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Materials

Readable downloads and prepared files

The files below are copied into a dedicated downloads folder with human-readable names, so links are easy to read and easy to share.

Downloads

English program files

Download all — ZIP archive

Program PDF, program DOCX, the reading lists, the Ivanov and Tuzhilin lecture files, the invited-talk slides and the TA session notes in one archive.

Program table — English PDF

Formatted English schedule in PDF.

Program table — English DOCX

Editable English version of the program.

Reading list — Tuzhilin minicourse (PDF)

All 45 works cited across the lecture series, formatted for download.

Reading list — Zhang Huichun minicourse (PDF)

All 21 works cited across the lecture series, formatted for download.

Reading list — web version

The same references on the course page, with clickable DOI/arXiv links.

Lecture files

Ivanov minicourse — Lectures 1–4

Lectures 1 and 2 — lecture text (PDF)

Hausdorff and Gromov–Hausdorff distances: the text of the first two lectures.

Lectures 1 and 2 — seminar problems (PDF)

Seminar problems for the first two lectures.

Lectures 1 and 2 — problem sheets (PDF)

Two-page problem sheet: lecture 1 on one side, lecture 2 on the other.

Lectures 2–4 — lecture text (PDF)

The text of the remaining lectures: properties of the GH-distance, compact spaces, local and global isometries.

Lectures 3–4 — seminar problems (PDF)

Seminar problems for lectures 3 and 4.

Lectures 3–4 — problem sheets (PDF)

One-page problem sheet for lectures 3–4: continuity and covering data, then local isometries.

Lecture files

Tuzhilin minicourse — Lectures 1–4

Lectures 1 and 2 — lecture text (PDF)

Lecture text of the Tuzhilin minicourse for lectures 1 and 2, with the chapter summaries.

Lectures 3 and 4 — lecture text (PDF)

Lecture text of the Tuzhilin minicourse for lectures 3 and 4: topological methods and the GH-distance to a normed space.

Lectures 1–2 — seminar problems (PDF)

Seminar problems on the GH-distance and partitions, Borsuk partitions and graph metrics, and ultrametrization.

Lectures 3–4 — seminar problems (PDF)

Seminar problems on the intersection property and on Jung's constant and its variations.

Lectures 1 and 2 — solutions of seminar problems (PDF)

Solutions of the seminar problems for lectures 1 and 2.

Lectures 3 and 4 — solutions of seminar problems (PDF)

Solutions of the seminar problems for lectures 3 and 4.

References

List of references for the lectures of the Tuzhilin minicourse.

Minicourses

Lecture slides of the minicourses

Jiang — Cheeger–Colding theory, lecture 1 (PDF)

Wenshuai Jiang (Zhejiang University). Volume, segments and elliptic estimates (31 slides).

Jiang — Cheeger–Colding theory, lectures 2–3 (PDF)

Wenshuai Jiang (Zhejiang University). Splitting maps and cones (29 slides).

Jiang — Cheeger–Colding theory, lectures 4–5 (PDF)

Wenshuai Jiang (Zhejiang University). Ricci limits and Einstein geometry (46 slides).

Zhang Huichun — L1: Alexandrov geometry (PDF)

Lecture 1 of the minicourse on Alexandrov geometry, RCD spaces and metric-valued analysis (28 slides).

Zhang Huichun — L2: RCD(K,N) spaces (PDF)

A brief introduction to RCD(K,N) spaces (20 slides, October 23, 2026).

Zhang Huichun — L3: Bochner inequality (PDF)

Bochner inequality on Alexandrov spaces and its applications (20 slides, October 28, 2026).

Zhang Huichun — L4: Sobolev maps (PDF)

Sobolev maps between metric spaces (22 slides, October 30, 2026).

Zhang Huichun — L5: harmonic map heat flows (PDF)

Gradient flows on CAT(0) spaces and harmonic map heat flows (25 slides, October 30, 2026).

Tuzhilin — Lecture 1: MST-spectrum and Borsuk problem (PDF)

Applications of the GH-distance: MST-spectrum, the Borsuk problem and combinatorial invariants (29 slides).

Tuzhilin — Lecture 2: Ultrametrization (PDF)

Ultrametrization and its applications in GH geometry (31 slides).

Tuzhilin — Lecture 3: Topological methods (PDF)

Topological methods in GH-distance estimates (31 slides).

Tuzhilin — Lecture 4: Jung constant (PDF)

Estimation of the GH-distance to a normed space through the Jung constant (27 slides).

Guest lectures

Level-set methods, medical imaging and knot theory

Samad Wali — guest lecture slides (PDF)

Dr. Samad Wali (Quanzhou University of Information Engineering). A robustly optimized level-set method for accurate cardiac image segmentation: level-set models, ADMM splitting, and cardiac MRI results (25 slides, October 30, 2026).

Danish Ali — guest lecture slides (PDF)

Danish Ali (Dalian University of Technology), “Using grid diagrams in knot theory to define Gromov–Hausdorff distance” — the complete slide deck (22 slides): grid diagrams, the metric space of a diagram, and a proposed GH lower bound for the Gordian distance.

Invited talks

Presentation files

Kushner — talk slides (PDF)

Alexei G. Kushner, “Geometric structures and second-order differential equations: results and open problems” — the complete slide deck, with its outline and comments.

