Proves geodesic connections on 2-torus without invariant tori.
problem Existence of geodesic connections on 2-torus without invariant tori.
method Uses J. Mather's result on connecting orbits for monotone twist maps.
result Proves existence of connecting geodesics on unit tangent bundle of 2-torus.
Given a closed, oriented surface, possibly with boundary, and a mapping class, we obtain sharp lower bounds on the number of fixed points of a surface symplectomorphism (i.e. area-preserving map) in the given mapping class, both with and without nondegeneracy assumptions on the fixed points. This generalizes the Poinca…
Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on solid tori, periodic flow-lines of which define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a braid Floer homology. This refinement of t…
We introduce large scale analogues of topological monotone and light maps, which we call coarsely monotone and coarsely light maps respectively. We show that these two classes of maps constitute a factorization system on the coarse category. We also show how coarsely monotone maps arise from a reflection in a similar w…
Study on biharmonic map heat flow with monotonicity formula.
problem Properties of biharmonic heat kernel and extrinsic biharmonic map heat flow.
method Derived an entropy type quantity exhibiting monotonicity behaviors.
result Monotonicity formula for extrinsic biharmonic map heat flow.
In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2 (2011)] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals, and having connected intersections with all translated coordinate cones in R^n. In this paper we develop this theory further by d…
The paper explores methods to decompose periodic maps into Dehn twists.
problem Factoring periodic maps into Dehn twists.
method Various methods for factoring periodic mapping classes into Dehn twists, up to conjugacy.
result Existence of conjugates of periodic maps whose product is pseudo-Anosov.
Constructs differential models for twisted Spin^c-bordism and its dual, defining a new anomaly map.
problem Modeling and understanding twisted Spin^c-bordism and its dual.
method Geometric construction using bundle gerbes, gerbe modules, and eta-invariants.
result Definition of a twisted anomaly map from differential twisted K-theory to differential Anderson dual of twisted Spin^c-bordism.
Solves generalized twisted rabbit problems for higher degree polynomials.
problem When a quadratic polynomial is twisted by a cyclic subgroup, what polynomial is equivalent?
method Uses d2-adic expansion instead of 4-adic for higher degree polynomials. result Provides a solution that depends on the d2-adic expansion of the power of the mapping class element. The paper explores the twisted Rokhlin property in mapping class groups of surfaces.
problem Classifying surfaces whose mapping class groups have the twisted Rokhlin property.
method Generalizing the Rokhlin property to the twisted version, the authors classify surfaces based on their mapping class groups' properties.
result The mapping class groups of connected orientable infinite-type surfaces without boundaries have the twisted Rokhlin property, while those of other surfaces do not.
In this paper, we introduce the stress-energy tensors of the partial energies E'(f) and E"(f) of maps between Kaehler manifolds. Assuming the domain manifolds poss some special exhaustion functions, we use these stress-energy tensors to establish some monotonicity formulae of the partial energies of pluriharmonic maps …
New quantity helps map homotopy classes in complex spaces.
problem Understanding homotopic classes of maps between complex spaces.
method Identified a new monotone quantity in mean curvature flows of maps between Riemannian manifolds.
result Sharp criteria for homotopic classes of maps between complex projective spaces and spheres.
Study of codimension-1 embeddings in 3-manifolds using twist maps and push maps.
problem Embedding 3-manifolds in specific topological spaces.
method Sphere twist maps and push maps to construct codimension-1 spun embeddings.
result Every closed orientable 3-manifold admits a codimension-1 spun embedding in specific spaces.
This work clarifies different transport map constructions and their causal interpretations.
problem Identifying distinct transport map constructions and their equivalence.
method Comparative analysis of three transport map constructions: cyclically monotone, quantile-preserving, and triangular monotone.
result Conditions for equivalence of different transport map constructions.
Paper studies involutions generating the twist subgroup of nonorientable surfaces.
problem Generating the twist subgroup by involutions on nonorientable surfaces.
method Analyzes involutions to find the smallest generating sets.
result Provides generating sets of involutions with minimal elements.
