Study groups of piecewise isometries in tessellations of Euclidean space.
problem Understanding the structure of groups formed by cutting and gluing tessellations.
method Proving structure results about groups of piecewise isometries of tessellations, including elementary amenability.
result Groups of piecewise isometries of tessellations are elementary amenable.
The paper solves the existence problem of sphere packings in higher dimensions.
problem Existence of crystallographic sphere packings in certain higher dimensions.
method Geometric doubling procedure and computations with Lorentzian quadratic forms.
result Solves the existence problem of crystallographic sphere packings in higher dimensions.
This paper is devoted to the problem of choosing the most suitable model of a geometrical system for describing the real crystallographic space. It has been shown that all 230 crystallographic groups used to describe the crystalline structures in a Euclidean space can be presented by elliptic motions in the closed spac…
We explicitly compute the lower algebraic K-theory of the split three-dimensional crystallographic groups; i.e., the groups G that act properly and cocompactly on three-dimensional Euclidean space by isometries, such that the natural map from G to O(3) is a split injection onto its image. There are 73 split three-dimen…
Study on braid groups' congruence subgroups and their crystallographic quotients.
problem Understanding congruence subgroups and crystallographic quotients of braid groups.
method Investigation of lower central series of congruence braid groups related to B3. result Quotients of congruence braid groups are almost crystallographic.
Derives representations invariant under crystallographic groups for functions.
problem Representing and learning functions invariant under crystallographic groups.
method Derives linear and nonlinear representations of functions invariant under crystallographic groups.
result Derives orthonormal crystallographically invariant basis functions and embedding maps.
Paper explores relations between braid groups and their quotients.
problem Understanding relations between braid groups and their quotients.
method Recalling and introducing elements of congruence braid groups, establishing isomorphisms between crystallographic and congruence braid groups.
result Established isomorphisms between crystallographic braid groups and quotients of congruence braid groups.
Study crystallographic groups for positive scalar curvature conditions.
problem Examining positive and negative results for Gromov-Lawson-Rosenberg Conjecture.
method Analyzing split extensions of free abelian by cyclic groups.
result Produce infinite counterexamples for the Gromov-Lawson-Rosenberg Conjecture.
Computes cohomology groups for NEC groups, focusing on Fuchsian groups.
problem Understanding the cohomology of non-Euclidean crystallographic groups.
method Computes cohomology groups for geometrically finite NEC groups, and determines the ring structure for Fuchsian groups.
result Determination of cohomology groups and ring structures for Fuchsian groups.
Construct special Lagrangian fibrations on abelian varieties using retraction techniques.
problem Constructing special Lagrangian fibrations on abelian varieties.
method Explicit construction using special techniques in non-Archimedean geometry.
result Solved a conjecture of Kontsevich-Soibelman for finite quotients of abelian varieties.
Following an idea of Gonçalvez, Guaschi and Ocampo on the usual braid group we construct crystallographic and Bieberbach groups as (sub)quotients of the generalized braid group associated to an arbitrary complex reflection group.
Generalizes crystallographic properties to all dimensions.
problem Analytic eigenfunctions in crystallographic groups.
method Algebraic, geometric, and analytic proofs.
result Equivalent conditions for real analytic eigenfunctions in crystallographic polytopes.
The paper studies congruence subgroups and crystallographic quotients of small Coxeter groups.
problem The congruence subgroup property for small Coxeter groups.
method Analyzes the properties of small Coxeter groups, proving the failure of the congruence subgroup property for certain groups.
result Proves the failure of the congruence subgroup property for infinite small Coxeter groups which are not virtually abelian.
New sphere packings found with unique arithmetic properties.
problem Finding sphere packings with specific arithmetic properties.
method Defined crystallographic sphere packings and used geometric and arithmetic methods.
result Infinite family of conformally inequivalent sphere packings with integer reciprocals of radii.
Study of commutator subgroups and crystallographic quotients of virtual groups.
problem Investigate commutator subgroups and crystallographic quotients of virtual groups.
method Derived explicit finite presentations and proved crystallographic properties.
result Explicit finite presentations of commutator subgroups and crystallographic quotients.
