Study of affine transformations on topological manifolds, focusing on local freeness and solvability.
problem Understanding the action of affine transformations on topological manifolds.
method Analyzing the subgroup of homeomorphisms that lift to affine transformations and studying the resulting foliation.
result The connected component of the subgroup acts locally freely and is solvable, with additional properties for polynomial manifolds.
Researchers classify invariant operators on weighted densities.
problem Classifying invariant differential operators on weighted densities.
method Investigated the aff(n∣1)-module structure and invariant binary differential operators. result Computed the first aff(n∣1)−relative differential cohomology. We show that aff(n), the Lie algebra of affine transformations of Rn, is formally and analytically nondegenerate in the sense of A. Weinstein. This means that every analytic (resp., formal) Poisson structure vanishing at a point with a linear part corresponding to aff(n) is locall…
We prove that in genus bigger than 2, the mapping class group action on Aff(C)-characters is ergodic. This implies that almost every representation π1S⟶Aff(C) is the holonomy of a branched affine structure on S, where S is a closed orientable surface of ge…
Extends Aff-Wild database for affect recognition in real-world settings.
problem Complex human emotional states in real-world settings.
method Developed deep neural architectures with attention mechanism for emotion recognition.
result Improved performance in emotion recognition using Aff-Wild2.
Affine rigidity theorem for special surfaces without integration.
problem Understanding affine transformations of surfaces with zero Gaussian curvature.
method Pure affine geometry, avoiding analysis tools.
result Affine rigidity theorem for CR-flat surfaces.
Researchers describe finite orbits in character varieties of a sphere.
problem Characterizing finite orbits in character varieties of the punctured sphere.
method Coalescence procedure and theory of finite complex reflection groups.
result Complete description of finite braid group orbits in Aff(C)-character varieties.
A simpler proof shows affine invariant points equal fixed points for convex bodies.
problem Proving Grünbaum's conjecture about affine invariant points and fixed points of convex bodies.
method Using topology of Aff(n) action on hyperspace of convex bodies.
result A shorter, simpler proof of Grünbaum's conjecture.
Unique birational structure proven on Inoue surfaces.
problem Proving uniqueness of birational structures on Inoue surfaces.
method Generalizing a result by Bruno Klingler, proving uniqueness of structures.
result The natural (Aff2(C),C2)-structure on an Inoue surface is the unique (Bir(P2),P2(C))-structure. Affine actions of Lie groups are studied and their K-theory is related to standard K-theory.
problem Understanding affine actions of Lie groups and their K-theory.
method Proved equivalence of categories and used monadic adjunction to relate affine actions to standard actions.
result Grothendieck groups of affine actions are isomorphic to standard K-theory groups.
Paper tackles multi-task learning for emotion recognition and generation using Aff-Wild dataset.
problem Developing a multi-task learning approach for emotion recognition and generation using Aff-Wild dataset.
method Deep neural network with shared hidden layers and GAN for multi-task learning and image generation.
result Good performance of the proposed approach on Aff-Wild dataset.
Classifies orbit closures of mapping class group action on affine character variety.
problem Classifying orbit closures of Mod(Σ)-action on χ(Aff(C)). method Classification up to exceptions of Mod(Σ)-action on affine character variety. result The only obstruction for a non-abelian representation to be the holonomy of a branched affine structure is to be Euclidean and not have positive volume.
Affine diffeomorphisms are undistorted in half-translation surfaces.
problem Undistorted nature of affine diffeomorphism groups.
method Proof of undistorted subgroup property and systole map embedding.
result Finitely generated subgroups of affine diffeomorphisms are undistorted.
Deep learning model predicts human emotions in real-world settings.
problem Automatic emotional state assessment in natural settings.
method End-to-end deep neural architecture (AffWildNet) combining convolutional and recurrent layers.
result AffWildNet achieved state-of-the-art results on Aff-Wild Challenge.
The paper identifies all flat CR Lie groups and their structures.
problem Identifying flat CR Lie groups and their structures.
method Analyzing Lie algebras and structures of Lie groups.
result Only specific Lie algebras are Cartan flat non-degenerate CR Lie algebras.
