arXiv research

Survey on affine Anosov representations and their implications.

The theory of flat Pseudo-Riemannian manifolds and flat affine manifolds is closely connected to the topic of prehomogeneous affine representations of Lie groups. In this article, we exhibit several aspects of this correspondence. At the heart of our presentation is a development of the theory of characteristic classes…

The paper defines cocycles for positive Anosov representations and constructs affine actions with bounded fundamental domains.

The paper generalizes deformation results for Fuchsian representations and shows proper affine actions.

We construct a new representation formula for indefinite improper affine spheres in terms of two para-holomorphic functions and study singularities which appear in this representation formula. As a result, it follows that cuspidal cross caps never appear as the singularities on indefinite improper affine spheres and so…

We present a representation formula for discrete indefinite affine spheres via loop group factorizations. This formula is derived from the Birkhoff decomposition of loop groups associated with discrete indefinite affine spheres. In particular we show that a discrete indefinite improper affine sphere can be constructed …

We define the notion of affine Anosov representations of word hyperbolic groups into the affine group $\mathsf{SO}^0(n+1,n)\ltimes\mathbb{R}^{2n+1}$. We then show that a representation $ρ$ of a word hyperbolic group is affine Anosov if and only if its linear part $\mathtt{L}_ρ$ is Anosov in $\mathsf{SO}^0(n+1,n)$ with …

Heegaard Floer homology connects to polynomial representations of Hecke algebras.

Constructs positive energy representations from Toda equations Stokes data.

In order to understand the structure of the cohomologies involved in the study of projectively equivariant quantizations, we introduce a notion of affine representation of a Lie algebra.We show how it is related to linear representations and 1-cohomology classes of the algebra. We classify the affine representations of…

Researchers study rational and pretzel knots using affine group representations.

Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper indefinite affine spheres with singularities, and, reciprocally, every indefini…

The complete classification of representations of the Trefoil knot group G in S^{3} and SL(2,R), their affine deformations, and some geometric interpretations of the results, are given. Among other results, we also obtain the classification up to conjugacy of the non cyclic groups of affine Euclidean isometries generat…

We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental for…

Random features enhance control of complex systems.

We define a fuchsian affine action of a surface group to be such that the linear part factors through a representation of $SL(2,{\mathbb R})$. We prove a fuchsian affine action of a surface group is never proper.

Neural networks can represent complex piecewise functions efficiently.

The paper reinterprets knot group invariants using affine transformations.

Study partially hyperbolic flows on flat bundles, proving equivalence for complete affine manifolds.

Study shows exponential growth of knot polynomial tied to Chern-Simons invariant.

Consider a lattice $Γ$ in a group $G = SL_2(\R), SO(1,n), SU(1,n)$, $SL_2(\Q_p)$. We discuss actions of $Γ$ by affine isometric transformations of Hilbert spaces. We show that for irreducible affine isometric action of $G$ its restriction to $Γ$ is irreducible. We prove the existence of canonical irreducible affine iso…

We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We characterize flat affine symplectic Lie groups using symplectic étale affine representations and as a consequence of this, we show that a …

Groups with hyperbolic properties don't have strong Property (T).

Method identifies latent variables from high-dimensional data with piecewise affine mixing.

New shape representation for airfoils improves design and manufacturing.

Study on faithfulness of Burau representation for Artin-Tits groups.

The identification of novel drug-target (DT) interactions is a substantial part of the drug discovery process. Most of the computational methods that have been proposed to predict DT interactions have focused on binary classification, where the goal is to determine whether a DT pair interacts or not. However, protein-l…

We give a conformal representation for indefinite improper affine spheres which solve the Cauchy problem for their Hessian equation. As consequences, we can characterize their geodesics and obtain a generalized symmetry principle. Then, we classify the helicoidal indefinite improper affine spheres and find a new family…

We study affine Jacobi structures on an affine bundle $π:A\to M$, i.e. Jacobi brackets that close on affine functions. We prove that there is a one-to-one correspondence between affine Jacobi structures on $A$ and Lie algebroid structures on the vector bundle $A^+=\bigcup_{p\in M}Aff(A_p,\R)$ of affine functionals. Som…

GEFA predicts drug-target affinity using graph neural networks.

We show that we can release the rigidity of the skew Howe duality process for ${\mathfrak sl}_n$ knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine ${\mathfrak sl}_m$ case, corresponding to looking at tan…

Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as $E^{3}$ (Euclidean 3-space), $H^{3}$ (hyperbolic 3-space) and $E^{2,1}$ (Minkowski 3-space), using quaternion algebra theory, are studied. We study the different representations of a 2-generator group in which th…

This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection. Then we specialize the characterization to the case of a left invariant connecti…

In exchange for large quantities of data and processing power, deep neural networks have yielded models that provide state of the art predication capabilities in many fields. However, a lack of strong guarantees on their behaviour have raised concerns over their use in safety-critical applications. A first step to unde…

We consider deep feedforward neural networks with rectified linear units from a signal processing perspective. In this view, such representations mark the transition from using a single (data-driven) linear representation to utilizing a large collection of affine linear representations tailored to particular regions of…

Properly discontinuous actions of a surface group by affine automorphisms of $\mathbb R^d$ were shown to exist by Danciger-Gueritaud-Kassel. We show, however, that if the linear part of an affine surface group action is in the Hitchin component, then the action fails to be properly discontinuous. The key case is that o…

MASC balances dataset representation using affinity clustering and distribution discrepancies.

Study on Hausdorff dimension of Anosov subgroup limit sets under specific affine complexity.

Inspired by the Weierstrass representation of smooth affine minimal surfaces with indefinite metric, we propose a constructive process producing a large class of discrete surfaces that we call discrete affine minimal surfaces. We show that they are critical points of an affine area functional defined on the space of qu…

We develop the HJM framework for forward rates driven by affine processes on the state space of symmetric positive matrices. In this setting we find a representation for the long-term yield and investigate the yield's asymptotic behaviour.

We give some examples of non-complete invariant affine connections on nilpotent and filiform Lie groups. This permits to describe non-nilpotent faithful representations on the model of filiform n-dimensional Lie algebras and, in particular, on the 3 dimensional Heisenberg algebra.

New method constructs proper affine actions of groups in higher dimensions.

Extend classical theory of affine processes to path-dependent setting

Motivation: Prediction of the interaction affinity between proteins and compounds is a major challenge in the drug discovery process. WideDTA is a deep-learning based prediction model that employs chemical and biological textual sequence information to predict binding affinity. Results: WideDTA uses four text-based inf…

Co-Diffusion predicts drug-target affinity by learning latent manifolds and diffusion, improving generalization.

Yau's Affine Normal Descent optimizes smooth unconstrained problems with geometrically adapted directions.

Constructs special Kähler structures on Lie groups.

Motivation: Drug discovery demands rapid quantification of compound-protein interaction (CPI). However, there is a lack of methods that can predict compound-protein affinity from sequences alone with high applicability, accuracy, and interpretability. Results: We present a seamless integration of domain knowledges and …