Study joins and intersections of subgroups in free groups, disproving and repairing a lemma.
problem Disproving and repairing a lemma on joins and intersections of subgroups in free groups.
method Analyzing graphs of joins and intersections of finitely generated subgroups of a free group.
result Showed how to disprove and repair a lemma of Imrich and Müller on these graphs.
Proves existence of a rank-preserving map for subgroup joins in free groups.
problem Determining the rank of subgroup joins in free groups.
method Uses epimorphisms and properties of free groups to prove the existence of a rank-preserving map.
result Existence of an epimorphism preserving rank for subgroup joins in free groups.
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,…
The paper generalizes the Hausdorff dimension of limit sets for self-joinings of hyperbolic groups.
problem Calculating the Hausdorff dimension of limit sets for self-joinings of hyperbolic groups.
method The paper generalizes a classical result by considering self-joinings of convex cocompact groups and proving new inequalities for the Hausdorff dimension of directional limit sets.
result For k≤3, the paper establishes bounds on the Hausdorff dimension of directional limit sets for self-joinings of convex cocompact groups. The study proves conditions for rigidity of Kleinian groups using measure theory and ergodic theory.
problem Conditions for rigidity of Kleinian groups via self-joinings.
method Ergodic theory for directional diagonal flows and conformal measure theory.
result Proves dichotomy conditions for Λ_f and Λ, with implications for the dimension and structure of limit sets.
The Hanna Neumann conjecture gives a bound on the intersection of finitely generated subgroups of free groups. We explore a natural extension of this result, which turns out to be true only in the finite index case, and provide counterexamples for the general case. We also see that the graph-based method of generating …
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 …
Study shows how to detect representation extendability using conformal measures.
problem Detecting extendability of representations using conformal measures.
method Using higher rank conformal measures and self-joinings of groups.
result Affirmative answer to detect extendability of representations.
The study proves theorems about quasiflats in hierarchically hyperbolic spaces.
problem Understanding the structure of hierarchically hyperbolic groups.
method Proving theorems about quasiflats and hierarchically hyperbolic groups.
result Proves that a group is hyperbolic if it contains no Z2 subgroups and contains a uniform-quality quasiflat. The paper proves rigidity and ergodicity of horospherical foliations.
problem Rigidity and ergodicity of horospherical foliations in higher rank.
method Establishes higher rank extensions of rigidity theorems for representations of discrete subgroups of divergence type, using conformal measures and boundary maps.
result Proves conformal measure rigidity and ergodicity of horospherical foliations for hypertransverse subgroups.
Given a positive and unitarily invariant Lagrangian L defined in the algebra of Hermitian matrices, and a fixed interval [a,b]⊂R, we study the action defined in the Lie group of n×n unitary matrices U(n) by S(α)=∫abL(α˙(t))dt, where α:[a,b]→U(n) is a …
Study optimal transport for stationary processes, estimating joinings and costs.
problem Optimal transport for stationary stochastic processes.
method Introduced estimators for optimal joinings and costs, established consistency and error rates.
result Consistent estimators of optimal joinings and costs under mild and stronger mixing assumptions.
We introduce the notion of controlled Floyd separation between geodesic rays starting at the identity in a finitely generated group G. Two such geodesic rays are said to be Floyd separated with respect to quasigeodesics if the (Floyd) length of c-quasigeodesics (for fixed but arbitrary c) joining points on the geodesic…
Smooth SE structures on Sasaki-joins and Bott orbifolds constructed.
problem Constructing smooth Sasaki-Einstein structures on Sasaki-joins.
method Categorical relationship between Sasaki-joins and Bott orbifolds, explicit construction.
result Infinitely many smooth SE structures up to dimension eleven, conjecture for all odd dimensions.
The paper bounds growth indicator functions for discrete subgroups in algebraic groups.
problem Bounding growth indicator functions for discrete subgroups in algebraic groups.
method Pointwise bound and equality conditions for growth indicator functions.
result Strict inequalities and equality conditions for growth indicator functions.
New findings on embedding simplicial complexes, showing instability under joins.
problem Conditions for embedding simplicial complexes into double dimension.
method Study of van Kampen obstructions and Smith classes.
result Smith index is not stable under joins, leading to new embeddability results.
Exploring Sasaki metrics in joined manifolds.
problem Existence of constant scalar curvature Sasaki metrics in joined manifolds.
method Investigating the Sasaki cone of joined manifolds and considering continuous families of extremal Sasaki twins.
result Conditions for the existence of constant scalar curvature Sasaki metrics in joined manifolds.
We recognize the Gromoll-Meyer sphere Sigma^7 as the geodesic join of a simple closed geodesic and a minimal subsphere Sigma^5, which can be equivariantly identified with the Brieskorn sphere W^5_3. As applications we in particular determine the full isometry group of Sigma^7, classify all closed subgroups that act fre…
Paper generalizes a theorem for real analytic singularities.
problem No specific problem stated; focuses on generalization.
method Generalization of a theorem for complex singularities.
result Generalized Join theorem for real analytic singularities.
Let G be a connected Lie group acting locally simply transitively on a manifold M. By connecting curves in M we mean the orbits of one-parameter subgroups of G. To block a pair of points m1,m2∈M is to find a finite set B⊂M∖m1,m2 such that every connecting curve joining m1 and $m_2…
S. Parsa proved embedding conditions for simplicial joins.
problem Embeddability conditions for simplicial joins.
method Constructing specific simplicial complexes K and L. result The join K∗L embeds into R2(k+l+1). The paper tackles fairness in overlapping populations using online learning techniques.
problem Improving fairness to subgroups in settings with overlapping populations and sequential predictions.
method The approach draws from the sleeping experts literature in online learning to achieve a goal of unweighted average of false negative and false positive rate for overlapping populations.
result It shows that satisfying the guarantee for multiple overlapping groups is not straightforward and can be statistically impossible even when predictors perform well separately on each subgroup.
