New braid representations using virtual knot theory.
problem No classical features in virtual knot theory.
method Construct new braid representations using virtual knot theory.
result New representations of classical braids.
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48 results for “representation theory”
Enhances knot Floer homology with algebraic representation theory.
problem Compatibility between summands in bordered knot Floer homology.
method Categorifies intertwining property of higher representations.
result New algebraic reformulation of compatibility property.
Study representation varieties of twisted Hopf links using combinatorial and Hodge theory.
problem Representation theory of Hopf link complements with n twists.
method Combinatorial problem and equivariant Hodge theory.
result Close formulas for E-polynomials of representation and character varieties for ranks 2 and 3.
Based on the analogies between knot theory and number theory, we study a deformation theory for SL_2-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-SL_2-representations, we prove the existence of the universal deformation of a given…
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.
New knot theory module shows torsion-ness in number theory.
problem Torsion-ness of Selmer modules in Galois representations.
method Introducing adjoint homological Selmer module for SL-representations of knot groups.
result Finitely generated torsion-ness of the new Selmer module.
Extends Heegaard Floer theory to surfaces of dimension one.
problem Developing a theory for surfaces of dimension one.
method 2-representation theory of gl(1|1)^+ and tensor product construction.
result Extension of Heegaard Floer theory to dimension one.
Maximal Laplacian algebras applied to invariant theory solved inverse problems.
problem Maximality of Laplacian algebras and their applications in invariant theory.
method Proof of maximality and applications to classical invariant theory.
result Introduction of generalized polarizations and if-and-only-if criterion.
Develops theory of relatively Anosov representations using flow examples.
problem Understanding relatively Anosov representations.
method Uses a contracting flow on a bundle to define Anosov representations and builds examples.
result Builds families of examples of relatively Anosov representations.
Quantum theory uses modular group representations to assign invariants to 3-manifolds.
problem Assigning invariants to 3-manifolds via modular group representations.
method Projective representations of the modular group derived from a noncommutative torus.
result Computed traces and determinants of matrices associated with modular group elements.
Weierstrass-type representations have been used extensively in surface theory to create surfaces with special curvature properties. In this paper we give a unified description of these representations in terms of classical transformation theory of -surfaces.
Theory of Θ-positive representations for real closed fields.
problem Generalizing positive representations to real closed fields.
method Developing theory for Fuchsian groups to linear groups over real closed fields.
result Theory encompasses many generalizations of positive or Anosov representations.
We introduce the idea of *representation stability* (and several variations) for a sequence of representations V_n of groups G_n. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical t…
Study on distinguishing mutant knots using specific representations.
problem Distinguishing mutant knots using colored HOMFLY-PT polynomials.
method Calculating polynomials and differences for mutant knot polynomials in specific representations.
result Properties of mutant knot polynomials in representations [3,1] and [4,2] were studied.
Theory of relatively Anosov representations using flow methods.
problem Developing a theory for relatively Anosov representations.
method Using the contracting flow on a bundle to define and study relatively Anosov representations.
result Definition and study of uniformly relatively Anosov representations and a stability result.
Study extends Vogel's universality to torus knots in adjoint representation.
problem Applying Vogel's universality to knot invariants in adjoint representation theory.
method Extending Vogel's parameters to include torus knots and focusing on with odd .
result Unified description of adjoint invariants for torus knots with odd .
The paper explores proper actions and their relation to representation theory, with new quantitative methods.
problem Understanding proper actions and their connection to representation theory.
method Geometric criteria, sharpness measure, and dynamical volume estimates.
result New quantitative methods have established temperedness criteria for unitary representations.
Researchers create projective representations of Hecke groups using TQFT.
problem Constructing projective representations of Hecke groups.
method Using Witten-Reshetikhin-Turaev topological quantum field theory of higher genus surfaces.
result The representation's image group is infinite at low levels in genus 2.
A Bornological Perspective on the Representability of Derived Moduli Stacks of Solutions to PDEsmath.AG
New theory proves representability of PDE solutions without complex machinery.
problem Proving representability of PDE solutions using traditional methods is difficult.
method Developed a new model of derived differential geometry using -bornological rings.
result Representability of derived moduli stacks of PDE solutions naturally follows from an Artin-Lurie style theorem.
GNNs learn graph representations, with new theory on their power and limitations.
problem Understanding the capabilities and limitations of GNNs.
method Theoretical analysis of GNNs, focusing on approximation and learning properties.
result New insights into the representation, generalization, and extrapolation of GNNs.
The paper reinterprets knot group invariants using affine transformations.
problem Alexander invariants of knots and their geometric interpretation.
method Representation varieties of knot groups into .
result Alexander polynomial as the singular locus of a coherent sheaf.
GQML uses symmetries from representation theory to improve quantum machine learning.
problem Creating quantum models with symmetries to improve performance.
method Introduction to representation theory for quantum learning, focusing on group actions and symmetries.
result Effective implementation of GQML requires knowledge of group representation theory.
We consider the problem of existence of representations of topological groupoids on a principal bundle and the classification of such representations up to gauge transformation. Such representations naturally occur in various contexts such as gauge theory, lattice gauge fields, equivariant bundles, etc. In the course o…
The paper studies conjugating complex representations into real ones.
problem Understanding representations of surface groups into complex Lie groups.
method Analyzes representations of finitely generated groups into and determines conjugacy conditions.
result Identifies representations in the larger variety that are conjugate in to a representation in .
