Study estimates gaps in semigroup products, proving embedding properties.
problem Estimating singular value gaps in semigroup products.
method Lower estimates for singular value gaps of free products of semigroups in ping-pong position.
result Groups generated by semigroups in ping-pong position are quasi-isometrically embedded.
Study ping-pong dynamics in hyperbolic-like groups with non-simple points.
problem Investigate the ping-pong dynamics of hyperbolic-like groups.
method Explicitly provide a proper ping-pong partition for any pair of non-cyclic point stabilizers.
result Existence of a proper ping-pong partition for any pair of non-cyclic point stabilizers.
New groups discovered with unique properties in a specific space.
problem Finding new discrete subgroups with special properties in a mathematical space.
method Proved by showing groups play ping-pong on cones, related to crooked surfaces.
result Infinite family of discrete subgroups with remarkable properties in Sp4(R). Geometric group theory explores groups through their geometric properties.
problem Understanding groups via geometric properties.
method Cayley and Schreier graphs, ping-pong lemma, quasi-isometries, growth of groups, hyperbolicity.
result Gromov's theorem on groups of polynomial growth and amenability.
We prove that all atoroidal automorphisms of Out(FN) act on the space of projectivized geodesic currents with generalized north-south dynamics. As an application, we produce new examples of non virtually cyclic, free and purely atoroidal subgroups of Out(FN) such that the corresponding free group extension is hyp…
In this paper, we prove a quantitative version of the Tits alternative for negatively pinched manifolds X. Precisely, we prove that a nonelementary discrete isometry subgroup of Isom(X) generated by two non-elliptic isometries g, f contains a free subgroup of rank 2 generated by isometries fN,h …
This note removes technical assumptions and characterizes relatively dominated representations.
problem Geometrically finiteness and Anosov conditions in higher-rank settings.
method Characterization using eigenvalue gaps and limit maps.
result Relatively dominated representations are characterized using eigenvalue gaps and limit maps.
We prove that if φ,ψ∈Out(FN) are hyperbolic iwips (irreducible with irreducible powers) such that <φ,ψ>≤Out(FN) is not virtually cyclic then some high powers of φ and ψ generate a free subgroup of rank two, all of whose nontrivial elements are again hyperbolic iwips. Being a hyperbolic iwip element of $…
Local-to-global principle for Morse actions on symmetric spaces.
problem Recognizing Morse actions on symmetric spaces.
method Equivariant Morse quasiisometric embeddings of trees into symmetric spaces.
result Algorithmic recognizability of Morse actions and construction of Morse Schottky subgroups.
We prove that every acylindrically hyperbolic group that has no non-trivial finite normal subgroup satisfies a strong ping pong property, the Pnaive property: for any finite collection of elements h1,…,hk, there exists another element γ=1 such that for all i, $\langle h_i, γ\rangle = \langle h_i …
Researchers discover all affinely homogeneous models for surfaces in 4D space.
problem Identifying all affinely homogeneous models for surfaces in 4D space.
method Improved power series method of equivalence, capturing invariants at the origin, creating branches, and infinitesimalizing calculations.
result Find several inequivalent terminal branches yielding each to some nonempty moduli space of homogeneous models.
We study subgroups of the mapping class group of the torus generated by powers generated by powers of Dehn twists. We give a criterion to show when a collection of powers Dehn twists generates a free group using the ping pong lemma. We show that the subgroup generated by three uniform powers of Dehn twists can be eithe…
We present a practical algorithm which, given a non-archimedean local field K and any two elements A,B∈SL2(K), determines after finitely many steps whether or not the subgroup ⟨A,B⟩≤SL2(K) is discrete and free of rank two. This makes use of the Ping Pong Lemma applied to the act…
Study matches two noisy point clouds with geometric transformations and relabeling.
problem Matching two noisy point clouds with orthogonal transformations and relabeling.
method Information-theoretic results and Ping-Pong algorithm for computational alignment.
result The Ping-Pong algorithm retrieves the planted signal after one step.
We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex…
Extends positive and almost positive links to successively almost positive ones.
problem Extending properties of positive and almost positive diagrams and links.
method Introducing successively almost positive diagrams and links, and analyzing their properties.
result Improves known results of positive and almost positive links.
