Mathematics Problem Archive
Problem 2 — Show that the inclusion of Proposition 5 is a bijection.
v1.3 research notesShow that the inclusion of Proposition 5 is a bijection. This will probably require more thought about the analytical and geometric constructions whic...
Problem 3 — Given a topological description of fk0,fk1 describe fk0+k1.
v1.3 research notesGiven a topological description of fk0,fk1 describe fk0+k1. A good understanding of this would enable one to drop the rather artificial introduction o...
Problem 4 — Reproduce the classification of manifolds with ω.K (ω)< 0 by studying the sets Cχ,θ for θ< 0.
v1.3 research notesReproduce the classification of manifolds with ω.K (ω)< 0 by studying the sets Cχ,θ for θ< 0. There is a network of interesting questions dealing with...
Problem 5 — Analyse the monodromy of Lefschetz fibrations on manifolds with K(ω) = 0.
v1.3 research notesAnalyse the monodromy of Lefschetz fibrations on manifolds with K(ω) = 0. Related to this is the general question of understanding the place of comple...
Problem 6 — Find special features of the monodromy of algebraic surfaces.
v1.3 research notesFind special features of the monodromy of algebraic surfaces. There is some good motivation for this coming from at least three directions • The probl...
Problem 1.1 — Investigate the dependence of Expρ,⟨f,M ⟩ on the marked Riemann surface ⟨f,M⟩.
v1.3 research notesInvestigate the dependence of Expρ,⟨f,M ⟩ on the marked Riemann surface ⟨f,M⟩....
Conjecture 2.1 — Let Ω∗(Hom(π,G )/G) be the de Rham algebra consisting of all measurable differential forms on Hom(π,G )/G.
v1.3 research notesLet Ω∗(Hom(π,G )/G) be the de Rham algebra consisting of all measurable differential forms on Hom(π,G )/G. Then the symplectic structures ωB generate t...
Conjecture 2.2 — Suppose C∞(Hom(π,G )/G) D− →C∞(Hom(π,G )/G) is a differential operator which commutes with the ModΣ-action on Hom(π,G…
v1.3 research notesSuppose C∞(Hom(π,G )/G) D− →C∞(Hom(π,G )/G) is a differential operator which commutes with the ModΣ-action on Hom(π,G )/G. Then D is a scalar multiple ...
Problem 2.2 — Decompose the representation on H0 into irreducible representations of ModΣ.
v1.3 research notesDecompose the representation on H0 into irreducible representations of ModΣ. When G = U(1), and Σ is the 2-torus, Hom( π,G )/G naturally identifies wi...
Problem 2.3 — Find a similar geometric interpretation for the irreducible constituents for compact nonabelian groups G.
v1.3 research notesFind a similar geometric interpretation for the irreducible constituents for compact nonabelian groups G....
Conjecture 2.3 — If r≥ 3, the action of Out(π) on Hom(π,G ) is ergodic.
v1.3 research notesIf r≥ 3, the action of Out(π) on Hom(π,G ) is ergodic. Using calculations in [ 43], this conjecture has been proved [ 47] when all of the simple facto...
Problem 2.4 — Determine necessary and sufficient conditions on a general representation ρ for its orbit ModΣ· [ρ] to be dense.
v1.3 research notesDetermine necessary and sufficient conditions on a general representation ρ for its orbit ModΣ· [ρ] to be dense. The case when G = SU(2) and Σ an n-hole...
Problem 2.5 — Construct an example of a pseudo-Anosov mapping class for a closed surface which is not ergodic on the SU(2)-characte…
v1.3 research notesConstruct an example of a pseudo-Anosov mapping class for a closed surface which is not ergodic on the SU(2)-character variety....
Conjecture 3.1 — Suppose that b = 0 (Σ is closed).
v1.3 research notesSuppose that b = 0 (Σ is closed). For each integer 1≤k≤ 2g +b− 2, the ModΣ-action on the component e−1(2− 2g +b +k) of Hom(π,G ) is ergodic. When b = ...
Problem 3.1 — Determine the smallest dimensional manifold homotopy-equivalent to Σ upon which ModΣ acts compatibly with the outer a…
v1.3 research notesDetermine the smallest dimensional manifold homotopy-equivalent to Σ upon which ModΣ acts compatibly with the outer action of ModΣ on π1(Σ)....