Holomorphic actions on complex spaces for nilpotent groups.
problem Understanding polynomial actions on complex spaces for nilpotent groups.
method Explicit construction of biholomorphisms by polynomial maps.
result Simply connected nilpotent Lie groups are biholomorphic to Cn. The paper characterizes crystallographic groups derived from virtual braid and twin groups.
problem Characterizing crystallographic groups from virtual braid and twin groups.
method Analyzing quotients of virtual braid and twin groups by their commutator subgroups.
result The quotients of virtual braid and twin groups by their commutator subgroups are crystallographic groups.
Paper introduces untangling number to quantify 3-periodic tangle complexity.
problem Quantifying the complexity of 3-periodic tangles in biological, chemical, and physical systems.
method Introduces untangling number, a measure of minimum distance to ground state through diagrammatic operations.
result For infinite open curves, generic ground states are crystallographic rod packings.
In this paper, we prove the K-theoretical and L-theoretical Farrell-Jones Conjecture with coefficients in an additive category for nearly crystallographic groups of the form Qn⋊Z, where Z acts on Qn as an irreducible integer matrix with determinant d, ∣d∣>1.
The paper analyzes quotient groups of Artin braid groups and proves they are almost-crystallographic.
problem Analyzing quotient groups of Artin braid groups.
method Analyzing the quotient group Bn/Γk(Pn) of the Artin braid group Bn by the subgroup Γk(Pn). result The quotient group Bn/Γk(Pn) is an almost-crystallographic group. Study on stable torsion length in groups, showing it vanishes in crystallographic groups and providing algorithms for computation.
problem Understanding the stable torsion length in groups, especially in crystallographic and free products of groups.
method Developed linear programming and exact algorithms to compute stable torsion length in free products of groups and finite groups.
result Showed that stable torsion length vanishes in crystallographic groups and provided exact computations for nontrivial examples.
Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups Γ in $\Aff (N)=N\rtimes \Aut (N)$ acting properly discontinuously and cocompactly on N. This situation is a natural generalization of the so-called affine crystallographic groups. We prove that for all dimensions…
Generalizes Lee's result to virtually polycyclic groups.
problem Understanding morphisms between virtually polycyclic groups.
method Study of translation-like actions and their behavior under subgroups and cosets.
result Generalization of Lee's result to virtually polycyclic groups.
We study properly discontinuous and cocompact actions of a discrete subgroup Γ of an algebraic group G on a contractible algebraic manifold X. We suppose that this action comes from an algebraic action of G on X such that a maximal reductive subgroup of G fixes a point. When the real rank of any simple subg…
We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial matroids and associated linear representation results that may be interesting in their own right. The same techniques immedi…
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.
We develop the foundations of the deformation theory of compact complete affine space forms and affine crystallographic groups. Using methods from the theory of linear algebraic groups we show that these deformation spaces inherit an algebraic structure from the space of crystallographic homomorphisms. We also study th…
The study analyzes surface braid groups to derive crystallographic groups and flat manifolds.
problem Analyzing surface braid groups to derive crystallographic groups and flat manifolds.
method Analyzing the quotient group Bn(M)/Γ2(Pn(M)) of surface braid groups Bn(M) by the commutator subgroup Γ2(Pn(M)). result Crystallographic groups and flat manifolds are constructed from surface braid groups.
Deformation K-theory associates to each discrete group G a spectrum built from spaces of finite dimensional unitary representations of G. In all known examples, this spectrum is 2-periodic above the rational cohomological dimension of G (minus 2), in the sense that T. Lawson's Bott map is an isomorphism on homotopy in …
An entirely new and independent enumeration of the crystallographic space groups is given, based on obtaining the groups as fibrations over the plane crystallographic groups, when this is possible. For the 35 ``irreducible'' groups for which it is not, an independent method is used that has the advantage of elucidating…
The complete classification of representations of the Trefoil knot group G in S^{3} and SL(2,R), their affine deformations, and some geometric interpretations of the results, are given. Among other results, we also obtain the classification up to conjugacy of the non cyclic groups of affine Euclidean isometries generat…
Defines a generalized string concept for abstract root systems.
