arXiv research

Develops a theory for equivariant networks with partial domain symmetry.

In this paper, we investigate an equivariant homeomorphism of the boundaries $\partial X$ and $\partial Y$ of two proper CAT(0) spaces $X$ and $Y$ on which a CAT(0) group $G$ acts geometrically. We provide a sufficient condition and an equivalent condition to obtain a $G$-equivariant homeomorphism of the boundaries $\p…

In this paper, we investigate an equivariant homeomorphism of the boundaries $\partial X$ and $\partial Y$ of two proper CAT(0) spaces $X$ and $Y$ on which a CAT(0) group $G$ acts geometrically. We provide a sufficient condition to obtain a $G$-equivariant homeomorphism of the two boundaries $\partial X$ and $\partial …

Proposes a topological model for partial equivariance in neural networks.

In this paper we first consider the Hamiltonian action of a compact connected Lie group on an $H$-twisted generalized complex manifold $M$. Given such an action, we define generalized equivariant cohomology and generalized equivariant Dolbeault cohomology. If the generalized complex manifold $M$ satisfies the $\bar{\pa…

We study equivariant contact structures on complex projective varieties arising as partial flag varieties $G/P$, where $G$ is a connected, simply-connected complex simple group of type $ADE$ and $P$ is a parabolic subgroup. We prove a special case of the LeBrun-Salamon conjecture for partial flag varieties of these typ…

We prove that for $k\ge 5$ there does not exist a continuous map $\partial CV(F_k)\to\mathbb PCurr(F_k)$ that is either $Out(F_k)$-equivariant or $Out(F_k)$-anti-equivariant. Here $\partial CV(F_k)$ is the "length-function" boundary of Culler-Vogtmann's Outer space $CV(F_k)$, and $\mathbb PCurr(F_k)$ is the space of pr…

The paper investigates how symmetry in models affects their performance and generalization.

Defines a new homomorphism for strongly invertible knots, proving equivariant algebraic concordance.

Novel threefold partitioning of $L^2$-spinor space on cone links.

The paper generalizes free boundary min-max theory to equivariant settings.

We compute the homotopy type of the space of T^n-equivariant symplectic embeddings from the standard 2n-dimensional ball of some fixed radius into a 2n-dimensional symplectic-toric manifold M, and use this computation to define a Z-valued step function on the positive real line which is an invariant of the symplectic-t…

Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.

Spherical data is found in many applications. By modeling the discretized sphere as a graph, we can accommodate non-uniformly distributed, partial, and changing samplings. Moreover, graph convolutions are computationally more efficient than spherical convolutions. As equivariance is desired to exploit rotational symmet…

It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this conjecture. We also show that it would give a new proof of the K-polystability of constant…

We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the $\mathfrak{sl}_N$ link homology categorifying the $\mathfrak{sl}_N$ link polynomial. We also provide connections to the equivarian…

Equivariant trisections for group actions on 4-manifolds are introduced and studied.

In this paper, we analyse the question of existence of a natural and projectively equivariant symbol calculus, using the theory of projective Cartan connections. We establish a close relationship between the existence of such a natural symbol calculus and the existence of an \sl(m+1,\R)-equivariant calculus over \R^{m}…

Geometric models improve feature extraction and equivariance in image generation.

We obtain an equivariant classification for orientable, closed, four-dimensional Alexandrov spaces admitting an isometric torus action. This generalizes the equivariant classification of Orlik and Raymond of closed four-dimensional manifolds with torus actions. Moreover, we show that such Alexandrov spaces are equivari…

We present a novel approach to the classification of conformally equivariant differential operators on spinors in the case of homogeneous conformal geometry. It is based on the classification of solutions for a vector-valued system of partial differential equations, associated to $\mathcal{D}$-modules for the homogeneo…

For suitable finite groups G, we construct contractible 4-manifolds C with an effective G-action on $\partial C$ whose associated pairs (C,g) for all $g \in G$ are distinct smoothings of the pair $(C,\partial C)$. Indeed C embeds in a 4-manifold so that cutting out C and regluing using distinct elements of G yield dist…

Stable blowup solutions found for supercritical Yang-Mills equations.

We study almost-calibrated, $O(n)$-equivariant Lagrangian mean curvature flow in $\mathbb{C}^n$, and prove structural theorems about the Type I and Type II blowups of finite-time singularities. In particular, we prove that any Type I blowup of such a flow must be a special Lagrangian pair of transversely intersecting p…

Study minimal rational curves on complex manifolds with isotropic VMRT.

