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.
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.
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.
Study on projective structures linked to Hitchin representations.
problem Understanding the topology and geometric properties of projective structures.
method Using gauge theory, flat bundles, Higgs bundles, and geometric constructions.
result Some projective structures are fibered in a standard way.
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.
This paper extends geometric structure theory to infinite type structures.
problem Calculating characteristic class relations in complex Cartan geometries.
method Improves representation theory for infinite type structures.
result Direct calculation of characteristic class relations from structure group representation.
New representation theory for closed geodesic subflows.
problem Classifying representations with good geometric properties.
method Restricting to invariant closed geodesic subflows.
result Equivalent characterizations and properties of new representations.
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…
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 …
We explain how, starting with a stack of D4-branes ending on an NS5-brane in type IIA string theory, one can, via T-duality and the topological-holomorphic nature of the relevant worldvolume theories, relate (i) the lattice models realized by Costello's 4d Chern-Simons theory, (ii) links in 3d analytically-continued Ch…
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.
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…
Springer varieties appear in both geometric representation theory and knot theory. Motivated by knot theory and categorification Khovanov provides a topological construction of (n/2,n/2) Springer varieties. We extend Khovanov's construction to all two-row Springer varieties. Using the combinatorial and diagrammatic …
From a geometrical point of view it is, so far, not sufficiently well understood what should be a "noncommutative principal bundle". Still, there is a well-developed abstract algebraic approach using the theory of Hopf algebras. An important handicap of this approach is the ignorance of topological and geometrical aspe…
I survey methods from differential geometry, algebraic geometry and representation theory relevant for the permanent v. determinant problem from computer science, an algebraic analog of the P v. NP problem.
Let G be a finite group. To every smooth G-action on a compact, connected and oriented surface we can associate its data of singular orbits. The set of such data becomes an Abelian group B_G under the G-equivariant connected sum. We will show that the map which sends G to B_G is functorial and carries many features of …
Based on representation theory of Clifford algebra, Ferus, Karcher and Münzner constructed a series of isoparametric foliations. In this paper, we will survey recent studies on isoparametric hypersurfaces of OT-FKM type and investigate related geometric constructions with mean curvature flow.
Shelstad's character identity is an equality between sums of characters of tempered representations in corresponding L-packets of two real, semisimple, linear, algebraic groups that are inner forms to each other. We reconstruct this character identity in the case of the discrete series, using index theory of elliptic…
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.
Simplified 3D Dijkgraaf-Witten theory with defects explained geometrically.
problem Constructing 3D Dijkgraaf-Witten theory with defects.
method Symmetric monoidal functor from defect cobordism category to vector spaces, using geometric and homotopy theoretic methods.
result Explicit construction of 3D untwisted Dijkgraaf-Witten theory with defects.
The paper studies heat kernel behavior on symmetric spaces.
problem Large-time behavior of heat operator traces on symmetric spaces.
method Uses representation theory and Carmona's proof of Vogan's lambda map.
result Provides an asymptotic formula for heat kernel behavior.
The paper connects knot representations and spherical quandle colorings.
problem Defining the Casson-Lin invariant for knots.
method One-to-one correspondence between SU(2)-representations and spherical quandle colorings. result Geometric interpretation of the trace-free condition for the Casson-Lin invariant.
New geometric variant of factorization homology for conformally flat manifolds.
problem Defining invariants of conformally flat manifolds.
method Introducing a metric-dependent geometric variant of factorization homology.
result Left Kan extensions of conformally flat d-disk algebras define invariants of conformally flat manifolds. Research classifies geometric structures on manifolds using surface group representations.
problem Classifying geometric structures on manifolds related to surface group actions on character varieties.
method Surveying results on surface groups, focusing on compact and non-compact target groups, discussing various representations and their dynamics.
result Dichotomy in dynamics of character varieties based on compactness of target groups.
New method learns disentangled representations using Gromov-Monge maps.
problem Learning disentangled representations from unlabelled data.
method Introduces a novel approach based on Gromov-Monge maps to preserve geometric features while aligning data distributions.
result Demonstrates effectiveness on four benchmarks, outperforming other methods.
