arXiv research

Extends optimal regularity and Uhlenbeck compactness to non-Riemannian manifolds.

In 1982, Uhlenbeck \cite {U2} established the well-known gauge fixing theorem, which has played a fundamental role for Yang-Mills theory. In this paper, we apply the idea of Uhlenbeck to establish a parabolic type of gauge fixing theorems for the Yang-Mills flow and prove existence of a weak solution of the Yang-Mills …

New conservation laws found for polyharmonic maps in critical dimension.

Study of mixed equation combining gauge theory and symplectic geometry.

Extends optimal regularity and compactness to vector bundles over non-Riemannian manifolds.

Survey on recent developments in isometric immersions using PDE techniques.

We consider the problem of extending functions φ:\to S^n to functions u:B^{n+1}\to S^n for n=2,3. We assume φto belong to the critical space W^{1,n} and we construct a W^{1,(n+1,\infty)}-controlled extension u. The Lorentz-Sobolev space W^{1,(n+1,\infty)} is optimal for such controlled extension. Then we use such resul…

In four and higher dimensions, we show that any stationary admissible Yang-Mills field can be gauge transformed to a smooth field if the $L^2$ norm of the curvature is sufficiently small. There are three main ingredients. The first is Price's monotonicity formula, which allows us to assert that the curvature is small n…

Introduces gauge theory for string algebroids, solving Calabi system.

In this paper we study the relationship between three compactifications of the moduli space of Hermitian-Yang-Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the Donaldson-Uhlenbeck-Yau theorem, this space is analytically isomorphic to the moduli spa…

New gauge fields modify Fokker-Planck dynamics without changing the stationary state.

This work extends Chern correspondence to higher gauge theory.

We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure. In particular, our Hitchin-Kobayashi theorem holds on any compact Sasakian manifold. We define the notion of stability for foliated Hermitian vector bundles with tr…

Let $X$ be a compact Hermitian surface, and $g$ be any fixed Gauduchon metric on $X$. Let $E$ be an Hermitian holomorphic vector bundle over $X$. On the bundle $E$, Donaldson's heat flow is gauge equivalent to a flow of holomorphic structures. We prove that this flow converges, in the sense of Uhlenbeck, to the double …

Extending isometric immersions with low regularity, especially supercritical.

We show the existence of Yang--Mills--Higgs (YMH) fields over a Riemann surface with boundary where a free boundary condition is imposed on the section and a Neumann boundary condition on the connection. In technical terms, we study the convergence and blow-up behavior of a sequence of Sacks-Uhlenbeck type $α$-YMH fiel…

An instanton $(E, D)$ on a (pseudo-)hyperkähler manifold $M$ is a vector bundle $E$ associated to a principal $G$-bundle with a connection $D$ whose curvature is pointwise invariant under the quaternionic structures of $T_x M, \ x\in M$, and thus satisfies the Yang-Mills equations. Revisiting a construction of solution…

Explains Uhlenbeck's work on maps and fields.

New proofs of Donaldson-Uhlenbeck-Yau theorem using geodesic rays.

Characterizes Lévy-driven Ornstein-Uhlenbeck processes linked to tempered stable distributions.

Sacks-Uhlenbeck's result on metric spaces expanded.

Pairs trading strategy improved using Ornstein-Uhlenbeck process.

This research announcement gives a brief report of the main results in our paper "PU(2) monopoles, I: Regularity, Uhlenbeck compactness, and transversality" (Journal of Differential Geometry, to appear). We describe the existence of perturbations for the PU(2) monopole equations, yielding both useful transversality pro…

Entropy-minimal measure calculated for a stochastic volatility model.

A new fast method simulates stochastic volatility models.

Develops European power option pricing under correlated interest rate and asset processes.

We will give a weak energy identity for Sacks-Uhlenbeck approximation of harmonic maps and calculate the length of the necks.

This paper proves a general Uhlenbeck compactness theorem for sequences of solutions of Yang-Mills flow on Riemannian manifolds of dimension $n \geq 4,$ including rectifiability of the singular set at finite or infinite time.

Constructs a moduli space for PDEs, linking stability to geometric metrics.

We prove sharp bounds for the growth rate of eigenfunctions of the Ornstein-Uhlenbeck operator and its natural generalizations. The bounds are sharp even up to lower order terms and have important applications to geometric flows.

Paper models non-maturing deposits using a Lévy-driven Ornstein-Uhlenbeck process.

Study optimal strategy for maximizing exponential utility in financial market with linear price impact.

The paper proves a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.

Model approximates market prices and returns without prior market dynamics.

We consider the Black--Scholes model of financial market modified to capture the stochastic nature of volatility observed at real financial markets. For volatility driven by the Ornstein--Uhlenbeck process, we establish the existence of equivalent martingale measure in the market model. The option is priced with respec…

When stock prices are observed at high frequencies, more information can be utilized in estimation of parameters of the price process. However, high-frequency data are contaminated by the market microstructure noise which causes significant bias in parameter estimation when not taken into account. We propose an estimat…

Method verifies if observed data fits Lévy-Driven Ornstein-Uhlenbeck process.

Study on gamma-related OU processes with simulation methods.

A result (Corollary 4.3) in an article by Uhlenbeck (1985) asserts that the $W^{1,p}$-distance between the gauge-equivalence class of a connection $A$ and the moduli subspace of flat connections $M(P)$ on a principal $G$-bundle $P$ over a closed Riemannian manifold $X$ of dimension $d\geq 2$ is bounded by a constant ti…

Study optimizes dividend payout strategies under fluctuating interest rates.

Karen Uhlenbeck's compactness theorem for sequences of connections with L2 bounds on curvature applies only to connections on principal bundles with compact structure group. This article states and proves an extension of Uhlenbecks theorem that describes sequences of connections on principal PSL(2;C) bundles over compa…

We propose a strategy for automated trading, outline theoretical justification of the profitability of this strategy and overview the hypothetical results in application to currency pairs trading. The proposed methodology relies on the assumption that processes reflecting the dynamics of currency exchange rates are in …

Modeling horse race betting odds with Ornstein-Uhlenbeck process.

Study prices energy derivatives using specific stochastic processes.

In this paper, we study an $α$-flow for the Sack-Uhlenbeck functional on Riemannian surfaces and prove that the limiting map by the $α$-flows is a weak solution to the harmonic map flow. By an application of the $α$-flow, we present a simple proof of an energy identity of a minimizing sequence in each homotopy class.

Uhlenbeck's compactness theorem can be used to analyze sequences of connections with anti-self dual curvature on principal SU(2) bundles over oriented 4-dimensional manifolds. The theorems in this paper give an extension of Uhlenbeck's theorem for sequences of solutions of certain SL(2,C) analogs of the anti-self dual …

We prove that any flat family $(\mathcal{ F}_u)_{u\in U}$ of rank 2 torsion-free sheaves on a Gauduchon surface defines a continuous map on the semi-stable locus $U^{\mathrm {ss}}:=\{u\in U \ |\ \mathcal{ F}_u\hbox{ is slope semi-stable}\}$ with values in the Donaldson-Uhlenbeck compactification of the corresponding in…

The aim of this paper is to construct the parabolic version of the Donaldson--Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algenraic surface with parabolic structures along a divisor with normal crossing singularities. We prove the non--emptiness of the moduli space of parabolic sta…