Proofs show embedding conditions for complex joins and factors.
problem Embeddability conditions for complex joins and factors.
method Configuration spaces, equivariant suspension theorem, join and cone properties.
result Embeddability conditions for K∗[3] and K. Paper shows how to join Milnor fibers of real analytic functions.
problem Understanding the topology of real analytic singularities.
method Analyzes tubular Milnor fibers and their homotopy equivalence.
result Tubular Milnor fibers of f_1 + f_2 are homotopy equivalent to the join of f_1 and f_2.
The ellipticity graph of a free group F was defined by I. Kapovich and M. Lustig in order to study the outer automorphism group of F, which acts on this graph. The graph was constructed to be analogous to the curve complex of a surface. It is a bipartite graph, whose vertices are conjugacy classes of nontrivial ele…
A new method joins two arcs with a degree of freedom.
problem Joining two arcs with a precise point.
method Geometric approach using tangent vectors and points.
result A novel method to determine the join point.
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…
New minimal surfaces in spheres with complex topologies from capillarity.
problem Constructing minimal surfaces in spheres with rich topologies.
method General construction of embedded minimal and constant mean curvature surfaces in Sn using capillary hypersurfaces. result Non-trivial sphere bundles over various base spaces, including Stiefel manifolds and complex quadrics.
Extending work of Kapouleas and Yang, for any integers N≥2, k,ℓ≥1, and m sufficiently large, we apply gluing methods to construct in the round 3-sphere a closed embedded minimal surface that has genus kℓm2(N−1)+1 and is invariant under a Dkm×Dℓm subgroup of O(4), where …
This work defines a categorical notion of principal bundles.
problem Different definitions of principal bundles in various categories.
method Formulated in join-restriction categories, which generalize partial maps.
result Shows the tangent bundle as the product of tangent space and group object.
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.
problem Understanding polynomial structure and tropicalization of b-Hurwitz numbers. method Introducing CJT-refinement of symmetric functions on Fock space.
result Derives tropicalization of b-Hurwitz numbers and solves an open problem. Holomorphic cylinders converge to disks joined by flow lines.
problem Convergence of perturbed holomorphic cylinders.
method Exponential estimates and flow line computation.
result Holomorphic cylinders converge to two disks joined by a flow line.
Generalizes Sasaki join construction for quasi-regular Sasakian structures.
problem Generalize Sasaki join construction for quasi-regular Sasakian structures.
method Inductive procedure for constructing Sasakian metrics of constant scalar curvature.
result Infinitely many smooth 7-manifolds with constant scalar curvature Sasaki metrics.
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…
New proof shows non-embeddable polyhedra and conditions for embedding products.
problem Non-embeddability of polyhedra in high-dimensional spaces.
method Geometric cohomology and embedding conditions for polyhedra products.
result Compact polyhedra can embed in higher dimensions under specific conditions.
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 S2×S5. 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 …
Characterizes homology d-manifolds with g2=3 for d≥3.
problem Classify homology d-manifolds with specific g2 values.
method Combinatorial characterization and operations like joins, retriangulations, and connected sums.
result Homology d-manifolds with g2=3 are spheres and can be derived from previous ones.
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…
Study counts and equidistributes tori in Kleinian group self-joinings.
problem Counting and equidistribution of tori in Kleinian group self-joinings.
method Analyzes d-dimensional torus packings invariant under a self-joining of a Kleinian group. result Equidistribution results for tori with small volume in a class of d-dimensional torus packings. The study shows acylindrical hyperbolicity for Artin groups not associated with joins or cones.
problem Proving acylindrical hyperbolicity for Artin groups of infinite type not associated with joins or cones.
method Developing and extending the clique-cube complex and action studies of Charney and Morris-Wright.
result Acylindrical hyperbolicity demonstrated for Artin groups of infinite type associated with graphs that are not cones.
Let U be an open subset of R^n. Let L^2=L^2(U,dx) and H^1_0=H^1_0(U) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group G of invertible operators on H^1_0 which preserve the L^2-inner product. When U is bounded and the border ∂U is smooth, this…
This paper studies posets associated with link diagrams and their algebraic properties.
problem Understanding the algebraic structure of posets derived from link diagrams.
method Associaed posets with link diagrams, proved distributivity, and described join irreducibles.
result Posets of Kauffman states are distributive lattices and isomorphic to coefficient quiver posets.
Robustifies tree learning algorithms for corrupted data.
problem Learning latent tree structures with corrupted vector observations.
method Presented robustified algorithms using truncated inner product.
result Optimalities of robust CLRG and NJ verified by sample complexities and impossibility results.
Complex projective structures can be joined via simple bubbling and debubbling.
problem Joining complex projective structures with quasi-Fuchsian holonomy.
method Performing (de)grafting via a sequence of one bubbling and one debubbling.
result Any complex projective structure can be joined to the uniformizing structure by a simple sequence of one bubbling and one debubbling.
Maximizes arcs on a sphere with constraints.
problem Finding the maximum number of arcs on a punctured sphere.
method Analyzing square annular diagrams and their dual curves.
result Proves the maximum size of arcs is \(\binom{n}{3}\).