The study examines Bergman kernels on complex manifolds with boundary and their asymptotic expansions.

It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized Kähler manifolds is in perfect agreement with the physical notion of general $(2,2)$ gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology t…

New method uses equivariant volume for gravitational extremization in holography.

Let $X, Y$ be complete, simply connected Riemannian surfaces with pinched negative curvature $-b^2 \leq K \leq -1$. We show that if $f : \partial X \to \partial Y$ is a Moebius homeomorphism between the boundaries at infinity of $X, Y$, then $f$ extends to an isometry $F : X \to Y$. This can be viewed as a generalizati…

In a recent paper, Belishev and Sharafutdinov consider a compact Riemannian manifold $M$ with boundary $\partial M$. They define a generalized Dirichlet to Neumann (DN) operator $Λ$ on all forms on the boundary and they prove that the real additive de Rham cohomology structure of the manifold in question is completely …

Homotopy equivalent boundaries of cube complexes are studied.

Researchers classify curvature homogeneous metrics on 4D manifolds.

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quo…

For any atoroidal iwip $φ\in Out(F_N)$ the mapping torus group $G_φ=F_N\rtimes_φ<t>e$ is hyperbolic, and the embedding $ι: F_N \overset{\lhd}{\longrightarrow} G_φ$ induces a continuous, $F_N$-equivariant and surjective {\em Cannon-Thurston map} $\hat ι: \partial F_N \to \partial G_φ$. We prove that for any $φ$ as above…

In this follow up work to [45, 33, 32, 46] we introduce and study a notion of geodesic stability restricted to rays with prescribed singularity types. A number of notions of interest fit into this framework, in particular algebraic- and transcendental K-polystability, equivariant K-polystability, and the geodesic K-pol…

Given a complex manifold $M$ equipped with a holomorphic action of a connected complex Lie group $G$, and a holomorphic principal $H$--bundle $E_H$ over $X$ equipped with a $G$--connection $h$, we investigate the connections on the principal $H$--bundle $E_H$ that are (strongly) adapted to $h$. Examples are provided by…

We partially describe equivariant Dirac and generalized complex structures on a homogeneous space $G/K$ by giving equivalent data involving only the Lie algebra. We consider real semisimple adjoint orbits in any semisimple Lie algebra over $\mathbb R$ and real nilpotent orbits in $sl_n (\mathbb R)$. We give a complete …

Generalizes Laudenbach-Poénaru theorem for group actions on 4-manifolds.

Develops a new approach to study nonlinear PDEs and their singularities.

The study computes Bergman kernels and point process asymptotics on Kähler manifolds.

We define an extended field theory in dimensions $1+1+1$, that takes the form of a `quasi 2-functor' with values in a strict 2-category $\widehat{\mathcal{H}am}$, defined as the `completion of a partial 2-category' $\mathcal{H}am$, notions which we define. Our construction extends Wehrheim and Woodward's Floer Field th…

Reconstruct flows and manifolds from their boundary actions on circles.

We prove that given a Hitchin representation in a real split rank 2 group $\mathsf G_0$, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrization of the Hitchin components by a Hermitian bundle over Teichmüller space. The proof goes through intr…

In this note we show that the property of having only vanishing triple Massey products in the equivariant cohomology is inherited by the set of fixed points of hamiltonian circle actions on closed symplectic manifolds. This result can be considered in a more general context of characterizing homotopic properties of Lie…

We consider the space of differential operators $\mathcal{D}_{λμ}$ acting between $λ$- and $μ$-densities defined on $S^{1|2}$ endowed with its standard contact structure. This contact structure allows one to define a filtration on $\mathcal{D}_{λμ}$ which is finer than the classical one, obtained by writting a differen…

Researchers create a new compactification of character varieties using geometric and algebraic methods.

Suppose a finitely generated group $G$ is hyperbolic relative to $\mathcal P$ a set of proper finitely generated subgroups of $G$. Established results in the literature imply that a "visual" metric on $\partial (G,\mathcal P)$ is "linearly connected" if and only if the boundary $\partial (G,\mathcal P)$ has no cut poin…

We classify compact simply-connected 5-dimensional manifolds which admit a metric of nonnegative curvature with a connected non-abelian group acting by isometries. We show that they are diffeomorphic to either S^5, S^3 x S^2, the nontrivial S^3-bundle over S^2 or the Wu-manifold, SU(3)/SO(3). This result is a consequen…