arXiv research

The paper explores mapping class groups and their quantum field theory representations.

We prove that there is no faithful finite-dimensional representation by skew-hermitian matrices of a ``basic algebra of observables'' B on a noncompact symplectic manifold M. Consequently there exists no finite-dimensional quantization of any Lie subalgebra of the Poisson algebra C^\infty(M) containing B.

Categorifies Jones polynomial using Lie theory.

Proves a representation that separates a subgroup in RAAGs.

We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming--Viot process is a particular example. The defining property of finite dimensional polynomial processes considered by Cuchiero et al. (2012) and Filipovic and Larsson (2016) is transferred to this infinite …

We investigate how one can twist L^2-invariants such as L^2-Betti numbers and L^2-torsion with finite-dimensional representations. As a special case we assign to the universal covering of a finite connected CW-complex X together with an element phi in H^1(X;R) a phi-twisted L^2-torsion function from R^{>0} to R, provid…

Proofs Lie's classification of certain vector field subalgebras.

Researchers create explicit representations for skein algebras of small surfaces, revealing their Azumaya loci.

We define an action of Artin's braid group on a finite dimensional algebra.

We study satellites of Legendrian knots in R^3 and their relation to the Chekanov-Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in R^3 and augmentations of its DGA by showing that the DGA has finite-dimensional represe…

With a view towards applications in the theory of infinite-dimensional representations of finite-dimensional Lie supergroups, we introduce a new category of supermanifolds. In this category, supermanifolds of `maps' and `fields' (fibre bundle sections) exist. In particular, loop supergroups can be realised globally in …

Extends Fried's result to arbitrary representations of compact hyperbolic manifolds.

Let G be a finite group. The unit sphere in a finite-dimensional orthogonal G-representation motivates the definition of homotopy representations, due to tom Dieck. We introduce an algebraic analogue, and establish its basic properties including the Borel-Smith conditions and realization by finite G-CW-complexes.

Infinite dimensional measure-valued processes modeled as polynomial diffusions.

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 …

Constructs CAT(0) actions for certain groups without unipotent elements.

We prove that there are no nontrivial finite-dimensional Lie representations of certain Poisson algebras of polynomials on a compact symplectic manifold. This result is used to establish the existence of a universal obstruction to quantizing a compact symplectic manifold, regardless of the dimensionality of the represe…

For a complete hyperbolic three manifold M, we consider the representations of its fundamental group obtained by composing a lift of the holonomy with complex finite dimensional representations of SL(2,C). We prove a vanishing result for the cohomology of M with coefficients twisted by these representations, using tech…

Essential curves in certain graph manifolds imply non-linear groups.

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…

Every finite dimensional real representation of a compact real semisimple Lie algebra determines a metric 2-step nilpotent Lie algebra and a corresponding simply connected metric 2-step nilpotent Lie group N. We study the differential geometry of N using representation theory of the complexified complex semisimple Lie …

We characterize conjugate nonparametric Bayesian models as projective limits of conjugate, finite-dimensional Bayesian models. In particular, we identify a large class of nonparametric models representable as infinite-dimensional analogues of exponential family distributions and their canonical conjugate priors. This c…

Infinite-activity completely random measures (CRMs) have become important building blocks of complex Bayesian nonparametric models. They have been successfully used in various applications such as clustering, density estimation, latent feature models, survival analysis or network science. Popular infinite-activity CRMs…

Characterizes Anosov reducible representations in terms of eigenvalues.

This work constructs a finite-dimensional projective representation for a quantum Teichmüller model.

The objective of this paper is to determine the finite dimensional, indecomposable representations of the algebra that is generated by two complex structures over the real numbers. Since the generators satisfy relations that are similar to those of the infinite dihedral group, we give the algebra the name iD-infinity.

In this paper we study the problem of approximation of the $L^2$-topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of Lück, dealing with towers of finitely sheeted normal coverings. We prove approximation theorems, establishing relations between the homological invar…

This paper solves nonparametric estimation of continuous DPPs using kernel methods.

