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Series: High Dimensional Seminar

TBANote the special time!

Series: School of Mathematics Colloquium

TBA

Series: Analysis Seminar

TBA

Series: High Dimensional Seminar

TBA

Series: High Dimensional Seminar

It has been known that when an equiangular tight frame (ETF) of size |Φ|=N exists, Φ ⊂ Fd (real or complex), for p > 2 the p-frame potential ∑i ≠ j | < φj, φk > |p achieves its minimum value on an ETF over all N sized collections of vectors. We are interested in minimizing a related quantity: 1/ N2 ∑i, j=1 | < φj, φk > |p . In particular we ask when there exists a configuration of vectors for which this quantity is minimized over all sized subsets of the real or complex sphere of a fixed dimension. Also of interest is the structure of minimizers over all unit vector subsets of Fd of size N. We shall present some results for p in (2, 4) along with numerical results and conjectures. Portions of this talk are based on recent work of D. Bilyk, A. Glazyrin, R. Matzke, and O. Vlasiuk.

Series: Analysis Seminar

Series: Analysis Seminar

Series: Stochastics Seminar

Series: Stochastics Seminar

Series: Combinatorics Seminar

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.