Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

The set of diffeomorphisms of the unit interval, or “warping functions,” plays an important role in many in functional data analysis applications. Most prominently, the problem of registering, or aligning, pairs of functions depends on solving for an element of the diffeomorphism group that, when applied to one function, optimally aligns it to the other.
The registration problem is posed as the unconstrained minimization of a cost function over the infinite dimensional diffeomorphism function space. We make use of its well-known Riemannian geometry to implement an efficient, limited memory Riemannian BFGS optimization scheme. We compare performance and results to the benchmark algorithm, Dynamic Programming, on several functional datasets. Additionally, we apply our methodology to the problem of non-parametric density estimation and compare to the benchmark performance of MATLAB’s built-in kernel density estimator ‘ksdensity’.

Phytoplankton are the base of the marine food web. They are also responsible for much of the oxygen we breathe, and they remove carbon dioxide from the atmosphere. The mechanisms that govern the timing of seasonal phytoplankton blooms is one of the most debated topics in oceanography. Here, we present a macroscale plankton ecology model consisting of coupled, nonlinear reaction-diffusion equations with spatially and temporally changing coefficients to offer insight into the causes of phytoplankton blooms. This model simulates biological interactions between nutrients, phytoplankton and zooplankton. It also incorporates seasonally varying solar radiation, diffusion and depth of the ocean’s upper mixed layer because of their impact on phytoplankton growth. The model is analyzed using seasonal oceanic data with the goals of understanding the model’s dependence on its parameters and of understanding seasonal changes in plankton biomass. A study of varying parameter values and the resulting effects on the solutions, the stability of the steady-states, and the timing of phytoplankton blooms is carried out. The model’s simulated blooms result from a temporary attraction to one of the model’s steady-states.

We have seen that Hankel index of varieties can be controlled by some invariants such as $$N_{2,p}$$ or p-base point free property. Also, we know that the Hankel index of (a linear join of) variety of minimal degree is infinity (and all invariant above are same as infinity). As next case, I will share some computations of invariants on a variety that projecting rational normal curve away from a point (which is a variety of almost minimal degree).

Like toric varieties, toric vector bundles are a rich class of algebraic varieties that can be described with combinatorial data.  Klyachko gave a classification of toric vector bundles in terms of certain systems of filtrations in a vector space.  I'll talk about some recent work with Kiumars Kaveh showing that Klyachko's data has an interesting interpretation in terms of tropical geometry.  In particular, we show that toric vector bundles can be classified by points on tropicalized linear spaces over a semifield of piecewise-linear functions.   I'll discuss how to use this recipe and a closely related tropicalization map to produce toric vector bundles and more general flat toric families.