The paper studies degenerations of rational maps and their limits as geometrically finite rational maps.
problem Understanding the limits of quasi post-critically finite degenerations of rational maps.
method Constructing limits as geometrically finite rational maps on a tree of Riemann spheres, proving boundedness, and giving convergence criteria.
result Progress towards Thurston's compactness theorem and double limit theorem in complex dynamics.
The paper connects Nahm's equations to rational maps between projective spaces.
problem Solving Nahm's equations with specific boundary conditions.
method Identifying moduli spaces with spaces of rational maps and using symplectic geometry.
result Dimensions of rational maps correspond to holomorphic charge.
The paper connects reflection groups to maps with specific dynamical properties.
problem Understanding the relationship between reflection groups and anti-rational maps.
method Established a correspondence using planar graphs.
result Complete answers to geometric mating problems for anti-rational maps.
Sharp estimate shows maps with small energy defect are close to rational maps.
problem Quantitative rigidity of maps from S2 to S2 of general degree. method Proved maps with small energy defect are essentially given by a collection of rational maps at different scales.
result Sharp quantitative rigidity estimate dist2≤Cδv(1+∣logδv∣), sharpness shown. Paper computes rational cohomology of spin hyperelliptic mapping class groups.
problem Computing rational cohomology of spin hyperelliptic mapping class groups.
method Computes the G-invariant part of the rational cohomology of the pure braid group. result Includes rational cohomology of spin hyperelliptic mapping class groups of genus g. In this paper, we study the dynamics of degenerating sequences of rational maps on Riemann sphere C^ using R-trees. Given a sequence of degenerating rational maps, we give two constructions for limiting dynamics on R-trees: one geometric and one algebraic. The geometric constructio…
The study finds rational points on specific types of hypersurfaces.
problem Identifying rational points on generic marked hypersurfaces.
method Analyzing hypersurfaces in projective spaces over various fields.
result Conditions for the existence of rational points on hypersurfaces.
Maps between classifying spaces for certain groups are studied, with rational cohomology results.
problem Analyzing maps between classifying spaces for specific groups.
method Study of maps Θ between null-components of homomorphism and based map spaces. result Surjectivity of map Θ in rational cohomology for certain groups, and non-surjectivity for others. The paper examines rational homology spheres that admit special generic maps into Euclidean spaces.
problem Whether rational homology n-spheres admit special generic maps into Rp for p<n. method Stein factorization technique to derive a necessary homological condition.
result New results on the (non-)existence of special generic maps for specific rational homology spheres.
We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We provide proof that a random backward iteration algorithm to draw the pictures of the Julia sets, previously proven to work in the context of iteration of a rational map of degree two or more, extends to finitely gene…
This paper shows similarities in deformation spaces of Kleinian groups and anti-holomorphic maps.
problem Comparing deformation spaces of Kleinian reflection groups and anti-holomorphic rational maps.
method Established an analogue of Thurston's compactness theorem for critically fixed anti-rational maps and characterized deformation space interactions.
result Deformation spaces of Kleinian reflection groups and anti-holomorphic rational maps share striking similarities.
We demonstrate that the question whether or not a given postcritically finite topological ramified covering map of the 2-sphere is Thurston equivalent to a rational map is algorithmically decidable.
Let X and Y be finite complexes. When Y is a nilpotent space, it has a rationalization Y→Y(0) which is well-understood. Early on it was found that the induced map [X,Y]→[X,Y(0)] on sets of mapping classes is finite-to-one. The sizes of the preimages need not be bounded; we show, however, that as…
Paper finds new realizable data for maps with three branch points.
problem Existence of rational maps with specific branch points.
method New families of branch data identified through football decomposition method.
result Identifies new realizable branch data and exceptional data.
Study on mapping classes of real rational surface automorphisms, focusing on reducible maps and pseudo-Anosov maps.
problem Investigating the mapping classes of real rational surface automorphisms and their restrictions.
method Analysis of reducible maps, determination of pseudo-Anosov mapping classes, and comparison with Penner's construction.
result Realized Lehmer's number as the stretch factor of a pseudo-Anosov map on a specific surface.
