ICM 2026 Special Joint Section Lecture - Tom Braden, Nicholas Proudfoot

Item: ICM 2026 Special Joint Section Lecture - Tom Braden, Nicholas Proudfoot
Rating: 5
Author: PostSingularity Institute

by Nicholas Proudfoot, Tom Braden · View the channel on PSI

Formal & Physical Sciences Mathematics PBMathematics

🎙 Nicholas Proudfoot, Tom Braden 👥 60K 📅 September 10, 2026 ⏱ 45 min 👁 7 📄 expert opinion 🧭 2026-09-10

Available in: English (current) Français

Keywords

intersection cohomologyKazhdan-Lusztig polynomialsmatroidstoric varietiesposets

Summary

The lecture by Nicholas Proudfoot and Tom Braden presents a unified combinatorial framework for constructing intersection cohomology groups without the need for geometric spaces. The talk is divided into three parts: first, an introduction to intersection cohomology and its properties; second, three families of examples: Kazhdan-Lusztig polynomials for Coxeter groups, g-polynomials for polytopes, and Kazhdan-Lusztig polynomials for matroids. For each, they show that positivity of the coefficients can be proven via geometric interpretations when the spaces exist, and then extended to non-geometric cases using algebraic constructions. The third part, presented by Braden, gives the sheaf-theoretic details: using equivariant intersection cohomology and sheaves on posets, they describe a recursive algorithm that computes these groups from the poset structure alone, highlighting the role of parity vanishing and the existence of a contracting torus. The talk emphasizes that these purely combinatorial constructions agree with the classical ones when the spaces exist, and open the way to deeper results such as the Dowling-Wilson conjecture.

160 words

Critical Evaluation

Value of the Information & Strength of the Argument

The value of this talk lies in its synthesis of three distinct areas in mathematics into a single framework, showing that intersection cohomology can be defined combinatorially. The argumentation is strong: they start from known geometric results, then abstract the essential properties needed (parity vanishing, existence of a one-dimensional torus action) and show how to replicate them algebraically. The reasoning is coherent and builds on established theorems, though the talk remains at the level of a survey and refers to original papers for complete details.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is high: the speakers are experts and cite many specific results (e.g., by Elias, Williamson, and others) without giving full references, but the audience is expected to be familiar with the literature. The title is an excellent match to the content, as it precisely describes the main theme. No conflicting sources were identified, and the presentation is internally consistent. Since no comments are provided, no public reception analysis is possible.

173 words

Title / Content Match

The title accurately reflects the content: the talk presents intersection cohomology and its algebraic generalizations without spaces.

Quality & Reliability

8/10

The talk is presented by established researchers in mathematics, covering well-known results. The presentation is rigorous, but as an overview, it does not contain full proofs. The claims are plausible and supported by references to prior work.

Key Moments

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

Contribution & Novelties

The main contribution is the proposal of a unified combinatorial framework for intersection cohomology, eliminating the need for underlying spaces. This allows the extension of classical results to cases where geometry is absent (e.g., non-rational polytopes, non-realizable matroids, arbitrary Coxeter groups). The talk also highlights the role of equivariant theory and sheaf theory on posets as a computational tool.

Pour aller plus loin :

Intersection homology — Wikipedia overview of intersection homology/cohomology.
Kazhdan–Lusztig polynomial — Definition and properties of these polynomials.
Matroid — Basic concept of matroids, relevant for the third example.
Toric variety — Used in the polytope example.

100 words

Radar Profile

The radar chart shows very high technical level (10) and good quantity/quality scores, with slightly lower but still solid global reliability. This suggests a dense, expert-level talk that is informative and trustworthy, though not exhaustive in its detail.

Reliability 8/10