dynamical simulations, scattering experiments of gels and elastic stress measurement of gels. The present talk is based on research results of the JST CREST project (2019 JPMJCR19T4) with the title: ``Construction of homological topology theory on the elasticity of polymer networks and creation of devices through mixing ring polymers in polymeric materials’’ and in collaboration with Jason Cantarella, Clayton Shonkwiler and Erica Uehara.
Recents observation of tensions in cosmology, such as the Hubble tension, may well be an indication that the experimental precision has reached a level that requires a more sophisticated framework than classical description. We consider finite-temperature one-loop renormalization of the Standard Model, coupled with dynamic metrics. The entire analysis is coherently carried out by using the refined background field method. It is evident that the high temperature QFT effects should be important in the history of the Universe. The implications of our findings for cosmology, particularly the Hubble tension and cosmological constant problem, are discussed. For the Hubble tension analysis, machine learning techniques will be useful.