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public 01:02:38

Robert Bryant : A Weierstrass representation for affine Bonnet surfaces

  -   Geometry and Topology ( 107 Views )

Ossian Bonnet (1819–1892) classified the surfaces in Euclidean 3-space that can be isometrically deformed without changing the mean curvature function H, showing that there are two types: the surfaces of constant mean curvature and a 4-dimensional ‘exceptional family’ (with variable mean curvature) that are now known as Bonnet surfaces. The corresponding problem in affine 3-space is much more difficult, and the full classification is still unknown. More than 10 years ago, I classified the affine surfaces that can isometrically deformed (with respect to the induced Blaschke metric) while preserving their affine mean curvature in a 3-dimensional family (the maximum dimension possible), showing that they depend on 2 functions of 1 variable in Cartan’s sense. When I gave a talk* in this seminar about these results on September 10, 2013, I only knew that these surfaces corresponded to pseudoholomorphic curves in a certain almost-complex surface. However, I have recently shown that the structure equations for these mysterious surfaces can be interpreted as describing holomorphic Legendrian curves in CP^3 subject to a natural positivity condition, and the integration corresponds to a flat sp(2,R) connection, i.e., they can be interpreted as a Lax pair, but of a very special kind, for which the integration can be effected explicitly. I’ll explain these results and use them to show how the classical problem of determining the affine surfaces with constant affine mean curvature and constant Gauss curvature of the Blaschke metric can be explicitly integrated, which, heretofore, was unknown. * https://www4.math.duke.edu/media/watch_video.php?v=6948e657e69cadbaa1a6915335e9ea87

public 51:31

Punit Gandhi : Conceptual modeling of dryland vegetation patterns across timescales

  -   Mathematical Biology ( 88 Views )

Strikingly regular, large-scale patterns of vegetation growth were first documented by aerial photography in the Horn of Africa circa 1950 and are now known to exist in drylands across the globe. The patterns often appear on very gently sloped terrain as bands of dense vegetation alternating with bare soil, and models suggest that they may be a strategy for maximizing usage of the limited water available. A particular challenge for modeling these patterns is appropriately resolving fast processes such as surface water flow during rainstorms while still being able to capture slow dynamics such as the uphill migration of the vegetation bands, which has been observed to occur on the scale of a band width per century. We propose a pulsed-precipitation model that treats rainstorms as instantaneous kicks to the soil water as it interacts with vegetation on the timescale of plant growth. We use a stochastic rainfall model with the influence of fast storm-level hydrology captured by the spatial distribution of the soil water kicks. The model allows for predictions about the influence of storm characteristics on the large-scale patterns. Analysis and simulations suggest that the distance water travels on the surface before infiltrating into the soil during a typical storm plays a key role in determining the spacing between the bands.