Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, October 21, 2008, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Uli Wagner

Hardness of Embedding Simplicial Complexes in R^d (Part I)

It is well known that one can test in linear time whether a given graph is planar. We consider the higher-dimensional generalization of this problem: Given a k-dimensional simplicial complex K and a target dimension d, does K embed into R^d? Surprisingly, rather little seems to be known about the algorithmic aspects of this problem.

In Part 1 of the talk, we will review and explain the relevant topological notions, at least on an intuitive level, and discuss some background, known results, and related problems. For instance, known results easily imply that the embeddability problem is solvable in polynomial time if k=d=2 or d=2k \geq 6. Moreover, we observe that the problem is algorithmically undecidable for k=d-1 and d \geq 5. This follows from a famous result of Novikov on the unsolvability of recognizing the 5-sphere.

In Part 2, we will prove our main result: It is at least NP-hard to decide if a 2-dimensional simplicial complex embeds into R^4. More generally, the problem is NP-hard if d \geq 4 and d \geq k \geq (2d-2)/3. These dimensions fall outside the so-called metastable range of a theorem of Haefliger and Weber, which characterizes embeddability in terms of the so-called deleted product obstruction. Our reductions are based on examples, due to Segal, Spiez, Freeman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range, the deleted product obstruction is insufficient.

Joint work with Jiří Matoušek and Martin Tancer.

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