Persistent homology is constrained to purely topological persistence, while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified low-dimensional multiscale paradigm for revealing topological persistence and extracting geometric shapes from high-dimensional datasets. For a point-cloud dataset, a filtration procedure is used to generate a sequence of chain complexes and associated families of simplicial complexes and chains, from which we construct persistent combinatorial Laplacian matrices. We show that a full set of topological persistence can be completely recovered from the harmonic persistent spectra, that is, the spectra that have zero eigenvalues, of the persistent combinatorial Laplacian matrices. However, non-harmonic spectra of the Laplacian
matrices induced by the filtration offer another powerful tool for data analysis, modeling, and prediction. In this work, fullerene stability is predicted by using both harmonic spectra and non-harmonic persistent spectra, while the latter spectra are successfully devised to analyze the structure of fullerenes and model protein flexibility, which cannot be straightforwardly extracted from the current persistent homology. The proposed method is found to provide excellent predictions of the protein B-factors for which current popular biophysical models break down.
Among all the interesting spaces in topology, the spheres are no doubt beautiful objects and of most consideration since antiquity. Any nontrivial observations of them are of course very important. From the categorical point of view, we should not just focus on the objects themselves, but the morphisms between them as well. For this purpose, in algebraic topology, we do want to classify the set of continuous morphisms between spheres under the equivalent relation named homotopy, which describes a continuous deformation between two continuous maps. Let Sn be the n-sphere and k (X) be the set of homotopic equivalent based maps from Sk to X. For the reason that Sk is a double suspension when k 2, the set is actually an abelian group. A natural question is, what are these abelian groups?
We prove that the 61-sphere has a unique smooth structure. Following results of Moise , Kervaire-Milnor , Browder  and Hill-Hopkins-Ravenel , we show that the only odd dimensional spheres with a unique smooth structure are S1, S3, S5 and S61.
We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the 2primary part of 5 1 is 8 8 2. This was the last unsolved 2extension problem left by the recent work of Isaksen and the authors through the 6 1stem.