problem Generalizing the concept of strings to abstract root systems.
method Introduces a new definition for Φ-strings in abstract root systems. result Defines a new set of elements in Σ based on a given λ and subset Φ of simple roots. The notion of limit roots of a Coxeter group W was recently introduced (see arXiv:1112.5415 and arXiv:1303.6710): they are the accumulation points of directions of roots of a root system for W. In the case where the root system lives in a Lorentzian space W admits a faithful representation as a discrete reflection grou…
Paper extends Fenchel's conjecture to non-Euclidean crystallographic groups.
problem Does every non-Euclidean crystallographic group have a specific subgroup?
method Examined various cases of non-Euclidean crystallographic groups and their orbit spaces.
result Affirmative answer in many cases, open in others.
In this paper, we prove that a normal subgroup N of an n-dimensional crystallographic group G determines a geometric fibered orbifold structure on the flat orbifold E^n/G, and conversely every geometric fibered orbifold structure on E^n/G is determined by a normal subgroup N of G, which is maximal in its commensurabili…
CrystalGAN generates novel stable chemical compounds using GANs.
problem Generating novel multi-element stable chemical compounds efficiently.
method Cross-domain Generative Adversarial Networks (GANs) with novel architecture and loss functions.
result CrystalGAN generates reasonable data with increased complexity.
We classify the systems of T-roots of the flag manifolds M of the exceptional compact simple Lie groups with the second Betti number b2(M)≥2.
Paper adapts causal analysis for time-dependent systems, especially energy management.
problem Challenges in root-cause analysis for systems with lagged time-dependencies, particularly in energy management.
method Adapts causal root-cause analysis method to time-dependent systems, discusses two truncation approaches.
result Extension effectively localizes root-causes in feature and time domain with enough lags.
Geometric models for Lie algebras from simple singularities.
problem Classifying simply-laced simple Lie algebras.
method Using polygonal wheels derived from Milnor fibers of simple singularities.
result Geometric root systems are isomorphic to Lie algebras.
New proof for symmetric spaces with rectangular lattices.
problem Characterizing symmetric spaces with rectangular unit lattices.
method Explicit construction of isometric embeddings and analysis of root systems.
result Symmetric spaces with rectangular unit lattices are symmetric R-spaces.
Develops Kleinian Sphere Packings and Bugs, proving their arithmetic origins.
problem Understanding sphere packings and their arithmetic origins in various dimensions.
method Introduces Kleinian Sphere Packings and Bugs, extending Arithmeticity Theorem.
result Kleinian packings and Bugs come from Q-arithmetic lattices of simplest type.
Paper defines invariants for elliptic Weyl groups and connects them to Frobenius structures.
problem Defining invariants for elliptic Weyl groups.
method Defines a set of good basic invariants and shows their connection to Frobenius structures.
result Good basic invariants give flat invariants and structure constants of Frobenius structures.
In this paper, we show that the quotient space of the domain by the reflection group for an elliptic root system has a structure of Frobenius manifold for the case of codimension 1. We also give a characterization of this Frobenius manifold structure under some suitable condition.
New framework for root-cause analysis in complex CPSs using spatiotemporal graphical modeling.
problem Anomaly detection and root-cause analysis in complex cyber-physical systems (CPSs).
method Spatiotemporal graphical modeling based on symbolic dynamics.
result Approaches S3 and A3 achieve high accuracy in root-cause analysis under various fault scenarios. We show that any n-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of Euclidean n -space by an action of a crystallographic group. We describe all such actions.
Paper tackles anomaly detection and RCA in dynamical systems using ICODE Networks.
problem Anomalies in dynamical systems impact performance and reliability.
method Proposes ICODE Networks for anomaly detection, RCA, and type classification.
result Demonstrates the ability to accurately detect anomalies, classify types, and pinpoint origins.
The paper proves properties of fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.
problem Properties of fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.
method Proves properties of fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth are finitely generated.
We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential varieties of homogeneous varieties already constructed. Our algorithms make no referenc…