Mathematics Problem Archive

Compute H 2 Q(X; A) for each X -module A....

Let the notation be as above. Then, extending the definition o f the quandle cocycle invariant, define a knot invariant assoc iated with α, which is, ...

(M. Polyak) Define a quantum quandle....

(C. Rourke, B. Sanderson) Is there a natural quandle space whose cohomology groups are the quandle cohomology groups?...

(R. Fenn, C. Rourke, B. Sanderson) H3(Rp)∼ = Z⊕ Z/pZ for p prime....

([188, Problem 3]) Is the representation of the braid group inside the Temperley-Lieb algebra faithful?...

Is the Burau representation of B4 faithful?...

(S.J. Bigelow) Is the action of B6 on V 6 2 faithful?...

(S.J. Bigelow) Generalise Lawrence’s construction to obtain the irreducible representations of the BMW algebra....

(S.J. Bigelow) Find a larger family of irreducible representa- tions of Bn which includes those coming from the BMW algebra....

Classify all irreducible representations of Bn....

(S.J. Bigelow) Is there a faithful representation of Bn into a group of matrices over ¯Q?...

(see [220, Problem 3.108]) Does there exist a closed 3-manifold M, other than S3, such that τ SO(3) r (M ) = τ SO(3) r (S3) for all odd r≥ 3?...

(S.K. Hansen, T. Takata) Find pairs of non-homeomorphic rational homology 3-spheres that can be distinguished by th eir quantum G invariants τ G r or ...

(S.K. Hansen, T. Takata) Do the family of quantum G invari- ants τ G r or the family of quantum P G invariants τ P G r, G running through all simply c...

Find a 3-dimensional topological interpretation of quantu m in- variants of 3-manifolds....

For non-vanishing $\tau_r^G(M)$, the absolute value $|\tau_r^G(M)|$ depends only on the fundamental group $\pi_1(M)$....

(The perturbative expansion conjecture) The asymptotic expansion of Z G k (M ) of a closed oriented 3-manifold M is given by Z G k (M ) ∼ k→∞ e−π√ −1(...

(The asymptotic expansion conjecture, J.E. Andersen [6]) Let{c0 = 0, c1,···, cm} be the set of values of the Chern-Simons functional of flat G connect...

(J.E. Andersen) If such an expansion exists, understand how it is related to the expansion of Ohtsuki and the expansion of Habiro. It will of course b...

(Topological interpretations of the dj ’s) Let Mj be the union of components of the moduli space of flat connections M which has Chern-Simons value cj...

(The growth rate conjecture) Let d = max{d0,..., dn}. Then|Z G r (M )| = O(rd). It is well known that the quantum invariants only grows like r to some...

There is a construct of the right measure, say τM (A)1/2 for A∈M i, from the square root of the Reidemeister torsion generaliz ing the non-degenerate ...

(H. Murakami [294]) For any closed 3-manifold M, 2π √ −1·o-lim N →∞ log τ SU (2) N (M ) N = CS(M ) + √ −1vol(M ), where vol(M ) and CS(M ) denote the ...

(H. Murakami) Calculate o- lim log τ SU (2) N (M ) N for Seifert fibered 3-manifolds M....

(D. Thurston) Find a series of invariants of a 3-manifold (de- pending on roots of unity) that grows as its hyperbolic volum e (or its simplicial volu...

(D. Thurston) Find a correct generalization of the volume conjecture to other non-compact Lie groups....

(S. Morita [228]) Define the Chern-Simons invariant CS(M ) as a topological invariant of any closed oriented 3-manifol d M, and of any knot (link) com...

(T. Ohtsuki) Give a “complex structure” to the set of 3- manifolds. More precisely, find an embedding (or, an immersi on) of the set of 3-manifolds to...

(S. Baseilhac, R. Benedetti) Generalize the construction of the QHI for flat principal G-bundles, for Lie groups G different from B. Section 7.4 was wr...

(S. Baseilhac, R. Benedetti) Fix (W, L) and vary ρ. Study KN as a function of the bundle, that is as a function defined on the character variety of W ...

(S. Baseilhac, R. Benedetti) Specialize Problem 7.19 to bun- dles coming from the ordinary cohomology as above. For real a dditive ones, analyze the b...

(S. Baseilhac, R. Benedetti) Understand the ‘phase factor’ (i.e. the ambiguity due to N -th roots of unity) of the state sum HN (T ). Possi- bly deriv...

(S. Baseilhac, R. Benedetti) Determine a suitable (2 + 1) ‘decorated’ cobordism theory supporting a (non purely topo logical) QFT con- taining the alr...

(S. Baseilhac, R. Benedetti) Develop a 4-dimensional theory of QHI based on Turaev’s shadow theory....

(S. Baseilhac, R. Benedetti) Determine the actual relation- ship between KN (S3,·) and the coloured Jones polynomial JN (·) (evaluated at ω = exp(2iπ/...

(S. Baseilhac, R. Benedetti) (Real Volume Conjecture for QHI) For any triple (W, L, ρ) one has: lim N →∞ (2π/N 2) log(|KN (W, L, ρ)|) = Im R ( cI (W, ...

For each rational homology 3-sphere M, calculate τ SO(3)(M ) and τ P SU (N )(M ) for all degrees....

(J. Roberts) Explain the appearance of modular forms in the Witten invariants....

Characterize those elements of Z[[q−1]] of the form τ SO(3)(M ) of integral homology 3-spheres M....

(K. Habiro, T. Le) For each g as above, there is a (unique) invariant I g(M )∈ R1 of an integral homology 3-sphere M such that for each root of unity ...

(K. Habiro) Suppose that Conjecture 7.29 would hold. For a new indeterminate t, set R′ 1 = lim←−nR1[t]/((t− q)(t− q2)···(t− qn)) Then there exists an ...

Characterize those elements of Habiro’s expansion (45) of τ SO(3)(M ) of integral homology 3-spheres M....

Find (and classify) all TQFT’s....

Find (and classify) all modular categories. For a TQFT ( V, Z), put P(V,Z )(t) =∑ ∞ g=0 ( dimV (Σ g) ) tg, where Σ g denotes a closed surface of genus...

(1) Characterize the power series of the form P(V,Z )(t). (2) For each power series P (t) (satisfying the characterization of (1)), classify all TQFT’...

Find other spin TQFT’s....

Formulate and find spin c TQFT’s....

(V. Turaev) (1) Extend HQFT’s to spin and spin c settings. (2) Find algebra structures behind spin and spin c HQFT’s in dimension 1+1....

(V. Turaev) Study (spin and spin c ) HQFT’s with the target space K(H, 2) in dimensions 1 + 1, 2 + 1, and 3 + 1 for H = ZN....