arXiv research

Geodesics on Kähler manifold potentials are paths of least action.

Machine Learning algorithms are typically regarded as appropriate optimization schemes for minimizing risk functions that are constructed on the training set, which conveys statistical flavor to the corresponding learning problem. When the focus is shifted on perception, which is inherently interwound with time, recent…

Let G be a connected semisimple Lie group without compact factors whose real rank is at least 2, and let Γ\subset G be an irreducible lattice. We provide a C^\infty classification for volume-preserving Cartan actions of Γand G. Also, if G has real rank at least 3, we provide a C^\infty classification for volume-preserv…

We prove that for each integer k of at least 2, there is an open neigborhood ν_k of the identity map of the 2-sphere S^2, in C^1-topology such that: if G is a nilpotent subgroup of Diff^1(S^2) with length k of nilpotency, generated by elements in ν_k, then the natural action on S^2 has non-empty fixed point set. Moreov…

Paper finds at least 6 fixed points for a specific circle action on a 10D manifold.

We classify symplectic actions of 2-tori on compact, connected symplectic 4-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a collection of invariants, which are invariants of the topology of the manifold, of th…

We classify compact homogeneous geometries of irreducible spherical type and rank at least 2 which admit a transitive action of a compact connected group, up to equivariant 2-coverings. We apply our classification to polar actions on compact symmetric spaces.

The paper finds infinite families of exotic spheres with free actions.

We construct new families of quasimorphisms on many groups acting on CAT(0) cube complexes. These quasimorphisms have a uniformly bounded defect of 12, and they "see" all elements that act hyperbolically on the cube complex. We deduce that all such elements have stable commutator length at least 1/24. The group actions…

In this paper we show that the Seiberg--Witten invariant is zero for all smooth 4--manifolds with $b_+{>}1$ which admit circle actions that have at least one fixed point. Furthermore, we show that all symplectic 4--manifolds which admit circle actions with fixed points are rational or ruled, and thus admit a symplectic…

There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical bui…

We consider a natural mechanical system on a Finsler manifold and study its \emph{curvature} using the intrinsic Jacobi equations (called \emph{Jacobi curves}) along the extremals of the least action of the system. The curvature for such a system is expressed in terms of the Riemann curvature and the Chern curvature (i…

Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain latti…

We prove that the number of distinct group actions on compact Riemann surfaces of a fixed genus $σ\geq 2$ is at least quadratic in $σ$. We do this through the introduction of a coarse signature space, the space $\mathcal{K}_σ$ of {\em skeletal signatures} of group actions on compact Riemann surfaces of genus $σ$. We di…

Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least 3 factors through a homomorphism onto a cocompact discrete subgroup of PSL(2,R). We also study actions of Kähler groups on infinite dimensional re…

Let $M$ be a compact oriented simply-connected manifold of dimension at least 8. Assume $M$ is equipped with a torsion-free semi-free circle action with isolated fixed points. We prove $M$ has a perfect invariant Morse-Smale function. The major ingredient in the proof is a new cancellation theorem for the invariant Mor…

We prove that any rigid representation of $π_1Σ_g$ in $\mathrm{Homeo}_+(S^1)$ with Euler number at least $g$ is necessarily semi-conjugate to a discrete, faithful representation into $\mathrm{PSL}(2,\mathbb{R})$. Combined with earlier work of Matsumoto, this precisely characterizes Fuchsian actions by a topological rig…

We study the problem of determining the least symmetric triangle, which arises both from pure geometry and from the study of molecular chirality in chemistry. Using the correspondence between planar $n$-gons and points in the Grassmannian of 2-planes in real $n$-space introduced by Hausmann and Knutson, this correspond…

We propose a method to extend submanifolds, singular Riemannian foliations and isometric actions from a boundary component of a noncompact symmetric space to the whole space. This extension method preserves minimal submanifolds, isoparametric foliations and polar actions, among other properties. One of the several appl…

We show that if $G\times M \to M$ is a cohomogeneity one action of a compact connected Lie group $G$ on a compact connected manifold $M$ then $H^*_G(M)$ is a Cohen-Macaulay module over $H^*(BG)$. Moreover, this module is free if and only if the rank of at least one isotropy group is equal to the rank of $G$. We deduce …

Study introduces constrained neural units over time for learning problems.

We show that for k at least 3, given any matrix in GL(k,Z), there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on Z^k is the given matrix.

For r at least 3, p at least 2, we classify all actions of the groups Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the circle. This is the same as describing all nontrivial group homomorphisms between groups of compactly supported diffeomorphisms on 1- manifolds. We show that all such actions…

The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. The resulting poset thus gives rise to a notion of the "best" hyperbolic action of a group as the largest element of this poset, if such an element exists. We call such an action a large…

New method constructs actions on R capturing foliations, proving left-orderability.

