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Algebraic techniques in incidence geometry

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If you have a question about this talk, please contact José Cañizo.

Joint Analysis-Combinatorics seminar

Recently, algebraic techniques have been introduced to count incidences between lines and points. The main idea behind these methods is the decomposition of the space we are working in—and therefore of our original set of points as well— by the zero set of a polynomial. This enriches our setting with extra structure, allowing us to understand it better. Such techniques were first used in incidence geometry by Dvir, for the solution of the Kakeya problem in finite field settings. The aim of this talk is to give a taste of these techniques (including Dvir’s basic argument), via the study of the joints problem in $\mathbb{R}n$.

More specifically, if $\mathfrak{L}$ is a finite set of lines in $\mathbb{R}^n$, we say that a point $x \in \mathbb{R}n$ is a joint formed by $\mathfrak{L}$ if at least $n$ lines of $\mathfrak{L}$ are passing through $x$, such that their directions span $\mathbb{R}n$. The joints problem asks for the optimal upper bound of the number of joints formed by $\mathfrak{L}$, depending only on the cardinality of $\mathfrak{L}$. The joints problem was solved in $\R3$ by Guth and Katz, and later in $\mathbb{R}n$ by Kaplan, Sharir and Shustin, and independently by Quilodran; all the solutions involved algebraic techniques. In particular, we will present Quilodran’s solution, which involves Dvir’s essential argument for his solution of the Kakeya problem in finite fields.

This talk is part of the Analysis Seminar series.

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