Friday, March 16, 2018
Recent Progress in Euclidean Geometry
Workshop on the occasion of the 65th birthday of Professor Paul Yiu
Titles and Abstracts
Robert Bosch: A New Proof of the Erdős-Mordell Inequality
Abstract: In this note we show a new proof of
the Erdős-Mordell inequality.
The new idea is to consider three interior points to the triangle,
then the resulting inequality becomes an Erdős-Mordell inequality
when the three before mentioned points coincide.
> Timothy Ford:
Sums of Squares in the Field of Rational
Abstract: When Paul Yiu came to our department, his research
was on problems related to sums of squares and employed methods
that were topological and combinatorial. By limiting ourselves
to questions about sums of squares
in the field of rational functions ℝ(x,y),
this presentation aims to illustrate
both the depth and breadth of this interesting subject.
We begin by sketching
contributions by Hilbert, Landau, Cassels, Ellison and Pfister
that prove that
the Pythagoras number (the minimum number of squares required to
an arbitrary sum of squares) of the field ℝ(x,y)
is four. We end with a theorem due to Colliot-Thélène
that shows the existence of many polynomials in
ℝ[x,y] that are positive, have even degree at least six,
and are not the sum of three squares in
ℝ(x,y). The proof is radically
different and uses the Noether-Lefschetz Theorem, Galois Cohomology
Brauer group criterion for sums of three squares.
Markov: The Search for Equable and m-Equable Triangles:
a Mathematical Delight
Abstract: Triangles with integer sides have fascinated us
since ancient times.
Let us call an integer-sided triangle equable, if its area
is numerically equal to the perimeter, and m-equable,
if the area is an integer multiple of the perimeter.
It has been known for some time that there are only five equable
In this talk, we will present an algorithm for finding
all m-equable triangles for a fixed m.
> Petra Surynková: Selected Planar Curves and their Constructions
Abstract: The contribution addresses the geometrical properties
of selected planar curves which are determined by moving points or
curves in the plane.
We will focus on curves such as conic sections, cycloids, epicycloids,
conchoids, involutes and evolutes.
We will show the examples of the constructions of centers of curvature,
osculating circles, cusps, inflection points.
Selected planar curves will be also mentioned with respect to
geometric problems of antiquity.
All constructions will be demonstrated using the dynamic geometry
> Xiao-Dong Zhang: On a Generalization of Bertrand’s Postulate
Abstract: A celebrated theorem in elementary number theory
known as Bertrand’s Postulate says that there is at
least one prime (number) between n and 2n for every
positive integer n ≥ 2. We then wonder if it is also
true that there are two primes, or three primes, or more primes
between n and 2n. In this talk we outline a
history of Bertrand’s Postulate and give the following generalization:
Given any positive integer k, there
are at least k primes between n and 2n
if n ≥ 2k2.
> Li Zhou:
Elementary Extrema in Euclidean Geometry
Abstract: We use examples from problem sections of math journals
to illustrate the value of Geometer's Sketchpad in experimenting
and discovering solutions.
This conference is partially
supported by a grant
from the Simons Foundation (Award #245848 to the first