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CATEGORIES:Analysis Seminar
SUMMARY:Algebraic techniques in incidence geometry - Marin
a Iliopoulou (University of Birmingham)
DTSTART:20131002T150000Z
DTEND:20131002T160000Z
UID:TALK1194AT
URL:/talk/index/1194
DESCRIPTION:Recently\, algebraic techniques have been introduc
ed to count incidences between lines and points. T
he main idea behind these methods is the decomposi
tion of the space we are working in --and therefor
e of our original set of points as well-- by the
zero set of a polynomial. This enriches our settin
g with extra structure\, allowing us to understand
it better. Such techniques were first used in inc
idence geometry by Dvir\, for the solution of the
Kakeya problem in finite field settings. The aim o
f this talk is to give a taste of these techniques
(including Dvir's basic argument)\, via the study
of the joints problem in $\\mathbb{R}n$.\n\nMore
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 $\\math
frak{L}$ if at least $n$ lines of $\\mathfrak{L}$
are passing through $x$\, such that their directio
ns span $\\mathbb{R}n$. The joints problem asks fo
r 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 solutio
ns involved algebraic techniques. In particular\,
we will present Quilodran's solution\, which invol
ves Dvir's essential argument for his solution of
the Kakeya problem in finite fields.
LOCATION:Room R17/18\, Watson building
CONTACT:José Cañizo
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