arXiv research

Solves modified conjecture for Fano manifolds using Ding stability.

We consider four extended Ricci flow systems---that is, Ricci flow coupled with other geometric flows---and prove dynamical stability of certain classes of stationary solutions of these flows. The systems include Ricci flow coupled with harmonic map flow (studied abstractly and in the context of Ricci flow on warped pr…

Paper solves vortex equations on complex surfaces, linking to Higgs bundle stability.

Extends martingale transport for robust finance problems.

We study partition functions of random Bergman metrics, with the actions defined by a class of geometric functionals known as `stability functions'. We introduce a new stability invariant - the critical value of the coupling constant - defined as the minimal coupling constant for which the partition function converges.…

We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, generalizing Kähler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler-Einstein Fano. In the Fano case we also prove that existe…

New stability criterion for Fano manifolds using anticanonically balanced metrics.

Graph-Coupled Oscillator Networks (GraphCON) tackles graph-based learning problems.

Motivated by the sigma model limit of multicomponent Ginzburg-Landau theory, a version of the Faddeev-Skyrme model is considered in which the scalar field is coupled dynamically to a one-form field called the supercurrent. This coupled model is investigated in the general setting where physical space is an oriented Rie…

We present a new implementation of anisotropic mean curvature flow for contour recognition. Our procedure couples the mean curvature flow of planar closed smooth curves, with an external field from a potential of point-wise charges. This coupling constrains the motion when the curve matches a picture placed as backgrou…

New rigidity results for scalar curvature with stabilized conditions.

We derive a system of equations governing the coupled gravitational and electromagnetic perturbations of Reissner-Nordström spacetime. The equations are derived in the context of global non-linear stability of Reissner-Nordström under axially symmetric polarized perturbations, as a generalization of the recent work on …

We prove boundedness and polynomial decay statements for solutions to the spin $\pm2$ generalized Teukolsky system on a Reissner-Nordström background with small charge. The first equation of the system is the generalization of the standard Teukolsky equation in Schwarzschild for the extreme component of the curvature $…

In these notes we give a shortened and more direct proof of Goto's generalized Kaehler stability theorem stating that if (J_1,J_2) is a generalized kaehler structure for which J_2 is determined by a nowhere vanishing closed form, then small deformations of J_1 can be coupled with small deformations of J_2 so that the p…

AI threatens financial stability through misuse and stealth adoption.

We study equations on a principal bundle over a compact complex manifold coupling a connection on the bundle with a Kahler structure on the base. These equations generalize the conditions of constant scalar curvature for a Kahler metric and Hermite-Yang-Mills for a connection. We provide a moment map interpretation of …

Study on spectral stability of an embedded annulus under curve shortening and Ricci flows.

FHRN uses continuous-time dynamics to stabilize reentrant neural computation.

The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background Kähler metric, to be constant. In this paper we introduce and study dHYM eq…

Gradient descent forces neural network eigenvalues to a specific threshold.

Global solutions found for a wave-Klein-Gordon system with strong couplings in divergence form.

Paper proposes a new method to optimize robot body structure and control policy.

The paper guarantees global stability for stochastic subgradient methods in nonsmooth nonconvex optimization.

Sharp pseudospectral bounds prevent transient amplification in coupled gradient descent.

A new RNN model based on coupled oscillators mitigates gradient issues.

New algorithms help machines forget old data efficiently.

The sectoral synchronization observed for the Japanese business cycle in the Indices of Industrial Production data is an example of synchronization. The stability of this synchronization under a shock, e.g., fluctuation of supply or demand, is a matter of interest in physics and economics. We consider an economic syste…

Momentum affects optimization differently at small vs large batch sizes near instability.

We introduce a notion of Gieseker stability for a filtered holomorphic vector bundle $F$ over a projective manifold. We relate it to an analytic condition in terms of hermitian metrics on $F$ coming from a construction of the Geometric Invariant Theory (G.I.T). These metrics are balanced in the sense of S.K. Donaldson.…

The variational calculus for the Faddeev-Hopf model on a general Riemannian domain, with general Kaehler target space, is studied in the strong coupling limit. In this limit, the model has key similarities with pure Yang-Mills theory, namely conformal invariance in dimension 4 and an infinite dimensional symmetry group…

A discussion is given of the conformal Einstein field equations coupled with matter whose energy-momentum tensor is trace-free. These resulting equations are expressed in terms of a generic Weyl connection. The article shows how in the presence of matter it is possible to construct a conformal gauge which allows to kno…

We identity a by-far-unrecognized problem of Adam-style optimizers which results from unnecessary coupling between momentum and adaptivity. The coupling leads to instability and divergence when the momentum and adaptivity parameters are mismatched. In this work, we propose a method, Laprop, which decouples momentum and…

Proposes a method to learn system dynamics and region of attraction from trajectories.

Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans $π_1, π_2, \ldots$ converges weakly to a transport plan $π$, then $π$ is also optimal (between its marginals). Alfonsi, Corbetta and Jourdain asked whether the same property is true for th…

Stability of Schwarzschild singularity in near-Schwarzschild black holes under perturbations.

Market makers optimize trading with a new implicit scheme for complex inequalities.

Let $P$ be a principal U(1)-bundle over a closed manifold $M$. On $P$, one can define a modified version of the Ricci flow called the Ricci Yang-Mills flow, due to these equations being a coupling of Ricci flow and the Yang-Mills heat flow. We use maximal regularity theory and ideas of Simonett concerning the asymptoti…

GraphShield uses dynamic graph learning to detect and visualize financial risks.

The paper develops a new algorithm for RBMs using dynamical mean-field theory.

Geometric formalism views optimization algorithms as discrete connections, revealing their algebraic curvature and flatness properties.

Study global solutions for Boussinesq systems on curved manifolds.

Asynchronous stochastic approximations (SAs) are an important class of model-free algorithms, tools and techniques that are popular in multi-agent and distributed control scenarios. To counter Bellman's curse of dimensionality, such algorithms are coupled with function approximations. Although the learning/ control pro…

The paper extends geometric inequalities for nearly spherical sets in various space forms.

Optimism stabilizes Thompson Sampling for adaptive inference in multi-armed bandits.

Study on stability of optimal transport problems for probability measures.

The goal of this paper is twofold: we study metric measure spaces $(X,d,m)$ with variable lower bounds for the Ricci curvature and we study pathwise coupling of Brownian motions. Given any lower semicontinuous function $k:X\to \mathbb R$ we introduce the curvature-dimension condition $CD(k,\infty)$ which canonically ex…

Multi-simulator training has contributed to the recent success of Deep Reinforcement Learning by stabilizing learning and allowing for higher training throughputs. We propose Gossip-based Actor-Learner Architectures (GALA) where several actor-learners (such as A2C agents) are organized in a peer-to-peer communication t…

We consider the Einstein flow on a product manifold with one factor being a compact quotient of 3-dimensional hyperbolic space without boundary and the other factor being a flat torus of fixed arbitrary dimension. We consider initial data symmetric with respect to the toroidal directions. We obtain effective Einsteinia…