Ten problems.(English)Zbl 0969.57001

Arnold, V. (ed.) et al., Mathematics: Frontiers and perspectives. Providence, RI: American Mathematical Society (AMS). 79-91 (2000).
Firstly, ten problems from the areas of mathematics in which the author has worked have been selected and listed. They are:
1. Find a nontrivial knot $$K$$ with $$V_K = 1$$.
2. Characterise those elements of $$\mathbb Z[t, t^{-1}]$$ of the form $$V_K(t).$$
3. Is the representation of the braid group inside the Temperley-Lieb algebra faithful?
4. Do Vassiliev invariants separate knots?
5. Calculate the set of indices of irreducible subfactors of the hyperfinite $${II}_1$$ factor.
6. What functions arise as the PoincarĂ© series of a subfactor?
In problems 7, 8 and 9, if $$(\cdot)$$ is a discrete group $$U(\cdot)$$ denote its von Neumann algebra.
7. Is $$U(F_n) \cong U(F_m)$$ for $$m \neq n (\geq 2)$$?
8. If $$(\cdot)$$ is a rigid discrete group and $$\alpha$$ an automorphism of $$U(\cdot)$$ is there a $$\beta\ni \operatorname{Aut}(\cdot)$$ with $$\alpha=\beta$$ modulo inner automorphism?
9. Calculate the fundamental group of $$U(SL(3,\mathbb Z)).$$
10. Solve the 2-dimensional square lattice Potts model.
Secondly, the problems are combined in four blocks:
1) Knot theoretic problems 1, 2, 3, 4 ;
2) Subfactor problems 5 and 6;
3) Problems 7, 8, 9. General $${II}_1$$ factors;
4) Problem 10: statistical mechanics, and, finally, they are discussed in detail.
There are given 57 references.
For the entire collection see [Zbl 1047.00015].

MSC:

 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 57M25 Knots and links in the $$3$$-sphere (MSC2010)

Keywords:

topology; invariants; knots