We give an elementary introduction to our papers relating the geometry of rational homogeneous varieties to representation theory. We also describe related work and recent progress.
Develops higher representation theory for odd Khovanov homology and rewriting theory.
problem Quantum topology and higher algebraic structures.
method Higher representation theory and rewriting theory applied to Khovanov homology.
result Established a basis theorem for graded -foams.
Explains how group representations behave under subgroup restrictions.
problem Behavior of irreducible representations when restricted to subgroups.
method Expository account of new directions in representation theory.
result Highlights recent advances in branching problems for real reductive groups.
We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generali…
Representation Stabilitymath.GT
Representation stability is a phenomenon whereby the structure of certain sequences of spaces can be seen to stabilize when viewed through the lens of representation theory. In this paper I describe this phenomenon and sketch a framework, the theory of FI-modules, that explains the mechanism behind it.
Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groupsmath.DG
Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.
problem Understanding geometrically finite Fuchsian groups and their representations.
method Theory of Anosov representations, type-preserving deformations, limit maps, relative Anosov and dominated representations.
result Cusped Hitchin representations are Borel Anosov, stable under deformations, and limit maps vary analytically.
These notes of a course given at IRMA in April 2009 cover some aspects of the representation theory of fundamental groups of manifolds of dimension at most 3 in compact Lie groups, mainly $\su$. We give detailed examples, develop the techniques of twisted cohomology and gauge theory. We review Chern-Simons theory and d…
Researchers define new quantum representations for a Lorentz algebra and study their Clebsch-Gordan decomposition.
problem Quantum representations of a Lorentz algebra and their Clebsch-Gordan decomposition.
method Defined new infinite-dimensional irreducible representations using quantum torus algebra and quantized Chern-Simons theory.
result The Clebsch-Gordan decomposition of tensor product representations reduces to problems in Fenchel-Nielson length operators in quantized Chern-Simons theory.
We adapt some of the methods of quantum Teichmüller theory to construct a family of representations of the pure braid group of the sphere.
-covering of a supermanifoldmath.DG
Theory of -graded manifolds and coverings of supermanifolds.
problem Developing a theory for -graded manifolds and coverings of supermanifolds.
method Using tools from representation theory, we introduce and investigate -graded coverings of supermanifolds.
result Theory of -graded coverings of supermanifolds introduced and investigated.
With this positional paper we present a representation learning view on predicate invention. The intention of this proposal is to bridge the relational and deep learning communities on the problem of predicate invention. We propose a theory reconstruction approach, a formalism that extends autoencoder approach to repre…
symmetryhep-th
Summarizes quantum field theories with discrete symmetry, classifying representations and anomalies.
problem Classifying representations and anomalies in quantum field theories with discrete symmetry.
method Classification of representations and anomalies using the ring of profinite integers.
result Rich and complex classification of representations and anomalies.
SSL theory improves representation learning from raw data.
problem Challenges in SSL, including instability and collapse.
method Precise analysis of generalization performance with a theory-friendly setup.
result Insights for SSL practitioners on data augmentation, network architecture, and training algorithm.
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.
Relation between generalized Weierstrass representation for conformal immersion of generic surfaces into three-dimensional space and Lax-Phillips scattering theory for automorphic functions is considered.
Fast algorithm for braid group Hecke representation, applied to knot invariants.
problem Computing topological invariants of knots efficiently.
method Representation-theoretic approach to braid group, leveraging quantum topology.
result Fast algorithm for Hecke representation of braid group, finding non-trivial braids.
Theory of smooth relative connections on quiver bundles developed.
problem Existence of smooth relative connections over quiver bundles.
method Developed a theory over on smooth twisted quiver bundles, provided obstructions and necessary/sufficient conditions.
result Established a necessary and sufficient condition for the existence of smooth relative connections on tree-type quiver bundles.
Springer varieties appear in both geometric representation theory and knot theory. Motivated by knot theory and categorification Khovanov provides a topological construction of Springer varieties. We extend Khovanov's construction to all two-row Springer varieties. Using the combinatorial and diagrammatic …
Positive energy representations of affine algebras and Stokes matrices of the affine Toda equationsmath.DG
Constructs positive energy representations from Toda equations Stokes data.
problem Creating positive energy representations of affine algebras.
method Using Stokes data of tt*-Toda equations to construct representations.
result Illustrates construction with examples in conformal field theory.
Neural networks are mathematically represented via quiver representations.
problem Understanding how neural networks process data and create representations.
method Representing neural networks as quiver representations with activation functions.
result Neural networks' computations can be studied algebraically and geometrically.
Ihara initiated to study a certain Galois representation which may be seen as an arithmetic analogue of the Artin representation of a pure braid group. We pursue the analogies in Ihara theory further, following after some issues and their inter-relations in the theory of braids and links such as Milnor invariants, John…
The paper introduces contexture theory to characterize representation learning from contexts.
problem Lack of systematic characterization of representation learning methods.
method Characterizes representation learning as learning from the association between input and context variable.
result Contexture theory shows that representations can be approximated by top singular functions of the context.
It is widely believed that learning good representations is one of the main reasons for the success of deep neural networks. Although highly intuitive, there is a lack of theory and systematic approach quantitatively characterizing what representations do deep neural networks learn. In this work, we move a tiny step to…
We set up foundations of representation theory over , the sphere spectrum, which is the `initial ring' of stable homotopy theory. In particular, we treat -Lie algebras and their representations, characters, -Verma modules and their duals, Harish-Chandra pairs and Zuckermann functors. As an application, w…