Study automorphisms and Torelli groups of right-angled Artin groups.
problem Characterize automorphisms and Torelli groups of right-angled Artin groups.
method Introduce palindromic automorphism and Torelli groups, relate to GL(n,Z) and mapping class groups, find generating sets.
result Finite generating sets for Pi A_G and centralizer, determine when groups coincide.
A palindrome in a free group F_n is a word on some fixed free basis of F_n that reads the same backwards as forwards. The palindromic automorphism group ΠA_n of the free group F_n consists of automorphisms that take each member of some fixed free basis of F_n to a palindrome; the group ΠA_n has close connections with h…
The braid group Bn, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism rev:Bn→Bn, v↦vˉ, defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation $τ:x \mapsto Δ^{…
We present and discuss some open problems formulated by participants of the International Workshop "Knots, Braids, and Auto\-mor\-phism Groups" held in Novosibirsk, 2014. Problems are related to palindromic and commutator widths of groups; properties of Brunnian braids and two-colored braids, corresponding to an amalga…
Graph of groups palindromic width mostly infinite.
problem Understanding palindromic widths in graph of groups structures.
method Proved infinite palindromic width for specific graph of groups structures.
result Palindromic width of graph of groups mostly infinite.
Let F=<a,b> be a rank two free group. A word W(a,b) in F is {\sl primitive} if it, along with another group element, generates the group. It is a {\sl palindrome} (with respect to a and b) if it reads the same forwards and backwards. It is known that in a rank two free group any primitive element is conjuga…
We give a unified geometric approach to some theorems about primitive elements and palindromes in free groups of rank 2. The geometric treatment gives new proofs of the theorems. Dedicated to Bill Harvey on his 65th birthday.
We consider non-elementary representations of two generator free groups in PSL(2,C), not necessarily discrete or free, G=<A,B>. A word in A and B, W(A,B), is a palindrome if it reads the same forwards and backwards. A word in a free group is {\sl primitive} if it is part of a minimal generating …
Product of shellable complexes yields shellable triangulations under tameness conditions.
problem Understanding shellability in products of simplicial complexes.
method Definition and proof of shellability properties for products of complexes under tameness conditions.
result Product of shellable complexes yields shellable triangulations under certain conditions.
Specialized knot theory theorems for strongly involutive links.
problem Classical Alexander and Markov theorems for links.
method Equivariant closure map for strongly involutive links.
result Surjective equivariant closure map up to equivalence of strongly involutive links.
Introduces q-transpose for q-deformed modular group matrices.
problem Understanding q-deformed rational numbers and their properties. method Introduces q-transpose and applies it to refine q-deformed modular group actions. result New proof and refinement of Leclere and Morier-Genoud's trace palindromicity theorem.
Shellable tilings on simplicial complexes help understand their structure.
problem Understanding the structure of simplicial complexes through tilings.
method Proving the existence of shellable h-tilings on finite simplicial complexes after stellar subdivisions.
result The h-vector of a tiling is determined by the critical vector, with palindromic properties for closed triangulated manifolds.
The braid group Bn, endowed with Artin's presentation, admits an antiautomorphism Bn→Bn, such that v↦vˉ is defined by reading braids in reverse order (from right to left instead of left to right). We prove that the map Bn→Bn, v↦vvˉ is injective. We also give s…
We study the restless bandit associated with an extremely simple scalar Kalman filter model in discrete time. Under certain assumptions, we prove that the problem is indexable in the sense that the Whittle index is a non-decreasing function of the relevant belief state. In spite of the long history of this problem, thi…
Machine learning identifies math sequences based on empirical laws.
problem Identifying interesting mathematical structures.
method Extract features from integer sequences using Benford's and Taylor's laws; experiment with classifiers.
result Machine learning can identify various mathematical properties in sequences.
We study the universal character ring of some families of one-relator groups. As an application, we calculate the universal character ring of two-generator one-relator groups whose relators are palindrome, and, in particular, of the (-2,2m+1,2n+1)-pretzel knot for all integers m and n. For the (-2,3,2n+1)-pretzel knot,…
Automorphisms of handlebodies arise naturally in the a classification of automorphisms of three-manifolds. Among automorphisms of handlebodies, there are certain automorphisms called irreducible (or generic), which are analogues of pseudo-Anosov automorphisms of surfaces. We show that irreducible automorphisms of handl…
The paper studies automorphisms of pure braid groups and their properties.
problem Investigating the structure and properties of automorphism groups of pure braid groups.
method Proving the generation of automorphism groups by specific subgroups and studying lifting and extension problems.
result No non-trivial central automorphism of Pn can be extended to an automorphism of Bn. Study shows RAAG automorphisms and outer automorphisms are not relatively hyperbolic.
problem Characterizing automorphism and outer automorphism groups of RAAGs.
method Analyzing groups of RAAGs with at least 3 vertices, categorizing based on graph structure.
result Automorphism and outer automorphism groups of RAAGs are not relatively hyperbolic.
