What the congress said was open, what fell since 2022, and which methods crossed fields — from primary sources only.
The International Congress of Mathematicians met in Philadelphia, 23–30 July 2026. This page is a working report built from the congress's own record: the IMU citation documents, the official speaker database, the published SIAM proceedings, and the speakers' lecture texts on arXiv. Press coverage was used only as a pointer, never as a source. Every quotation below was checked verbatim against the document it cites.
The 2026 medal-level story: kinetic theory rigorized (Yu Deng), symplectic geometry's foundations rebuilt and applied (John Pardon), o-minimality established as the transfer method into arithmetic and Hodge theory (Jacob Tsimerman), and the multiscale/decoupling toolkit breaking the hardest problems of geometric measure theory (Hong Wang) — with the algorithms medal (Shayan Oveis Gharan) awarded for importing the geometry of polynomials into the analysis of Markov chains.
The proceedings show a congress organized around travelling methods more than subfield walls: the same machinery appears in the lectures of two or three different sections. The AI presence is real but reflective — theory of machine learning in three sections plus one special plenary about mathematical practice, and no section lecture claiming machine-assisted results. A political dimension stayed at the level of institutional statements, reported below without comment. The proceedings themselves were published by SIAM in the same month the congress ended — so this report is built from the lectures as their authors wrote them.
Short citations quoted in full from the IMU's citation documents (linked per prize) — the canonical texts. A citation is the best compressed statement of what the field decided mattered this cycle; each clause names a specific theorem cluster.
“For his work in partial differential equations, including the rigorous derivation of the Boltzmann equation from hard-sphere dynamics for rarefied gases, the derivation of wave kinetic equations from nonlinear dispersive systems, and probabilistic approaches to nonlinear Schrödinger dynamics.”
The long citation names the collaborations: the Boltzmann derivation with Zaher Hani and Xiao Ma — valid for as long as the Boltzmann solution exists; wave kinetic equations with Hani; random averaging operators and random tensor theory with Andrea Nahmod and Haitian Yue. His invited lecture: Hilbert's Sixth Problem: Particles and Waves.
“For his achievements in symplectic geometry including new approaches to virtual fundamental cycles, Fukaya categories of certain manifolds and counting holomorphic curves, and for his contributions to other areas of geometry and topology, including group actions on 3-manifolds and knot theory.”
Specifics from the long citation: Gromov's question on distortion of torus knots; the Hilbert–Smith conjecture for n = 3; virtual fundamental cycles via algebraic topology; wrapped Fukaya categories with Ganatra–Shende; and the Gromov-Witten / Donaldson-Thomas correspondence of Maulik–Nekrasov–Okounkov–Pandharipande “when the anti-canonical bundle is nef.”
“For his contribution in the recasting of o-minimality as a fundamental method of arithmetic and complex algebraic geometry, and his role in the proof of many central conjectures including Griffiths' conjecture on the algebraicity of images of the period maps, and the André-Oort conjecture for Siegel modular varieties.”
The long citation traces the line: o-minimal GAGA with Bakker–Brunebarbe; Griffiths' conjecture via Bakker–Klingler–Tsimerman; André-Oort through Galois lower bounds (average Colmez) plus Ax-Lindemann with Pila; Ax-Schanuel for Shimura varieties with Mok and Pila; a functional André-Grothendieck conjecture with Bakker. His plenary text: Baily–Borel Compactifications for Period Images.
“For her work in harmonic analysis and geometric measure theory, including applications of multiscale and decoupling techniques to the local smoothing conjecture for the planar wave equation, and major advances in Fourier restriction, Falconer distance sets, Furstenberg sets in the plane, and the Kakeya problem in three dimensions.”
Named collaborations: local smoothing with Larry Guth and Ruixiang Zhang; Falconer with Guth, Iosevich, Ou; Furstenberg sets with Kevin Ren; Kakeya in ℝ³ with Joshua Zahl — “new bounds and structural techniques for collections of thin tubes,” which the citation says are “reshaping current approaches.” Her invited lecture was on a different front: Restricted Families of Orthogonal Projections in ℝⁿ; Zahl's companion survey covers the Kakeya side.
“…for his landmark contributions to the theory of algorithms, taking the analysis of algorithms beyond combinatorics and probabilistic analysis into the theory of the geometry of polynomials… Oveis Gharan's work improved the approximation of the metric travelling salesperson problem, breaking the limit on the approximation factor in Christofides' 3/2-algorithm from 50 years ago, and led to a revolution in the understanding of Markov Chain Monte Carlo sampling algorithms… through spectral independence.”
“…for his visionary mathematical insights, that have had an enduring influence in a wide range of fields, including topology, mathematical physics, representation theory and category theory.”
The long citation: classifying spaces, the McDuff–Segal scanning method, the Atiyah–Segal completion theorem and the Segal conjecture, delooping machines (“Segal space,” “Segal category”), the axiomatic formulation of 2d conformal field theory — the seed of Atiyah's TQFT axioms — and loop groups with Pressley and Wilson.
“…for his groundbreaking work on mathematical optimization that provided the theoretical foundation and algorithmic backbone for many numerical and data-driven fields, enabling practical computations that were previously too time-consuming to be feasible.”
Self-concordant barriers and polynomial-time interior-point methods (with Nemirovski); the 1983 accelerated gradient method with its optimal first-order rate — “Nesterov momentum” — which the citation explicitly ties to the training of modern AI models.
“Hannah Fry is one of the foremost global ambassadors for mathematical thinking… Through a wide range of media — including books, videos, and television programs — she has used outstanding creativity and originality to translate mathematics into a language of wonder and relevance for the public without diminishing its scope and importance.”