The paper extends positivity results from vector bundles to Kobayashi positive ones.
problem Extending positivity results from vector bundles to Kobayashi positive ones.
method Using convexity of Kobayashi positive Finsler metrics and duality for convex Finsler metrics.
result The quotient and tensor product of Kobayashi positive vector bundles are also Kobayashi positive.
The study establishes conditions for positive and quasi-positive links.
problem Characterizing and testing positive and quasi-positive links.
method Proves necessary conditions for link concordance and positivity.
result Characterizes positive links with unlinking number 1 and 2, and tests positive links as closures of positive braids.
The paper defines new types of positivity and proves properties of Schur forms for vector bundles.
problem Defining and characterizing new types of positivity for vector bundles.
method Introducing and characterizing two types of strongly decomposable positivity, proving properties of Schur forms.
result Schur forms of strongly decomposable positive vector bundles are positive or weakly positive, answering a question of Griffiths.
New characterizations of partial positivity using Hörmander's L2-estimate.
problem Characterizing partial positivity in complex geometry.
method Using a twisted version of Hörmander's L2-estimate. result New characterizations of partial positivity, including uniform q-positivity and RC-positivity. Introduces Θ-positivity in Lie groups, generalizing Lusztig's positivity.
problem Generalizing Lusztig's total positivity to a broader class of Lie groups.
method Introduces and studies Θ-positivity in real simple Lie groups. result Four families of Lie groups admit Θ-positive structures. New bounds on Jones polynomial positivity for specific links.
problem Determining when Jones polynomial of positive links is non-negative.
method Developed new bounds and used them to obstruct positivity for infinitely many almost-positive diagrams.
result Infinitely many knots are classified as almost-positive.
Uniform RC-positivity results for direct image bundles.
problem Understanding the relation between rational connectedness and RC-positivity.
method Analyzing vector bundles and their direct images, using weak RC-positivity as a starting point.
result Uniform RC-positivity of direct image bundles under weak RC-positivity conditions.
We introduce the notion of a positive opetope and positive opetopic cardinals as certain finite combinatorial structures. The positive opetopic cardinals to positive-to-one polygraphs are like simple graphs to free omega-categories over omega-graphs, c.f. [MZ]. In particular, they allow us to give an explicit combinato…
Characterizes a subset of links using quasipositive and homogeneous properties.
problem Understanding the properties of T-positive links.
method Characterization through strongly quasipositive and T-homogeneous braids.
result T-positive links are precisely the strongly quasipositive links that are closures of T-homogeneous braids.
Positive braid knots have simple knot Floer homology.
problem Computing knot Floer homology for positive braids.
method Computed the next-to-top term of knot Floer homology.
result Rank is 1 for any prime positive braid knot.
Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.
problem Generalizing Lusztig's total positivity to the setting of general real semisimple Lie groups.
method Classifying Lie groups admitting a positive structure and establishing key properties of unipotent positive semigroups.
result Establishes key geometric properties of elements in the positive semigroup.
Extends Perelman's theorem to positive intermediate curvature conditions.
problem Positive intermediate curvature conditions and their implications.
method Generalization of Perelman's gluing theorem to positive intermediate curvature conditions.
result Observer moduli space can have non-trivial higher homotopy groups.
An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, 4-genus and 3-genus of a positive knot are equal. In this paper, we p…
Satellite links of fully positive braids are characterized.
problem Characterizing satellite links of fully positive braids.
method Analyzing fully positive braids and their satellites.
result Satellite links of fully positive braids are characterized by specific conditions.
New Θ-positive representations of surface groups discovered.
problem Generalizing Lusztig's total positivity to surface groups.
method Introducing Θ-positivity and proving properties of Θ-positive representations. result Discrete and faithful Θ-positive representations exist and form open sets in representation varieties. Workshop notes on positivity in Lie groups and its applications.
problem Understanding total and Θ-positivity in semisimple Lie groups. method Discussion and analysis of existing theories and recent developments.
result Progress in classifying higher Teichmüller spaces through Θ-positivity. Positive representations on surfaces have positive cross-ratios and satisfy a collar lemma.
problem Characterizing representations of surface groups with positive properties.
method Proving a collar lemma and showing positivity of cross-ratios for Θ-positive representations. result Closed subsets of representation varieties are characterized by Θ-positive representations. New upper bound on Jones polynomial for fibered positive links.
problem Classifying positive and non-positive knots of crossing number ≤ 12.
method Proved a new upper bound on the maximum degree of Jones polynomial for fibered positive knots.
result Maximum degree of Jones polynomial for fibered positive knots is at most four times the minimum degree.