Develops moment map theory for twisted scalar curvature in Kähler geometry.
problem Understanding the geometry of holomorphic submersions and foliations.
method Introduces a coupled system of equations on a holomorphic submersion.
result The coupled system appears as a moment map, generalizing to foliations.
Calculates Dehn twist actions on conformal blocks for modular categories.
problem Order of Dehn twists on spaces of conformal blocks.
method Quantum representations of mapping class groups, ribbon twists, monoidal powers.
result Order of Dehn twists generalizes previous results for small quantum group.
Study connects surface twists to curve invariants.
problem Understanding fractional Dehn twists on Riemann surfaces.
method Characterization of pseudo-periodic maps and bounds on fractional twists.
result Established relationship between twists and curve invariants.
We give a small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twists. The difference between the number of the generators and a lower bound of numbers of generators for the twist subgroup by Dehn twists is one. The lower bounds is obtained from an argument of Hiros…
The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.
New method calibrates neural network predictions for better reliability.
problem Improper probability estimates from deep networks leading to unreliable predictions.
method Proposes a constrained optimization approach for a monotonic calibration map.
result Achieves state-of-the-art performance across various datasets and models.
We establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable proper map f:X→Y (not necessarily K-oriented). The push-forward map generalizes the push-forward map in ordinary K-theory for any K-oriented differentiable…
We obtain a finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface. The generating set consists of crosscap pushing maps along non-separating two-sided simple loops and squares of Dehn twists along non-separating two-sided simple closed curves. We also prove t…
New natural presentation of supergravity c-map using Hodge structures.
problem Presenting a new natural presentation of the supergravity c-map.
method Explicit description of correspondence between projective special Kähler manifolds and variations of Hodge structure, and twist construction.
result General isomorphisms can be naturally lifted along the deformed c-map.
The paper derives Liouville theorems for various generalized maps on Riemannian manifolds.
problem Deriving Liouville theorems for generalized maps on Riemannian manifolds.
method Using conservation laws and monotonicity formulas, the paper derives Liouville theorems for different types of maps under various conditions.
result The paper establishes Liouville theorems for several types of generalized maps, including φ-F harmonic maps, φ-F symphonic maps, and φ-F-V-harmonic maps. Generates spin structure stabilizers using Dehn twists.
problem Determining generating sets for spin structure stabilizers.
method Using Dehn twists on specific curves.
result Explicit generating sets for stabilizers of spin structures.
We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant PU(H)−principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of …
Locally maximizing orbits studied in twist maps and billiards.
problem Characterize orbits in locally maximizing class for twist maps.
method Geometric and variational analysis of orbits in the cotangent bundle of a torus or ball bundle over a sphere.
result Two generating functions for the Birkhoff billiard map have the same class of locally maximizing orbits.
We consider a twisted version of the Hurewicz map on the complement of a hyperplane arrangement. The purpose of this paper is to prove surjectivity of the twisted Hurewicz map under some genericity conditions. As a corollary, we also prove that a generic section of the complement of a hyperplane arrangement has non-tri…
The authors study in detail new types of varieties with degenerate Gauss maps: varieties with multiple foci and their particular case, the so-called twisted cones. They prove an existence theorem for twisted cones and describe their structure.
Shifts are not type-preserving on surface graphs.
problem Understanding the type-preserving property of shift maps on surface graphs.
method Analyzing Dehn twists and shift maps on arc, curve, and relative arc graphs of surfaces.
result Shift maps are not type-preserving on surfaces with isolated punctures.
The paper explores mapping class group quotients by Dehn twists and their representations.
problem Finite quotients and representations of mapping class groups by powers of Dehn twists.
method Construction of finite quotients using representations with Zariski dense images into semisimple Lie groups, and Long and Moody's method.
result The Fibonacci TQFT representation is a specialization of the Jones representation in genus 2.
