arXiv research

New Smith-Gysin sequence for non-semi-free actions without semi-free condition.

In this paper we describe a method to establish when a symplectic manifold $M$ with semi-free Hamiltonian $S^{1}$-action is unique up to isomorphism (equivariant symplectomorphism). This will rely on a study of the symplectic topology of the reduced spaces. We prove that if the reduced spaces satisfy a rigidity conditi…

In this paper, we study smooth, semi-free actions on closed, smooth, simply connected manifolds, such that the orbit space is a smoothable manifold. We show that the only simply connected $5$-manifolds admitting a smooth, semi-free circle action with fixed-point components of codimension $4$ are connected sums of $S^3$…

Stable approach solves equivariant Hopf theorem for G-manifolds.

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…

John Lott defined an integer-valued signature $σ_{S^1}(M)$ for the orbit space of a compact orientable manifold with a semi-free $S^1$-action but he did not construct a Dirac-type operator which has this signature as its index. We construct such operator on the orbit space and we show that it is essentially unique and …

John Lott has computed an integer-valued signature for the orbit space of a compact orientable $(4k+1)$ manifold with a semi-free $S^1$-action, which is a homotopy invariant of that space, but he did not construct a Dirac type operator which has this signature as its index. In this Thesis, we construct such operator on…

In this paper we introduce invariants of semi-free Hamiltonian actions of $S\sp 1$ on compact symplectic manifolds (which satisfy some technical conditions related to positivity) using the space of solutions to certain gauge theoretical equations. These equations generalize at the same time the vortex equations and the…

We show that, on the connected sum of complex projective planes, any toric LeBrun metric can be identified with a Joyce metric admitting a semi-free circle action through an explicit conformal equivalence. A crucial ingredient of the proof is an explicit connection form for toric LeBrun metrics.

This paper contains several results concerning circle action on almost-complex and smooth manifolds. More precisely, we show that, for an almost-complex manifold $M^{2mn}$(resp. a smooth manifold $N^{4mn}$), if there exists a partition $λ=(λ_{1},...,λ_{u})$ of weight $m$ such that the Chern number $(c_{λ_{1}}... c_{λ_{…

We establish a necessary and sufficient condition for pairs of integers to arise as the weights at the fixed points of an effective circle action on a compact almost complex 4-manifold with a discrete fixed point set. As an application, we provide a necessary and sufficient condition for a pair of integers to arise as …

We study topological T-duality for spaces with a semi-free $S^1-$action with isolated fixed points. Physically, these correspond to spacetimes containing Kaluza-Klein monopoles. We demonstrate that the physical dyonic coordinate of such spaces has an analogue in our formalism. By analogy with the Dirac monopole, we stu…

The study simplifies complex functions on surfaces using a special transformation.

We explore a connection between the Finslerian area functional based on the Busemann-Hausdorff-volume form, and well-investigated Cartan functionals to solve Plateau's problem in Finsler 3-space, and prove higher regularity of solutions. Free and semi-free geometric boundary value problems, as well as the Douglas probl…

We investigate U(1)-equivariant deformations of C. LeBrun's self-dual metric with torus action. We explicitly determine all U(1)-subgroups of the torus for which one can obtain U(1)-equivariant deformation that do not preserve semi-free U(1)-action. This gives many new self-dual metrics with U(1)-action which are not c…

Kawakubo and Uchida showed that, if a closed oriented $4k$-dimensional manifold $M$ admits a semi-free circle action such that the dimension of the fixed point set is less than $2k$, then the signature of $M$ vanishes. In this note, by using $G$-signature theorem and the rigidity of the signature operator, we generaliz…

Przytycki and Sokolov proved that a three-manifold admits a semi-free action of the finite cyclic group of order $p$ with a circle as the set of fixed points if and only if $M$ is obtained from the three-sphere by surgery along a strongly $p-$periodic link $L$. Moreover, if the quotient three-manifold is an integral ho…

Geometrically solves differentiating simplicial manifolds.

We construct geometric generators of the effective $S^1$-equivariant Spin- (and oriented) bordism groups with two inverted. We apply this construction to the question of which $S^1$-manifolds admit invariant metrics of positive scalar curvature. It turns out that, up to taking connected sums with several copies of the …

In this paper we study the topological T-dual of spaces with a non-free circle action mainly using the stack theory method of Bunke and co-workers \cite{Bunke1}. We first compare three formalisms for obtaining the Topological T-dual of a semi-free $S^1$-space in a simple example. Then, we calculate the T-dual of genera…

Study circle actions on unitary manifolds with discrete fixed points.

We construct metrics of positive scalar curvature on manifolds with circle actions. One of our main results is that there exist $S^1$-invariant metrics of positive scalar curvature on every $S^1$-manifold which has a fixed point component of codimension 2. As a consequence we can prove that there are non-invariant metr…

Formula for Lefschetz number of knot branched covers.

We formulate a refinement of SU(N) Chern-Simons theory on a three-manifold via the refined topological string and the (2,0) theory on N M5 branes. The refined Chern-Simons theory is defined on any three-manifold with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refi…

Generates samples conditioned on labels using optimal transport.

The paper classifies Finsler surfaces satisfying the T-condition or σT-condition.

The paper develops a new approach to conditional risk measures using modular convex analysis.

Paper constructs solutions to Bogomolny equations with specific boundary and asymptotic conditions.

We extend probabilistic programming to handle conditioning on marginal distributions.

New tests for conditional copulas based on decision trees.

Paper finds necessary condition for logarithmic Minkowski problem in higher dimensions.

This paper introduces a neural operator for probabilistic conditioning.

CSI method learns conditional distributions by estimating flow equations.

New conditional risk measures called conditional generalized quantiles defined and characterized.

A new method for learning conditional distributions using ODEs and neural networks.

Sharp statistical theory for conditional diffusion models.

An analysis is made of reality conditions within the context of noncommutative geometry. We show that if a covariant derivative satisfies a given left Leibniz rule then a right Leibniz rule is equivalent to the reality condition. We show also that the matrix which determines the reality condition must satisfy the Yang-…

New conditions prevent gaps in optimal control problems.

We identify conditional parity as a general notion of non-discrimination in machine learning. In fact, several recently proposed notions of non-discrimination, including a few counterfactual notions, are instances of conditional parity. We show that conditional parity is amenable to statistical analysis by studying ran…

We consider families of strongly consistent multivariate conditional risk measures. We show that under strong consistency these families admit a decomposition into a conditional aggregation function and a univariate conditional risk measure as introduced Hoffmann et al. (2016). Further, in analogy to the univariate cas…

Proposes a new method for interpreting feature importance and effects in dependent feature models.

New boundary conditions solve Cauchy problem for Dirac operators on spacetimes.

We extend CS divergence to conditional distributions and show its advantages in time series data and sequential decision making.

The Samuelson condition is not satisfied by tangent lines of quadratic curves.

We describe a Groebner basis of relations among conditional probabilities in a discrete probability space, with any set of conditioned-upon events. They may be specialized to the partially-observed random variable case, the purely conditional case, and other special cases. We also investigate the connection to generali…

A new method tests conditional independence by transforming it into an unconditional problem using transport maps.

DG algorithms often fail to generalize well in limited domains, highlighting necessary vs. sufficient conditions.

In this paper, we examine higher order difference problems. Using the "squeezing" argument, we derive both Euler's condition and the transversality condition. In order to derive the two conditions, two needed assumptions are identified. A counterexample, in which the transversality condition is not satisfied without th…