Study subgroup actions on mapping class groups using Heisenberg representations.
problem Untwisting representations of mapping class groups on Heisenberg subgroups.
method Restrict and analyze twisted representations of mapping class groups to Heisenberg subgroups.
result Untwisting representations on Torelli group for any Heisenberg representation.
The symplectic representation of mapping classes is not surjective for certain types of mapping classes.
problem The surjectivity of the symplectic representation of mapping classes, particularly pseudo-Anosov ones, is not always preserved.
method Explicit construction of symplectic matrices with a bi-Perron leading eigenvalue that cannot be represented by orientable pseudo-Anosov mapping classes.
result The symplectic representation of orientable pseudo-Anosov mapping classes is not surjective.
The paper explores mapping class groups and their quantum field theory representations.
problem Understanding finite dimensional representations of mapping class groups.
method Survey of topological quantum field theory aspects.
result Discussion of finite dimensional representations in quantum field theory.
Polynomial representations found in surface braid and mapping class groups.
problem Homological representations of surface braid and mapping class groups.
method Study of homological representation functors and short exact sequences.
result Many homological representation functors are polynomial.
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.
Study mapping class groups and their representations linking to algebraic structures.
problem Understanding projective representations of mapping class groups and their Morita classes.
method Defined projective representations of mapping class groups using modular fusion categories and analyzed their irreducibility.
result Irreducible representations imply unique Morita-class of simple algebras.
Study Heisenberg homology on surface configurations, revealing new representations of mapping class groups.
problem Homology of surface configurations with Heisenberg group representations.
method Analysis of unordered configurations in a surface, using Heisenberg group actions and representations.
result Obtained genuine and projective representations of mapping class groups from Heisenberg group actions.
The paper proposes a deep learning technique for structured and composable representations.
problem Learning structured and composable representations from input images and discrete labels.
method End-to-end deep learning to learn representations based on distance estimates between class label and contextual information.
result The representations have a clear structure allowing for class and environment decomposition.
Learn class-invariant and symmetry-equivariant representations for multi-class data.
problem Deep neural networks learn opaque representations; we aim to make them more transparent.
method Probabilistic modelling with two separate latent variables: invariant and equivariant.
result Qualitative and quantitative performance competitive with other methods, with little tuning.
New representations for surface groups expand known Anosov classes.
problem Understanding new types of representations for surface groups.
method Introducing and studying simple Anosov representations of closed hyperbolic surface groups.
result Simple Anosov representations strictly contain Anosov representations.
The paper computes characteristic classes for Lie group representations.
problem Computing characteristic classes for Lie group representations.
method The paper outlines a procedure to compute characteristic classes of irreducible representations of Lie groups, expressing them as polynomial functions in the highest weight.
result The paper expresses characteristic classes of Lie group representations as polynomial functions in the highest weight.
Study of low dimensional representations of mapping class groups of surfaces, focusing on genus ≥ 7.
problem Classifying (2g+1)-dimensional complex linear representations of mapping class groups. method Using twisted 1-cohomology groups and Morita's computation, a complete classification is given for g≥7. result No irreducible linear representations of dimension 2g+1 for g≥7. Quantum representations of mapping class groups are locally rigid at prime levels.
problem Locally rigid properties of quantum representations of mapping class groups.
method Proving local rigidity for Fibonacci representations of mapping class groups at prime levels.
result Local rigidity of Fibonacci representations of mapping class groups at prime levels.
Classifies representations up to dimension 3g-3 for surface mapping class groups.
problem Classifying representations of mapping class groups up to a certain dimension.
method Direct sum of a 2g or 2g+1 dimensional representation and a trivial one.
result Any representation up to dimension 3g-3 is a direct sum of a 2g or 2g+1 dimensional representation and a trivial one.
The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it i…
In this paper, we introduce two new classes of representations of the framed braid groups. One is the homological representation constructed as the action of a mapping class group on a certain homology group. The other is the monodromy representation of the confluent KZ equation, which is a generalization of the KZ equ…
Unified approach to homological representations of topological groups.
problem Constructing homological representations of topological groups.
method Functorial approach using topological enrichment of Quillen bracket construction.
result Unified construction of homological representations for mapping class groups and motion groups.
The paper studies Atiyah classes for Lie algebroid representations and homotopies.
problem Understanding Atiyah classes for Lie algebroid representations and homotopies.
method Constructing cocycles from higher connection forms of representations up to homotopy and studying their cohomology.
result There exists a cohomology class independent of extensions, vanishing for compatible extensions, and related to quasi-isomorphisms of Atiyah classes.
Several classes of irreducible orthogonal representations of compact Lie groups that are of importance in Differential Geometry have the property that the second osculating spaces of all of their nontrivial orbits coincide with the representation space. We say that representations with this property are of class O^2. O…
New findings on mapping class groups and their subgroups in homological representations.
problem Understanding subgroups within mapping class groups through homological representations.
method Analyzing the Johnson filtration and finite covers of mapping class groups.
result No faithful homological representations of mapping class groups exist.
Restricts quantum representations of mapping class groups to integral coefficients.
problem Integrality of non-semisimple quantum representations of mapping class groups.
method Exhibits explicit bases of states spaces that span Z[ζ]-lattices invariant under mapping class groups. result Restricts quantum representations to integral coefficients from Q(ζ) to Z[ζ]. Study shows mapping class group action is ergodic on specific representations.
problem Ergodicity of mapping class group action on specific representations.
method Applied symplectic methods developed by Goldman and Xia.
result The action is ergodic.
The Soft Nearest Neighbor Loss improves representation quality and generalization.
problem Improving the quality of representations in machine learning models.
method Explored and expanded the Soft Nearest Neighbor Loss to measure class entanglement.
result Maximizing the entanglement of different class representations leads to better generalization and uncertainty estimation.
