We show that there is no analog of Kirszbraun's extension theorem for Almgren's multiple valued functions.
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48 results for “Almgren's Theorem”
Explains basic ideas of two area minimizing surface theorems.
problem Regularity theory of area minimizing surfaces.
method Simplified proof of De Giorgi's and Almgren's theorems.
result Illustrates fundamental PDE estimates.
Proves regularity for semicalibrated currents in smooth manifolds.
problem Regularity of semicalibrated currents in smooth manifolds.
method Analogy with Almgren's Theorem for area-minimizing currents, proving singular set dimensionality.
result Singular set of Hausdorff dimension at most \( m-2 \) for semicalibrated currents.
Study improves optimal execution model with trading volume considerations.
problem Optimizing trading strategies in models with varying market volumes.
method Introduced a penalization method for an adaptive optimization problem in the Almgren-Chriss model.
result Verified the optimality of the volume-weighted average-price strategy and derived a second-order asymptotic expansion of the optimal strategy.
Analytic saddle spheres in S^3 are equators.
problem Characterizing saddle-shaped minimal surfaces in 3-sphere.
method Purely geometric approach, no PDE imposed.
result Analytic saddle spheres in S^3 are equators.
The nonlocal Almgren problemmath.AP
Investigates convexity of minimizers under mass constraint using nonlocal perimeter and potential.
problem Convexity of minimizers under mass constraint.
method Nonlocal free energy with nonlocal perimeter and convex potential.
result Quantitative stability theorem for nonlocal free energy assuming symmetry on the potential.
Revisits and proves a reparametrization theorem for multi-valued graphs in higher codimension.
problem Analyzing multi-valued sections of vector bundles and proving a reparametrization theorem.
method Develops properties of -multisections and provides a geometric proof.
result Elementary and purely geometric proof of a reparametrization theorem for multi-valued graphs.
We show that wealth processes in the block-shaped order book model of Obizhaeva/Wang converge to their counterparts in the reduced-form model proposed by Almgren/Chriss, as the resilience of the order book tends to infinity. As an application of this limit theorem, we explain how to reduce portfolio choice in highly-re…
This research proves that two min-max theories for hypersurfaces are equivalent.
problem Comparing two min-max theories for hypersurfaces.
method Developed and proved the equivalence of Almgren-Pitts and Allen-Cahn min-max theories.
result The Almgren-Pitts widths and Allen-Cahn widths are equivalent.
Two methods find exact solutions to optimal portfolio execution, linking to Riccati equations.
problem Optimal execution of portfolio transactions.
method Two methods: rewriting and reparametrization.
result Equivalence to Riccati equations and exact solutions found.
Constructs area-minimizing submanifolds with fractal singularities.
problem Area-minimizing submanifolds with fractal singular sets.
method Integral currents, mod v currents, stable stationary varifolds.
result Sharp dimensionwise solution to Almgren's conjecture.
Proves spectra equivalence for Riemannian manifolds.
problem Equivalence of Almgren-Pitts and phase-transition half-volume spectra.
method Proof of spectra equivalence for Riemannian manifolds.
result Confirms conjecture about spectra equivalence.
On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesicsmath.DS
An example from Almgren and Federer shows geodesics that are not always the shortest.
problem Illustrating the subtleties of geodesic minimization in complex metrics.
method Exposition of a specific example in to clarify definitions.
result Found geodesics that are not minimizers in their homotopy classes.
Solves optimal liquidation problem for stock price following geometric Brownian motion.
problem Optimal liquidation problem for stock price process following geometric Brownian motion.
method Functional analysis tools; working in terms of cash.
result Explicit solution to the problem, extending to stochastic drift.
Study confirms a 2-sphere metric with three geodesics of minimal length.
problem Understanding the systolic, width, and Gromov-Guth metrics on a 2-sphere.
method Classical min-max and hyperbolic geometry tools.
result Figure-eight geodesics achieve the systolic, width, and Gromov-Guth metrics on a 2-sphere.
Study proves uniform ellipticity implies uniform polyconvexity for anisotropic energy functionals.
problem Investigating uniform ellipticity and polyconvexity for anisotropic geometric energy functionals.
method Proves a variant of a recent result using real polyhedral chains.
result Uniform ellipticity of an anisotropic energy functional implies uniform polyconvexity of the integrand.
