Let $X$ and $Y$ be domains of $\mathbb{R}^n$ equipped with respective probability measures $\mu$ and $ \nu$. We consider the problem of optimal transport from $\mu$ to $\nu$ with respect to a cost function $c: X \times Y \to \mathbb{R}$. To ensure that the solution to this problem is smooth, it is necessary to make several assumptions about the structure of the domains and the cost function. In particular, Ma, Trudinger, and Wang established regularity estimates when the domains are strongly \textit{relatively $c$-convex} with respect to each other and cost function has non-negative \textit{MTW tensor}. For cost functions of the form $c(x,y)= \Psi(x-y)$ for some convex function $\Psi$, we find an associated K\"ahler manifold whose orthogonal holomorphic bisectional curvature is proportional to the MTW tensor. We also show that relative $c$-convexity geometrically corresponds to geodesic convexity with respect to a dual affine connection. Taken together, these results provide a geometric framework for optimal transport which is complementary to the pseudo-Riemannian theory of Kim and McCann.

We provide several applications of this work. In particular, we find a complete K\"ahler surface with non-negative orthogonal bisectional curvature that is not a Hermitian symmetric space or biholomorphic to $\mathbb{C}^2$. We also address a question in mathematical finance raised by Pal and Wong on the regularity of \textit{pseudo-arbitrages}, or investment strategies which outperform the market.

Encouraging sustainable mobility patterns is at the forefront of policymaking at all scales of governance as the collective consciousness surrounding climate change continues to expand. Not every community, however, possesses the necessary economic or socio-cultural capital to encourage modal shifts away from private motorized vehicles towards active modes. The current literature on `soft' policy emphasizes the importance of tailoring behavior change campaigns to individual or geographic context. Yet, there is a lack of insight and appropriate tools to promote active mobility and overcome transport disadvantage from the local community perspective. The current study investigates the promotion of walking and cycling adoption using a series of focus groups with local residents in two geographic communities, namely Chicago's (1) Humboldt Park neighborhood and (2) suburb of Evanston. The research approach combines traditional qualitative discourse analysis with quantitative text-mining tools, namely topic modeling and sentiment analysis. The analysis uncovers the local mobility culture, embedded norms and values associated with acceptance of active travel modes in different communities. We observe that underserved populations within diverse communities view active mobility simultaneously as a necessity and as a symbol of privilege that is sometimes at odds with the local culture. The mixed methods approach to analyzing community member discourses is translated into policy findings that are either tailored to local context or broadly applicable to curbing automobile dominance. Overall, residents of both Humboldt Park and Evanston envision a society in which multimodalism replaces car-centrism, but differences in the local physical and social environments would and should influence the manner in which overarching policy objectives are met.

A set of data with positive values follows a Pareto distribution if the log-log plot of value versus rank is approximately a straight line. A Pareto distribution satisfies Zipf's law if the log-log plot has a slope of -1. Since many types of ranked data follow Zipf's law, it is considered a form of universality. We propose a mathematical explanation for this phenomenon based on Atlas models and first-order models, systems of positive continuous semimartingales with parameters that depend only on rank. We show that the stable distribution of an Atlas model will follow Zipf's law if and only if two natural conditions, conservation and completeness, are satisfied. Since Atlas models and first-order models can be constructed to approximate systems of time-dependent rank-based data, our results can explain the universality of Zipf's law for such systems. However, ranked data generated by other means may follow non-Zipfian Pareto distributions. Hence, our results explain why Zipf's law holds for word frequency, firm size, household wealth, and city size, while it does not hold for earthquake magnitude, cumulative book sales, the intensity of solar flares, and the intensity of wars, all of which follow non-Zipfian Pareto distributions.