Classifies special homogeneous curves with polynomial equations.

A hyperkähler 4-metric with a triholomorphic SU(2) action gives rise to a family of confocal quadrics in Euclidean 3-space when cast in the canonical form of a hyperkähler 4-metric metric with a triholomorphic circle action. Moreover, at least in the case of geodesics orthogonal to the U(1) fibres, both the covariant S…

New findings on mapping class group actions on the circle, improving critical regularity.

If $Γ$ is a discrete subgroup of $PSL(3,\Bbb{C})$, it is determined the equicontinuity region $Eq(Γ)$ of the natural action of $Γ$ on $\Bbb{P}^2_\Bbb{C}$. It is also proved that the action restricted to $Eq(Γ)$ is discontinuous, and $Eq(Γ)$ agrees with the discontinuity set in the sense of Kulkarni whenever the limit s…

If $G$ is a semisimple Lie group of real rank at least 2 and $Γ$ is an irreducible lattice in $G$, then every homomorphism from $Γ$ to the outer automorphism group of a finitely generated free group has finite image.

Let p be a finite regular covering on a 2-sphere with at least three branch points. In this paper, we construct a local signature for the class of fibrations whose general fibers are isomorphic to the covering p.

We characterize Lie group actions for which there exists, at least locally, an evaluation map that defines a cochain map from the differential complex of invariant forms on a manifold to the De Rham complex for the quotient.

It is proved that an arbitrary finite group acting locally linearly, homologically trivially, and pseudofreely on a closed, simply connected 4-manifold must in fact be cyclic and act semifreely, provided the second betti number of the manifold is at least three.

Let S be an orientable surface of finite type and let Mod(S) be its mapping class group. We consider actions of Mod(S) by semisimple isometries on complete CAT(0) spaces. If the genus of S is at least 3, then in any such action all Dehn twists act as elliptic isometries. The action of Mod(S) on the completion of Teichm…

Fix a free, orientation-preserving action of a finite group G on a 3-dimensional handlebody V. Whenever G acts freely preserving orientation on a connected 3-manifold X, there is a G-equivariant imbedding of V into X. There are choices of X closed and Seifert-fibered for which the image of V is a handlebody of a Heegaa…

Let $(\mathcal{V},\mathcal{W};F)$ be a weakly reducible, unstabilized, Heegaard splitting of genus at least three in an orientable, irreducible $3$-manifold $M$. Then $Mod(M,F)$ naturally acts on the disk complex $\mathcal{D}(F)$ as a group action. In this article, we prove if $F$ is topologically minimal and its topol…

Develops statistical framework for resolving reward function ambiguity in inverse reinforcement learning.

The main result of this paper is the conformal flatness of real-analytic compact Lorentz manifolds of dimension at least $3$ admitting a conformal essential (i.e. conformal, but not isometric) action of a Lie group locally isomorphic to PSL(2,R). It is established by using a general result of M. Gromov on local isometr…

Let $\cT$ be Teichmüller space of a closed surface of genus at least 2. For any point $c\in \cT$, we describe an action of the circle on $\cT\times \cT$, which limits to the earthquake flow when one of the parameters goes to a measured lamination in the Thurston boundary of $\cT$. This circle action shares some of the …

New rigidity result for CAT(0) spaces of higher rank.

Recently, Gareth Jones observed that every finite group $G$ can be realized as the group of automorphisms of some dessin d'enfant ${\mathcal D}$. In this paper, complementing Gareth's result, we prove that for every possible action of $G$ as a group of orientation-preserving homeomorphisms on a closed orientable surfac…

Study on finiteness property of right-angled Artin groups actions on extension graphs.

Sublinear LSVI via LSH reduces runtime to sublinear in actions.

New research shows exponential lower bounds for planning in MDPs with linearly-realizable optimal action-value functions.

Multicurve stabilizers' extensions are hierarchically hyperbolic.

In a strengthening of the G-Signature Theorem of Atiyah and Singer, we compute, at least in principle (modulo certain torsion of exponent dividing a power of the order of G), the class in equivariant K-homology of the signature operator on a G-manifold, localized at a prime idea of R(G), in terms of the classes in non-…

Galois action on manifold structures of complex varieties is abelian.

Let $J$ be a semisimple Lie group with all simple factors of real rank at least two. Let $Γ<J$ be a lattice. We prove a very general local rigidity result about actions of $J$ or $Γ$. This shows that almost all so-called "standard actions" are locally rigid. As a special case, we see that any action of $Γ$ by toral aut…