arXiv research

Study optimal transport for stationary processes, estimating joinings and costs.

Smooth SE structures on Sasaki-joins and Bott orbifolds constructed.

Exploring Sasaki metrics in joined manifolds.

New findings on embedding simplicial complexes, showing instability under joins.

Paper generalizes a theorem for real analytic singularities.

S. Parsa proved embedding conditions for simplicial joins.

Proofs show embedding conditions for complex joins and factors.

We study graphs of (generalized) joins and intersections of finitely generated subgroups of a free group. We show how to disprove a lemma of Imrich and Müller on these graphs and how to repair this lemma.

A new method joins two arcs with a degree of freedom.

We will show that if a proper complete CAT(0) space X has a visual boundary homeomorphic to the join of two Cantor sets, and X admits a geometric group action by a group containing a subgroup isomorphic to Z^2, then its Tits boundary is the spherical join of two uncountable discrete sets. If X is geodesically complete,…

In this paper we show that the matrix of chromatic joins and the Gram matrix of the Temperley-Lieb algebra are similar (after rescaling), with the change of basis given by diagonal matrices.

We give a partial characterization of bordered Floer homology in terms of sutured Floer homology. The bordered algebra and modules are direct sums of certain sutured Floer complexes. The algebra multiplication and algebra action correspond to a new gluing map on SFH. It is defined algebraically, and is a special case o…

This work defines a categorical notion of principal bundles.

Let f_1 and f_2 be real analytic germs of independent variables. In this paper, we assume that f_1, f_2 and f = f_1 + f_2 satisfy a_f -condition. Then we show that the tubular Milnor fiber of f is homotopy equivalent to the join of tubular Milnor fibers of f_1 and f_2.

We introduce the functor * which assigns to every metric space X its symmetric join *X. As a set, *X is a union of intervals connecting ordered pairs of points in X. Topologically, *X is a natural quotient of the usual join of X with itself. We define an Isom(X)-invariant metric d* on *X. Classical concepts known for H…

Refines Hurwitz numbers with a two-parameter theory.

Holomorphic cylinders converge to disks joined by flow lines.

We give a survey of our recent work describing a method which combines the Sasaki join construction with the admissible Kähler construction of to obtain new extremal and new constant scalar curvature Sasaki metrics, including Sasaki-Einstein metrics. The constant scalar curvature Sasaki metrics also provide explicit so…

Let $H, K$ be two finitely generated subgroups of a free group, let $\langle H, K \rangle$ denote the subgroup generated by $H, K$, called the join of $H, K$, and let neither of $H$, $K$ have finite index in $\langle H, K \rangle$. We prove the existence of an epimorphism $ζ: \langle H, K \rangle \to F_2$, where $F_2$ …

New proof shows non-embeddable polyhedra and conditions for embedding products.

Study shows how to detect representation extendability using conformal measures.

The paper generalizes the Hausdorff dimension of limit sets for self-joinings of hyperbolic groups.

We describe various constructions in Sasakian geometry. First we generalize the join construction of the first two authors to arbitrary Sasakian manifolds. We then give several examples, including ones which prove the existence of Sasakian-Einstein metrics on manifolds homeomorphic to $S^2\times S^5.$ Then we use a gen…

We propose a primitive called PJOIN, for "predictive join," which combines and extends the operations JOIN and LINK, which Valiant proposed as the basis of a computational theory of cortex. We show that PJOIN can be implemented in Valiant's model. We also show that, using PJOIN, certain reasonably complex learning and …

By combining the join construction from Sasakian geometry with the Hamiltonian 2-form construction from Kähler geometry, we recover Sasaki-Einstein metrics discovered by physicists. Our geometrical approach allows us to give an algorithm for computing the topology of these Sasaki-Einstein manifolds. In particular, we e…

Characterizes homology d-manifolds with g2=3 for d≥3.

Let $S$ be an $n$-punctured sphere, with $n \geq 3$. We prove that $\binom{n}{3}$ is the maximum size of a family of pairwise non-homotopic simple arcs on $S$ joining a fixed pair of distinct punctures of $S$ and pairwise intersecting at most twice. On the way, we show that a square annular diagram $A$ has a corner on …

The study shows acylindrical hyperbolicity for Artin groups not associated with joins or cones.

Study counts and equidistributes tori in Kleinian group self-joinings.

This paper studies posets associated with link diagrams and their algebraic properties.

Robustifies tree learning algorithms for corrupted data.

Globally hyperbolic spacetimes admitting infinitely many causal (and timelike) homotopy classes of curves joining two prescribed points, are exhibited and discussed.

New Sasaki metrics with constant scalar curvature on sphere bundles are constructed.

The study shows how to construct $d$-spheres from $(d-1)$-spheres and $d$-balls without additional vertices.

Let H and K be subgroups of a free group of ranks h and k \geq h. We prove the following strong form of Burns' inequality: rank(H \cap K) - 1 \leq 2(h-1)(k-1) - (h-1)(rank(H \vee K) -1). A corollary of this, also obtained by L. Louder and D. B. McReynolds, has been used by M. Culler and P. Shalen to obtain information …

The paper explores generalizations of Mirzakhani's recursion and computes volumes for physical gravity models.

The paper studies Sasakian geometry on sphere bundles, focusing on extremal metrics and cohomology.

We prove that knowing the length of geodesics joining points on the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.

We establish a boundary connected sum theorem for asymptotically hyperbolic Einstein metrics; this requires no nondegeneracy hypothesis. We also show that if the two metrics have scalar positive conformal infinities, then the same is true for this boundary join.

New theorem proves rigidity of Kleinian group representations under specific conditions.

Classifies orbit closures in translation surface strata.

We present a baseline approach for cross-modal knowledge fusion. Different basic fusion methods are evaluated on existing embedding approaches to show the potential of joining knowledge about certain concepts across modalities in a fused concept representation.

Merging datasets is a key operation for data analytics. A frequent requirement for merging is joining across columns that have different surface forms for the same entity (e.g., the name of a person might be represented as "Douglas Adams" or "Adams, Douglas"). Similarly, ontology alignment can require recognizing disti…

We construct embedded closed minimal surfaces in the round three-sphere, resembling two parallel copies of the Clifford torus, joined by m^2 small catenoidal bridges symmetrically arranged along a square lattice of points on the torus.

Introduces zebra structures on surfaces for directional foliation.

Study on stability of Sasaki structures under deformations.

The space of all probability measures having positive density function on a connected compact smooth manifold $M$, denoted by $\mathcal{P}(M)$, carries the Fisher information metric $G$. We define the geometric mean of probability measures by the aid of which we investigate information geometry of $\mathcal{P}(M)$, equ…

We prove a continuity property for ending invariants of convergent sequences of Kleinian surface groups. We also analyze the bounded curve sets of such groups and show that their projections to non-annular subsurfaces lie a bounded Hausdorff distance from geodesics joining the projections of the ending invariants.