Walk algebras, distinguished subexpressions, and point counting in Kac-Moody flag varieties

We study walk algebras and Hecke algebras for Kac-Moody root systems. Each choice of orientation for the set of real roots gives rise to a corresponding "oriented" basis for each of these algebras. We show that the notion of distinguished subexpression naturally arises when studying the transition matrix between oriented bases. We then relate these notions to the geometry of Kac-Moody flag varieties and Bott-Samelson varieties. In particular, we show that the number of points over a finite field in certain intersections of these varieties is given by change of basis coefficients between oriented bases of the Hecke algebra. Using these results we give streamlined derivations of Deodhar's formula for $R$-polynomials and point-counting formulas for specializations of nonsymmetric Macdonald polynomials $E_\lambda(\mathsf{q},t)$ at $\mathsf{q}=0,\infty$

An Investigation of Disaccharide-Induced Osmoprotection in Sinorhizobium Meliloti

This project sought to investigate the mechanism of disaccharide-induced osmoprotection in S. meliloti. It has been shown that sucrose, trehalose, cellobiose, gentiobiose, palatinose, maltose, turanose and maltotriose can act as non-accumulated osmoprotectants in several strains of S. meliloti (8). Sucrose was shown to provide full recovery of growth in salt shocked cultures of S. meliloti strain 1021, quantified by measuring light scattering at 600 nm. However, cellobiose and maltose were found to be ineffective osmoprotectants for strain 1021. Osmoprotection by sucrose may proceed via a separate pathway than that of cellobiose and maltose. A mutant strain, 307620, which can grow on neither cellobiose nor glucose, was also studied. In preliminary experiments the mutant strain displayed osmosensitivity and was not rescued in 0.45M NaCI by the sucrose, cellobiose, or maltose. In the absence of salt, strain 307620 had significantly stunted growth but recovered with supplemental amounts of sucrose, maltose, or glucose. Further experiments are needed for conclusive identification of strain 307620

Multiscale Predictive Modeling of Wave Velocity and Its Distribution Across a Rocky Intertidal Landscape

Much importance has been placed on understanding and quantifying how diversity, community structure and mechanistic processes change over different scales. There is a need in ecology for multiscale approaches quantitatively linking scales to characterize, quantify and better understand how ecological patterns and processes functioning at different scales interact. Many of the structural mechanisms established in rocky intertidal systems are driven or influenced by larger scale oceanographic processes that scale down to sub-meter scale effects. Here I develop and integrate two models that link these two scales. Velocity was measured in a swath along shore using an array of dynamometers. Intertidal and subtidal landscapes were mapped using field survey techniques. Digital elevation models were created using GIS, from which landscape characteristics such as slope, curvature and aspect were quantified at a fine scale. A large scale model (p-value \u3c 0.001, R2 = 0.803) uses buoy data, local bathymetry and landscape characteristics to predict the mean maximum velocity experienced on rocky intertidal shorelines. A fine scale model (p-value \u3c 0.001, R2 = 0.633) uses the mean maximum velocity and fine scale landscape characteristics to predict the fine scale distribution of flow speeds across rocky intertidal landscapes. Their integration (p-value \u3c 0.001, R2 = 0.6849) links large scale oceanographic conditions to the fine scale distribution of wave velocity across rocky intertidal landscapes. The multiscale models presented here address the need in ecology for multiscale modeling approaches to quantify and link multiscale patterns to better understand how processes and mechanisms interact at different scales