This paper generalizes the Maurer--Pontil framework of finite-dimensional lossy coding schemes to the setting where a high-dimensional random vector is mapped to an element of a compact set of latent representations in a lower-dimensional Euclidean space, and the reconstruction map belongs to a given class of nonlinear…

Geometrically represents the Jacobian for a mechanical system with symmetry.

We produce a sequence of finite dimensional representations of the fundamental group $π_1(S)$ of a closed surface where all simple closed curves act with finite order, but where each non--simple closed curve eventually acts with infinite order. As a consequence, we obtain a representation theoretic algorithm which deci…

Compute central extension of mapping class group from stated skein algebra

Cochran, Orr and Teichner introduced $L^2$--eta--invariants to detect highly non--trivial examples of non slice knots. Using a recent theorem by Lück and Schick we show that their metabelian $L^2$--eta--invariants can be viewed as the limit of finite dimensional unitary representations. We recall a ribbon obstruction t…

Proposes a neural network autoencoder for smoothing and representation learning of functional data.

Kernel methods are powerful tools to capture nonlinear patterns behind data. They implicitly learn high (even infinite) dimensional nonlinear features in the Reproducing Kernel Hilbert Space (RKHS) while making the computation tractable by leveraging the kernel trick. Classic kernel methods learn a single layer of nonl…

The paper explores mapping class group quotients by Dehn twists and their representations.

We prove a uniqueness result for finite-dimensional representations of the Kauffman skein algebra $\mathcal{S}_A(S)$ of a surface $S$, when $A$ is a root of unity and when the surface $S$ is a sphere with at most four punctures or a torus with at most one puncture. We show that, if two irreducible representations of $\…

Study Anosov representations of reducible suspensions of hyperbolic groups.

New structure for quantum algebra representations.

Let $Y=Γ\backslash H^n$ be a quotient of the hyperbolic space by the action of a discrete convex-cocompact group of isometries. We describe certain spaces of $Γ$-invariant currents on the sphere at infinity of $H^n$ with support on the limit set of $Γ$. These spaces are finite-dimensional. The main result identifies th…

In this paper, we introduce a study of prolongations of representations of Lie groups. We obtain a faithful (one-to-one) representation of TG where G is a finite-dimensional Lie group and TG is the tangent bundle of G, by using (not necessarily faithful) representations of G. We show that tangent functions of Lie group…

In this paper we show how to approximate a Heath-Jarrow-Morton dynamics for the forward prices in commodity markets with arbitrage-free models which have a finite dimensional state space. Moreover, we recover a closed form representation of the forward price dynamics in the approximation models and derive the rate of c…

We develop a functional encoder-decoder approach to supervised meta-learning, where labeled data is encoded into an infinite-dimensional functional representation rather than a finite-dimensional one. Furthermore, rather than directly producing the representation, we learn a neural update rule resembling functional gra…

The main goal of this note is to provide evidence that the first rational homology of the Johnson subgroup $K_{g,1}$ of the mapping class group of a genus g surface with one marked point is finite-dimensional. Building on work of Dimca-Papadima, we use symplectic representation theory to show that, for all $g > 3$, the…

Hidden symmetry of a G'-space X is defined by an extension of the G'-action on X to that of a group G containing G' as a subgroup. In this setting, we study the relationship between the three objects: (A) global analysis on X by using representations of G (hidden symmetry); (B) global analysis on X by using representat…

Developed a diffusion model on spherical data, addressing geometric and stochastic challenges.

We investigate the representation theory of the polynomial core of the quantum Teichmuller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finite-dimensional represent…

The global symmetry algebras of partially-massless (PM) higher-spin (HS) fields in (A)dS$_{d+1}$ are studied. The algebras involving PM generators up to depth $2\,(\ell-1)$ are defined as the maximal symmetries of free conformal scalar field with $2\,\ell$ order wave equation in $d$ dimensions. We review the constructi…