Problem 3.2 — Find general conditions which ensure that (10) is proper.
v1.3 research notesFind general conditions which ensure that (10) is proper. The level set R3∩κ−1(2) consists of characters of abelian representations, and ModΣ is ergod...
Problem 3.3 — Determine the ergodic behavior of the ModΣ-action on the level sets ( iR× R×iR ) ∩κ−1(t) wheret> 2.
v1.3 research notesDetermine the ergodic behavior of the ModΣ-action on the level sets ( iR× R×iR ) ∩κ−1(t) wheret> 2. The level sets for t> 6 contains wandering domains...
Problem 3.4 — Find a point ρ∈ Hom(π, SL(2, C)) such that the closure of its orbit ModΣ· [ρ] meets both the image of the unitary cha…
v1.3 research notesFind a point ρ∈ Hom(π, SL(2, C)) such that the closure of its orbit ModΣ· [ρ] meets both the image of the unitary characters Hom(π, SU(2)) and the clo...
Problem 3.6 — Find a substitute for convex cocompactness in higher rank which includes the above examples of proper ModΣ-actions, a…
v1.3 research notesFind a substitute for convex cocompactness in higher rank which includes the above examples of proper ModΣ-actions, and for which Eρ is proper. The wo...
Conjecture 3.2 — If k = 1, then U is onto.
v1.3 research notesIf k = 1, then U is onto. In general a PSL(2, R)-representation with dense image lies in Image(U). 220 W. Goldman...
Problem 2: — For a fixed number R > 0, is there is compact subset K(R) of moduli space containing the projection of every Teichm¨…
v1.3 research notesFor a fixed number R > 0, is there is compact subset K(R) of moduli space containing the projection of every Teichm¨ uller geodesic γ which satisfies ...
Problem 3: — Describe the space of geodesic currents for Mg,m.
v1.3 research notesDescribe the space of geodesic currents for Mg,m. Is the set of weighted sums of Dirac masses at the pairs of fixed points of pseudo-Anosov elements d...
Problem 4: — Does the above definition of a convex cocompact subgroup of Mg,m coincide with the definition of Farb and Mosher in […
v1.3 research notesDoes the above definition of a convex cocompact subgroup of Mg,m coincide with the definition of Farb and Mosher in [FMo]? Is the natural extension of...
Problem 5: — Develop a theory of geometrically finite subgroups of Mg,m which include the groups defined by Veech surfaces.
v1.3 research notesDevelop a theory of geometrically finite subgroups of Mg,m which include the groups defined by Veech surfaces. 14. Geometric properties of the mapping...
Problem 8: — Is the mapping class group a-T-menable?
v1.3 research notesIs the mapping class group a-T-menable? 14. Geometric properties of the mapping class group 245...
Problem 1 — (Geodesics on general flat surfaces).
v1.3 research notes(Geodesics on general flat surfaces). Describe the behavior of geodesics on general flat surfaces. Prove (or disprove) the conjecture that the geodesi...
Problem 2 — (Billiards in general polygons).
v1.3 research notes(Billiards in general polygons). Does every billiard table have at least one regular periodic trajectory? If the answer is affirmative, does this trajec...
Problem 3 — (Renormalization of billiards in polygons).
v1.3 research notes(Renormalization of billiards in polygons). Is there a natural dynamical system acting on the space of billiards in polygons so as to allow a useful r...
Problem 4 — (Characterization of Veech surfaces).
v1.3 research notes(Characterization of Veech surfaces). Characterize all Veech surfaces (for each stratum of each genus). This problem is trivial in genus one; in genus...
Problem 5 — (Fuchsian groups).
v1.3 research notes(Fuchsian groups). Which Fuchsian groups are realized as Veech groups? Which subgroups of the mapping class group appear as Veech groups? This is equi...
Problem 6 — (Purely cyclic).
v1.3 research notes(Purely cyclic). Is there a Veech group that is cyclic and generated by a single hy- perbolic element? Equivalently, is there a pseudo-Anosov map such...
Problem 7 — (Algorithm for Veech groups).
v1.3 research notes(Algorithm for Veech groups). Is there an algorithm for determining the Veech group of a general translation surface or quadratic differential? An inte...
Problem 8 — (Orbits of square-tiled surfaces).
v1.3 research notes(Orbits of square-tiled surfaces). Classify the SL(2, R) orbits of square-tiled sur- faces in any stratum. Describe their Teichm¨ uller discs. A parti...