Abstract: Rational decomposition of homeomorphism spaces for manifolds.
problem Constructing rational homotopy pullback decompositions for homeomorphism spaces.
method Rational homotopy pullback decomposition, nullhomotopy of stabilisation maps, tensor products of truncated operads.
result Rational section of the stabilisation map for homeomorphisms of R^d.
Survey explores interactions between four conformal dynamics branches.
problem Understanding complex dynamics through different mathematical concepts.
method Examples and general results with technical tools.
result Dynamical relations between Schwarz reflection parameter spaces and anti-rational maps/ reflection groups.
Rational maps structure theorem with geometric decomposition and realizability proof.
problem Realizability of rational maps branch data.
method Geometric decomposition of pullback metric into footballs and application to realizability.
result Realizability of branch data for rational maps when k>l+1. Overview of dynamics in algebraic correspondences and their connections.
problem Understanding dynamics in algebraic correspondences and their connections.
method Focus on matings between rational maps and Kleinian groups, highlighting unifying structures.
result Rich dynamics and connections between moduli spaces of rational maps and Kleinian groups.
Study contact structures on lens spaces, classifying rational knots.
problem Classify rational knots in lens spaces.
method One-parametric convex surface theory to classify Legendrian and transverse rational unknots.
result Determine the contact mapping class group of lens spaces.
Given a sub-hyperbolic semi-rational branched covering which is not CLH-equivalent a rational map, it must have the non-empty canonical Thurston obstruction. By using this canonical Thurston obstruction, we decompose this dynamical system in this paper into several sub-dynamical systems. Each of these sub-dynamical sys…
Unified and generalized mating frameworks for Kleinian groups and rational maps.
problem Combining two frameworks for mating Kleinian groups with rational maps.
method Extended mating framework to genus zero hyperbolic orbifolds, constructed correspondences, defined parameter space.
result Explicit description and construction of conformal matings and correspondences.
We discuss a connection between the lantern relation in mapping class groups and the rational blowing down process for 4-manifolds. More precisely, if we change a positive relator in Dehn twist generators of the mapping class group by using a lantern relation, the corresponding Lefschetz fibration changes into its rati…
Generalizes Lefschetz fibrations with rational homology disk smoothings.
problem Understanding rational homology disk smoothings of surface singularities.
method Introduces a genus to generic fibers of Lefschetz fibrations.
result Families of relations in mapping class groups represent smoothings.
We explore a relationship between topological properties of orbits of 2-cycles in the symplectomorphism group Symp(M) and the existence of rational curves in M. Under the absence of rational curves hypothesis, we show that evaluation map vanishies on the second homotopy group and obtain a Gottlieb-type vanishing theore…
Characterizes mappings preserving Pythagorean-hodograph curves.
problem Preserving Pythagorean-hodograph curves in various dimensions.
method Proves conformal functions with square rational dilation are PH-preserving.
result Conformal functions with square rational dilation are the only PH-preserving mappings.
Maps from rational homology solid tori yield rank inequalities in Heegaard Floer homology.
problem Rank inequalities in Heegaard Floer homology.
method Using Hanselman-Rasmussen-Watson's bordered Floer homology, we extend their proof to rational homology solid tori.
result We provide rank inequalities for Heegaard Floer homology.
We study the Abel-Jacobi map for bisections of a certain rational elliptic surface. As an application, we construct examples of Zariski N-plets for conic arrangements.
Instanton Floer homology matches Heegaard Floer for almost-rational plumbings.
problem Matching instanton Floer homology with Heegaard Floer for specific 3-manifolds.
method Utilizes lattice homology and a recent cobordism map decomposition theorem.
result Isomorphism between framed instanton Floer homology and Heegaard Floer for almost-rational plumbings.
The Epstein deformation space parameterizes marked rational maps with prescribed combinatorial and dynamical structure. For the family of quadratic rational maps with a periodic critical cycle of order 4 and an extra critical point not lying in this cycle, S. Koch and I recently showed that the deformation space has in…
We consider the interplay of point counts, singular cohomology, étale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree d rational curves in Pn. We deduce as special cases algebro-geome…
Every negative amphichiral knot is rationally slice.
problem Proving every negative amphichiral knot is rationally slice.
method Systematic analysis of the action induced by the negative amphichiral map on the JSJ decomposition of the link exterior.
result Every negative amphichiral link is rationally slice.
