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Mikhail Feldman

Bio: Mikhail Feldman is an academic researcher from University of Wisconsin-Madison. The author has contributed to research in topics: Potential flow & Free boundary problem. The author has an hindex of 20, co-authored 62 publications receiving 1792 citations. Previous affiliations of Mikhail Feldman include Australian National University & Northwestern University.


Papers
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Journal ArticleDOI
TL;DR: In this article, the authors constructed new families of Kahler-Ricci solitons on complex line bundles over ℂℙn−1, n ≥ 2, and exhibited a noncompact Ricci flow that shrinks smoothly and self-similarly for t 0.
Abstract: We construct new families of Kahler-Ricci solitons on complex line bundles over ℂℙn−1, n ≥ 2. Among these are examples whose initial or final condition is equal to a metric cone ℂn/ℤk. We exhibit a noncompact Ricci flow that shrinks smoothly and self-similarly for t 0; this evolution is smooth in space-time except at a single point, at which there is a blowdown of a ℂℙn−1. We also construct certain shrinking solitons with orbifold point singularities.

300 citations

Journal ArticleDOI
TL;DR: In this article, the authors give a third existence proof for optimal mappings, which requires no continuity or separation of the mass distributions, yet their explicit construction yields more geometrical control than the abstract method of Sudakov.
Abstract: The Monge-Kantorovich problem is to move one distribution of mass onto another as efficiently as possible, where Monge's original criterion for efficiency [19] was to minimize the average distance transported. Subsequently studied by many authors, it was not until 1976 that Sudakov showed solutions to be realized in the original sense of Monge, i.e., as mappings from Rn to Rn [23]. A second proof of this existence result formed the subject of a recent monograph by Evans and Gangbo [7], who avoided Sudakov's measure decomposition results by using a partial differential equations approach. In the present manuscript, we give a third existence proof for optimal mappings, which has some advantages (and disadvantages) relative to existing approaches: it requires no continuity or separation of the mass distributions, yet our explicit construction yields more geometrical control than the abstract method of Sudakov. (Indeed, this control turns out to be essential for addressing a gap which has recently surfaced in Sudakov's approach to the problem in dimensions n > 3; see the remarks at the end of this section.) It is also shorter and more flexible than either, and can be adapted to handle transportation on Riemannian manifolds or around obstacles, as we plan to show in a subsequent work [13]. The problem considered here is the classical one: Problem 1 (Monge). Fix a norm d(x, y) = llx-yll on Rn, and two densities -non-negative Borel functions f+, fE Ll (Rn) satisfying the mass balance condition

197 citations

Journal ArticleDOI
TL;DR: In this paper, the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids was studied. But the authors focused on the stability of the multidisensor shocks for a single-dimensional transonic wave.
Abstract: We are concerned with the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as the following second-order nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential φ : Ω ⊂ R → R: (1.1) div (ρ(|Dφ|)Dφ) = 0, where the density function ρ(q) is

191 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of optimally transporting mass on a complete, connected, Riemannian manifold, assuming absolute continuity of the Borel probability of the mass.
Abstract: Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures μ + ¬= μ - , find the measurepreserving map s: M → M between them which minimizes the average distance transported Set on a complete, connected, Riemannian manifold M - and assuming absolute continuity of μ + - an optimal map will be shown to exist Aspects of its uniqueness are also established

101 citations

Journal ArticleDOI
TL;DR: In this paper, the authors developed a rigorous mathematical approach to overcome these difficulties and establish a global theory of existence and stability for shock reflection by large-angle wedges for potential flow.
Abstract: When a plane shock hits a wedge head on, it experiences a reflectiondiffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. Experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection may occur, including regular and Mach reflection. However, most of the fundamental issues for shock reflection have not been understood, including the global structure, stability, and transition of the different patterns of shock reflection. Therefore, it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, there has been no rigorous mathematical result on the global existence and structural stability of shock reflection, including the case of potential flow which is widely used in aerodynamics. Such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic-hyperbolic type, free boundary problems, and corner singularity where an elliptic degenerate curve meets a free boundary. In this paper we develop a rigorous mathematical approach to overcome these difficulties i nvolved and establish a global theory of existence and stability for shock reflection by large-angle wedges for potential flow. The techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties.

92 citations


Cited by
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Book
02 Jan 2013
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
Abstract: Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

5,524 citations

Book
01 Jan 2005
TL;DR: In this article, Gradient flows and curves of Maximal slopes of the Wasserstein distance along geodesics are used to measure the optimal transportation problem in the space of probability measures.
Abstract: Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).

3,401 citations

Journal ArticleDOI
TL;DR: To the best of our knowledge, there is only one application of mathematical modelling to face recognition as mentioned in this paper, and it is a face recognition problem that scarcely clamoured for attention before the computer age but, having surfaced, has attracted the attention of some fine minds.
Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

3,015 citations

Book
01 Jan 1996
TL;DR: A review of the collected works of John Tate can be found in this paper, where the authors present two volumes of the Abel Prize for number theory, Parts I, II, edited by Barry Mazur and Jean-Pierre Serre.
Abstract: This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

2,014 citations

Posted Content
TL;DR: This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.
Abstract: Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

1,355 citations