Dmitri Burago

Bio: Dmitri Burago is an academic researcher from Pennsylvania State University. The author has contributed to research in topics: Geodesic & Metric (mathematics). The author has an hindex of 17, co-authored 58 publications receiving 3622 citations. Previous affiliations of Dmitri Burago include University of Pennsylvania & Russian Academy of Sciences.

A Course in Metric Geometry

Abstract: Metric Spaces Length Spaces Constructions Spaces of Bounded Curvature Smooth Length Structures Curvature of Riemannian Metrics Space of Metric Spaces Large-scale Geometry Spaces of Curvature Bounded Above Spaces of Curvature Bounded Below Bibliography Index.

Hard Ball Systems and the Lorentz Gas

Abstract: Part I. Mathematics: 1. D. Burago, S. Ferleger, A. Kononenko: A Geometric Approach to Semi-Dispersing Billiards.- 2. T. J. Murphy, E. G. D. Cohen: On the Sequences of Collisions Among Hard Spheres in Infinite Spacel- 3. N. Simanyi: Hard Ball Systems and Semi-Dispersive Billiards: Hyperbolicity and Ergodicity.- 4. N. Chernov, L.-S. Young: Decay of Correlations for Lorentz Gases and Hard Balls.- 5. N. Chernov: Entropy Values and Entropy Bounds.- 6. L. A. Bunimovich: Existence of Transport Coefficients.- 7. C. Liverani: Interacting Particles.- 8. J. L. Lebowitz, J. Piasecki and Ya. G. Sinai: Scaling Dynamics of a Massive Piston in an Ideal Gas .- Part II. Physics: 1. H. van Beijeren, R. van Zon, J. R. Dorfman: Kinetic Theory Estimates for the Kolmogorov-Sinai Entropy, and the Largest Lyapunov Exponents for Dilute, Hard-Ball Gases and for Dilute, Random Lorentz Gases.- 2. H. A. Posch and R. Hirschl: Simulation of Billiards and of Hard-Body Fluids.- 3. C. P. Dettmann: The Lorentz Gas: a Paradigm for Nonequilibrium Stationary States.- 4. T. Tl, J. Vollmer: Entropy Balance, Multibaker Maps, and the Dynamics of the Lorentz Gas.- Appendix: 1. D. Szasz: Boltzmanns Ergodic Hypothesis, a Conjecture for Centuries?

Riemannian tori without conjugate points are flat

Abstract: This statement is known as the Hopf conjecture and it has been proved by E. Hopf ([Ho]) for the case n = 2. The proof of Theorem 1 is contained in sections 1–5. The main idea of our proof is that the limit norm of such a metric (see section 1.2) is a Euclidean norm. For two unit vectors p, q in a Banach space with its unit sphere having a unique supporting linear function −Bp at p one can define something like inner product 〈p, q〉 = −Bp(q). To show that it actually is an inner product we prove that Euclidean norms possess some extremal property of integral type which makes them distinguishable among all Banach norms. Then we note that the functions Bp(q) for the limit norm of our metric can be drawn from the infinitesimal inner product by the means of integral geometry, and we check the property above for the limit norm. This proves that the limit norm is Euclidean and our inequalities for the integrals turn out to be equalities almost everywhere. Then a rather simple additional argument shows that in this case our metric is flat. Section 6 contains a brief discussion and the volume growth theorem. The history of the subject will not be touched upon in this short paper. We express our gratitude to Prof. V. Bangert for thoroughly reading the manuscript, his valuable remarks and his help in finding references. The first author also would like to use this opportunity to thank Prof. V. Bangert for his kind invitation to the University of Freiburg and actually interesting discussions.

Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus

Abstract: We show that partially hyperbolic diffeomorphisms of the 3-torus are dynamically coherent.

Conjugation-invariant norms on groups of geometric origin

Abstract: A group is said to be bounded if it has a finite diameter with respect to any bi-invariant metric. In the present paper we discuss boundedness of various groups of diffeomorphisms.

A Course in Metric Geometry

Abstract: Metric Spaces Length Spaces Constructions Spaces of Bounded Curvature Smooth Length Structures Curvature of Riemannian Metrics Space of Metric Spaces Large-scale Geometry Spaces of Curvature Bounded Above Spaces of Curvature Bounded Below Bibliography Index.

On the geometry of metric measure spaces. II

Abstract: We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} \right)} > K$ (introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD(K, ∞). The additional real parameter N plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD(K, N) is equivalent to $${\text{Ric}}_{M} {\left( {\xi ,\xi } \right)} > K{\left| \xi \right|}^{2} $$ and dim(M) ⩽ N. The curvature-dimension condition CD(K, N) is stable under convergence. For any triple of real numbers K, N, L the family of normalized metric measure spaces (M, d, m) with CD(K, N) and diameter ⩽ L is compact. Condition CD(K, N) implies sharp version of the Brunn–Minkowski inequality, of the Bishop–Gromov volume comparison theorem and of the Bonnet–Myers theorem. Moreover, it implies the doubling property and local, scale-invariant Poincare inequalities on balls. In particular, it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.

Ricci curvature for metric-measure spaces via optimal transport

Abstract: We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdor limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X;d) in which the distance between two points equals the inmum of the lengths of curves joining the points. In the rest of this introduction we assume that X is a compact length space. Alexandrov gave a good notion of a length space having \curvature bounded below by K", with K a real number, in terms of the geodesic triangles in X. In the case of a Riemannian manifold M with the induced length structure, one recovers the Riemannian notion of having sectional curvature bounded below by K. Length spaces with Alexandrov curvature bounded below by K behave nicely with respect to the GromovHausdor topology on compact metric spaces (modulo isometries); they form