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Register a new userFrom uncountable abelian groups to uncountable nonabelian groups
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Katsuya Eda
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The present note surveys my research related to generalizing notions of abelian group theory to non-commutative case and applying them particularly to investigate fundamental groups.
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We propose a new method to count objects of specific categories that are significantly smaller than the ground sampling distance of a satellite image. This task is hard due to the cluttered nature of scenes where different object categories occur. Target objects can be partially occluded, vary in appearance within the same class and look alike to different categories. Since traditional object detection is infeasible due to the small size of objects with respect to the pixel size, we cast object counting as a density estimation problem. To distinguish objects of different classes, our approach combines density estimation with semantic segmentation in an end-to-end learnable convolutional neural network (CNN). Experiments show that deep semantic density estimation can robustly count objects of various classes in cluttered scenes. Experiments also suggest that we need specific CNN architectures in remote sensing instead of blindly applying existing ones from computer vision.
This article studies Paleys theory for lacunary Fourier series on (nonabelian) discrete groups. The results unify and generalize the work of Rudin for abelian discrete groups and the work of Lust-Piquard and Pisier for operator valued functions, and provide new examples of Paley sequences and $Lambda(p)$ sets on free groups.
In this paper we are interested in finding and evaluating cardinal characteristics of the continuum that appear in large-scale topology, usually as the smallest weights of coarse structures that belong to certain classes (indiscrete, inseparated, large) of finitary or locally finite coarse structures on $omega$. Besides well-known cardinals $mathfrak b,mathfrak d,mathfrak c$ we shall encounter two new cardinals $mathsf Delta$ and $mathsf Sigma$, defined as the smallest weight of a finitary coarse structure on $omega$ which contains no discrete subspaces and no asymptotically separated sets, respectively. We prove that $max{mathfrak b,mathfrak s,mathrm{cov}(mathcal N)}lemathsf Deltalemathsf Sigmalemathrm{non}(mathcal M)$, but we do not know if the cardinals $mathsf Delta,mathsf Sigma,mathrm{non}(mathcal M)$ can be distinguished in suitable models of ZFC.
Given a Lie group $G$ with finitely many components and a compact Lie group A which acts on $G$ by automorphisms, we prove that there always exists an A-invariant maximal compact subgroup K of G, and that for every such K, the natural map $H^1(A,K)to H^1(A,G)$ is bijective. This generalizes a classical result of Serre [6] and a recent result in [1].
Let $A$ be a finite abelian $p$ group of rank at least $2$. We show that $E^0(BA)/I_{tr}$, the quotient of the Morava $E$-cohomology of $A$ by the ideal generated by the image of the transfers along all proper subgroups, contains $p$-torsion. The proof makes use of transchromatic character theory.
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