Undecidability in Algebra and Topology

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Course <dramatic pause> why?

The point of the course is to understand the limits of computational topology by showing that some simple and natural questions in topology are undecidable.

We will consider topological spaces with a combinatorial description(like a topological space with a finite simplicial complex) and discuss the proof of undecidablity of certain problems in group theory.

Then we will finally we would show how that can be used to prove the undecidability of homeomorphism of two topological spaces.

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Notes

• Pre-recs
  • Free Group
  • Presentation of a Group
  • Turing Machines
• Notes
  • Some Problems in Computational Algebra
    • Conjugation Problem on a Free group
      • Cyclically Reduced Term
  • Modular Machines
    • Modular Machines are Turing Complete
  • HNN - Extensions
  • Britton’s Lemma
  • Word Problem for Finitely Presented Groups is Undecidable

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MOCs

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References

• Notes on Computational Topology by Francis Lazarus and Arnaud de Memsey
• Modular Machines