In current paper we refer to the geometrical classification of the Einstein equations which has been developed by one of the authors of this paper. This classification was based on the classical theory for decomposition of the tensor product of representations into irreducible components, which is studied in the elemen…
Motivated by recent developments in the AdS/CFT correspondence, we provide several alternative bulk descriptions of an arbitrary Wilson loop operator in Chern-Simons theory. Wilson loop operators in Chern-Simons theory can be given a description in terms of a configuration of branes or alternatively anti-branes in the …
Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander's theorem), Mathematical Physics (Yang-Baxter's equation) and Algebraic Geometry (monodromy invariants). In this note we will focus on their algebraic-geometric aspects, explaining how the representation the…
These notes discuss various aspect of the ``representation theory'' of Poisson manifolds, with focus on Morita equivalence and Picard groups. We give a brief introduction to Poisson geometry (including Dirac and twisted Poisson structures) and algebraic Morita theory before presenting the geometric Morita theory of Poi…
Geometric structures on manifolds became popular when Thurston used them in his work on the geometrization conjecture. They were studied by many people and they play an important role in higher Teichmüller theory. Geometric structures on a manifold are closely related with representations of the fundamental group and w…
Foundation models fail to preserve continuous geometry, identified as the Geometric Alignment Tax.
problem Continuous geometry is lost in foundation models due to discrete categorical bottlenecks.
method Controlled ablations on synthetic systems and evaluation of 14 biological models using rate-distortion theory and MINE.
result Replacing cross-entropy with a continuous head reduces geometric distortion by up to 8.5x.
The paper connects Apollonian packings to knot theory and improves link representations.
problem Realizing algebraic links in Apollonian packings.
method Introducing new representations of links in tangency graphs of sphere packings, proving link realizability, and improving upper bounds.
result Any algebraic link can be realized in the cubic section of the orthoplicial Apollonian packing.
This article is a survey on the braid groups, the Artin groups, and the Garside groups. It is a presentation, accessible to non-experts, of various topological and algebraic aspects of these groups. It is also a report on three points of the theory: the faithful linear representations, the cohomology, and the geometric…
Geometrically reformulates elasticity theory using exterior calculus.
problem Formulating nonlinear elasticity theory geometrically.
method Using exterior calculus and bundle-valued differential forms.
result Equivalence to standard tensor calculus formulations.
Unified approach to data processing using gauge theory.
problem Data representation and analysis with consistent symmetry.
method Geometric gauge theory for discrete vector bundles.
result Unified understanding of heat kernel properties and data transformation.
Study geometric and representation theory of statistical transformation models.
problem Understand relationships between induced structures and actions on measure spaces.
method Investigate geometric properties and symplectic actions on induced structures.
result Show equivariance of action and relationships between tangent bundles and projectivizations.
Geometric approach connects Burau representation to sphere metrics, identifying kernels.
problem Faithfulness of the Burau representation for the 4-strand case.
method Geometric and orbifold theory.
result Identifies the kernel of the Burau representation for some cases.
We study the representation theory of the quantum Teichmueller space when going to infinity in the classical Teichmueller space. The geometric ingredients are the extension of Thurston's shear coordinates to the augmented Teichmueller space and the study of the Weil-Petersson Poisson structure for this extension. The r…
Representations of C∗-algebras are realized on section spaces of holomorphic homogeneous vector bundles. The corresponding section spaces are investigated by means of a new notion of reproducing kernel, suitable for dealing with involutive diffeomorphisms defined on the base spaces of the bundles. Applications of th…
This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (QSF), geometric simple connectivity (GSC), topological inverse-representations, and the notion of easy groups. As we will explain, these properties are central in the theory of d…
Develops reduction method for strong Dirac maps.
problem Generalizing Poisson momentum maps.
method General procedure for reduction along strong Dirac maps.
result Recover and introduce new Poisson, quasi-Poisson, and Dirac reduced structures.
We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line …
Starting from a higher Courant bracket associated to exceptional generalized geometry, we provide a systematic derivation of all types of fluxes and their Bianchi identities for four-dimensional compactifications of M-theory. We show that these fluxes may be understood as generalized Wess-Zumino terms in certain topolo…
This paper develops a geometric framework for Wilson surfaces in higher gauge theory.
problem Quantum field theory of Wilson surfaces in higher gauge theory.
method Higher coadjoint orbit theory and derived geometric framework.
result Identification of derived coadjoint orbits and their quantization.
The paper outlines future work in random sets theory.
problem Developing a theory of statistical reasoning with random sets.
method Generalizing logistic regression, probability laws, and geometric uncertainty.
result A new geometric approach to uncertainty with general random sets.
Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles. Bounding chains are realized, and used to obtain equivalence moves for homologous c…
New proof and description of commutator subgroups for free and surface groups.
problem Understanding commutator subgroups of free and surface groups.
method Geometric proof and representation-theoretic description.
result New free generating sets and structure descriptions for commutator subgroups.
Let M be a surface (possibly nonorientable) with punctures and/or boundary components. The paper is a study of ``geometric subgroups'' of the mapping class group of M, that is subgroups corresponding to inclusions of subsurfaces (possibly disconnected). We characterise the subsurfaces which lead to virtually abelian ge…