Aff-Wild database expands facial expression recognition to real-world conditions.
problem Lack of spontaneous facial expression databases in real-world conditions.
method Collects spontaneous facial expressions from YouTube, annotates with valence and arousal, uses deep learning techniques.
result Developed an end-to-end DNN model achieving 0.555 CCC for valence and 0.499 CCC for arousal.
Two-stream model recognizes affect from audio and video.
problem Human affect recognition in real-world settings.
method Two-stream aural-visual analysis model with separate audio and visual processing.
result Model achieves promising results on Aff-Wild2 database.
The paper connects knot homology to JM elements in affine braid groups.
problem Understanding knot homology through affine braid group elements.
method Constructing a homomorphism from affine braid group to knot homology.
result Derives a relation between knot homologies of specific braid group elements.
Let S be a (topological) compact closed surface of genus two. We associate to each translation surface (X,ω)∈H(2)⊔H(1,1) a subgraph C^cyl of the curve graph of S. The vertices of this subgraph are free homotopy classes of curves which can be represented either …
We prove that the ring $\Aff{\R}{M}$ of all polynomials defined on a real algebraic variety M⊂Rn is dense in the Hilbert space $L^2(M,e^{-|x|^2}\deμ)$, where $\deμ$ denotes the volume form of M and $\deν=e^{-|x|^2}\deμ$ the Gaussian measure on M.
First ABAW 2020 Competition analyzes affective behavior tasks.
problem Automatic analysis of valence-arousal, basic expressions, and action units in real-world scenarios.
method Provided Aff-Wild2 database, described Challenges, evaluation metrics, and top-performing systems.
result Demonstrated the feasibility of automatic affective behavior analysis in real-world settings.
Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups Γ in $\Aff (N)=N\rtimes \Aut (N)$ acting properly discontinuously and cocompactly on N. This situation is a natural generalization of the so-called affine crystallographic groups. We prove that for all dimensions…
New insights into 5D Sasakian Lie algebras with trivial center.
problem Characterizing five-dimensional Sasakian Lie algebras with trivial center.
method Analyzing φ-symmetry and constructing contact metric structures. result Every 5D Sasakian Lie algebra with trivial center is φ-symmetric. The present work is devoted to compact completely solvable solvmanifolds which admit Kahlerian metrics whose Kahler forms are homogeneous. In particular, we show that such manifolds are diffeomorphic to flat tori. Our proof is based on Dynkin diagrams associated to left invariant closed 2-forms in completely solvable L…
Paper presents a CNN-RNN method for multi-dimensional emotion recognition in-the-wild.
problem Dimensional emotion recognition in real-world scenarios.
method Pre-training with Aff-Wild and Aff-Wild2, extracting low-, mid-, and high-level features, using RNN subnets in a multi-task framework, and fusion of networks.
result Our approach outperformed state-of-the-art methods using only visual information.
This paper studies actions of solvable Lie groups on nilpotent Lie groups.
problem Characterizing which solvable Lie groups can act simply transitively on nilpotent Lie groups.
method Using Lie algebra properties and semisimple splitting, the paper provides methods to check for such actions.
result A full description of possibilities for actions up to dimension 4.
Sei G eine zusammenhängende reduktive komplexe algebraische Gruppe, die auf einer glatten affinen komplexen Varietät M wirke, und bezeichne $\Diff[G]{M}$ die G-invarianten algebraischen Differentialoperatoren auf M. Zerlegt man den Koordinatenring $\Aff{\C}{M}$ in G-isotypische Komponenten, so zeigen wir, daß die hierb…
No non-product Hessian rank 1 affine homogeneous hypersurfaces exist in dimensions 5 and above.
problem Identifying non-product Hessian rank 1 affine homogeneous hypersurfaces in higher dimensions.
method Developed a normal form for hypersurfaces under the affine group, up to order ≤ n+5, in any dimension n ≥ 2.
result Non-existence of non-product Hessian rank 1 affine homogeneous hypersurfaces in dimensions 5 and above.
