Mathematics Problem Archive

(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....

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

(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) [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 C1 with a degenerate S -matrix as in Problem 9.6. By the method in [288], we can also ...

(N. Sato) Find a subfactor which can distinguish lens spaces L(7, 1) and L(7, 2). Moreover, find a subfactor to classify 3-manifolds as well as possib...

(N. Sato) Let us consider the Turaev-Viro-Ocneanu invariant from a subfactor with a degenerate braiding. Then, find a desc ription of this invariant a...

(V. Turaev) Relate this surgery formula for the Casson-Wal- ker-Lescop invariant with that of Lescop [251]. 32The normalization here is that λ CW(M ) ...

(M. Polyak) Define an invariant λ of a pair (M, σ) of a closed 3-manifold M and a spin structure σ on M such that λCWL(M ) = ∑ σ λ(M, σ) for any close...

(M. Polyak) Is there a “Rokhlin invariant” of a pair (M, α) of a closed 3-manifold M and a spin c structure α on M? (See Question 10.21.)...

(M. Polyak) By presenting 3-manifolds by surgery along framed links in S3, we can regard an invariant of 3-manifolds as an invari- ant of framed links...

F as d (ZM)/F as d+1(ZM) (resp.F b d (ZM)/F b d+1(ZM)) is torsion free for each d....

A(∅; Z) is torsion free. 10.2.2 Do finite type invariants distinguish homology 3-sph eres?...

Finite type invariants distinguish integral homology 3-spheres. (See Conjecture 11.2.) 10.2.3 Dimensions of spaces of finite type invariants A finite ...

Describe Vogel’s algebra Λ, say, by giving complete sets of generators and relations of Λ....

Find a constructive combinatorial presentation of each fini te type invariant of integral homology 3-spheres, and, in part icular, of the Casson invar...

(J. Roberts) What is the space of 3-manifolds?...

The map (50) is an isomorphism. 35The Yd -equivalence is also called the ( d − 1)-equivalence (due to Goussarov) in some literatures. This conjecture ...

{M ∼ Y2d S3}/ ∼ Y2d+1 is torsion free for each d....

(T. Ohtsuki) Define a product M1◦ M2 of integral homol- ogy 3-spheres M1 and M2 which is related, by (50), to the product of Jacobi diagrams given by ...