The center of a quotient group of piecewise linear homeomorphisms is trivial.
problem Understanding the structure of a specific group of homeomorphisms.
method Analyzing a quotient of piecewise linear homeomorphisms of the real line.
result The center of the quotient group is trivial.
New proof shows smallest non-cyclic automorphism group quotients are linear 2-groups.
problem Identifying the smallest non-cyclic quotients of free group automorphism groups.
method Elementary proof using standard projection and automorphisms.
result All minimal quotients are linear groups over the field of two elements.
We consider quotients of spheres by linear actions of real tori. To each quotient we associate a matroid built out of a diagonalization of the torus action. We find the integral homology groups of the resulting quotient spaces in terms of the Tutte polynomial of the matroid. We also find the homotopy type and homology …
Investigates intrinsic Lipschitz sections in nonlinear quotient maps.
problem Analyzing intrinsic Lipschitz sections in non-linear quotient maps.
method Introduced Leibniz formula for intrinsic slope under weaker conditions, used properties of intrinsic dilations in Carnot groups, and provided conditions for sum of sections.
result Found conditions for sum of intrinsically Lipschitz sections in Carnot groups of step 2.
Researchers classify quotients of lens spaces using linear actions and topological tools.
problem Classifying quotients of lens spaces under linear actions of (Z/p)2. method Postnikov towers and surgery theory.
result Quotients are classified up to homotopy by k-invariants and up to homeomorphism by Pontrjagin classes. New proof shows extendable shellability for simple complexes.
problem Proving extendable shellability for specific simplicial complexes.
method Considering chordal graph structure and linear quotients.
result All d-dimensional complexes with d+3 vertices are extendably shellable. Stratifies singular hyperkahler quotients into smooth manifolds.
problem Singular hyperkahler quotients by non-free actions.
method Topological stratification, global Poisson structures, local normal form.
result Quotients are locally isomorphic to linear complex-symplectic reductions.
We characterize finite groups G generated by orthogonal transformations in a finite-dimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient piecewise linear structure.
Paper generalizes sub-slope definition and solves complex equations on compact manifolds.
problem Solving complex equations on compact almost Hermitian manifolds.
method Generalized sub-slope definition and proved existence of solutions for a class of equations.
result Solved complex Hessian quotient and deformed Hermitian-Yang-Mills equations.
The study classifies isotopy types of 3-periodic nets and their embeddings.
problem Classifying isotopy types of 3-periodic nets and their embeddings.
method Definition of entangled embedded periodic nets, classification methodology using linear graph knots.
result Enumeration and classification of isotopy classes for various 3-periodic nets.
The paper studies symmetry reduction of control systems and its implications for feedback linearization.
problem Understanding symmetry reduction and its effects on feedback linearizability of control systems.
method Generalizing the notion of transversality of Lie group actions, analyzing the geometry of invariant distributions, and extending the S-G-S test.
result Classification of SFL quotients based on geometric properties of the control system and symmetry group.
We develop a graphical representation of polynomial invariants of unitary gauge groups, and use it to find the algebraic curve corresponding to a hyperkahler quotient of a linear space. We apply this method to four dimensional ALE spaces, and for the A_k, D_k, and E_6 cases, derive the explicit relation between the def…
We show that for any positive integer m≥1, m-relator quotients of the modular group M=PSL(2,Z) generically satisfy a very strong Mostow-type \emph{isomorphism rigidity}. We also prove that such quotients are generically "essentially incompressible". By this we mean that their "absolute T-invariant…
The moduli space of jets of certain G-structures (basically those which admit a canonical linear connection) is shown to be isomorphic to the quotient of a natural G-module by G.
In this paper we provide some stability criteria for systems of linear subspaces of V⊗W and for systems of quotient coherent sheaves, using, respectively, the Hilbert-Mumford numerical criterion and moment map. Along the way, we generalize the Gelfand-MacPherson correspondence [11] from point sets to sets of …
In this article, we give a survey of Geometric Invariant Theory for Toric Varieties, and present an application to the Einstein-Weyl Geometry. We compute the image of the Minitwistor space of the Honda metrics as a categorical quotient according to the most efficient linearization. The result is the complex weighted pr…
New rigidity results for complex and quaternionic moment-angle manifolds.
problem Equivariant topological rigidity of complex and quaternionic moment-angle manifolds.
method Reduction to equivariant rigidity of quasitoric (or quoric) quotients and principal bundles.
result Full equivariant rigidity for manifolds with four-dimensional quoric quotients and primary rigidity for higher dimensions.
