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.
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48 results for “Almgren's center manifold”
This is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents. Here we perform the second main step in the analysis of the singularities, namely the construction of a center manifold, i.e. an approximate average of the sheets of an almost flat area minimizing cu…
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.
Upper bound on singular set dimension for area-minimizing currents.
problem Bounding the dimension of singular points in area-minimizing currents.
method Using upper Minkowski dimension and properties of blow-up scales.
result Upper Minkowski bound of for the interior singular set.
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.
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.
In analogy with Almgren's Theorem for area minimizing currents of general dimension and codimension, we prove that an -dimensional semicalibrated current in a -dimensional manifold, semicalibrated by a -form, has singular set of Hausdorff dimension at most .
Analyzes branch points of area-minimizing currents with non-2 planar frequency.
problem Understanding the structure of area-minimizing currents near branch points.
method Intrinsic frequency function and geometric arguments avoiding center manifolds.
result Establishes higher order asymptotics and topological control near branch points.
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 .
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.
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.
We propose two methods to obtain exact solutions for the Almgren-Chriss model about optimal execution of portfolio transactions. In the first method we rewrite the Almgren-Chriss equation and find two exact solutions. In the second method, employing a general reparametrized time, we show that the Almgren-Chriss equatio…
We show that there is no analog of Kirszbraun's extension theorem for Almgren's multiple valued functions.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 this work we prove the existence of embedded closed minimal hypersurfaces in non-compact manifolds containing a bounded open subset with smooth and strictly mean-concave boundary and a natural behavior on the geometry at infinity. For doing this, we develop a modified min-max theory for the area functional following…
The paper proves the existence of a center in disks within Riemannian manifolds.
problem Proving the existence of a center in disks within Riemannian manifolds.
method Analyzing the existence of a center in embedded -disks in Riemannian -manifolds.
result Existence and equivariance of centers in certain conditions, non-existence in others.
This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren's partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis deducing the regularity of area minimizing currents from that of Dir-minimizing multip…
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…
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.
I will talk about my recent work with Fernando Marques where we used Almgren-Pitts Min-max Theory to settle some open questions in Geometry: The Willmore conjecture, the Freedman-He-Wang conjecture for links (jointly with Ian Agol), and the existence of infinitely many minimal hypersurfaces in manifolds of positive Ric…
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.
Paper introduces new center of mass for flat manifolds.
problem Defining center of mass for asymptotically flat manifolds.
method Using double forms of Kulkarni and Labbi to prove existence and well-definedness.
result Existence and well-definedness of the Gauss-Bonnet-Chern center of mass.
Proves branch set dimension for stationary varifolds with ε-regularity.
problem Analyzing the structure of stationary varifolds with ε-regularity.
method Utilizes planar frequency function and geometric analysis.
result Hausdorff dimension of branch set is at most n-2 for certain varifolds.
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.
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-…
Diameters of ball intersections decrease as centers move apart.
problem Behavior of intersections of moving balls in Riemannian manifolds.
method Continuous decrease of intersection diameter as centers move apart.
result Diameter of intersections decreases continuously.
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.
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…
Study partially hyperbolic dynamics on 3-manifolds with quasi-isometric center.
problem Characterize dynamics on 3-manifolds with specific center properties.
method Analyzes partially hyperbolic diffeomorphisms with quasi-isometric center under non-wandering conditions.
result Volume-preserving diffeomorphisms are ergodic without -tori, confirming a conjecture.
Minimal surfaces in 8D smooth and nondegenerate.
problem Generic regularity of minimal hypersurfaces in 8D.
method Analysis of -generic metrics.
result All minimal hypersurfaces are smooth and nondegenerate.
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.
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…
Paper improves Morse index bound for hypersurfaces.
problem Improving Morse index bound for hypersurfaces.
method Construction of hierarchical deformations and restrictive min-max theory.
result Generalizes a result by X. Zhou for .