Related papers: The Bourgain-Tzafriri conjecture and concrete cons…

This paper explores the intriguing connections between the invariant subspace problem, the Kadison-Singer problem, and the Borel conjecture. The Kadison-Singer problem, originally formulated in terms of pure states on C*-algebras, was later…

We establish formulas for the Hilbert series of the Chow ring of a polymatroid using arbitrary building sets. For braid matroids and minimal building sets, our results produce new formulas for the Poincar\'e polynomial of the moduli space…

A conjecture of Voisin states that two points on a smooth projective complex variety whose algebra of holomorphic forms is generated in degree 2 are rationally equivalent to each other if and only if their difference lies in the third step…

An old conjecture in non-K\"ahler geometry states that, if a compact Hermitian manifold has constant holomorphic sectional curvature, then the metric must be K\"ahler (when the constant is non-zero) or Chern flat (when the constant is…

We prove that Brauer's Height Zero Conjecture holds for p-blocks of finite groups with metacyclic defect groups. If the defect group is nonabelian and contains a cyclic maximal subgroup, we obtain the distribution into p-conjugate and…

The study of infinitesimal deformations of a variety embedded in projective space requires that of deformations of a collection of points, as specified by a zero-dimensional scheme. Further, basic problems in infinitesimal interpolation…

Using the structure of Singer cycles in general linear groups, we prove that a conjecture of Zeng, Han and He (2007) holds in the affirmative in a special case, and outline a plausible approach to prove it in the general case. This…

The geometry conjecture, which was posed nearly a quarter of a century ago, states that the fixed point set of the composition of projectors onto nonempty closed convex sets in Hilbert space is actually equal to the intersection of certain…

We construct log canonical pairs $(X,B)$ with $B$ a nonzero reduced divisor and $K_X+B$ ample that have the smallest known volume. We conjecture that our examples have the smallest volume in each dimension. The conjecture is true in…

We prove the Nonvanishing conjecture for uniruled projective log canonical pairs of dimension $n$, assuming the Nonvanishing conjecture for smooth projective varieties in dimension $n-1$. We also show that the existence of good minimal…

We prove some new equivalences of the paving conjecture and obtain some estimates on the paving constants. In addition we give a new family of counterexamples to one of the Akemann-Anderson conjectures.

We prove that Bourgain's hyperplane conjecture and the Kannan-Lov\'asz-Simonovits (KLS) isoperimetric conjecture hold true up to a factor that is polylogarithmic in the dimension.

In the early 2000's the first and second named authors worked for a period of six years in an attempt of proving the Compositional Shuffle Conjecture [1]. Their approach was based on the discovery that all the Combinatorial properties…

The Bourque-Ligh conjecture states that if $S=\{x_1,x_2,\ldots,x_n\}$ is a gcd-closed set of positive integers with distinct elements, then the LCM matrix $[S]=[\hbox{lcm}(x_i,x_j)]$ is invertible. It is well known that this conjecture…

The Kochen-Specker no-go theorem established that hidden-variable theories in quantum mechanics necessarily admit contextuality. This theorem is formally stated in terms of the partial Boolean algebra structure of projectors on a Hilbert…

We find a sharp combinatorial bound for the metric entropy of sets in R^n and general classes of functions. This solves two basic combinatorial conjectures on the empirical processes. 1. A class of functions satisfies the uniform Central…

Akemann and Anderson made a conjecture about ``paving'' projections in finite dimensional matrix algebras which, if true, would settle the well-known Kadison-Singer problem. We falsify their conjecture by an explicit seqence of…

A metric projective structure is a manifold equipped with the unparametrised geodesics of some pseudo-Riemannian metric. We make acomprehensive treatment of such structures in the case that there is a projective Weyl curvature nullity…

The restricted invertibility theorem was originally introduced by Bourgain and Tzafriri in $1987$ and has been considered as one of the most celebrated theorems in geometry and analysis. In this note, we present weighted versions of this…

We construct polynomial conformal invariants, the vanishing of which is necessary and sufficient for an $n$-dimensional suitably generic (pseudo-)Riemannian manifold to be conformal to an Einstein manifold. We also construct invariants…