We prove asymptotic faithfulness for the quantum Sp(4) mapping class group representation. This provides the first example of asymptotic faithfulness lying outside of the An family. The methods used are generalized from the proof of asymptotic faithfulness for skein SU(2)k mapping class group represent…
New unitary representations for mapping class groups without almost invariant vectors.
problem Understanding unitary representations of mapping class groups.
method Space of measured foliations and Teichmüller space.
result None of the representations has almost invariant vectors.
Researchers study Prym representations for handlebody groups, focusing on cyclic cases.
problem Understanding Prym representations for handlebody groups and their images.
method Restrict Prym representations to handlebody groups and twist groups, focusing on cyclic cases.
result Determined the image of Prym representations in the cyclic case for handlebody groups.
We study the TQFT mapping class group representations for surfaces with boundary associated with the SU(2) gauge group, or equivalently the quantum group $U_q(\Sl(2))$. We show that at a prime root of unity, these representations are all irreducible. We also examine braid group representations for transcendental valu…
Unified study of homological representations of mapping class groups.
problem Fundamental relationships between different homological representations.
method Analysis of twisted homology of configuration spaces.
result Unified account of non-degenerate pairings, embeddings, and isomorphisms.
We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU(2) representations. In particular, they may be used to give an explicit check that spherical braid grou…
Researchers compute cohomology of mapping class groups with Prym representations, showing instability for large genus.
problem Computing the cohomology of mapping class groups with level structures and Prym representations.
method Using twisted cohomology and Prym representations for any positive integer r.
result Cohomology exhibits instability for large genus, but remains stable for r=0 or r=1.
Any nontrivial homomorphism from the mapping class group of an orientable surface of genus g≥3 to $\GL(2g,\C)$ is conjugate to the standard symplectic representation. It is also shown that the mapping class group has no faithful linear representation in dimensions less than or equal to 3g−3.
We use quantum invariants to define an analytic family of representations for the mapping class group of a punctured surface. The representations depend on a complex number A with |A| <= 1 and act on an infinite-dimensional Hilbert space. They are unitary when A is real or imaginary, bounded when |A|<1, and only densel…
We describe a construction of the modular class associated to a representation up to homotopy of a Lie groupoid. In the case of the adjoint representation up to homotopy, this class is the obstruction to the existence of a volume form, in the sense of Weinstein's "The volume of a differentiable stack".
Study on mapping class groups for low genus surfaces.
problem Arithmetic image of homological representations for low genus mapping class groups.
method Examined centralizers of finite groups acting on surfaces with genus at most 3.
result Complete answer to the arithmetic image question for low genus surfaces.
Minimal action of mapping class group on character variety.
problem Character variety of Deroin-Tholozan representations.
method Geometric perspective using symplectic structure.
result Infinite mapping class group orbits are dense.
ZSL-KG learns class representations from common sense knowledge graphs.
problem Predicting classes without labeled examples using semantic class representations.
method TrGCN, a novel transformer graph convolutional network, embeds nodes from common sense knowledge graphs in a vector space.
result ZSL-KG improves over existing methods on five out of six zero-shot benchmark datasets.
This paper introduces a new feature learning technique based on error representation.
problem Learning high-level features for classification from diverse and imbalanced data.
method Inverse feature learning using error representation approach.
result Significantly better performance compared to state-of-the-art techniques.
Paper shows ergodicity and irreducibility of mapping class group boundary representation.
problem Ergodicity and irreducibility of mapping class group boundary representation.
method Statistical hyperbolicity and classical result of Masur generalization.
result Boundary representation of mapping class group is ergodic and irreducible.
Let e denote the Euler class on the space Hom(Γg,PSL(2,R)) of representations of the fundamental group Γg of the closed surface Σg of genus g. Goldman showed that the connected components of Hom(Γg,PSL(2,R)) are precisely the inverse images e−1(k), for 2−2g≤k≤2g−2, and t…
The SU(2) TQFT representation of the mapping class group of a closed surface of genus g, at a root of unity of prime order, is shown to be irreducible. Some examples of reducible representations are also given.
In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer representation of the braid group B_n, which was shown to be faithful by Bigelow and Krammer. We obtain a faithful repres…
Study mapping class groups' action on surface homology.
problem Characterize mapping class groups' homology representation.
method Precise characterization of induced homology representation.
result Characterized the image of the homology representation.
Proves rigidity of SU(2) and SO(3) quantum representations at prime levels.
problem Quantum representations of mapping class groups at prime levels.
method Ocneanu rigidity of modular categories and harmonic representatives in Hodge theory.
result Rigidity of SU(2) and SO(3) quantum representations at all prime levels for closed surfaces of genus at least 7.
Study of Anosov representations with Lipschitz limit set and applications to rigidity.
problem Characterizing Anosov representations with specific limit set properties.
method Introducing an unstable Jacobian and analyzing its orbit growth rate.
result Many higher rank representations belong to the studied class.
Open set recognition problems exist in many domains. For example in security, new malware classes emerge regularly; therefore malware classification systems need to identify instances from unknown classes in addition to discriminating between known classes. In this paper we present a neural network based representation…
We study the action of the mapping class group on the real homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen-Thurston …
We show that the bounded Borel class of any dense representation $ρ: G\to \PSL_n\bC$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group G. When n=2, the Borel class is equal to the 3-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, …
Proposes a new framework for image generation using classification latent space representations.
problem Combining discriminative and dense representations for image generation and reconstruction.
method Discriminative modeling framework using manipulated supervised latent representations.
result Higher classification accuracy and visually realistic image generation compared to existing models.