Department of Computer Science | Institute of Theoretical Computer Science | CADMO
Prof. Emo Welzl and Prof. Bernd Gärtner
Theory of Combinatorial Algorithms
Teaching and Research Group Emo Welzl
Institut für Theoretische Informatik
Departement Informatik
ETH Zürich
CH–8092 Zürich
phone | +41–44–632 73 92 |
fax | +41–44–632 10 63 |
(Financed by the Innovedum Fund at ETH Zurich).
In technical fields, we write our documents almost exclusively in TeX/LaTeX.
The goal of this project is to provide the means of integrating interactive
elements (such as multiple choice questions with automatic feedback, but also
more advanced and complex features) in TeX-documents using native syntax already
familiar to instructors, and resulting in a look and feel already familiar to students.
The three main software components of this project are as follows:
Contact: B. Gärtner, M. Wettstein
Duration: Nov 2020 – Oct 2021.
(Financed by the Swiss National Science Foundation).
The classical Falconer distance conjecture says that for any compact set
E ⊂ R^{d}, if the Hausdorff dimension of E is greater than d/2,
then the Lebesgue measure of the distance set Δ(E) is positive.
The recent breakthrough paper of
Guth, Iosevich, Ou, and Wang tells us that in two dimensions, for any E ⊂ R^{2},
if the Hausdorff dimension of E is greater than 5/4, then the distance set is
of positive Lebesgue measure. This improves the some 20 year old estimate
dim_{H}(E)> 4/3 due to Wolff.
Let q be a prime power, and F_{q} be the finite field of order q.
The finite field version of this problem is known as the Erdős-Falconer
distance problem, which asks for the smallest number s such that for any
E ⊂ F_{q}^{d}, if |E| >= q^{s}, then
|Δ(E)| >= cq for some positive constant c. In even dimensions, it
is conjectured that s= d/2.
In the recent paper, Murphy, Petridis,
Pham, Rudnev, Stevens proved that in the plane over prime fields F_{p}^{2},
one has s <= 5/4. More precisely, let E be a set in F_{p}^{2},
if |E| >= p^{5/4}, then we have |Δ(E)| >= cp. This is directly
in line with Guth et al.'s result on the Falconer distance conjecture.
The main purpose of this project is to improve the exponent 5/4 and to study related questions.
Contact: Th. Pham
Duration: Sep 2020 – Sep 2021.
(Financed by the Swiss National Science Foundation).
Arrangements and drawings are two fundamental concepts that play a
central role in discrete and computational geometry: Given a set L of
planar curves, the arrangement of L is the subdivision of the plane
into faces, edges, and vertices, each defined as a maximal connected
component contained in the intersection of 0, 1, or 2 elements in L,
respectively. Given a graph G, a planar drawing of G assigns a point
in the plane to each vertex of G and represents the edges of G by
simple Jordan arcs that connect the corresponding
endpoints.
Arrangements and drawings pop up in almost every corner of discrete
and computational geometry. For example, they can be used to encode
order relationships between points, to compute convex hulls, or to
find ham-sandwich cuts. Due to the fundamental nature and the
simplicity of the concepts, a vast number of variants, special cases,
generalizations, and abstractions have been developed. The goal of
this project is to undertake a systematic study of the different
structures and their representations, and of their relevant
combinatorial and algorithmic properties.
The project will be carried out as an international collaboration with
partners from Berlin and Graz.
Contact: M. Hoffmann
Duration: Jan 2018 – Sep 2021.