Problems

What is the S_n -module structure of tildeH_3k-4(NM_k(n)) ?

Assuming that a minimal partition extender exists, is it unique?

If Δ is a simplicial complex such that depth k[Δ] ≥ dim k[Δ]-1 for all fields k , then Δ has a shelling extender.

What are the homotopy types of Q_n(Π) ? (in general their order complexes aren't necessarily spheres, or even Cohen-Macaulay)

Does a Nim-basis, if it exists, necessarily consist of the disjoint unions of circuits of the complex?

(i) For every triangulation K of a Witt space with vanishing middle intersection homology GIN(K) ⊂ GIN(C(d,n)). (2.1) (ii) The strong upper bound conjecture holds for arbitrary polyhedral complexes (and even for all regular cell complexes…

Let Δ be a Nim-regular complex, F a nonempty face, D_i a minimal cover of F by circuits and D = uplus_i D_i. Is it necessarily true that D - F is not a DUOC?

Give a bound R depending on some invariants of the simplicial complex Δ such that for r ≥R the polynomial h^sd^r(Δ)(t) has only real roots.

The magnitude homology of a graph obtained by gluing two cycle graphs C_3 along single edges to a single cycle graph C_4 has diagonal magnitude homology provided those triangles are not attached to opposite sides of the 4-cycle.

Given a finite CW complex X which is not contractible and two ρ-immersed matroids (M, l) and (N, l') such that T_X(M, l) ≃ T_X(N, l'), if there exists a surjective weak map τ: M → N, then τ^# is an isomorphism.

Can occurrences of mesh patterns be used to compute the Betti numbers of permutation complexes? Or can we define a set of mesh patterns P such that if π avoids P its permutation complex is contractible?

For central arrangements whose underlying matroid is connected, the homotopy type of the complement determines the underlying matroid.