Localized min-max method proves minimal hypersurface existence.
problem Existence of minimal hypersurfaces in complete manifolds.
method Localized min-max approach to prove existence.
result Existence of complete embedded minimal hypersurface with index at most one.
Paper proves existence of area-minimizing submanifolds on almost any manifold with fractal singular sets.
problem Existence of area-minimizing submanifolds with fractal singular sets.
method Constructing and proving existence on almost any smooth manifold.
result Existence of area-minimizing submanifolds with fractal singular sets on almost any smooth manifold.
In this paper we discuss various minimality properties for the orthogonal product of two 1-dimensional $\Y$ sets, and some related problems. This is motivated by an attempt to give the classification of singularities for 2-dimensional Almgren-minimal sets in .
Minimizing the free energymath.AP
Sharp stability in Almgren problem solved in any dimension.
problem Quantitative stability in the radial isotropic Almgren problem.
method Developed a theory for estimating the sharp modulus under minimal assumptions.
result Sharp in any dimension, solving the critical mass problem.
Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated…
The paper solves a thermodynamics problem about crystal shape.
problem Understanding if minimizing free energy with convex potential and mass constraint generates a convex crystal.
method Utilized a stability theorem, convexity, and a new maximum principle approach to prove a three-dimensional convexity theorem.
result Completely settled the Almgren problem in under generic conditions.
One-dimensional crystals have convex shapes under certain conditions.
problem Determining if one-dimensional crystals have convex shapes.
method Analyzing the free energy under mass constraints and convexity assumptions.
result In one dimension, crystals have convex shapes under given conditions.
Optimizes liquidation strategies for assets with Levy process price dynamics.
problem Maximizing cash received from asset sale with price impact.
method Almgren-Chriss framework, constant absolute risk aversion, Levy process approximation.
result Explicit expression for optimal liquidation trajectories.
Derives Weyl law for volume spectrum using parametric inequalities.
problem Deriving the Weyl law for the volume spectrum in compact Riemannian manifolds.
method Proves parametric generalizations of isoperimetric and coarea inequalities to derive the Weyl law.
result Derives the Weyl law for 1-cycles in 3-manifolds.
Proves monotonicity of parabolic frequency on all manifolds without curvature assumptions.
problem Monotonicity of parabolic frequency on manifolds.
method Analyzes parabolic frequency function on manifolds, proving monotonicity without curvature assumptions.
result Monotonicity of parabolic frequency on all manifolds, no curvature assumption needed.
Study motion of discrete interfaces on triangular lattice using Almgren, Taylor, and Wang's approach.
problem Motion of discrete interfaces on triangular lattice driven by ferromagnetic interactions.
method Coupling Almgren, Taylor, and Wang's minimizing movements approach with Braides, Gelli, and Novaga's discrete-to-continuum analysis.
result Limit motion of origin-symmetric convex hexagons compared to crystalline curvature evolution.
This paper optimizes brokerage contracts for multiple clients trading a single asset.
problem Optimizing brokerage contracts for multiple clients trading a single asset.
method Endogenously determines clients' reservation values and strategically chooses clients. Characterizes optimal portfolios computationally.
result Characterizes optimal portfolios of clients and their profits, showing dependence on price impact coefficients.
Proves existence of a third minimal hypersurface with fixed boundary.
problem Existence of minimal hypersurfaces with fixed boundary.
method Develops min-max methods adapted to discrete setting.
result Proves existence of a third embedded minimal hypersurface.
The classical literature on optimal liquidation, rooted in Almgren-Chriss models, tackles the optimal liquidation problem using a trade-off between market impact and price risk. Therefore, it only answers the general question of the optimal liquidation rhythm. The very question of the actual way to proceed with liquida…
We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-…
In a series of papers, including the present one, we give a new, shorter proof of Almgren's partial regularity theorem for area minimizing currents in a Riemannian manifold, with a slight improvement on the regularity assumption for the latter. This note establishes a new a priori estimate on the excess measure of an a…
The paper constructs optimal hedging strategies for options with price impact.
problem Optimal hedging strategies for options with temporary price impact.
method Combining analytic and probabilistic tools to establish feedback representation of the optimal strategy and derive utility indifference price.
result Explicit asymptotic expansion of utility indifference price quantifying price impact.
