In a complete Riemannian manifold (M,g) if the hessian of a real valued function satisfies some suitable conditions then it restricts the geometry of (M,g). In this paper we characterize all compact rank-1 symmetric spaces, as those Riemannian manifolds (M,g) admitting a real valued function u such that the hessian of u has atmost two eigenvalues −u and − 2 , under some mild hypothesis on (M,g)...

An isometric path is merely any shortest path between two vertices. If the vertices of the hypercube Qn are represented by the set of 0–1 vectors of length n, an isometric path is obtained by changing the coordinates of a vector one at a time, never changing the same coordinate more than once. The minimum number of isometric paths required to cover the vertices of Qn is at least 2=(n+1). We sho...

Ring geometry and the geometry of matrices meet naturally at the ring R := Kn×n of n× n matrices with entries in a (not necessarily commutative) field K. Our aim is to strengthen the interaction between these disciplines. Below we sketch some results from either side, even though not in their most general form, but in a way which is tailored for our needs. Let us start with ring geometry, where...

For odd m, I write down tensors in C m ⊗C m ⊗C m of border rank at least 2m − 1, showing the non-triviality of the Young-flattening equations of [6] that vanish on the matrix multiplication tensor. I also study the border rank of the tensors of [1] and [3]. I show the tensors T 2 k ∈ C k ⊗C 2 k ⊗C 2 k , of [1], despite having rank equal to 2 k+1 − 1, have border rank equal to 2 k. I show the eq...

We analyze the asymptotic cones of Teichmüller space with the Teichmüller metric, pT pSq, dT q. We give a new proof of a theorem of Eskin-Masur-Rafi [EMR13] which bounds the dimension of quasiisometrically embedded flats in pT pSq, dT q. Our approach is an application of the ideas of Behrstock [Beh06] and Behrstock-Minsky [BM08] to the quasiisometry model we built for pT pSq, dT q in [Dur13].

In this paper we characterize the optimization geometry of a matrix factorization problem where we aim to find n×r and m×r matrices U and V such that UV T approximates a given matrixX. We show that the objective function of the matrix factorization problem has no spurious local minima and obeys the strict saddle property not only for the exact-parameterization case where rank(X) = r, but also f...

We show (under mild topological assumptions) that small oscillation of the unit normal vector implies Reifenberg flatness. We then apply this observation to the study of chord-arc domains and to a quantitative version of a two-phase free boundary problem for harmonic measure previously studied by Kenig-Toro [KT06].

In a previous paper an incomplete investigation into regular two-graphs on 36 vertices established the existence of at least 227. Using a more efficient algorithm, the two authors have independently verified that in fact these 227 comprise the complete set. An immediate consequence of this is that all strongly regular graphs with parameters (35,16,6,8), (36,14,4,6), (36,20,10,12) and their comp...

A k-fault-tolerant gossip graph is a (multiple) graph whose edges are linearly ordered such that for any ordered pair of vertices u and v, there are k + 1 edge-disjoint ascending paths from u to v. Let τ(n, k) denote the minimum number of edges in a k-fault-tolerant gossip graph with n vertices. In this paper, we present upper and lower bounds on τ(n, k) which improve the previously known bound...

We propose a new Riemannian geometry for fixed-rank matrices that is specifically tailored to the low-rank matrix completion problem. Exploiting the degree of freedom of a quotient space, we tune the metric on our search space to the particular least square cost function. At one level, it illustrates in a novel way how to exploit the versatile framework of optimization on quotient manifold. At ...