Problem 11 — (Minimal sets).
v1.3 research notes(Minimal sets). Describe the minimal sets for the SL(2, R)-action on H1(α) and Q1(β). Since Veech surfaces give rise to minimal sets, this problem gen...
Problem 13 — (Kernel foliation).
v1.3 research notes(Kernel foliation). IsN a complex-analytic (complex-algebraic) orbifold? When is dimCN = dim CO +n−m? On the other hand when does N coincide with the ...
Problem 14 — (Decomposition of surfaces).
v1.3 research notes(Decomposition of surfaces). Given a connected component of the stratum H(α) of Abelian differentials (or of quadratic differentials Q(β) find those con...
Problem 15 — (Lyapunov exponents).
v1.3 research notes(Lyapunov exponents). Study individual Lyapunov exponents of the Teichm¨ uller geodesic flow: – for all known SL(2; R)-invariant subvarieties; – for s...
Problem 16 — (Dynamical Hodge decomposition).
v1.3 research notes(Dynamical Hodge decomposition). Study properties of distributions of the La- grangian subspaces in H 1(S; R) defined by the Teichm¨ uller geodesic fl...
Problem 17 — (Converse to dichotomy).
v1.3 research notes(Converse to dichotomy). Characterize translation surfaces for which (1) the set of minimal directions coincides with the set of uniquely ergodic dire...
Problem 18 — (Quadratic asymptotics for any surface).
v1.3 research notes(Quadratic asymptotics for any surface). Is it true that every translation surface or quadratic differential has exact quadratic asymptotics for the nu...
Problem 19 — (Error term for counting functions).
v1.3 research notes(Error term for counting functions). What can be said about the error term in the quadratic asymptotics for counting functions N ((X,ω ),L )∼c·L2 on a...
Problem 20 — (Topology of strata).
v1.3 research notes(Topology of strata). Is it true that the connected components of the strata H(α) and of the strata Q(β) areK(π, 1)-spaces (i.e. their universal cover...
Problem 21 — (Exceptional Strata).
v1.3 research notes(Exceptional Strata). Find a geometric invariant which distinguishes differ- ent connected components of the four exceptional strata Q(−1, 9),Q(−1, 3, ...
Problem 1.1 — Given a subgroup G <MCG (S), how is the geometry of ΓG related to the dynamics of the action of G on Thurston ’s comp…
v1.3 research notesGiven a subgroup G <MCG (S), how is the geometry of ΓG related to the dynamics of the action of G on Thurston ’s compactification of Teichm¨ uller spa...
Problem 2.2 — Does the converse hold in the above theorem without the assumption that G is free?
v1.3 research notesDoes the converse hold in the above theorem without the assumption that G is free? The gist of Problem 2.2 is to find an extension of the Bestvina-Fei...
Problem 3.3 — Does there exist an algorithm which produces the integer M in Theorem 3.1, given φ1,...,φ n?
v1.3 research notesDoes there exist an algorithm which produces the integer M in Theorem 3.1, given φ1,...,φ n?...
Problem 3.4 — If H⊂MCG (S) is finite rank free subgroup whose nonidentity elements are pseudo-Anosov, is H a Schottky group?
v1.3 research notesIf H⊂MCG (S) is finite rank free subgroup whose nonidentity elements are pseudo-Anosov, is H a Schottky group? For specific examples on which to test ...
Problem 3.5 — Is every finite rank subgroup of Whittlesey’s group a Schottky subgroup of MCG (S)?
v1.3 research notesIs every finite rank subgroup of Whittlesey’s group a Schottky subgroup of MCG (S)? As a consequence of Theorem 2.1, if H <MCG (S) has a finite index ...
Problem 3.6 — Give examples and constructions of virtual Schottky subgroups of MCG (S).
v1.3 research notesGive examples and constructions of virtual Schottky subgroups of MCG (S). One such construction is due to Honglin Min, currently a doctoral candidate ...
Problem 4.1 — Do there exist two surfaces S,S ′, closed and of genus ≥ 2, such that MCG (S) contains a subgroup isomorphic to π1(S′…
v1.3 research notesDo there exist two surfaces S,S ′, closed and of genus ≥ 2, such that MCG (S) contains a subgroup isomorphic to π1(S′) all of whose nontrivial element...