Related papers: Bridgeland Stability on Blow Ups and Counterexampl…
For any divergence free initial datum $u_0$ with $\|u_0\|_\infty+\|\nabla u_0\|_{L^p}+\|\nabla^2 u_0\|_{L^p}<\infty$ for some $p>d\ (d\ge 2)$, the well-posedness and smoothness are proved for incompressible Navier-Stokes equations on…
We introduce the notions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. We prove that for any projective K3 surface, there exists a constant C depending only on the rank and…
We give an explicit proof of a Bogomolov-type inequality for $c_3$ of reflexive sheaves on $\mathbb{P}^3$. Then, using resolutions of rank-two reflexive sheaves on $\mathbb{P}^3$, we prove that some strata of the moduli of rank-two…
In this work we investigate symmetry breaking in the presence of a turbulent environment. The transition from a symmetric state to a symmetry-breaking state is demonstrated using two examples: (i) the transition of a two-dimensional flow to…
Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result…
We study the twisted ampleness criterion due to Collins, Jacob and Yau on surfaces, which is equivalent to the existence of solutions to the deformed Hermitian-Yang-Mills (dHYM) equation. When $X$ is a Weierstrass elliptic K3 surface, and…
This paper first generalises the Bogomolov-Tian-Todorov unobstructedness theorem to the case of Calabi-Yau threefolds with canonical singularities. The deformation space of such a Calabi-Yau threefold is no longer smooth, but the general…
We study the three-dimensional Navier-Stokes equations in the presence of the axisymmetric linear strain, where the strain rate depends on time in a specific manner. It is known that the system admits solutions which blow up in finite time…
We study the Hilbert scheme of twisted cubics in the three-dimensional projective space by using Bridgeland stability conditions. We use wall-crossing techniques to describe its geometric structure and singularities, which reproves the…
A pure and incompressible material is confined between two plates such that it is heated from below and cooled from above. When its melting temperature is comprised between these two imposed temperatures, an interface separating liquid and…
We deal with a divisorial contraction in dimension 3 which contracts its exceptional divisor to a smooth point. We prove that any such contraction can be obtained by a suitable weighted blow-up.
We study Bridgeland moduli spaces of semistable objects of $(-1)$-classes and $(-4)$-classes in the Kuznetsov components on index one prime Fano threefold $X_{4d+2}$ of degree $4d+2$ and index two prime Fano threefold $Y_d$ of degree $d$…
We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3…
A model predicting the structure of repulsive, spherically symmetric, monodisperse particles confined between two walls is presented. We study the buckling transition of a single flat layer as the double layer state develops. Experimental…
Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is called the BCOV invariant. In this paper, we extend the BCOV invariant to such pairs $(X,D)$, where $X$ is a compact K\"ahler…
We construct stable bundle extensions on elliptically fibered Calabi-Yau threefolds. We show that these bundles can solve the topological anomaly constraint in heterotic string theory without the need of invoking background fivebranes.
We provide a systematic way of calculating a quiver region associated to a given exceptional collection, which as an application is used to prove that $\mu$-stable sheaves represented by $2$-step complexes are Bridgeland stable. In the…
For a smooth bounded domain $\Omega \subset \mathbb R^3$ and smooth functions $a$ and $V$, we consider the asymptotic behavior of a sequence of positive solutions $u_\epsilon$ to $-\Delta u_\epsilon + (a+\epsilon V) u_\epsilon =…
We study the Clifford type inequality for a particular type of curves $C_{2,2,5}$, which are contained in smooth quintic threefolds. This allows us to prove some stronger Bogomolov-Gieseker type inequalities for Chern characters of stable…
We survey recent developments on Donaldson-Thomas theory, Bridgeland stability conditions and wall-crossing formula. We emphasize the importance of the counting theory of Bridgeland semistable objects in the derived category of coherent…