Uhlenbeck–Donaldson compactification for framed sheaves on projective surfaces

We construct a compactification $M^{\mu ss}$ of the Uhlenbeck–Donaldson type for the moduli space of slope stable framed bundles. This is a kind of a moduli space of slope semistable framed sheaves. We show that there exists a projective morphism $\gamma :M^{ss} \rightarrow M^{\mu ss}$ , where $M^{ss}$ is the moduli space of S-equivalence classes of Gieseker-semistable framed sheaves. The space $M^{\mu ss}$ has a natural set-theoretic stratification which allows one, via a Hitchin–Kobayashi correspondence, to compare it with the moduli spaces of framed ideal instantons.     

Geometric aspects of the Kapustin–Witten equations

This expository paper introduces the Kapustin?CWitten equations to mathematicians. We discuss the connections between the complex Yang?CMills equations and the Kapustin?CWitten equations. In addition, we show the relation between the Kapustin?CWitten equations, the moment map condition and the gradient Chern?CSimons flow. The new results in the paper correspond to estimates on the solutions to the Kapustin?CWitten equations given an estimate on the complex part of the connection. This leaves open the problem of obtaining global estimates on the complex part of the connection.     

Parity sheaves on the affine Grassmannian and the Mirković–Vilonen conjecture

We prove the Mirkovi?–Vilonen conjecture: the integral local intersection cohomology groups of spherical Schubert varieties on the affine Grassmannian have no p-torsion, as long as p is outside a certain small and explicitly given set of prime numbers. (Juteau has exhibited counterexamples when p is a bad prime.) The main idea is to convert this topological question into an algebraic question about perverse-coherent sheaves on the dual nilpotent cone using the Juteau–Mautner–Williamson theory of parity sheaves.     

The construction of numerically Calabi–Yau orders on projective surfaces

In this paper, we construct a vast collection of maximal numerically Calabi–Yau orders utilising a noncommutative analogue of the well-known commutative cyclic covering trick. Such orders play an integral role in the Mori program for orders on projective surfaces and although we know a substantial amount about them, there are relatively few known examples.     

Existence of Kähler–Einstein metrics and multiplier ideal sheaves on del Pezzo surfaces

We apply Nadel’s method of multiplier ideal sheaves to show that every complex del Pezzo surface of degree at most six whose automorphism group acts without fixed points has a Kähler–Einstein metric. In particular, all del Pezzo surfaces of degree 4, 5, or 6 and certain special del Pezzo surfaces of lower degree are shown to have a Kähler–Einstein metric. These existence statements are not new, but the proofs given in the present paper are less involved than earlier ones by Siu, Tian and Tian–Yau.     

Noncommutative geometry and conformal geometry. III. Vafa–Witten inequality and Poincaré duality

This paper is the third part of a series of papers whose aim is to use the framework of twisted spectral triples to study conformal geometry from a noncommutative geometric viewpoint. In this paper we reformulate the inequality of Vafa–Witten [42] in the setting of twisted spectral triples. This involves a notion of Poincaré duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa–Witten, in the sense of an explicit control of the Vafa–Witten bound under conformal changes of metrics. This result has several noncommutative manifestations for conformal deformations of ordinary spectral triples, spectral triples associated with conformal weights on noncommutative tori, and spectral triples associated with duals of torsion-free discrete cocompact subgroups satisfying the Baum–Connes conjecture.     

Families of 4-Manifolds with Nontrivial Stable Cohomotopy Seiberg–Witten Invariants,and Normalized Ricci Flow

In this article, we produce infinite families of 4-manifolds with positive first Betti numbers and meeting certain conditions on their homotopy and smooth types so as to conclude the non-vanishing of the stable cohomotopy Seiberg–Witten invariants of their connected sums. Elementary building blocks used in Ishida and Sasahira (arXiv:0804.3452, 2008) are shown to be included in our general construction scheme as well. We then use these families to construct the first examples of families of closed smooth 4-manifolds for which Gromov’s simplicial volume is nontrivial, Perelman’s \(\bar{\lambda}\) invariant is negative, and the relevant Gromov–Hitchin–Thorpe type inequality is satisfied, yet no non-singular solution to the normalized Ricci flow for any initial metric can be obtained. Fang et al. (Math. Ann. 340:647–674, 2008) conjectured that the existence of any non-singular solution to the normalized Ricci flow on smooth 4-manifolds with non-trivial Gromov’s simplicial volume and negative Perelman’s \(\bar{\lambda}\) invariant implies the Gromov–Hitchin–Thorpe type inequality. Our results in particular imply that the converse of this fails to be true for vast families of 4-manifolds.     

The spectrum of the twisted Dirac operator on K?hler submanifolds of the complex projective space

We establish an upper estimate for the small eigenvalues of the twisted Dirac operator on K?hler submanifolds in K?hler manifolds carrying K?hlerian Killing spinors. We then compute the spectrum of the twisted Dirac operator of the canonical embedding ${{\mathbb C}P^d\rightarrow {\mathbb C}P^n}$ in order to test the sharpness of the upper bounds.     