Study knot invariants using automorphism groups of free nilpotent groups.
problem Developing knot invariants using automorphism groups.
method Nilpotently p-localization of knot groups and automorphism groups of free nilpotent groups. result Maps from outer automorphism groups yield knot invariants.
Constructs Cartan geometries from automorphism behaviors.
problem Determining Cartan geometries from automorphism local behavior.
method Introduces a construction for Cartan geometries capturing automorphism local behavior.
result The sprawl uniquely characterizes Cartan geometries with equivalent local behavior.
Study finds all specific hyperbolic manifolds with high automorphism groups.
problem Classifying hyperbolic manifolds with high-dimensional automorphism groups.
method Examined homogeneous Kobayashi-hyperbolic manifolds of dimension n≥2. result Determined all such manifolds with holomorphic automorphism group dimension n2−2. Automorphisms of pants complex are shown to be inner.
problem Understanding automorphisms of pants complexes.
method Proving automorphisms are inner for specific groups.
result Automorphism groups are naturally isomorphic to pants complex.
Automorphism groups of Hopf manifolds are finite and have a bounded order.
problem Understanding the structure of automorphism groups of Hopf manifolds.
method Proved that the automorphism groups of Hopf manifolds are Jordan.
result Automorphism groups of Hopf manifolds are finite and have a bounded order.
Classifies and constructs all extendable automorphisms of closed surfaces over the 3-sphere.
problem Identifying extendable automorphisms of closed surfaces over the 3-sphere.
method Classification and construction of extendable automorphisms through embeddings and Heegaard surfaces.
result All extendable automorphisms of closed surfaces can be induced by automorphisms of the 3-sphere on Heegaard surfaces.
Finite groups can be automorphism groups of translation surfaces with poles.
problem Existence of finite automorphism groups on translation surfaces with poles.
method Analyzing translation surfaces with poles and extending results to branched projective structures.
result Finite groups can be automorphism groups of translation surfaces with poles.
New f-vectors reveal geometric Lefschetz-like decompositions of flag spheres.
problem Understanding f-vectors of balanced simplicial complexes and flag spheres. method Analyzing h-vectors and f-vectors of flag spheres and balanced simplicial complexes. result Found f-vectors leading to geometric Lefschetz-like decompositions. Automorphisms of Lie algebras and their root systems are fully lifted.
problem Understanding automorphisms of real semisimple Lie algebras and their root systems.
method Proving every automorphism of the restricted root system can be lifted to a Lie algebra automorphism.
result Automorphisms of restricted root systems can be fully lifted to Lie algebras.
Parabolic automorphisms on hyperkahler manifolds act ergodically on fibers.
problem Understanding the dynamics of parabolic automorphisms on hyperkahler manifolds.
method Analyzing the action of parabolic automorphisms on the second cohomology group and fibers of Lagrangian fibrations.
result Parabolic automorphisms preserving Lagrangian fibrations act ergodically on the fibers.
Paper studies Engel structures and automorphisms on 4-manifolds.
problem Understanding Engel structures and their automorphisms.
method Developing maps and Cartan prolongations of contact 3-orbifolds.
result Automorphism groups of Engel manifolds are embedded into automorphism groups of contact 3-orbifolds.
Defines biquandle structures and relates automorphism groups of biquandles and quandles.
problem Understanding the structure and automorphism groups of biquandles.
method Define biquandle structures, determine isomorphisms, and compute automorphism groups.
result Establishes a relationship between biquandle and quandle automorphism groups.
Study finds all hyperbolic manifolds with specific automorphism group dimensions.
problem Classifying homogeneous Kobayashi-hyperbolic manifolds based on automorphism group size.
method Analyzing holomorphic automorphism groups of dimension n2−3 for manifolds of dimension n≥2. result All connected homogeneous Kobayashi-hyperbolic manifolds of dimension n≥2 with automorphism group of dimension n2−3 are identified. Automorphism groups of parabolic geometries are Lie groups.
problem Topology of automorphism groups of parabolic geometries.
method Proved C0 and C∞ topologies coincide and structure as Lie groups. result Automorphism group is closed in the homeomorphism group.