New relation found in 4D symplectic mapping class group.
problem Relation between Dehn twists in symplectic 4-manifolds.
method Holomorphic curve techniques, symplectic isotopy problem solution.
result Relation between two products of Dehn twists.
This is a PhD thesis in low-dimensional topology. Its main purpose is to examine so-called tête-à-tête twists. Those were defined by A'Campo and give an easy combinatorial description of certain mapping classes on surfaces with boundary. Whereas the well-known Dehn twists are twists around a simple closed curve, tête-à…
New generators found for twist subgroup of nonorientable surfaces.
problem Identifying generators for twist subgroup of nonorientable surfaces.
method Analyzing mapping class group of nonorientable surfaces of genus g≥13. result Twist subgroup can be generated by two involutions and an element of order g or g−1. Paper examines Dehn twists on non-orientable surfaces and their limitations.
problem Limitations of generating Dehn twists on non-orientable surfaces.
method Analyzes the level 2 mapping class group of non-orientable surfaces and their subgroups.
result Dehn twist subgroup of M2(Ng) cannot be generated by squares of Dehn twists about non-separating curves. Holomorphic maps are a special case of Hermitian pluriharmonic maps between almost Hermitian manifolds.
problem Characterizing maps between almost Hermitian manifolds.
method Introducing Hermitian pluriharmonic maps and proving their properties.
result Holomorphic or anti-holomorphic maps are Hermitian pluriharmonic.
We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating cur…
We construct an infinitely exchangeable process on the set $\cate$ of subsets of the power set of the natural numbers N via a Poisson point process with mean measure Λ on the power set of N. Each $E\in\cate$ has a least monotone cover in $\catf$, the collection of monotone subsets of $\cate$, an…
Study on realizing subgroup twists in 3-manifolds.
problem Realizing subgroups of twist groups in 3-manifolds.
method Analyzing Nielsen realization problem for Twist(M) subgroups, applying to Burnside problem.
result Nontrivial subgroups of Twist(M) are realized by diffeomorphisms if and only if they are cyclic and M is a connected sum of lens spaces.
This paper provides an infinite presentation for a subgroup of mapping class groups of non-orientable surfaces.
problem Finite presentations for mapping class groups of non-orientable surfaces.
method Using Stukow's finite presentation and Birman exact sequences.
result An infinite presentation for the twist subgroup of the mapping class group of a compact non-orientable surface.
We show that our earlier work in \cite{LT05} extends to the twisted case, that is, we defined a notion of moment map and reduction in both twisted generalized complex geometry and twisted generalized Kähler geometry.
Study shows Dehn twist coefficients are consistent across different actions on surfaces.
problem Consistency of fractional Dehn twist coefficients under various actions.
method Analyzes left orderings of mapping class groups and uses cofinality properties.
result Fractional Dehn twist coefficients are independent of the underlying action for surfaces with genus > 1.
Characterizes continuity of monotone functionals in mixed topology.
problem Continuity of monotone functionals in mixed topology.
method Characterization through lower semicontinuity and dual representations.
result Continuity in mixed topology is equivalent to dual representation in terms of countably additive measures.
Boundary Dehn twists become trivial after abelianization.
problem Understanding the behavior of boundary Dehn twists in smooth mapping classes.
method Constructing diffeomorphisms and showing commutators represent boundary Dehn twists.
result Boundary Dehn twists become trivial after abelianization.
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…
Study spherical twists on K3 surfaces, compute their centers.
problem Understanding autoequivalence groups of K3 surfaces.
method Introduced spherical twists, studied their intersection numbers, and classified subgroups.
result Computed the center of autoequivalence groups of K3 surfaces.
The paper extends T-duality and Jacobi forms to Witten gerbe modules.
problem Extending T-duality and Jacobi forms to Witten gerbe modules.
method Constructing graded Hori maps and showing their isomorphisms on T-dual circle bundles, and constructing Witten gerbe modules.
result Graded twisted Chern characters of Witten gerbe modules are Jacobi forms under certain conditions.