Quantifying Patterns of Supraglacial Debris Thickness and Their Glaciological Controls in High Mountain Asia

Mapping patterns of supraglacial debris thickness and understanding their controls are important for quantifying the energy balance and melt of debris-covered glaciers and building process understanding into predictive models. Here, we find empirical relationships between measured debris thickness and satellite-derived surface temperature in the form of a rational curve and a linear relationship consistently outperform two different exponential relationships, for five glaciers in High Mountain Asia (HMA). Across these five glaciers, we demonstrate the covariance of velocity and elevation, and of slope and aspect using principal component analysis, and we show that the former two variables provide stronger predictors of debris thickness distribution than the latter two. Although the relationship between debris thickness and slope/aspect varies between glaciers, thicker debris occurs at lower elevations, where ice flow is slower, in the majority of cases. We also find the first empirical evidence for a statistical correlation between curvature and debris thickness, with thicker debris on concave slopes in some settings and convex slopes in others. Finally, debris thickness and surface temperature data are collated for the five glaciers, and supplemented with data from one more, to produce an empirical relationship, which we apply to all glaciers across the entire HMA region. This rational curve: 1) for the six glaciers studied has a similar accuracy to but greater precision than that of an exponential relationship widely quoted in the literature; and 2) produces qualitatively similar debris thickness distributions to those that exist in the literature for three other glaciers. Despite the encouraging results, they should be treated with caution given our relationship is extrapolated using data from only six glaciers and validated only qualitatively. More (freely available) data on debris thickness distribution of HMA glaciers are required.

A Topological View of Reed–Solomon Codes

We studied a particular class of well known error-correcting codes known as Reed–Solomon codes. We constructed RS codes as algebraic-geometric codes from the normal rational curve. This approach allowed us to study some algebraic representations of RS codes through the study of the general linear group GL(n,q). We characterized the coefficients that appear in the decompostion of an irreducible representation of the special linear group in terms of Gromov–Witten invariants of the Hilbert scheme of points in the plane. In addition, we classified all the algebraic codes defined over the normal rational curve, thereby providing an algorithm to compute a set of generators of the ideal associated with any algebraic code constructed on the rational normal curve (NRC) over an extension Fqn of Fq.

Approximation and interpolation of regular maps from affine varieties to algebraic manifolds

We consider the analogue for regular maps from affine varieties to suitable algebraic manifolds of Oka theory for holomorphic maps from Stein spaces to suitable complex manifolds. The goal is to understand when the obstructions to approximation or interpolation are purely topological. We propose a definition of an algebraic Oka property, which is stronger than the analytic Oka property. We review the known examples of algebraic manifolds satisfying the algebraic Oka property and add a new class of examples: smooth nondegenerate toric varieties. On the other hand, we show that the algebraic analogues of three of the central properties of analytic Oka theory fail for all compact manifolds and manifolds with a rational curve; in particular, for projective manifolds.

Automorphism Groups of Compact Complex Surfaces

We study automorphism groups and birational automorphism groups of compact complex surfaces. We show that the automorphism group of such a surface $X$ is always Jordan, and the birational automorphism group is Jordan unless $X$ is birational to a product of an elliptic and a rational curve.

Kontsevich spaces of rational curves on Fano hypersurfaces

We investigate the spaces of rational curves on a general hypersurface. In particular, we show that for a general degree d hypersurface in {\mathbb{P}^{n}} with {n\geq d+2} , the space {\overline{\mathcal{M}}_{0,0}(X,e)} of degree e Kontsevich stable maps from a rational curve to X is an irreducible local complete intersection stack of dimension {e(n-d+1)+n-4} . This resolves all but one case of a conjecture of Coskun, Harris and Starr, and also proves that the Gromov–Witten invariants of these hypersurfaces are enumerative.