A new Tverberg theorem with negative coefficients.
problem Proving a Tverberg type theorem with negative coefficients.
method General position set and partition into r sets with affine combination constraints.
result The unique point in the intersection of affine hulls can be expressed as a weighted sum with exactly k negative weights.
A (flat) affine 3-manifold is a 3-manifold with an atlas of charts to an affine space R3 with transition maps in the affine transformation group Aff(R3). We will show that a connected closed affine 3-manifold is either an affine Hopf 3-manifold or decomposes canonically to conca…
This paper shows how post-Lie algebra structures can be induced by simply transitive NIL-affine actions.
problem Understanding which solvable Lie groups can act simply transitively on nilpotent Lie groups.
method Introducing post-Lie algebra structures and showing their correspondence with simply transitive actions.
result Simply transitive NIL-affine actions induce complete post-Lie algebra structures in the 2-step nilpotent case.
This paper presents a geometric description of Lagrangian and Hamiltonian systems on Lie affgebroids subject to affine nonholonomic constraints. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee that the dynamics of the system can be obtained as a suitabl…
The paper studies third-order PDEs invariant under affine transformations and connects them to the Fubini-Pick invariant.
problem Investigating third-order PDEs invariant under affine transformations.
method Using a general method introduced in [D.V. Alekseevsky, J. Gutt, G. Manno, and G. Moreno: A general method to construct invariant PDEs on homogeneous manifolds].
result Derives third-order PDEs from the Fubini-Pick invariant.
An (flat) affine 3-manifold is a 3-manifold with an atlas of charts to an affine space R3 with transition maps in the affine transformation group Aff(R3). Equivalently an affine 3-manifold is a 3-manifold with a flat torsion-free affine connection. We show that a closed affine 3-mani…
Project creates new dataset for 'in the wild' emotions recognition.
problem Recognition of emotions in real-life unpredictable situations.
method Design and implement a new dataset and deep learning model.
result Demonstrates effectiveness of the new deep learning model for real-life emotions recognition.
We study affine Jacobi structures on an affine bundle π:A→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+=⋃p∈MAff(Ap,R) of affine functionals. Som…
To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence questio…
The group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus is the action of the affine group $\Aff(2)$ on $\bbR^2$. Since this action has non-compact stabiliser $\GL(2,\bbR)$, the underlying locally homogeneous geometry is highly non-Riemannian. In this articl…
The paper finds a correspondence between invariant PDEs and hypersurfaces.
problem Constructing invariant PDEs on projective and affine spaces.
method Applying a general method to specific cases of projective and affine spaces.
result Projectively or affinely invariant PDEs correspond to CO(d, n-d) invariant hypersurfaces.
Study rigid Lie affine foliations on compact manifolds.
problem Cohomological criterion for rigidity of Lie foliations.
method Detailed study of cohomology groups, Morse-Novikov cohomology.
result Many examples of rigid Lie affine foliations on compact manifolds.
We consider the standard contact structure on the supercircle, S^{1|1}, and the supergroups E(1|1), Aff(1|1) and SpO(2|1) of contactomorphisms, defining the Euclidean, affine and projective geometry respectively. Using the new notion of (p|q)-transitivity, we construct in synthetic fashion even and odd invariants chara…
Develops a real-analytic embedding for diffeomorphisms of the line, linking to Fisher-Rao geometry.
problem Embedding diffeomorphisms of the line in a geometric framework.
method Real-analytic embedding, Lp Fisher-Rao geometry, Schwarzian curvature. result Establishes a connection between diffeomorphisms and Fisher-Rao geometry, providing explicit geodesics and connections.
Attention tokens are group elements, negated algebra norms.
problem Attention mechanism for matrix Lie groups
method Lie-Algebra Attention
result Score matches learned kernel on invariant poses, outperforms vector-token baseline
Study of recurrent Lorentzian Weyl spaces with detailed local and global structures.
problem Characterizing and classifying non-closed Lorentzian Weyl manifolds.
method Analyzing curvature tensors, Lie group actions, and differential invariants.
result Locally homogeneous non-closed recurrent Lorentzian Weyl manifolds are classified.