We characterise positive braid links with positive Seifert form via a finite number of forbidden minors. From this we deduce a one-to-one correspondence between prime positive braid links with positive Seifert form and simply laced Dynkin diagrams, as well as a simple classification of alternating positive braid knots.
Proves positive Euler characteristic for certain manifolds with positive second intermediate Ricci curvature.
problem Hopf's conjecture on positive sectional curvature and its failure under relaxed conditions.
method Non-trivial extension of the Four Periodicity Theorem to higher degrees.
result Proves positive Euler characteristic for specific manifolds with positive second intermediate Ricci curvature.
Curvature of Bazaikin spaces fully characterized, with unique quasi-positive metric.
problem Characterizing sectional curvature of Bazaikin spaces.
method Completely characterized the sectional curvature of all 13-dimensional Bazaikin spaces.
result Unique quasi-positively curved Riemannian metric on Bazaikin spaces, with a unique almost positively curved but not positively curved space.
We prove new results about unknotting fibered positive knots and braids.
problem Proving the unknotting number equals genus for fibered positive knots and braids.
method Analyzing positive braid diagrams and fibered positive knots, proving new constraints and conjectures.
result We found fibered positive knots that cannot be unknotted optimally, contradicting Stoimenow's conjecture.
Positive Thompson links are proven for oriented subgroup elements.
problem Proving properties of Thompson links with positive elements.
method Analyzing elements of the oriented subgroup of the Thompson group.
result Positive oriented Thompson links are established.
The paper proves positivity of third Chern form for certain vector bundles.
problem Proving positivity of third Chern form for Griffiths positive vector bundles.
method Analyzing mixed discriminants and Schur forms.
result Positivity of third Chern form for Griffiths positive vector bundles.
Study on quantum sl3 invariant for positive links.
problem Characterizing and understanding the quantum sl3 invariant of positive links. method Skein theory of sl3-webs, explicit formulae, diagrammatic quantities, obstructions. result Positive links are fibered if and only if the second coefficient of the polynomial is 1.
We introduce Θ-positivity, a new notion of positivity in real semisimple Lie groups. The notion of Θ-positivity generalizes at the same time Lusztig's total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of …
Preserves positive intermediate curvature on manifolds.
problem Obstructs positive intermediate curvature on partial tori.
method Shows smooth interpolation of metrics with positive intermediate curvature.
result Proves non-existence of certain manifolds with positive intermediate curvature.
Study k-positive surface group representations and their degenerations.
problem Understanding the behavior of surface group representations under degenerations.
method Introduced k-positive representations and studied their degenerations using a limit theorem for positively ratioed representations.
result Degenerations of k-positive representations can lead to limits that are at least (k-3)-positive and irreducible limits are (k-1)-positive.
The paper explores criteria for positivity of forms and proves their strong positivity in specific cases.
problem Understanding positivity of exterior forms on complex vector spaces.
method Dimensionality reduction and criteria based on Hermitian matrices.
result Strong positivity of certain forms proven by duality.
New methods learn from PU data with non-representative positives.
problem Learning from PU data with non-representative positive classes.
method Integrates negative-unlabeled and unlabeled-unlabeled learning, or uses a recursive risk estimator.
result Effective across various real-world datasets and forms of positive bias.
Abstract reviews hyperbolic positive energy theorems.
problem Analyzing positive energy theorems for hyperbolic spaces.
method Review of existing literature on asymptotically hyperbolic manifolds.
result Summarizes positive energy theorems for hyperbolic spaces.
A complex disproves a curvature property.
problem Proving non-equivalence of curvature properties.
method Analyzing a specific 2-complex presentation.
result The contracting non-positive immersions property is not equivalent to positive maximal irreducible curvature.