Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Tuesday, September 29, 2009, 12:15 pm

**Duration**: This information is not available in the database

**Location**: OAT S15/S16/S17

**Speaker**: Anna Gundert (FU/TU Berlin)

It is easy to prove that any $k$-dimensional simplicial complex embeds into $\R^{2k+1}$, so the maximal number of $k$-simplices one can get when embedding into $\R^{2k+1}$ is ${n \choose {k+1}} = \Theta(n^{k+1})$ where $n$ is the number of vertices.

For the case $k=2$ this yields $\Theta(n^3)$ for embeddability into $\R^5$. One can also show that a 2-complex which embeds into $\R^3$ can have at most $n(n-3) = \Theta(n^2)$ triangles. What happens in $\R^4$ is an open question.

This talk will address the general question of the maximal number of $k$-simplices for a complex which is embeddable into $\R^{d}$ for some $k \leq d \leq 2k$. A lower bound of $f_k(C_{d+ 1}(n)) = \Omega(n^{\lceil\frac{d}{2}\rceil})$, which might even be sharp, is given by the cyclic polytopes. To find upper bounds for the cases $d=2k$ we look for forbidden subcomplexes. A generalization of the theorem of van Kampen and Flores yields those. Then the problem can be tackled with the methods of extremal hypergraph theory, which gives an upper bound of $O(n^{d+1-\frac{1}{3^d}})$.

Upcoming talks | All previous talks | Talks by speaker | Upcoming talks in iCal format (beta version!)

Previous talks by year: 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996

Information for students and suggested topics for student talks

Automatic MiSe System Software Version 1.4803M | admin login