arXiv research

We show that there exists an infinite family of knots, each of which has, for each integer k>=0, a destabilized (2k+5)-bridge sphere. We also show that, for each integer n>=4, there exists a knot with a destabilized 3-bridge sphere and a destabilized n-bridge sphere.

The paper explores maximal destabilizers for both K-stability and Chow-stability in unstable situations.

Study on transverse knots and their neighborhoods, proving unique standard neighborhoods and destabilization results.

We give a generalisation of the theory of optimal destabilizing 1-parameter subgroups to non-algebraic complex geometry. Consider a holomorphic action $G\times F\to F$ of a complex reductive Lie group $G$ on a finite dimensional (possibly non-compact) Kähler manifold $F$. Using a Hilbert type criterion for the (semi)st…

We show that the ``classical'' Harder-Narasimhan filtration associated to a non semistable vector bundle $E$ can be viewed as a limit object for the action of the gauge group in the direction of an optimal destabilizing vector. This vector appears as an extremal value of the so called "maximal weight function". We give…

Equivalence proven between algebraic stability and geometric stability.

Characterizes solutions to Z-critical equations on surfaces using effective conditions.

This paper describes how, in the case of algebraic surfaces, the well-known theorem of Donaldson-Uhlenbeck-Yau can be proved in a framework of generalized 'multiplier ideal sheaves', following the ideas of Siu. The key concept is that the destabilizing sheaf satisfies a differential inclusion relation. This relation is…

A virtual $n$-string $α$ is a collection of $n$ oriented smooth generic loops on a surface $M$. A stabilization of $α$ is a surgery that results in attaching a handle to $M$ along disks avoiding $α$, and the inverse operation is a destabilization of $α$. We consider virtual $n$-strings up to virtual homotopy, i.e., seq…

When calculating the index of a minimal surface, the set of smooth functions on a domain with compact support is the standard setting to describe admissible variations. We show that the set of admissible variations can be widened in a geometrically meaningful manner by considering the difference of area functional, lea…

We construct a sequence of pairs of 3-manifolds each with torus boundary and with the following two properties: 1) For the result of a carefully chosen glueing of the nth pair of 3-manifolds along their boundary tori, the ratio of the genus of the resulting 3-manifold to the sum of the genera of the pair of 3-manifolds…

Proves properness of K-moduli spaces for Fano varieties.

In this note we introduce the (homologically essential) arc complex of a surface as a tool for studying properties of open book decompositions and contact structures. After characterizing destabilizability in terms of the essential translation distance of the monodromy of an open book we given an application of this re…

The Markov Theorem Without Stabilization (MTWS) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive there are three key isoto…

The Markov Theorem Without Stabilization (MTWS) (see math.GT/0310279) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive the…

We prove some criteria for uniform K-stability of log Fano pairs. In particular, we show that uniform K-stability is equivalent to $β$-invariant having a positive lower bound. Then we study the relation between optimal destabilization conjecture and the conjectural equivalence between uniform K-stability and K-stabilit…

We study the J-flow from the point of view of an algebro-geometric stability condition. In terms of this we give a lower bound for the natural associated energy functional, and we show that the blowup behavior found by Fang-Lai is reflected by the optimal destabilizer. Finally we prove a general existence result on com…

The paper proves conditions for minimal surfaces to be holomorphic and stable.

It is known that the automorphism group of a K-polystable Fano manifold is reductive. Codogni and Dervan construct a canonical filtration of the section ring, called Loewy filtration, and conjecture that the Loewy filtration destabilizes any Fano variety with non-reductive automorphism group. In this note, we give a co…

We show that the presence of a plastikstufe induces a certain degree of flexibility in contact manifolds of dimension 2n+1>3. More precisely, we prove that every Legendrian knot whose complement contains a "nice" plastikstufe can be destabilized (and, as a consequence, is loose). As an application, it follows in certai…

Dynamic Vocabulary Pruning stabilizes LLM training by removing low-probability tokens.

Quantizes Kähler-Ricci flow for Fano manifolds.