The abstract discusses new 3-manifold invariants and ETQFTs from Lie superalgebra representations.
problem Developing new 3-manifold invariants and ETQFTs from Lie superalgebra representations.
method Examining two m-traces in the category of representations over quantum sl(m∣n), considering quotients, and conjecturing generalizations. result Quotients of perturbative modules over quantum sl(m∣n) lead to 3-manifold invariants and ETQFTs. Study of foliations' geometric and topological structures.
problem Analyzing the geometric and topological properties of transversely affine foliations.
method Attach holonomy group and quotient stack, identify reparametrisations, classify them, and study the Kato-Nakayama space.
result Holonomy group controls the geometric part, while the Kato-Nakayama space captures the topological and dynamical aspects.
Linear bound on Betti numbers of negatively curved orbifolds.
problem Bounding Betti numbers of negatively curved orbifolds.
method Quantitative bound on the homology of spherical quotients.
result Linear growth of Betti numbers with volume, arbitrary field coefficients.
Unified algebraic framework for virtual braid structures with strong structural consequences.
problem Unified algebraic framework for virtual braid structures with various types of crossings.
method Introducing the universal virtual braid group UVn(c) and proving its properties. result Strong structural consequences including residual finiteness, linearity, and solvability of conjugacy problems.
In this survey we collect all results regarding the construction of the Framization of the Temperley-Lieb algebra of type A as a quotient algebra of the Yokonuma-Hecke algebra of type A. More precisely, we present all three possible quotient algebras the emerged during this construction and we discuss their dimensi…
The paper examines gradient Ricci solitons on orbifolds and proves their rigidity properties.
problem The rigidity of positively curved gradient Ricci solitons on orbifolds.
method Analyzes scalar curvature, uses nonnegative curvature operator, κ-noncollapsed condition, and asymptotic quotient cylindrical properties.
result Steady gradient Ricci solitons on orbifolds with positive curvature are rigid and must be quotients of the Bryant soliton.
The square root velocity function (SRVF), introduced by Srivastava et al, has proved to be an effective way to compare absolutely continuous curves in RN modulo reparametrization. Several computational papers have been published based on this method. In this paper, we carefully establish the theoretical foundations …
We demonstrate the agreement between the Higgs branches of two N=2 theories proposed by Argyres and Seiberg to be S-dual, namely the SU(3) gauge theory with six quarks, and the SU(2) gauge theory with one pair of quarks coupled to the superconformal theory with E_6 flavor symmetry. In mathematical terms, we demonstrate…
We say that a collection Gamma of geodesics in the hyperbolic plane H^2 is a modular pattern if Gamma is invariant under the modular group PSL_2(Z), if there are only finitely many PSL_2(Z)-equivalence classes of geodesics in Gamma, and if each geodesic in Gamma is stabilized by an infinite order subgroup of PSL_2(Z). …
We study non-linear sigma models whose target spaces are the Higgs phases of supersymmetric SO and USp gauge theories by using the Kahler and hyper-Kahler quotient constructions. We obtain the explicit Kahler potentials and develop an expansion formula to make use of the obtained potentials from which we also calculate…
Normal forms and moduli stacks for flat connections on complex manifolds.
problem Understanding singular flat connections on complex manifolds.
method Introducing homogeneous Lie groupoids and studying their representation theory to prove normal form theorems and moduli space structures.
result Moduli spaces of singular flat connections admit the structure of algebraic quotient stacks.
Study of symplectic cross-ratios in moduli spaces of line configurations.
problem Understanding geometric and combinatorial properties of line configurations in symplectic spaces.
method Analysis of moduli spaces, geometric and combinatorial problems, isomorphism to quotient spaces of linear difference operators.
result Moduli space of Lagrangian configurations is parametrized by n+1 symplectic cross-ratios, satisfying a remarkable relation.
Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.
problem Understanding non-linear Hopf manifolds and their properties.
method Holomorphic embeddings and LCK metrics.
result Non-linear Hopf manifolds admit LCK metrics.
