相关论文: Morse theory of the moment map for representations…
We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble…
Recent developments have extended the concept of global symmetries in several directions, offering new perspectives across a wide range of physical systems. This work shows that generalized global symmetries naturally emerge in shallow…
This Ph.D. thesis studies the relation between the Harder-Narasimhan filtration and a notion of GIT maximal unstability. When constructing a moduli space by using Geometric Invariant Theory (GIT), a notion of GIT stability appears, which is…
A geometric flow on $(2,2)$-forms is introduced which preserves the balanced condition of metrics, and whose stationary points satisfy the anomaly equation in Strominger systems. The existence of solutions for a short time is established,…
The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of…
In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows…
Consider the holomorphic Hamiltonian action of a compact Lie group $K$ on a compact K\"ahler manifold $M$ with a moment map $\Phi: M\rightarrow \mathfrak{k}^*$. Assume that $0$ is a regular value of the moment map. Weitsman raised the…
We refine Kirwan's surjectivity and formality theorems for a Hamiltonian G-action on a compact symplectic manifold M. For a regular value of the moment map, we show that the Kirwan map is surjective and additively split after inverting the…
The equation of a motion of curves in the projective plane is deduced. Local flows are defined in terms of polynomial differential functions. A family of local flows inducing the Kaup-Kupershmidt hierarchy is constructed. The integration of…
We prove the longtime existence for the mean curvature flow problem with a perpendicular Neumann boundary condition in a Generalized Robertson Walker (GRW) spacetime that obeys the null convergence condition. In addition, we prove that the…
Some turbulent flows self-organize into large-scale structures, rather than breaking up into ever-smaller scales. Underpinning this phenomenon is the existence of two sign-definite quantities which are conserved by the dynamics.…
This report is actually the in-class project of MATH489 FA2022, Dynamics and Differential Equations in University of Illinois Urbana-Champaign. The report is mainly about gradient flow of a Morse-type functions (Morse functions, Morse--Bott…
This is the second in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology of a smooth polarized complex projective variety with the action of a connected complex reductive…
We combine Gromov's amenable localization technique with the Poincar\'{e} duality to study the traversally generic vector flows on smooth compact manifolds $X$ with boundary. Such flows generate well-understood stratifications of $X$ by the…
A new monotone quantity in graphical mean curvature flows of higher codimensions is identified in this work. The submanifold deformed by the mean curvature flow is the graph of a map between Riemannian manifolds, and the quantity is…
We characterize the rate of convergence of a converging volume-normalized Yamabe flow in terms of Morse theoretic properties of the limiting metric. If the limiting metric is an integrable critical point for the Yamabe functional (for…
We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on $S^{2}$ conformal to the round metric. A key tool is to employ the smooth Cheeger-Gromov compactness theorem to…
The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow $\Phi: \mathbb{R}\times X\to X$ on pfaffian set $X$ is tame if the graph of $\Phi$ is a pfaffian subset of…
We complete the theoretical framework required for the construction of a Morse homology theory for certain types of forced mean curvature flows. The main result of this paper describes the asymptotic behaviour of these flows as the forcing…
We investigate the effects of given pressure gradients on hydrodynamic flow equations. We obtain results in terms of implicit solutions and also in the framework of an extra-dimensional formalism involving the diffusion/Schrodinger…