Study on cohomology of spin hyperelliptic mapping class groups.
problem Cohomology of spin hyperelliptic mapping class groups.
method Study of G-invariant part of rational cohomology of pure braid groups. result Independence of cohomology dimensions in low degrees and formulas for dimensions.
Researchers determine the rational abelianization of a subgroup of mapping class groups.
problem Understanding the structure of the Chillingworth subgroup of mapping class groups.
method Using Johnson homomorphism and Casson-Morita homomorphism, they compute the abelianization and order of related Euler classes.
result They find the rational abelianization of the Chillingworth subgroup as a full mapping class group module.
We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute t…
Study homology groups of mapping and Torelli groups for surfaces with abelian covers.
problem Understanding the homology of mapping and Torelli groups for surfaces with specific topologies.
method Examined the first homology group of mapping and Torelli groups with coefficients in the first rational homology group of the universal abelian cover of the surface.
result For surfaces with one boundary component, the twisted homology groups are finite-dimensional, but for surfaces with one puncture, they are infinite-dimensional.
Constructs chiral rational homology spheres with hyperbolic groups.
problem Existence of strongly chiral rational homology spheres with hyperbolic fundamental groups.
method Construction of rational homology spheres using r-spins and investigation of self-map degrees. result Strongly chiral rational homology spheres with hyperbolic fundamental groups constructed.
In this paper, we show that one can naturally associate a limiting dynamical system F:T⟶T on an R-tree to any degenerating sequence of rational maps $f_n: \hat\C \longrightarrow \hat\C$ of fixed degree. The construction of F is in 2 steps: first we use barycentric extension to get $\E f_n : \Hy…
Compute group cohomology for mapping class group with non-symplectic coefficients.
problem Compute group cohomology for mapping class group with non-symplectic coefficients.
method Compute the invariant subspace of the rational group ring of a surface, truncated by powers of the augmentation ideal, under the action of the mapping class group.
result First group cohomology computation for the mapping class group with non-symplectic coefficients.
For a nullhomologous Legendrian knot in a closed contact 3-manifold Y we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of Y is mapped by a surgery cobordism to the contact invariant of the result of contact surgery. In ad…
If there exists a diffeomorphism f on a closed, orientable n-manifold M such that the non-wandering set Ω(f) consists of finitely many orientable (±) attractors derived from expanding maps, then M must be a rational homology sphere; moreover all those attractors are of topological dimension n−2. Expandi…
Study the monodromy and center-focus problems for rational maps defined by products of generic lines.
problem Monodromy and center-focus problems for rational maps defined by products of generic lines.
method Analyze the 1-homology group and meromorphic 1-forms to characterize vanishing Abelian integrals.
result Characterize meromorphic 1-forms whose Abelian integrals vanish on cycles around a center singularity.
Suppose X and Y are finite complexes, with Y simply connected. Gromov conjectured that the number of mapping classes in [X,Y] which can be realized by L-Lipschitz maps grows asymptotically as Lα, where α is an integer determined by the rational homotopy type of Y and the rational cohomology of X. Thi…
Novikov theorem extended to rational Pontryagin classes for cyclic group C4.
problem Classifying stable Cp-smoothings of high-dimensional manifolds. method Computing equivariant homotopy groups and applying to C4. result Novikov's theorem extended to rational Pontryagin classes for C4. Proves existence of maps with controlled small curvatures.
problem Existence of locally distance-increasing maps with controlled curvatures.
method Proves existence using controlled small curvatures.
result Existence of locally distance-increasing maps with controlled small curvatures.
In the first part of the paper we construct a ring structure on the rational cobordism classes of Morin maps (i. e. smooth generic maps of corank 1). We show that associating to a Morin map its singular strata defines a ring homomorphism to $Ω_* \otimes \Q$, the rational oriented cobordism ring. This is proved by analy…
We investigate random complex dynamics of rational or polynomial maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, generically, the chaos of the averaged system disappears at any point in the Riemann sphere due to the automatic coopeartion of many kinds of maps in the system, ev…
The paper explores rational functions with 3 branching points on the Riemann sphere.
problem Existence of rational functions with specific branching points.
method Utilizes complex analysis to establish properties of rational functions.
result Identifies new types of exceptional branching data.