arXiv research

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type was established by the authors in the context of manifolds with corners; the canonical construction induces fibrations on the boundary faces of the resolution…

The `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a `resolution tower' which projects …

We show that there is an one-to-one correspondence between resolutions (equivariant w.r.t. a Lie groupoid action) of a singular subset of a manifold, and substacks (of a certain type) of the differential stack associated to the Lie groupoid in question. In particular, we show how to build an equivariant resolution out …

Non-Abelian actions are resolved using equivariant K-theory and delocalized cohomology.

Proves representability of complex semigroup systems.

The smooth action of a compact Lie group on a compact manifold can be resolved to an iterated space, as made explicit by Pierre Albin and the second author. On the resolution the lifted action has fixed isotropy type, in an iterated sense, with connecting fibrations and this structure descends to a resolution of the qu…

We prove the existence of steady Kähler-Ricci solitons on equivariant crepant resolutions of $\mathbb{C}^n/G$, where $G$ is a finite subgroup of $SU(n)$.

New compact manifolds with closed G2 structures found.

Unique soliton found on resolved cones.

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 …

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a `resolution stru…

We provided two explicit formulas for the intersection cohomology (as a graded vector space with pairing) of the symplectic quotient by a circle in terms of the $S^1$ equivariant cohomology of the original symplectic manifold and the fixed point data. The key idea is the construction of a small resolution of the symple…

New minimal hypersphere found in 4-sphere solving Bernstein problem.

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces... In this work, we prove existence o…

Framework learns surrogates for molecular dynamics across multiple time-scales.

Solves Tian's stabilization problem for toric Fano manifolds.

Proves invariance of weighted extremal Kähler metrics under smooth blowups.

We give a proof of Kontsevich's formality theorem for a general manifold using Fedosov resolutions of algebras of polydifferential operators and polyvector fields. The main advantage of our construction of the formality quasi-isomorphism is that it is based on the use of covariant tensors unlike Kontsevich's original p…

Study of $D_2$ ALF manifolds via hyperkahler quotients of affine spaces.

We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler-Ricci soliton. In particular, it follows that for any $n\in\mathbb{N}_{0}$ and…

Latent MoS learns multiple symmetries for efficient dynamic learning.

We exploit the Fedosov-Weinstein-Xu (FWX) resolution proposed in q-alg/9709043 to establish an isomorphism between the ring of Hochschild cohomology of the quantum algebra of functions on a symplectic manifold M and the ring H(M, C((h))) of De Rham cohomology of M with the coefficient field C((h)) without making use of…

Let $Γ$ be a finite group acting linearly on $\C^n$, freely outside the origin, and let $N$ be the number of conjugacy classes of $Γ$ minus one. A construction of Kronheimer of moduli spaces $X_ζ$ of translation-invariant $Γ$-equivariant instantons on $\C^2$ is generalised to $\C^n$. The moduli spaces $X_ζ$ depend on a…

SR-NAM maps low-res images to multiple high-res images realistically.

Deep learning speeds up whole heart MRI to 30 seconds.

New model predicts molecular wavefunctions and densities with unprecedented accuracy.

Geodesic rays and chordal distances link algebraic and geometric properties of positive metrics.

Study of invariants on manifolds with boundary involving equivariant spectral flow and η-invariants.

Group-equivariant subsampling layers improve CNNs' equivariance.

The paper proposes a method to improve fine-resolution predictions using coarse-resolution data.

Study of equivariant ribbon concordance using Khovanov homology.

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

This study prioritizes temporal resolution over spatial in energy systems models due to higher influence.

Deep learning improves 3D microscopy resolution without matched target images.

There are 2^n possible resolutions of a smooth pseudodiagram with n precrossings. If we consider piecewise-linear (PL) pseudodiagrams and resolutions that themselves are PL, certain resolutions of the pseudodiagram may not exist in three-space. We investigate this situation and its impact on the weighted resolution set…

For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ri…

Recently M. Kreck introduced a class of stratified spaces called p-stratifolds [M. Kreck, Stratifolds, Preprint]. He defined and investigated resolutions of p-stratifolds analogously to resolutions of algebraic varieties. In this note we study a very special case of resolutions, so called optimal resolutions, for p-str…

Defines an equivariant Ruelle dynamical zeta function for flows on manifolds.

Study of equivariant movie moves for involutive links.

We define the equivariant holonomy of an invariant connection on a principal U(1)-bundle. The properties of the ordinary holonomy are generalized to the equivariant setting. In particular, equivariant U(1)-bundles with connection are shown to be classified by its equivariant holonomy modulo isomorphisms. We also show t…

The paper presents a method to recover high-resolution signals from low-resolution measurements.

A bound on knot unknotting using equivariant signature.

New proof for 4D symplectic manifolds: equivariant cohomology determines diffeotype.

A resolution of the St. Petersburg paradox is presented. In contrast to the standard resolution, utility is not required. Instead, the time-average performance of the lottery is computed. The final result can be phrased mathematically identically to Daniel Bernoulli's resolution, which uses logarithmic utility, but is …

The paper develops methods for calculating equivariant homology from Morse functions.

Efficiently samples and learns densities with symmetries using equivariant methods.

Study contact resolutions for Jacobi structures, providing examples and impossibility results.

Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.