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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 Poincaré 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)
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