Study growth rates of automorphisms of special groups.
problem Understanding the growth rates of automorphisms of special groups.
method Analyzing outer automorphisms of virtually special groups, showing polynomial or exponential growth, and constructing Nielsen-Thurston decompositions.
result Outer automorphism groups of virtually special groups are boundary amenable, have finite virtual cohomological dimension, and satisfy the Tits alternative.
Proves strong Tits alternative for 3D automorphisms over zero char fields.
problem Tits alternative for 3D tame automorphisms over zero char fields.
method Proved strong Tits alternative using 3D tame automorphisms over zero char fields.
result Strong Tits alternative proven for 3D tame automorphisms over zero char fields.
Free groups' automorphisms have bounded orbits.
problem Understanding automorphisms of infinite rank free groups.
method Proved coarsely bounded automorphism groups via metric space actions.
result Free groups' automorphism groups have quasi-isometry type of a point.
Automorphisms of symplectic connections are studied, with examples showing not all are symplectic.
problem Characterizing infinitesimal automorphisms of symplectic connections.
method Analyzing conditions for infinitesimal automorphisms to be symplectic vector fields.
result Examples show that not all infinitesimal automorphisms are symplectic.
Study on automorphisms of K3 and Enriques surfaces, proving entropy gaps and achirality.
problem Entropy norms and achirality of automorphisms on K3 and Enriques surfaces.
method Proves gap theorems for entropy norms and studies achirality in terms of genus-one fibrations.
result Entropy gaps and achirality results for automorphisms of K3 and Enriques surfaces.
Study finds conditions for finite-dimensional Lie group structure in basic automorphism groups of certain Cartan foliations.
problem Characterizing the basic automorphism groups of Cartan foliations.
method Analyzes sufficient conditions and estimates dimensions for basic automorphism groups of Cartan foliations covered by fibrations.
result Identifies sufficient conditions for the existence of a finite-dimensional Lie group structure in basic automorphism groups.
Study automorphic forms on bounded domains, proving spanning results and estimating norms.
problem Understanding automorphic forms on bounded symmetric domains and their norms.
method Proving spanning results for vector-valued Poincaré series and analyzing holomorphic automorphic forms.
result Found different asymptotic behaviors of norms for certain submanifolds.
We consider automorphisms of homogeneous parabolic geometries with a fixed point. Parabolic geometries carry the distinguished distributions and we study those automorphisms which enjoy natural actions on the distributions at the fixed points. We describe the sets of such automorphisms on homogeneous parabolic geometri…
Research examines octonionic slice regular functions and their automorphisms and invariants.
problem Analyzing slice regular functions in the octonionic algebra.
method Investigates automorphisms and invariants of octonionic slice regular functions.
result Characterizes the automorphisms and invariants of octonionic slice regular functions.
Study automorphisms of regular Courant algebroids using dissections.
problem Computing automorphisms of regular Courant algebroids.
method Using dissections and proposing an infinitesimal version.
result Recover examples of the literature.
Homology of automorphism groups stabilizes with product of RAAGs.
problem Stability of automorphism group homology under product of RAAGs.
method Analyzes homology of automorphism groups of RAAGs and their products.
result Homology of automorphism groups stabilizes under product with any RAAG.
Classifies hyperbolic manifolds with specific automorphism groups.
problem Classifying Kobayashi-hyperbolic manifolds with high-dimensional automorphism groups.
method Analyzes manifolds of dimension n≥2 with automorphism groups of dimensions n2−7 or n2−8. result Completes the classification for automorphism groups n2−7 and n2−8. Outer automorphism groups of Coxeter groups are trivial except for small ranks.
problem Triviality of outer automorphism groups of Coxeter groups.
method Using Guirardel-Levitt outer space for free products, proving triviality and cyclic order for specific ranks.
result Outer automorphism groups are trivial except for small ranks.
We classify isotopy classes of automorphisms (self-homeomorphisms) of 3-manifolds satisfying the Thurston Geometrization Conjecture. The classification is similar to the classification of automorphisms of surfaces developed by Nielsen and Thurston, except an automorphism of a reducible manifold must first be written as…
Paper describes invariants of slice regular functions' automorphism group.
problem Understanding invariants of slice regular functions' automorphism group.
method Analyzes automorphism group of slice regular functions over Clifford algebras.
result Describes invariants of the automorphism group of slice regular functions.