Title: Radical Sylvester-Gallai theorem for cubics - and beyond
| Speaker: | Rafael Oliveira |
| Affiliation: | University of Waterloo |
| Location: | MC 5501 or please contact Emma Watson for Zoom link |
Abstract:
In 1893, Sylvester asked a basic question in combinatorial geometry: given a finite set of distinct points $v_1, \ldots, v_m \in \R^N$ such that the line defined by any pair of distinct points $v_i, v_j$ contains a third point $v_k$ in the set, must all points in the set be collinear?
Generalizations of Sylvester's problem, which are known as Sylvester-Gallai type problems, have found applications in algebraic complexity theory (in Polynomial Identity Testing - PIT) and coding theory (Locally Correctable Codes).
The
underlying
theme
in
all
these
types
of
questions
is
the
following:
Are
Sylvester-Gallai
type
configurations
always
low-dimensional?
In 2014, Gupta, motivated by such applications in algebraic complexity theory, proposed wide-ranging non-linear generalizations of Sylvester's question, with applications on the PIT problem.
In this talk, we will discuss these non-linear generalizations of Sylvester's conjecture, their intrinsic relation to algebraic computation, and a recent theorem proving that radical Sylvester-Gallai configurations for cubic polynomials must have small dimension.
Joint work with Akash Kumar Sengupta