Study block mapping class groups and their finiteness properties.
problem Investigate the finiteness properties of block mapping class groups.
method Consider Cantor surfaces and block mapping class groups with local action prescribed by subgroups.
result Prove finiteness properties of block mapping class groups for spheres and tori.
Study shows mapping class groups differ for h-cobordant manifolds.
problem Mapping class groups are invariant under h-cobordism. method Introduced moduli spaces of h-block bundles to distinguish manifolds. result Mapping class groups of h-cobordant manifolds can differ. Semisimplicity proven for conformal blocks representations.
problem Semisimplicity of conformal blocks representations.
method Theory of extensions in non-Abelian Hodge theory and Ocneanu rigidity.
result Semisimplicity of braid group and mapping class group representations.
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.
Abstract: Mapping class groups act on cohomology of surfaces via Hochschild cohomology.
problem Understanding the action of mapping class groups on cohomology of surfaces.
method Associate cochain complexes to surfaces, with mapping class groups acting projectively on cohomology.
result Projective action of mapping class groups on Hochschild cohomology of Hopf algebras.
The goal of this paper is to find a close to isomorphic presentation of 3-manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3-dimensional topology. The first is the category \Cob of connect…
There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fit…
AANets balance stability and plasticity in CIL.
problem Stability-plasticity dilemma in class-incremental learning.
method Adaptive Aggregation Networks (AANets) with stable and plastic residual blocks.
result AANets improve performance on CIL benchmarks.
Conventional application of convolutional neural networks (CNNs) for image classification and recognition is based on the assumption that all target classes are equal(i.e., no hierarchy) and exclusive of one another (i.e., no overlap). CNN-based image classifiers built on this assumption, therefore, cannot take into ac…
We will survey the work on the topology of Out(Fn) in the last 20 years or so. Much of the development is driven by the tantalizing analogy with mapping class groups. Unfortunately, Out(Fn) is more complicated and less well-behaved. Culler and Vogtmann constructed Outer Space Xn, the analog of Teichmüller spac…
The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. What makes this study tractable is that there is a surjective map from the hyperelliptic mapping class group to a mapping class group of a punctured sphere. The more general family of superelliptic mapping…
A formula connects two algebraic structures derived from a category.
problem Connecting two algebraic structures derived from a category.
method Using differential graded modular functors and Calabi-Yau structures.
result The action of a specific mapping class group element transforms one algebraic structure into another.
Recently it has become popular to learn sparse Gaussian graphical models (GGMs) by imposing l1 or group l1,2 penalties on the elements of the precision matrix. Thispenalized likelihood approach results in a tractable convex optimization problem. In this paper, we reinterpret these results as performing MAP estimation u…
Classifies manifolds with dense conjugacy classes in their mapping class groups.
problem Classifying manifolds based on conjugacy classes in their mapping class groups.
method Analyzing connected orientable 2-manifolds and their mapping class groups.
result Mapping class groups of certain manifolds have dense conjugacy classes.
Explicit presentations found for asymptotically rigid mapping class groups.
problem Understanding the structure of asymptotically rigid mapping class groups.
method Using a graph of groups structure, we compute explicit presentations.
result Computed explicit presentations for asymptotically rigid mapping class groups of surfaces.
New model detects hidden group structures in criminal networks.
problem Challenges in identifying group structures in criminal networks with noisy data.
method Developed an extended stochastic block model (ESBM) to infer group structures.
result Unveiled complex block structures in an Italian mafia network.
The article explores the mapping class group using unicellular maps and provides filtrations.
problem Understanding the structure of the mapping class group.
method Using unicellular maps and surgeries, the article describes the mapping class group.
result Provides filtrations of the mapping class group.
We establish an effective version of the classical Lie--Kolchin Theorem. Namely, let A,B∈GLm(C) be quasi--unipotent matrices such that the Jordan Canonical Form of B consists of a single block, and suppose that for all k≥0 the matrix ABk is also quasi--unipotent. Then A and B have a…
Study of hyperelliptic mapping class groups with applications and profinite completions.
problem Understanding hyperelliptic mapping class groups and their properties.
method Defined and studied hyperelliptic mapping class groups, applied theory to counterexamples, and examined profinite completions.
result Found a counterexample to a conjecture about mapping class groups and extended congruence subgroup property.
Overview of infinite surface mapping class groups.
problem Understanding mapping class groups of infinite surfaces.
method Survey of recent research findings.
result Recent developments in mapping class groups of infinite surfaces.
In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show th…
Continuous epimorphisms between certain mapping class groups are induced by homeomorphisms.
problem Understanding continuous epimorphisms between specific mapping class groups.
method Analyzing subgroups of mapping class groups of infinite-genus 2-manifolds with no planar ends.
result Continuous epimorphisms are induced by homeomorphisms for the specified subgroups.
Paper shows mapping class groups are not extremely amenable except for specific cases.
problem Determining when mapping class groups are extremely amenable.
method Utilized Kechris-Pestov-Todorčević machinery.
result Big mapping class groups are not extremely amenable unless the surface is a sphere or a once-punctured sphere.
This thesis introduces big mapping class groups and their structure.
problem Understanding mapping class groups of infinite-type surfaces.
method Systematic introduction and analysis of structure and topological generation.
result Key differences from finite-type mapping class groups.
The study finds abundant normal generators for mapping class groups.
problem Understanding normal generation in mapping class groups.
method Analyzing restrictions on invariant subsurfaces and Teichmüller spaces.
result Reducible mapping classes can normally generate mapping class groups based on their asymptotic translation lengths.
