Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

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)

An Extremal Problem for Embeddable Simplicial Complexes

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:   2024  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