Mathematics Problem Archive

(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....

Find a geometric construction of a TQFT using H 0(MΣ,L⊗k). Namely, find a geometric way to associate a vector in H 0(MΣ,L⊗k) to a 3- manifold M with ∂...

(G. Masbaum) Study this action of the finite group E(Σ) on H 0(MΣ,L⊗k), and describe the induced decompositions of this vector spa ce according to the...

For a given TQFT (V, Z), determine whether the image of ˜Mg in End ( V (Σ g) ) is finite....

(G. Masbaum) Is there a relation between the Nielsen-Thurs- ton classification of mapping classes of Σ g and their images on V (Σ g) for TQFT’s (V, Z)...

(T. Kerler) [Cyclotomic integer TQFT’s] (1) Find explicit/computable bases for the Vp(Σ g) as free modules over Z[ζp]. (2) Show that Vp can be extende...

(T. Kerler) [Homological TQFT’s] (1) Find the irreducible components and ring structure (w.r.t ⊕ and⊗) of Q∗. (2) Determine whether all strictly homol...

(T. Kerler) [Length = 1 TQFT’s] (1) Describe and construct algebraic L = 1 -extensions of Γ g -representations to TQFT’s, preferably as “simple” gener...

(T. Kerler) [ q/l -solvable and Casson TQFT’s] (1) Lift the 1/1-solvable TQFT’s of Casson type over Fp to a universal 1/1- solvable TQFT’s of Casson t...

(T. Kerler) [3-dim cobordisms from Hopf algebras] (1) Find further relations on Alg, besides the ones arising from the axiomat- ics of Hopf algebras, ...

(T. Kerler) [Extended and half-projective TQFT’s] (1) Describe in how far an ETQFT V with circle category C can differ from V C, thus introducing a equ...

(T. Kerler) [Non-semisimple vs. semisimple TQFT’s, the dou - ble conjecture] (1) Clarify the difference in the content of VC andVC! Are there homologic...

(1) Find (and classify) all semi-simple monoidal categories (w ith finitely many isomorphism classes of simple objects). (2) Find (and classify) (fini...

(Y. Kawahigashi) Suppose we have a three-dimensional TQFT. Can we determine whether it arises from a fusion rule algebra and 6j -symbols? If yes, can ...

(Y. Kawahigashi) Suppose we have two fusion rule algebras with 6j -symbols and that two TQFT’s arising from them are isomorphi c. What relation do we ...

(Y. Kawahigashi) Suppose we have a TQFT arising from a fusion rule algebra with 6j -symbols. Using a fusion rule subalgebra and 6j - symbols restricte...

(Y. Kawahigashi) Suppose we have a semisimple ribbon cat- egory C with finitely many isomorphism classes of simple objects. If the S - matrix is inver...

(Y. Kawahigashi) Suppose we have a semisimple ribbon cat- egory C1 with finitely many isomorphism classes of simple objects, bu t the S -matrix is not...

(Y. Kawahigashi) Suppose we have a semisimple ribbon cat- egory C1 with a degenerate S -matrix as in Problem 9.6. By the method in [288], we can also ...