School on Metric Geometry and Gromov–Hausdorff Distance

Alexander O. Ivanov

Lomonosov Moscow State University

Introduction and basic properties of Gromov–Hausdorff distance

A four-lecture series on the foundations of GH geometry and the structure of compact metric spaces.

• Hausdorff and Gromov–Hausdorff distances
• Properties of the Gromov–Hausdorff distance
• Compact metric spaces in GH-topology
• Local and global isometries of space ℳ

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Alexey A. Tuzhilin

Lomonosov Moscow State University

Gromov–Hausdorff distance: applications and lower bounds

A four-lecture series on applications, lower bounds, ultrametrization, and geometric methods.

• Applications of GH distance: MST-spectrum, Borsuk problem and combinatorial invariants
• Ultrametrization and its applications in GH geometry
• Topological methods in GH-distance estimates
• Estimation of the GH-distance to a normed space through the Jung constant

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Zhang Huichun

Sun Yat-sen University · School of Mathematics

Alexandrov geometry, RCD spaces, and metric-valued analysis

A five-lecture series on Alexandrov spaces, RCD spaces, Sobolev mappings, and harmonic map regularity, presented by a professor of geometric analysis at Sun Yat-sen University.

• Introduction to Alexandrov geometry
• A brief introduction to RCD spaces
• Bochner inequality and linear geometric analysis on Alexandrov/RCD spaces
• Sobolev spaces with metric targets
• Regularity of harmonic maps and heat flows into CAT(0) spaces

Research area: geometric analysis on nonsmooth spaces, Alexandrov geometry, and geometric PDEs · Contact: zhanghc3@mail.sysu.edu.cn

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Jiang Wenshuai

Zhejiang University · School of Mathematical Sciences

Ricci limit spaces, splitting, and geometric analysis

A five-lecture series on volume comparison, splitting, almost cones, and regularity of Ricci limit spaces, presented by a professor of geometric analysis at Zhejiang University.

• Volume comparison, segment inequality, Poincaré and Sobolev inequalities, elliptic estimates
• GH-splitting theorem, volume convergence, almost volume cone and almost metric cone
• Almost volume cone, almost metric cone, and related applications
• Regularity theorem and structure of Ricci limit spaces
• Open problems and further topics on the geometry of Ricci curvature

Research area: geometric analysis · Contact: wsjiang@zju.edu.cn

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Semeon A. Bogatyi

Lomonosov Moscow State University

Analogues of the Gromov–Hausdorff distance

Monday, October 26 · 10:00–10:50

The talk compares the two classical approaches to the Gromov–Hausdorff distance: the geometric (external) approach, based on isometric embeddings of the two metric spaces into a common ambient space and the Hausdorff distance between their images, and the functional (internal) approach, based on pairs of maps between the spaces and the distortion they produce. Modifications of the classical definition are surveyed — restricting or extending the class of isometric embeddings, replacing the Hausdorff distance by the Charatonik or Borsuk metrics on the exponent exp Z, and the resulting Gromov–Borsuk distance — together with parallels to the Banach–Mazur distance, to Borsuk’s theory of shapes (ε-maps and multivalued maps) and to the topology of the Minkowski compact of n-dimensional normed spaces.

Arsen Kh. Galstyan

Harbin Institute of Technology · School of Mathematics

On the existence of minimal parametric networks

Monday, October 26 · 11:10–12:00

Finding a minimal (shortest) network connecting a prescribed finite boundary in a metric space is a classical problem of the calculus of variations and geometric optimization, best known in the form of the Steiner problem. Fixing the tree structure turns it into a parameter and leads to the problem of minimal parametric networks. The talk discusses which spaces and hyperspaces are suitable for solving this problem, which topological and functional-analytic conditions are responsible for solvability, and whether algorithms that find the true optimum can be expected at all.

Alexei G. Kushner

Lomonosov Moscow State University

Geometric structures and second-order differential equations: results and open problems

Tuesday, October 27 · 10:00–10:50

From a geometric viewpoint, second-order differential equations are submanifolds of 2-jet spaces; following V. Lychagin, a wide class of such equations — including all linear and quasilinear equations and the Monge–Ampère equations — is described by differential 2-forms on 1-jet spaces, which makes it possible to reduce non-degenerate nonlinear equations to linear ones by symplectic and contact transformations. The talk discusses the geometric structures associated with second-order equations (distributions, tensor fields, complex structures, almost-product structures and Riemannian structures), illustrates them with examples from physics, formulates open problems, and touches upon the prospects of applying the theory of the continuous Gromov–Hausdorff distance to such equations.

Li Yan

Invited speaker

Gromov–Hausdorff convergence with group action

Tuesday, October 27 · 11:10–12:00

In many algebro-geometrical problems, a sequence of algebraic varieties with a reductive group G-action converges (in a certain sense analogous to Gromov–Hausdorff convergence) to a limit variety that also admits a G-action. Moreover, there are also results on properties of the structure of orbits in the limit variety. In this talk, we will first introduce some algebraic background, then we will show that for a bounded sequence of compact metric spaces with compact torus action, a subsequence converges in the Gromov–Hausdorff topology to a limit space which also has a compact torus action. This result partially generalizes a result of Lim–Memoli.