We give a parametrization of test configurations in the sense of Donaldson via spherical buildings, and show the existence of "optimal" destabilizing test configurations for unstable varieties, in the wake of Mumford and Kempf. We also give an account of the recent slight amendment to definition of K-stability after Li…

We prove a conjecture of Gromov's to the effect that manifolds with isotropic curvature bounded below by 1 (after possibly rescaling) are macroscopically 1-dimensional on the scales greater than 1. As a consequence we prove that compact manifolds with positive isotropic curvature have virtually free fundamental groups.…

Our analysis of financial data, in terms of super-exponential growth, suggests that the seed of the 2002/03 crisis of the Dutch supermarket giant AHOLD was planted in 1996. It became quite visible in 1999 when the post-bubble destabilization regime was well-developed and acted as the precursor of an inevitable collapse…

We extend to the multi-asset case the framework of a discrete time model of a single asset financial market developed in Ghoulmie et al (2005). In particular, we focus on adaptive agents with threshold behavior allocating their resources among two assets. We explore numerically the effect of this diversification as an …

S. K. Donaldson asked whether the lower bound of the Calabi functional is achieved by a sequence the normalized Donaldson-Futaki invariants. We answer the question for the Ricci curvature formalism, in place of the scalar curvature. The principle is that the stability indicator is optimized by the multiplier ideal shea…

New inequality helps map stability in minimal surfaces.

We first review empirical evidence that asset prices have had episodes of large fluctuations and been inefficient for at least 200 years. We briefly review recent theoretical results as well as the neurological basis of trend following and finally argue that these asset price properties can be attributed to two fundame…

Using numerical simulations of the axisymmetric Navier-Stokes equations with swirl on a no-slip flat boundary, Hsu-Notsu-Yoneda [J. Fluid Mech. 2016] observed the creation of a high-vorticity region on the boundary near the axis of symmetry. In this paper, using a differential geometric approach, we prove that such flo…

Projection maps virtual Legendrian knots to classical ones.

We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that gr…

Let M_1 and M_2 be compact, orientable 3-manifolds with incompressible boundary, and M the manifold obtained by gluing with a homeomorphism $φ:\bdy M_1 \to \bdy M_2$. We analyze the relationship between the sets of low genus Heegaard splittings of M_1, M_2, and M, assuming the map φis "sufficiently complicated." This a…

AIS corrects rollout-training mismatch in quantized RL, improving speed and stability.

This paper investigates - on the basis of the Cont-Bouchaud model - whether a Tobin tax can stabilize foreign exchange markets. Compared to earlier studies, this paper explicitly recognizes that a transaction tax-induced reduction in market depth may increase the price responsiveness of a given order. We find that the …

This Chapter reviews statistical models for the probability distribution of money developed in the econophysics literature since the late 1990s. In these models, economic transactions are modeled as random transfers of money between the agents in payment for goods and services. Starting from the initially equal distrib…

The Tobin tax is an often discussed method to tame speculation and get a source of income. The discussion is especially heated when the financial markets are in crisis. In this article we refer to foreign exchange markets. The Tobin tax should be a small international tax affecting all currency transactions and thus co…

Modeling HFT interactions reveals market instability.

A flat virtual link is a finite collection of oriented closed curves $\mathfrak L$ on an oriented surface $M$ considered up to virtual homotopy, i.e., a composition of elementary stabilizations, destabilizations, and homotopies. Specializing to a pair of curves $(L_1,L_2)$, we show that the minimal number of intersecti…

Investment diversification affects financial stability, depending on network connectivity.

We expect manifolds obtained by Dehn filling to inherit properties from the knot manifold. To what extent does that hold true for the Heegaard structure? We study four changes to the Heegaard structure that may occur after filling: (1) Heegaard genus decreases, (2) a new Heegaard surface is created, (3) a non-stabilize…

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature Kähler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture…

We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type K, we analyze the Legendrian knots in knot types obtained from K by cabling, in terms of Legendrian knots in the knot type K. As a corollary of this analysis, we show that the (2,3)-cable of the (2,3)-torus knot is not transver…

Recently, the notion of cryptocurrencies has come to the fore of public interest. These assets that exist only in electronic form, with no underlying value, offer the owners some protection from tracking or seizure by government or creditors. We model these assets from the perspective of asset flow equations developed …

A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] a transver…

Actor-critic methods can achieve incredible performance on difficult reinforcement learning problems, but they are also prone to instability. This is partly due to the interaction between the actor and critic during learning, e.g., an inaccurate step taken by one of them might adversely affect the other and destabilize…

As a generalization of Kahler-Einstein metrics for Fano manifolds with nonvanishing Futaki invariant, Mabuchi solitons are critical points of a Calabi-type energy functional. We study their existence on toric Fano varieties and the underlying algebraic stability notion: relative Ding stability. As a toy model for a YTD…

We introduce the inverse Monge-Ampere flow as the gradient flow of the Ding energy functional on the space of Kahler metrics in $2 πλc_1(X)$ for $λ=\pm 1$. We prove the long-time existence of the flow. In the canonically polarized case, we show that the flow converges smoothly to the unique Kahler-Einstein metric with …