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We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry. 

The relationship between equiangular tight frames and strongly regular graphs has been known for several years. This relationship has been exploited to construct many of the latest examples of new strongly regular graphs. Recently it was shown that there is a similar relationship between two-distance tight frames and strongly regular graphs. In this paper we present a new tensor like construction of two-distance tight frames, and hence a family of strongly regular graphs. While graphs with these parameters were known to exist, this new construction is very simple, requiring only the existence of an affine plane, whereas the original constructions often require more complicated objects such as generalized quadrangles. 

It is often of interest to identify a given number of points in projective space such that the minimum distance between any two points is as large as possible. Such configurations yield representations of data that are optimally robust to noise and erasures. The minimum distance of an optimal configuration not only depends on the number of points and the dimension of the projective space, but also on whether the space is real or complex. For decades, Neil Sloane’s online Table of Grassmannian Packings has been the go-to resource for putatively or provably optimal packings of points in real projective spaces. Using a variety of numerical algorithms, we have created a similar table for complex projective spaces. This paper surveys the relevant literature, explains some of the methods used to generate the table, presents some new putatively optimal packings, and invites the reader to competitively contribute improvements to this table. 

The geological record encodes the relationship between climate and atmospheric carbon dioxide (CO₂) over long and short timescales, as well as potential drivers of evolutionary transitions. However, reconstructing CO₂beyond direct measurements requires the use of paleoproxies and herein lies the challenge, as proxies differ in their assumptions, degree of understanding, and even reconstructed values. In this study, we critically evaluated, categorized, and integrated available proxies to create a high-fidelity and transparently constructed atmospheric CO₂record spanning the past 66 million years. This newly constructed record provides clearer evidence for higher Earth system sensitivity in the past and for the role of CO₂thresholds in biological and cryosphere evolution.