Convex Equipartitions via Equivariant Obstruction Theory

We describe a regular cell complex model for the configuration space F(\R^d,n). Based on this, we use Equivariant Obstruction Theory to prove the prime power case of the conjecture by Nandakumar and Ramana Rao that every polygon can be partitioned into n convex parts of equal area and perimeter.Comment: Revised and improved version with extra explanations, 20 pages, 7 figures, to appear in Israel J. Mat

The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes

We compute the complete Fadell-Husseini index of the 8 element dihedral group D_8 acting on S^d \times S^d, both for F_2 and for integer coefficients. This establishes the complete goup cohomology lower bounds for the two hyperplane case of Gr"unbaum's 1960 mass partition problem: For which d and j can any j arbitrary measures be cut into four equal parts each by two suitably-chosen hyperplanes in R^d? In both cases, we find that the ideal bounds are not stronger than previously established bounds based on one of the maximal abelian subgroups of D_8.Comment: new version revised according to referee's comments, 44 pages, many diagrams; a shorter version of this will appear in Topology and its Applications (ATA 2010 proceedings

Tverberg plus constraints

Many of the strengthenings and extensions of the topological Tverberg theorem can be derived with surprising ease directly from the original theorem: For this we introduce a proof technique that combines a concept of "Tverberg unavoidable subcomplexes" with the observation that Tverberg points that equalize the distance from such a subcomplex can be obtained from maps to an extended target space. Thus we obtain simple proofs for many variants of the topological Tverberg theorem, such as the colored Tverberg theorem of Zivaljevic and Vrecica (1992). We also get a new strengthened version of the generalized van Kampen-Flores theorem by Sarkaria (1991) and Volovikov (1996), an affine version of their "j-wise disjoint" Tverberg theorem, and a topological version of Soberon's (2013) result on Tverberg points with equal barycentric coordinates.Comment: 15 pages; revised version, accepted for publication in Bulletin London Math. Societ

Correlations Estimates in the Lattice Anderson Model

We give a new proof of correlation estimates for arbitrary moments of the resolvent of random Schr\"odinger operators on the lattice that generalizes and extends the correlation estimate of Minami for the second moment. We apply this moment bound to obtain a new $n$-level Wegner-type estimate that measures eigenvalue correlations through an upper bound on the probability that a local Hamiltonian has at least $n$ eigenvalues in a given energy interval. Another consequence of the correlation estimates is that the results on the Poisson statistics of energy level spacing and the simplicity of the eigenvalues in the strong localization regime hold for a wide class of translation-invariant, selfadjoint, lattice operators with decaying off-diagonal terms and random potentials.Comment: 16 page

Optimal bounds for the colored Tverberg problem

We prove a "Tverberg type" multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Barany et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Barany & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.Comment: 17 pages, 3 figures; revised version (February 2013), to appear in J. European Math. Soc. (JEMS