ICM 2026 Invited Section Lecture - Richard Montgomery

Value of the Information & Strength of the Argument

The talk provides a comprehensive overview of recent advances, with clear explanations of key ideas and the role of sublinear expansion. The argumentation is rigorous, relying on theorem statements and proof sketches. The presenter highlights the practical utility of sublinear expansion in resolving long-standing problems, such as the existence of cycles with power-of-two lengths and the asymptotic minimization of reciprocal cycle sum. The value is high for researchers in graph theory, as it serves both as a literature review and as a presentation of fresh results, though the details of proofs are largely omitted. The logical flow is clear: motivation, definition, applications, and outlook.

Scientific Rigor, Source Quality, Title Accuracy

The talk is based on published results and collaborations, citing the foundational work of Komlós and Szemerédi, as well as later contributions by the speaker and his colleagues. However, no external links are provided in the description, so the audience cannot directly access the referenced papers. The title is generic but faithful to the event; the description adequately frames the content, and the lecture matches the expected scope. Adequation between the title and the content is satisfactory, though a more descriptive title would have better highlighted the graph theory focus.

The title is generic but accurate; it identifies the event (ICM 2026) and the speaker, while the description clarifies the focus on recent graph theory using expansion. The lecture content matches the described scope.

8/10

The talk is delivered by a recognised expert in extremal graph theory, presenting results that are largely published or in submission. The technical depth is high and the reasoning is rigorous, though the transcript is generated by automatic speech recognition and contains minor mispronunciations (e.g., 'Erdish' for Erdős). The presentation covers multiple results without external verification by the audience, but the content aligns with known research directions.

Markers derived by PSI from the transcript: the creator did not define chapters.

• 00:00Introduction to the lecture and overview of expansion in graph theory.
• 03:00Definition of expander graphs and their historical context, including early work by Bassalygo and Pinsker.
• 05:20Introduction of sublinear expansion and its definition, emphasizing its weakness and utility.
• 08:00Discussion of the connectivity properties of sublinear expanders and their proof via iterative expansion.
• 12:00First main application: proof that every graph with large average degree contains a cycle with length a power of two.
• 18:00Discussion on minimizing the sum of reciprocals of cycle lengths, including results with Hong and with Proski and Sudakov.
• 22:00Second main application: decomposing graphs into equal-disjoint cycles, introducing the Erdős–Gallai conjecture.
• 28:00Progress on the decomposition problem: from O(n log n) cycles to O(n log* n) cycles, highlighting the role of sublinear expansion.
• 35:00Summary of the lecture and concluding remarks on the impact of sublinear expansion.

• Bassalygo and Pinsker (1970s) - Expanders — Mentioned as the origin of expander graphs in communication networks.
• Komlós and Szemerédi (1994) - Sublinear expansion — Introduced sublinear expansion and applied it to prove a result on subdivisions.
• Montgomery and Hong - Power-of-two cycles — Proved the existence of cycles of length a power of two in graphs with large average degree.
• Montgomery, Proski and Sudakov - Reciprocal sum of cycle lengths — Showed asymptotic optimality of complete bipartite graphs for the sum of reciprocals of cycle lengths.
• Erdős and Gallai (1966) - Conjecture on graph decomposition — Proposed that every graph can be decomposed into O(n) cycles and edges.
• Conlon, Fox and Sudakov - Decomposition result — Improved the bound to O(n log log n) cycles.
• Montgomery and Mati - Decomposition result — Further improved the bound to O(n log* n) cycles using sublinear expansion.