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

Lecture slides: Lectures 1–2: Hausdorff and GH distances (PDF) · Lectures 2–4: compactness and isometries (PDF)

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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

Lecture slides: L1: MST-spectrum and Borsuk problem (PDF) · L2: Ultrametrization (PDF) · L3: Topological methods (PDF) · L4: Jung constant (PDF)

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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

Lecture slides: L1: Alexandrov geometry (PDF) · L2: RCD(K,N) spaces (PDF) · L3: Bochner inequality (PDF) · L4: Sobolev maps (PDF) · L5: Harmonic map heat flows (PDF)

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

Lecture slides: Lecture 1: volume, segments, elliptic estimates (PDF) · Lectures 2–3: splitting maps and cones (PDF) · Lectures 4–5: Ricci limits and Einstein geometry (PDF)

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

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Alexei G. Kushner

Lomonosov Moscow State University

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

Monday, October 26 · 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.

Slides (PDF) · Lecture text (PDF)

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.

Slides (PDF)

Semeon A. Bogatyi

Lomonosov Moscow State University

Analogues of the Gromov–Hausdorff distance

Tuesday, October 27 · 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.

Slides (PDF)

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.

Slides (PDF, 15 slides) · Exercises (PDF)

Yu-Sheng Wang

Beijing Normal University

A proof of the globalization theorem in Alexandrov geometry

Thursday, October 29 · 14:00–14:50

An Alexandrov space of curvature bounded below is defined through local comparison triangles in a model plane, and the Globalization Theorem of Burago–Gromov–Perelman states that such local curvature bounds propagate to the whole space whenever it is complete, or more generally geodesic. The talk presents a new and more elementary proof of this theorem. The argument weakens the classical definition of Alexandrov spaces, phrasing the local conditions through imaginary comparison angles and one-sided second-order bounds in the support sense, and it also yields a new proof of the Doubling Theorem. Consequences for the structure theory of spaces with curvature bounded below are discussed.

Published paper: A proof of Toponogov's theorem in Alexandrov geometry (PDF) · DOI:10.1090/proc/16192

Wancheng Zhang

Zhejiang University · School of Mathematical Sciences

Cheeger–Colding theory — TA session 1

Wednesday, October 28 · teaching-assistant session

The session collects the metric and analytic ingredients behind the Cheeger–Colding theory. The first part proves Gromov's precompactness theorem: a family of compact metric spaces with a common diameter bound and a uniform bound on the number of r-balls needed to cover every member is precompact in the Gromov–Hausdorff metric, the proof running through uniform finite ε-nets, bounded distance matrices and total boundedness of the isometry classes. The second part proves the mean-value inequality for subharmonic functions on a complete manifold with Ric ≥ −(n−1)λg: a normalized Neumann–Sobolev inequality and a Caccioppoli inequality feed Moser iteration, which gives an L²-to-L∞ bound, and the L¹ endpoint is then obtained by absorption with Young's inequality and a geometric series of radii. These two estimates are the technical core of the compactness and regularity arguments used later in the course.

Session 1 notes (PDF) · arXiv:2610.10575 · arXiv:2610.32734

Research area: geometric analysis — Ricci flow, Cheeger–Colding theory and Gromov–Hausdorff convergence, complete Kähler metrics of nonnegative Ricci curvature.

Wancheng Zhang

Zhejiang University · School of Mathematical Sciences

Cheeger–Colding theory — TA session 2

Thursday, October 29 · teaching-assistant session

The second session proves the almost splitting theorem of Cheeger and Colding: a complete pointed manifold with Ric ≥ −(n−1)g whose unit ball has small excess is close, in the pointed Gromov–Hausdorff sense, to a metric product R^k × Y. The proof is assembled from the standard analytic toolkit — the Bochner formula, Laplacian comparison, elliptic estimates and the maximum principle — together with harmonic coordinates and the notion of a (k,ε)-splitting map. Busemann functions are replaced by their harmonic replacements, and the argument is split into four claims: comparison for the diagonal directions, the gradient estimate, the polarization identity and the Hessian bound. The final part treats the second theorem, where a small excess function defined through signed distances yields a one-dimensional splitting map and hence the splitting itself.

Session 2 notes (PDF)

Wancheng Zhang

Zhejiang University · School of Mathematical Sciences

Cheeger–Colding theory — TA session 3: volume convergence

Friday, October 30 · teaching-assistant session

The third session proves Colding's volume convergence theorem: if a sequence of n-dimensional manifolds with Ric ≥ −(n−1)Λ converges in the pointed Gromov–Hausdorff sense to a Riemannian manifold, then the volumes of unit balls converge. The upper bound is Bishop–Gromov; the lower bound is the hard part. An n-splitting map supplies n harmonic coordinates, a metric lemma turns the Gram and Hessian errors into a maximal-function estimate whose good set has almost full measure, and a degree-mod-2 argument combined with the area formula produces an actual preimage and hence a lower volume bound. The second part derives the local curvature estimate: assuming a curvature blow-up, one selects points by a weighted maximum, rescales to curvature 1 on an expanding ball, transfers the almost maximal volume and uses compactness of Ricci-flat metrics to reach a Euclidean limit that contradicts the normalised curvature — the first step towards the volume rigidity of Ricci-flat manifolds.

Session 3 notes (PDF)

Samad Wali

Quanzhou University of Information Engineering (Fujian, China) · General Education Centre

A robustly optimized level-set method for accurate cardiac image segmentation

Friday, October 30 · guest lecture

The lecture follows a variational route from image restoration to medical image segmentation. Classical models — total variation (ROF) and the distance-regularised level set (DRLSE) — are recast as constrained inverse problems, and the bias-field model I = b·C_k + η of an observed cardiac image motivates a level-set formulation with a bias-field estimate. The proposed variant is split with the alternating direction method of multipliers (ADMM), which yields closed-form updates for the Heaviside-consistency variable, the bias field and the level-set function itself, and converges far more robustly than plain gradient descent. The method is tested on synthetic images and on short-axis cine-MRI cardiac images — including myocardial infarction — against EDRLSE, RESLS and DRLSU-Net, and the talk closes with limitations (dataset generalisation, parameter sensitivity) and ongoing work on coupling the model with CNN-based feature extraction.

Slides (PDF, 25 slides)

Research area: variational and PDE-based image processing — level-set evolution, total variation, ADMM solvers, medical image segmentation and image denoising. His publication record runs through Nankai University, Tianjin Normal University and the University of Electronic Science and Technology of China; before moving to Quanzhou he was on the faculty of Namal University (Mianwali, Pakistan).

Danish Ali

Dalian University of Technology · School of Mathematical Sciences

Using grid diagrams in knot theory to define Gromov–Hausdorff distance: connecting knot theory, metric geometry, and the Gordian distance

Friday, October 30 · guest lecture

The talk builds a bridge between knot theory and metric geometry: a grid diagram of a knot is turned into a finite metric space, so that knots can be compared by the Gromov–Hausdorff distance. The construction keeps the combinatorial data of the diagram — the X/O markings, the horizontal and vertical segments, the endpoints and the crossing information — in a weighted graph, and the talk discusses which choices of points and weights prevent all knots from collapsing to the same space. The goal is a lower bound for the Gordian distance, and hence for the unknotting number, in terms of the Gromov–Hausdorff distance between the associated spaces.

Slides (PDF, 22 slides) · ORCID · Faculty page

Assistant Professor, School of Mathematical Sciences, Dalian University of Technology; PhD in Mathematical Sciences (2023, supervisor Lei Fengchun). Research area: low-dimensional topology, knot theory and knot invariants · Contact: danishali@dlut.edu.cn