arXiv research

Classifies 6-manifolds with maximal symmetry up to diffeomorphism.

Classifies symplectic torus actions up to equivariant symplectomorphism.

Proves conjectures about maximal antipodal sets in symmetric and generalised symmetric spaces.

Study fibrations of projective spaces for maximal representations.

Let Y=G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define an equivariant differential operator on G/(L cap K) playing the role of an equivariant Dolbeault Laplacian when restricted to the complex manifold G/L, using a distribution transverse to the fibers and satisfying the Hormand…

Projections from flats to maximal flats defined and studied.

The paper introduces models to learn generalized transformation equivariant representations.

Study of strongly invertible Legendrian links in contact 3-space.

The elaboration of new quantization methods has recently developed the interest in the study of subalgebras of the Lie algebra of polynomial vector fields over a Euclidean space. In this framework, these subalgebras define maximal equivariance conditions that one can impose on a linear bijection between observables tha…

Maximal index vanishes for certain spin manifolds with positive scalar curvature.

The paper classifies surfaces in a 3-torus with maximal symmetry.

Study quantisation of geometric operators on manifolds with group actions.

The paper defines a ring structure in twisted equivariant $KK$-theory for Lie groups.

Optimizes the first $p$-Laplacian eigenvalue on hypersurfaces.

Infinite $G$-invariant minimal hypersurfaces found in Riemannian manifolds.

Optimizes eigenvalues on surfaces with symmetries.

New neural network architecture for auction design exploiting permutation symmetry.

We consider the T-equivariant cohomology of Bott-Samelson desingularisations of Schubert varieties in the flag manifold of a connected semi-simple complex algebraic group of adjoint type with maximal torus T. We construct a combinatorially pure (in the sense of T. Braden and R. Macpherson) sheaf on the Bruhat graph of …

RCNPs extend equivariant neural processes to higher dimensions, improving performance on tasks with inherent symmetries.

Upper bound found for torus actions on certain manifolds.

We study an integration theory in circle equivariant cohomology in order to prove a theorem relating the cohomology ring of a hyperkahler quotient to the cohomology ring of the quotient by a maximal abelian subgroup, analogous to a theorem of Martin for symplectic quotients. We discuss applications of this theorem to q…

Harmonic Networks make images equivariant to translation and rotation.

Unstable minimal surfaces in n-space link to hyperbolic products.

For an equivariant Morse stratification which contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincare series achieves the maximal possible value (instead of the minimal possible value 0 in the equivari…

Characterizes invariant and equivariant linear layers for graphs.

New classification for curved manifolds with specific symmetries.

New method trains any neural network as a generative model.

Permutation-equivariant neural networks improve auction mechanisms by reducing regret and sample complexity.

Graphical theory of binary Hamiltonian forms and their values.

The paper finds rotationally symmetric critical metrics for Laplace eigenvalues on tori.

Defines an equivariant index for proper actions by $H imes G$.

Study equivariant embeddings into Anti-de Sitter space and their relation to Lagrangian submanifolds.

We construct a gerbe over a complex reductive Lie group G attached to an invariant bilinear form on a maximal diagonalizable subalgebra which is Weyl group invariant and satisfies a parity condition. By restriction to a maximal compact subgroup K, one then gets a gerbe over K. For a simply-connected group, the parity c…

Frame Averaging makes neural networks invariant or equivariant to new symmetries.

In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an a…

Let G be a connected compact Lie group acting on a manifold M and let D be a transversally elliptic operator on M. The multiplicity of the index of D is a function on the set of irreducible representations of G. Let T be a maximal torus of G with Lie algebra Lie(T). We construct a finite number of piecewise polynomial …

We show an equivariant bordism principle for constructing metrics of positive scalar curvature that are invariant under a given group action. Furthermore, we develop a new codimension-2 surgery technique which removes singular strata from fixed point free $S^1$-manifolds while preserving equivariant positive scalar cur…

Let $S$ be a complex reductive group acting holomorphically on a complex Lie group $N$ via holomorphic automorphisms. Let $K(S)\subset S$ be a maximal compact subgroup. The semidirect product $G := N\rtimes K(S)$ acts on $N$ via biholomorphisms. We give an explicit description of the isomorphism classes of $G$-equivari…

The paper proves the existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.

New spectral estimates for minimal surfaces with boundary conditions.

Kähler-Einstein metrics found on compactifications of groups.

Extends fixed point theorem to noncompact manifolds using KK-theory.

Infinitesimal conformal transformations of $R^n$ are always polynomial and finitely generated when $n>2$. Here we prove that the Lie algebra of infinitesimal conformal polynomial transformations over $R^n$, $n>1$, is maximal in the Lie algebra of polynomial vector fields. When $n$ is greater than 2 and $p,q$ are such t…

We investigate representations of Kähler groups $Γ= π_1(X)$ to a semisimple non-compact Hermitian Lie group $G$ that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor--Wood inequality similar to those found by Burger--Iozzi and Koziarz--Maubon. Thanks…

Let $ρ$ be a maximal representation of a uniform lattice $Γ\subset{\rm SU}(n,1)$, $n\geq 2$, in a classical Lie group of Hermitian type $H$. We prove that necessarily $H={\rm SU}(p,q)$ with $p\geq qn$ and there exists a holomorphic or antiholomorphic $ρ$-equivariant map from complex hyperbolic space to the symmetric sp…

We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kahler manifolds and show the necessity of the Kahler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying…

Optimizes metrics on surfaces for eigenvalues.

Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.