Let X be a Kahler manifold that is presented as a Kahler quotient of C^n by the linear action of a compact group G. We define the hyperkahler analogue M of X as a hyperkahler quotient of the cotangent bundle T^*C^n by the induced G-action. Special instances of this construction include hypertoric varieties and quiver v…
We consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the groups of flat virtual braids and virtual Gauss braids and study some of their properties, in particular their linearity.
We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge-Ampère, Hessian and inverse Hessian equations. As an application we solve a class of He…
The study connects lattices, Garside structures, and weakly modular graphs.
problem Exploring combinatorial non-positive curvature in various simplicial complexes.
method Analyzing lattices with Z-actions and their quotients. result Lattices and their quotients give rise to weakly modular graphs.
New DQ based on expectiles improves portfolio diversification.
problem Improving diversification in financial portfolios.
method Diversification quotient based on expectiles, offering simple formulas and pseudo-convexity.
result The expectile-based DQ is efficient and effective in portfolio optimization.
Formula calculates higher moments of Siegel-Veech transform over Hecke triangle groups.
problem Computing higher moments of Siegel-Veech transform over specific groups.
method Geometric results and linear algebra to create integration formulas.
result Explicit integration formulas for densities of vector orbits.
Simplified presentation of symplectic fillings of lens spaces.
problem Symplectic fillings of lens spaces.
method Visual presentation of rational blowdown algorithm using triangulations of convex polygons.
result Organization of symplectic fillings into a graph.
Study examines quotients of affine connection control systems.
problem Existence of quotients preserving mechanical structures.
method Local and global sufficient and necessary conditions for geodesically accessible systems.
result Quotients can be constructed for geodesically accessible systems.
The paper studies hyperbolic quotients of projection complexes and their actions.
problem Understanding the structure and properties of quotients of projection complexes.
method Analyzing the quotient of projection complexes by normal subgroups and studying the resulting actions.
result The quotient complex is δ-hyperbolic under certain conditions, and the quotient group is acylindrically hyperbolic.
Thurston's fibered face theory allows us to partition the set of pseudo-Anosov mapping classes on different compact oriented surfaces into subclasses with related dynamical behavior. This is done via a correspondence between the rational points on fibered faces in the first cohomology of a hyperbolic 3-manifold and the…
The paper introduces pseudo-quotients for algebraic actions and applies them to character varieties.
problem Characterizing algebraic actions and their quotients.
method Introducing pseudo-quotients as a weak version of quotients for algebraic actions, focusing on purely topological properties.
result Pseudo-quotients are unique up to virtual class in characteristic zero and can be used to compute character varieties.
Study non-arithmetic orbifold ball quotients using Jacobian of Bolza curve.
problem Identify and study non-arithmetic orbifold ball quotients.
method Use Jacobian of Bolza curve and birational transformations.
result Obtain orbifold ball quotient surfaces with interesting configurations.
We study the partial resolutions of singularities related to Hilbert schemes of points on an affine space. Consider a quotient of a vector space V by an action of a finite group G of linear transforms. Under some additional assumptions, we prove that the partial desingularization of Hilbert type is smooth only if t…
Sharp bounds found on nonabelian quotients of surface braid groups.
problem Finding the smallest nonabelian quotients of surface braid groups.
method Sharp lower bounds and classification of quotients.
result Quotients of minimum order are either symmetric groups or 2-step nilpotent p-groups.
Finite quotients of braid groups have large sizes.
problem Understanding the size of finite quotients of braid groups.
method Derived lower bounds on the size of quotients.
result Lower bounds are superexponential in the number of strands.
Study quotients of curve complex actions by mapping class group.
problem Understanding actions of mapping class group on curve complex quotients.
method Cone off uniformly quasi-convex subspaces to form symmetric curve sets, non-maximal train track sets, and compression body disc sets. Analyze actions of mapping class group on these quotients.
result Actions of mapping class group on quotients are strongly WPD, non-elementary, and have infinite diameter.
Study finds smallest non-trivial quotients of braid groups and commutator subgroups.
problem Identifying smallest non-trivial quotients of braid groups and commutator subgroups.
method Analyzing symmetric and alternating groups for quotients of braid groups and commutator subgroups.
result Proved smallest quotients for specific braid groups and commutator subgroups.
New infinite discrete group without finite quotients found.
problem Finding discrete groups without finite quotients.
method Constructing an infinite discrete subgroup of H3. result Infinite discrete subgroup with no finite non-trivial quotients.