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.
Study geometry and PDEs from group-determinants and representation theory.
problem Geometry and PDEs from group-determinants and representation theory.
method Analysis of group-determinants and representation theory.
result Spectral theory of operators linked to finite Fourier transform theory.
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…
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.
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.
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.
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.
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.
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 PGL(k,C) and determines conjugacy conditions. result Identifies representations in the larger variety that are conjugate in PGL(k,C) to a representation in PGL(k,R). Surveying Hitchin representations of Fuchsian groups.
problem Understanding representations of Fuchsian groups.
method Survey and conjectural geometric description.
result Conjectural geometric picture of an augmented Hitchin component.
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…
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 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.
Characterizes Coxeter groups with convex cocompact representations in projective space.
problem Understanding representations of Coxeter groups as convex cocompact reflection groups.
method Investigates representations of Coxeter groups into GL(n,R) as geometric reflection groups in projective space.
result Characterizes Coxeter groups that admit convex cocompact representations and describes the spaces of such representations.
We adapt some of the methods of quantum Teichmüller theory to construct a family of representations of the pure braid group of the sphere.
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 SL2-representations of knot groups. result Finitely generated torsion-ness of the new Selmer module.
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 extrmAGL1(C). 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.
Representation stability is a theory describing a way in which a sequence of representations of different groups is related, and essentially contains a finite amount of information. Starting with Church-Ellenberg-Farb's theory of FI-modules describing sequences of representations of the symmetric groups, we now have …
We express the real connective K theory groups of the quaternion QL group of order 2j≥8 in terms of the representation theory of by showing ko4k−1(BQL)=KSp(S4k+3/τQL) where tau is any fixed point free representation of QL in U(2k+2)
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…
In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-gr…
Quantum groups created from disk configuration space homologies.
problem Creating quantum groups from algebraic structures.
method Reconstructing quantum groups from homologies of configuration spaces of disks.
result New combinatorics and actual submanifolds of configuration spaces.
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.
Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on …
Theory of H-graded manifolds and coverings of supermanifolds.
problem Developing a theory for H-graded manifolds and coverings of supermanifolds. method Using tools from representation theory, we introduce and investigate H-graded coverings of supermanifolds. result Theory of H-graded coverings of supermanifolds introduced and investigated. 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.
The paper constructs representations of flat virtual braids by free group automorphisms.
problem Representing flat virtual braids by automorphisms of free groups.
method Construction of representations of flat virtual braid groups FVBn by automorphisms of free groups of rank 2n. result Established conditions of faithfulness and properties of the kernel for n≥3. Study classifies Lie group representations with non-empty boundary orbit space.
problem Classifying representations of Lie groups with non-empty boundary orbit space.
method Detailed calculations based on previous work.
result Classification of Lie group representations with non-empty boundary orbit space.
We show that representations of the loop braid group arise from Aharonov-Bohm like effects in finite 2-group (3+1)-dimensional topological higher gauge theory. For this we introduce a minimal categorification of biracks, which we call W-bikoids (welded bikoids). Our main example of W-bikoids arises from finite 2-groups…
Group equivariant neural networks simplify complex tasks with group representation theory.
problem Challenging tasks requiring input transformations like rotations.
method Group representation theory, non-commutative harmonic analysis, differential geometry.
result A neural network is group equivariant if and only if it has a convolutional structure.
Extending braid group representations to singular braid monoids and groups.
problem Extending braid group representations to singular braid monoids and groups.
method Investigating the extension of representations from braid groups to singular braid monoids and groups, and computing defects.
result Constructing a linear representation of the singular braid group that is an extension of the Lawrence-Krammer-Bigelow representation and computing its defect.
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 …
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…
Consider a finite, regular cover Y→X of finite graphs, with associated deck group G. We relate the topology of the cover to the structure of H1(Y;C) as a G-representation. A central object in this study is the {\em primitive homology} group $H_1^{\mathrm{prim}}(Y;\mathbb{C})\subseteq H_1(Y;\mathbb{…
Deformation K-theory associates to each discrete group G a spectrum built from spaces of finite dimensional unitary representations of G. In all known examples, this spectrum is 2-periodic above the rational cohomological dimension of G (minus 2), in the sense that T. Lawson's Bott map is an isomorphism on homotopy in …
In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these repr…
We review recent developments in the theory of Thompson group representations related to knot theory.
These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal …
Paper discusses groups where twisted Alexander polynomials vanish.
problem Understanding groups with vanishing twisted Alexander polynomials.
method Introduced and studied twisted Alexander vanishing groups, constructed knots, and analyzed representations.
result Every faithful irreducible representation of a TAV group causes the twisted Alexander polynomial to be zero.
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.
Autoencoder learns group representations from actions, improving future prediction accuracy.
problem Learning internal models of interactions with the real world.
method Homomorphism autoencoder with group representation trained on equivariance-derived loss.
result Agents can predict future actions with improved accuracy.
Study projective representations of infinite-dimensional Hilbert-Lie groups.
problem Characterize and classify representations of Hilbert-Lie groups.
method Use covariance with respect to one-parameter groups of automorphisms and implement perturbation theory.
result Explicit determination of central extensions for projective representations.
The paper calculates Alexander polynomials for knots using finite group representations.
problem Calculating Alexander polynomials for knots using specific group representations.
method Defined twisted Alexander polynomials associated with regular representations of finite groups.
result Several formulas for the twisted Alexander polynomial are provided.
Quantum theory constructs a group and skein module for knot complements.
problem Understanding the fundamental group of knot complements using quantum methods.
method Using bottom tangles, the universal space of quantum representations is constructed, then factored by the skein relation to get the skein module.
result Derives recurrence relation for the colored Jones polynomial, known as Aq polynomial. We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups GL(n,C) and U(p,q). Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott--Morse theory on the corresponding moduli spaces.
Motivated by the study of the interrelation between functorial and algebraic quantum field theory, we point out that on any locally trivial bundle of compact groups, representations up to homotopy are enough to separate points by means of the associated representations in cohomol- ogy. Furthermore, we observe that the …
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.