Georgia Institute of Technology College of Sciences

A recent extension by Guth (2015) of the basic polynomial partitioning technique of Guth and Katz (2015) shows the existence of a partitioning polynomial for a given set of k-dimensional varieties in R^d, such that its zero set subdivides space into open cells, each meeting only a small fraction of the given varieties. For k > 0, it is unknown how to obtain an explicit representation of such a partitioning polynomial and how to construct it efficiently. This, in particular, applies to the setting of n algebraic curves, or, in fact, just lines, in 3-space. In this work we present an efficient algorithmic construction for this setting almost matching the bounds of Guth (2015); For any D > 0, we efficiently construct a decomposition of space into O(D^3\log^3{D}) open cells, each of which meets at most O(n/D^2) curves from the input. The construction time is O(n^2), where the constant of proportionality depends on the maximum degree of the polynomials defining the input curves. For the case of lines in 3-space we present an improved implementation using a range search machinery. As a main application, we revisit the problem of eliminating depth cycles among non-vertical pairwise disjoint triangles in 3-space, recently been studied by Aronov et al. Joint work with Boris Aronov and Josh Zahl.

The boundary value and mixed value problems for linear and nonlinear degenerate abstract elliptic and parabolic equations are studied. Linear problems involve some parameters. The uniform L_{p}-separability properties of linear problems and the optimal regularity results for nonlinear problems are obtained. The equations include linear operators defined in Banach spaces, in which by choosing the spaces and operators we can obtain numerous classes of problems for singular degenerate differential equations which occur in a wide variety of physical systems. In this talk, the classes of boundary value problems (BVPs) and mixed value problems (MVPs) for regular and singular degenerate differential operator equations (DOEs) are considered. The main objective of the present talk is to discuss the maximal regularity properties of the BVP for the degenerate abstract elliptic and parabolic equation We prove that for f∈L_{p} the elliptic problem has a unique solution u∈ W_{p,α}² satisfying the uniform coercive estimate ∑_{k=1}ⁿ∑_{i=0}²|λ|^{1-(i/2)}‖((∂^{[i]}u)/(∂x_{k}^{i}))‖_{L_{p}(G;E)}+‖Au‖_{L_{p}(G;E)}≤C‖f‖_{L_{p}(G;E)} where L_{p}=L_{p}(G;E) denote E-valued Lebesque spaces for p∈(1,∞) and W_{p,α}² is an E-valued Sobolev-Lions type weighted space that to be defined later. We also prove that the differential operator generated by this elliptic problem is R-positive and also is a generator of an analytic semigroup in L_{p}. Then we show the L_{p}-well-posedness with p=(p, p₁) and uniform Strichartz type estimate for solution of MVP for the corresponding degenerate parabolic problem. This fact is used to obtain the existence and uniqueness of maximal regular solution of the MVP for the nonlinear parabolic equation.

The Whitney-Graustein theorem classifies immersions of the circle in the plane by their turning number. In this talk, I will describe a proof of this theorem, as well as a related result due to Hopf.