Kushner — lecture text (PDF)

Alexei G. Kushner, “Geometric Structures and Second-Order Differential Equations: Results and Open Problems” — a detailed 19-page write-up of the talk: jets and the Cartan distribution, differential equations and multivalued solutions, the Khokhlov–Zabolotskaya equation and exact solutions, and open problems.

Galstyan — talk slides (PDF)

Arsen Kh. Galstyan, “On the existence of minimal parametric networks” — the complete slide deck, with its outline and comments.

Bogatyi — talk slides (PDF)

Semeon A. Bogatyi, “Analogues of the Gromov–Hausdorff distance” — the complete slide deck (61 slides).

Wang — published paper (PDF)

Shengqi Hu, Xiaole Su and Yusheng Wang, “A proof of Toponogov's theorem in Alexandrov geometry”, Proc. Amer. Math. Soc. 151 (2023), 1743–1748.

Li Yan — talk slides (PDF)

Li Yan (Beijing Institute of Technology), joint with Lang Lang. “Gromov–Hausdorff convergence with group actions” — the complete slide deck (15 slides, 27 October 2026).

Li Yan — exercises (PDF)

Exercises for the talk: Hausdorff convergence of graphs, group actions on limit spaces and related problems (2 pages, 27 October 2026).

Detailed contents

Talk outlines

Kushner — outline

  1. Jets
  2. Coordinate notation
  3. The jet bundle
  4. Cartan's distribution
  5. Differential equations: invariant definition
  6. Examples of multivalued solutions
  7. The Khokhlov–Zabolotskaya equation
  8. Open problems

Galstyan — outline

  1. Introduction
  2. Minimal networks
  3. Hyperspaces
  4. Compactly equipped spaces
  5. Dual spaces
  6. Networks in incomplete spaces
  7. Generalisation of Bednov's result
  8. Vietoris topology
  9. Kuratowski limit
  10. Lower semicontinuity of the Hausdorff distance
  11. Examples of compactly equipped spaces

Bogatyi — outline

  1. Introduction: two approaches to the Gromov–Hausdorff distance
  2. Distance functions on 2^Z and the Hausdorff distance
  3. The Gromov–Hausdorff distance and the directed distance
  4. The continuous Borsuk distance on 2^Z
  5. The Borsuk problem and Cook continua
  6. The Gromov–Borsuk distance and its homological, homotopical and shape properties
  7. The homotopical metrics
  8. The fundamental metric
  9. The shape metric
  10. The metric defined by S-morphisms; distances on subclasses
  11. Whitney functions (maps)
  12. Applications: the Jalavd distance and Charatonik’s problem
  13. Modifications of the definition and the functional–internal approach
  14. References

Hu–Su–Wang — paper outline

  1. Introduction: Alexandrov spaces of curvature ≥ k and Toponogov's theorem
  2. Strategy: splitting bad triangles and transporting “curvature ≥ k” information
  3. Basic lemmas on comparison angles and Alexandrov's Lemma
  4. Bad angles: the splitting lemma and its corollary
  5. Proof of the theorem for locally compact spaces
  6. Proof of the theorem for non-locally compact spaces
  7. Acknowledgement and references

Seminar sessions

Teaching-assistant session notes

Cheeger–Colding theory — TA session 1 (PDF)

Wancheng Zhang (Zhejiang University). Gromov's precompactness theorem and the elliptic mean-value estimate, with complete proofs.

Cheeger–Colding theory — TA session 2 (PDF)

Wancheng Zhang (Zhejiang University). The almost splitting theorem: harmonic coordinates, splitting maps, Bochner's inequality and the small-excess theorem (26 slides).

Cheeger–Colding theory — TA session 3 (PDF)

Wancheng Zhang (Zhejiang University). Volume convergence theorem: splitting coordinates, the metric lemma, the degree-mod-2 argument, and the local curvature estimate (26 slides, 30 October 2026).

About the session and the speaker

Short description on the home page, together with research interests and recent preprints.

Detailed contents

Session outlines

Wancheng Zhang — session 1 outline

  1. Gromov's precompactness theorem: statement
  2. Proof, step 1: uniform finite ε-nets
  3. Proof, step 2: bounded distance matrices and total boundedness
  4. Mean-value inequality for subharmonic functions: statement
  5. Step 1: the normalized Neumann–Sobolev input
  6. Step 2: a Caccioppoli inequality
  7. Step 3: Moser iteration from L² to L∞
  8. Step 4: the L¹ endpoint by absorption

Wancheng Zhang — session 2 outline

  1. Almost splitting theorem: statement
  2. Tools: Bochner formula, Laplacian comparison, elliptic estimates
  3. Elliptic estimates: L¹ controls L∞; the maximum principle
  4. Local regularity: harmonic coordinates
  5. Splitting maps: definition and basic properties
  6. Busemann functions and their harmonic replacements
  7. Diagonal poles and Claim 1
  8. Claims 2–4: upper and lower comparison, gradient estimate, Hessian bound
  9. Finishing the proof: cutoff functions and the Bochner formula
  10. Small excess implies almost splitting: from signed distances to a one-dimensional splitting map