Finite presentations for mapping class groups of surfaces and surfaces with points/boundaries.
problem Finding finite presentations for balanced superelliptic mapping class groups.
method Construct finite presentations for corresponding liftable mapping class groups in a different generating set.
result Finite presentations for balanced superelliptic mapping class groups of various surfaces.
Study maps surface configurations to Heisenberg homologies for mapping class groups.
problem Understanding Mapping Class Groups of punctured surfaces.
method Action of mapping classes on Heisenberg homologies of surface configurations.
result Representations of Mapping Class Groups derived from Heisenberg homologies.
Three elements generate balanced superelliptic mapping class groups.
problem Generating balanced superelliptic mapping class groups.
method Proving groups are generated by three elements through normalizers and liftable mapping class groups.
result Balanced superelliptic mapping class groups are generated by three elements.
We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that ma…
Two elements generate extended mapping class groups of certain surfaces.
problem Generating extended mapping class groups with specific elements.
method Analyzing finite order elements and isotopy classes of homeomorphisms.
result Extended mapping class groups of certain surfaces are generated by two elements of finite order.
We prove that both the hyperelliptic mapping class group and the extended hyperelliptic mapping class group are generated by two torsion elements. We also compute the index of the subgroup of the hyperelliptic mapping class group which is generated by involutions and we prove that the extended hyperelliptic mapping cla…
Generic pseudo-Anosov mapping classes in mapping class groups.
problem Understanding the prevalence of pseudo-Anosov mapping classes.
method Proving genericity with respect to specific notions of genericity.
result Pseudo-Anosov mapping classes are generic in mapping class groups.
Computes mapping class groups of 4-manifolds with boundary.
problem Computing mapping class groups for 4-manifolds with boundary.
method Topological and smooth methods applied to compact, simply connected 4-manifolds.
result Description of topological and stable smooth mapping class groups.
Proves asymptotic mapping class groups of Cantor manifolds are of type F_infinity.
problem Finiteness properties of asymptotic mapping class groups.
method General theorem deducing asymptotic mapping class groups of Cantor manifolds are of type F_infinity under certain hypotheses.
result Asymptotic mapping class groups of Cantor manifolds are of type F_infinity.
Article provides conditions for embedding braid groups into mapping class groups.
problem Embedding braid groups into mapping class groups.
method Necessary and sufficient condition for embedding finite index subgroups.
result Conditions for embedding braid groups into mapping class groups.
New descriptions of a subgroup in mapping class groups.
problem Understanding the normal subgroup generated by genus n bounding pair maps. method Using Chillingworth and Casson-Morita invariants.
result Two descriptions of the subgroup generated by genus n bounding pair maps. Infinitely many unique ways to generate a surface's mapping class group.
problem Identifying unique ways to generate mapping class groups.
method Analyzing Nielsen equivalence classes of two-element generators.
result There are infinitely many unique ways to generate a surface's mapping class group.
Existing MAP inference algorithms for determinantal point processes (DPPs) need to calculate determinants or conduct eigenvalue decomposition generally at the scale of the full kernel, which presents a great challenge for real-world applications. In this paper, we introduce a class of DPPs, called BwDPPs, that are char…
Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.
problem Understanding asymptotic dimension of big mapping class groups of infinite type surfaces.
method Analyzing big mapping class groups with coarsely bounded generating sets and essential shifts.
result Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.
Every normal subgroup of Cantor tree's mapping class group is geometric.
problem Characterizing normal subgroups of mapping class groups.
method Generalized curve graph study and adaptation of Brendle-Margalit strategy.
result All normal subgroups of Cantor tree's mapping class group are geometric.
The paper studies liftable mapping class groups of cyclic covers of spheres.
problem Understanding liftable mapping class groups of cyclic covers of spheres.
method Derived finite generating sets, provided algorithms, determined isomorphism classes, derived presentations, and calculated normalizers and centralizers.
result Presentations and isomorphism classes of liftable mapping class groups for various covers.
LieTransformer extends self-attention to Lie groups for improved deep learning tasks.
problem Improving deep learning performance through group equivariant self-attention.
method LieSelfAttention layers that are equivariant to arbitrary Lie groups and their discrete subgroups.
result Competitive experimental results on various tasks.
Study mapping class groups of infinite type surfaces with noncompact boundaries.
problem Classify pure mapping class groups of infinite type surfaces.
method Developed a method to cut surfaces into simpler ones and combined recent results.
result Complete classification of perfect and uniformly perfect pure mapping class groups.
New findings on generating mapping class groups with specific torsion elements.
problem Understanding the structure of mapping class groups through torsion elements.
method Analyzing the orders of torsion elements required to generate mapping class groups of surfaces of varying genus.
result For specific genera, mapping class groups can be generated by two torsion elements of particular orders.
The paper computes the mapping class group of certain 6-manifolds.
problem Computing the mapping class group of specific 6-manifolds.
method Algebraic properties and homology groups of the mapping class group were determined.
result The mapping class group is residually finite and virtually torsion-free.
We show that the mapping class group of a compact orientable surface with higher complexity has the following extreme rigidity in the sense of measure equivalence: if the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernel. Moreover, we describe all lattice emb…
We show that finite index subgroups of the handlebody group are rigid in their ambient mapping class group: any injective map of a finite index subgroup of the genus g handlebody group into the genus g mapping class group is conjugation by a mapping class group element. On the other hand, we construct an injection …
We show that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups, induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses automorphisms of free groups with boundaries which play the role of mapping class groups o…