Optimal trading and liquidation strategies with signals and regulatory constraints.
problem Optimal trading and liquidation in models with price predictions and regulatory limits.
method Almgren-Chriss model with general signals, target zone models, and lookback option analysis.
result Explicit formulas for optimal liquidation rates in Bachelier and Black-Scholes dynamics.
Study proves orientability of specific hypersurfaces in positive Ricci curvature manifolds.
problem Proving orientability of min-max hypersurfaces in manifolds with positive Ricci curvature.
method Analyzes Almgren-Pitts width and uses index 1 minimal hypersurfaces with multiplicity 1.
result Extends previous results to dimensions .
For a market impact model, price manipulation and related notions play a role that is similar to the role of arbitrage in a derivatives pricing model. Here, we give a systematic investigation into such regularity issues when orders can be executed both at a traditional exchange and in a dark pool. To this end, we focus…
The paper establishes a sub-additive inequality for volume and ε-phase-transition spectra of Riemannian manifolds.
problem Analyzing the volume and ε-phase-transition spectra of Riemannian manifolds.
method Using the Almgren-Pitts width and Allen-Cahn approach.
result Proves sub-additive inequalities for volume and ε-phase-transition spectra.
In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a Riemannian manifold of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension . We characterize the …
Modeling market makers' quoting strategies to understand price impact.
problem Understanding how price impact arises from market makers' quoting strategies.
method Modeling market making as a dynamic auction using Stochastic Differential Games and finding Nash Equilibrium.
result The price impact function derived from market makers' strategies matches the Almgren-Chriss model.
In this paper, we develop a general existence theory for properly embedded minimal surfaces with free boundary in any compact Riemannian 3-manifold with boundary . The main feature of our result is that no convexity assumption is required on . Our proof uses a variant of the min-max construc…
Study on the nodal set of Dirac equation solutions on manifolds.
problem Understanding the structure of nodal sets of solutions to Dirac equations.
method Proved Hausdorff dimension of nodal sets, extended to locally Lipschitz coefficients, provided stratification results.
result Stratification result for nodal sets, providing new insights even in the smooth case.
We consider the optimal trade execution strategies for a large portfolio of single stocks proposed by Almgren (2003). This framework accounts for a nonlinear impact of trades on average market prices. The results of Almgren (2003) are based on the assumption that no shares of assets per unit of time are trade at the be…
The paper shows how Sobolev maps affect currents in metric spaces.
problem Understanding how Sobolev maps affect currents in metric spaces.
method Proving that a Sobolev map pushes almost every compactly supported integral current to an Ambrosio-Kirchheim integral current.
result The paper proves an isoperimetric inequality for Sobolev mappings relative to bounded, closed, and additive cochains.
In illiquid markets, option traders may have an incentive to increase their portfolio value by using their impact on the dynamics of the underlying. We provide a mathematical framework within which to value derivatives under market impact in a multi-player framework by introducing strategic interactions into the Almgre…
Anisotropic min-max theory constructs stable minimal surfaces in 3-manifolds.
problem Constructing stable anisotropic minimal surfaces in 3-manifolds.
method Anisotropic min-max theory, removable singularity theorems.
result Constructs stable anisotropic minimal surfaces in 3-manifolds without singularities.
Paper solves trade-off between internalisation and externalisation in stochastic trade flows.
problem Managing risk in stochastic trade flows between internalisation and externalisation.
method Derives almost-closed-form solutions using Almgren-Chriss framework for quadratic execution costs. Uses numerical methods for more general cases. Proposes reinforcement learning as an alternative.
result Almost-closed-form solutions and numerical methods for optimal strategies.
New bounds on singular set size for harmonic maps into 2-sphere in higher dimensions.
problem Bounding the size of singular set for harmonic maps into 2-sphere.
method Extending previous results to higher dimensions, proving new inequalities.
result Stable bounds on singular set size under small perturbations.
Two years ago, F.C. Marques and A.A. Neves implemented, in the framework of closed rectifiable 2-dimensional currents of the 3-dimensional sphere, a min-max method in geometric measure theory due to F. Almgren and J. Pitts. Using this approach they succeeded in proving that the famous Clifford torus minimizes the area …