Compact kernel sections of long-wave–short-wave resonance equations on infinite lattices

In the present paper, we consider the nonautonomous long-wave–short-wave resonance equations on infinite lattices. We first prove the existence of compact kernel sections for the associated process. Then we give an upper bound of the Kolmogorov ε$\epsilon$-entropy and verify the upper semicontinuity of these kernel sections.     

On the Ma–Trudinger–Wang curvature on surfaces

We investigate the properties of the Ma–Trudinger–Wang nonlocal curvature tensor in the case of surfaces. In particular, we prove that a strict form of the Ma–Trudinger– Wang condition is stable under C ⁴ perturbation if the nonfocal domains are uniformly convex; and we present new examples of positively curved surfaces which do not satisfy the Ma–Trudinger–Wang condition. As a corollary of our results, optimal transport maps on a “sufficiently flat” ellipsoid are in general nonsmooth.     

Y spaces and global smooth solution of fractional Navier–Stokes equations with initial value in the critical oscillation spaces

For fractional Navier–Stokes equations and critical initial spaces X, one used to establish the well-posedness in the solution space which is contained in $C({\mathbb{R}}_{+},X)$. In this paper, for heat flow, we apply parameter Meyer wavelets to introduce Y spaces $Y^{m,\beta }$ where $Y^{m,\beta }$ is not contained in $C({\mathbb{R}}_{+},{\dot{B}}_{\infty }^{1?2\beta ,\infty })$. Consequently, for $\frac{1}{2}<\beta <1$, we establish the global well-posedness of fractional Navier–Stokes equations with small initial data in all the critical oscillation spaces. The critical oscillation spaces may be any Besov–Morrey spaces ${({\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\mathbb{R}}^n))}^n$ or any Triebel–Lizorkin–Morrey spaces ${({\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\mathbb{R}}^n))}^n$ where $1\le p,q\le \infty ,0\le {\gamma }_2\le \frac{n}{p},{\gamma }_1?{\gamma }_2=1?2\beta$. These critical spaces include many known spaces. For example, Besov spaces, Sobolev spaces, Bloch spaces, Q-spaces, Morrey spaces and Triebel–Lizorkin spaces etc.     

$$L^p$$-Analysis of the Hodge–Dirac Operator Associated with Witten Laplacians on Complete Riemannian Manifolds

We prove R-bisectoriality and boundedness of the \(H^\infty \)-functional calculus in \(L^p\) for all \(1<p<\infty \) for the Hodge–Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry–Emery Ricci curvature on k-forms.     

Relationship between the Udwadia–Kalaba equations and the generalized Lagrange and Maggi equations

In their paper “A New Perspective on Constrained Motion,” F. E. Udwadia and R. E. Kalaba propose a new form of matrix equations of motion for nonholonomic systems subject to linear nonholonomic second-order constraints. These equations contain all of the generalized coordinates of the mechanical system in question and, at the same time, they do not involve the forces of constraint. The equations under study have been shown to follow naturally from the generalized Lagrange and Maggi equations; they can be also obtained using the contravariant form of the motion equations of a mechanical system subjected to nonholonomic linear constraints of second order. It has been noted that a similar method of eliminating the forces of constraint from differential equations is usually useful for practical purposes in the study of motion of mechanical systems subjected to holonomic or classical nonholonomic constraints of first order. As a result, one obtains motion equations that involve only generalized coordinates of a mechanical system, which corresponds to the equations in the Udwadia–Kalaba form.     

Fundamental results on the reaction–diffusion equations associated with a PEPA model

This paper presents an extension of the fluid approximation of a PEPA model by augmenting with diffusion to take spatial information into account, which is described by a reaction–diffusion system with homogeneous Neumann boundary conditions. The existence and uniqueness of the solution are given, positivity and boundedness of the solution to the system are also established. Moreover, sufficient conditions for the convergence are discussed under different cases. Our results show that the action rates determine the behavior of positive solutions. Numerical simulations are presented to illustrate the analytical results.     

Decay of solutions for 2D Navier–Stokes equations posed on Lipschitz and smooth bounded and unbounded domains

Initial–boundary value problems for 2D Navier–Stokes equations posed on bounded and unbounded rectangles as well as on bounded and unbounded smooth domains were considered. The existence and uniqueness of regular global solutions in bounded rectangles and bounded smooth domains as well as exponential decay of solutions on bounded and unbounded domains were established.     

Biderivations of the twisted Heisenberg–Virasoro algebra and their applications

In this paper, the biderivations without the skew-symmetric condition of the twisted Heisenberg–Virasoro algebra are presented. We find some non-inner and non-skew-symmetric biderivations. As applications, the characterizations of the forms of linear commuting maps and the commutative post-Lie algebra structures on the twisted Heisenberg–Virasoro algebra are given. It is also proved that every biderivation of the graded twisted Heisenberg–Virasoro left-symmetric algebra is trivial.