Named lectures: Abel Lecture — László Lovász; Emmy Noether Lecture — Karen Vogtmann. A curiosity for the record: the congress website's own display texts of two Fields citations differ from the IMU PDFs (Pardon's “certain manifolds” became “Liouville manifolds”; Tsimerman's André-Oort clause was dropped). The PDFs above are the canonical versions.
In the speakers' own words, from their lecture texts (each linked). The marker [math] stands for a formula lost in text extraction — consult the linked paper for the exact statement. This lists what the congress itself chose to emphasize, not a survey of 2022–26 mathematics.
“In 2024, Delcourt and Postle [18] developed the methodology of refined absorption, giving a one-step, combinatorial proof of the Existence Conjecture. Later in 2024, Keevash [47] provided a more concise proof…” — and the high-girth version: “in 2024 Delcourt and Postle [17] settled the High Girth Existence Conjecture in full.” Luke Postle, arXiv:2510.19978
Nevanlinna theory has no lecture and no proceedings chapter at this congress — checked against the full tables of contents. And the MLC conjecture (local connectivity of the Mandelbrot set) appears not as reported progress but reframed as a structured renormalization program with named conjectures — see the roster below.
Entry criterion: a primary source states the problem — no famous-problem padding from general knowledge. The value is what this congress said was open, and how it framed it in 2026. “Why hard” appears only where the source states an obstruction; quotations are verbatim from the linked texts; [math] marks a formula lost in extraction.
In higher dimensions the situation becomes even worse, as the number and complexity of infinitely ruled surfaces increases.
An interesting open problem is whether the arguments in [52, 50, 51] can be performed in a more efficient manner… to prevent the losses described above— and
to what extent the ideas from [51] can be used to understand the Fourier restriction problem in [math].
Identifying the limit for general polynomial ergodic averages is a well-known open problem in ergodic theory, even in the linear case.The multilinear state of knowledge is
dramatically worse.
The presence or not of singular continuous part… is a very interesting open problem.
it remains an open question whether or not such an expansion holds in many interesting cases.
ruling out local minimizers as being global ones using a case-by-case analysis of the cell-incidence graphs… becomes intractableas the number of bubbles grows.
wide open.
Is the Fourier critical index [math] greater than [math] for every [math]? This would solve Connes' rigidity… at least for infinitely many pairs.
a very subtle problem which puts together highly singular operators from Euclidean harmonic analysis with Connes' rigidity.
When [math] is not big, even the following weaker statement is open: [math] contains a closed positive current… with vanishing Lelong numbers;
Conjecture 3.7 is open even on [math] surfaces;
Can the Gromov-Hausdorff limit be identified explicitly?
For general compact Kähler manifolds though, the transcendental Morse inequality is still a conjecture; the cone duality remains conjectural.
It remains an open question whether there exist compact nonprojective Oka manifolds which fail to be elliptic;
The problem whether every open Riemann surface embeds as a closed nonsingular complex curve in [math] is still wide open;
we do not know which groups arise as fundamental groups of Oka manifolds.
The next open cases concern stable rationality of cubic fourfolds and rationality of cubic fivefolds.
Does [math] consist of algebraic classes?— following his theorem for abelian fourfolds of Weil type and sixfolds of split Weil type with CM.
Establish pseudo-Siegel a priori bounds in the remaining unbounded satellite ql cases— the most explicit roadmap in the corpus.
What is the exact value of [math]? It is also interesting to know its lower/upper bounds.
Classify correspondence realizing matings of rational maps and Kleinian groups;
Construct matings of arbitrary Fuchsian groups with polynomials; mate polynomials with Bers boundary groups,
possibly with Peano curve limit sets.
Tangent-flows are independent of the choice of sequence of rescaling factors(Conjecture 5.2); the multiplicity-one conjecture; and: give a mean curvature flow proof of the Smale conjecture.
De Giorgi's splitting result does not hold without graphicality; in higher dimensions
very little is known.
Even in the simplest case… the conjecture remains open; infinite-genus surfaces are
an almost unexplored research topic.
remains elusive;
all uniqueness statements remain conjectural.
It is a wide open question whether the strong closing lemmas hold for high-dimensional Hamiltonian dynamics.
Is the group of area-preserving diffeomorphisms of [math] perfect?— a longstanding question; and
Is there anything like the above story in higher dimensions?
even though Hamiltonian isotopic exact Lagrangian submanifolds are Fukaya isomorphic, it is not clearthe converse holds — and whether Fukaya objects
even behave in the geometric manner suggested by the algebra… is harder than one might think, already for surfaces.
An interesting open question is whether or not the contact push-off of a Legendrian submanifold determines the Legendrian submanifold.
Does there exist an exotic smooth structure on [math] for some [math]?;
Can skein lasagna modules detect exotic smooth structures on some closed four-manifolds?; the slice-ribbon conjecture, stated as open.
remains open, with the main obstacle being the absence of a local mixing theorem in this broader setting.
Is there a dense subset of [math] formed by dynamics with finitely many attractors, whose basin union covers [math] a.e.?— asked after his own wandering-domain constructions show what fails.
Are there integrable outer billiards in the plane other than ellipses?; the Minkowski-billiard total-integrability question, communicated by Ostrover.
held back by the same issues preventing a rigorous RG analysis of logarithmic corrections in six-dimensional percolation.
perhaps the most difficult problem will be to prove the effectively short-range part of Sak's prediction.
The grand challenge is to understand the probabilistic approach of the (non compact) imaginary Liouville theory.
The Hausdorff dimension [math] of a [math]-LQG surface is not known for general [math]; scaling limits
without using the mating-of-trees frameworkwould be of significant interest.
The uniqueness of the martingale problem remains open, which probably requires a more explicit characterisation of the quadratic variation;
it is an open question how different is the SHF from a GMC.
this “un-inverted” formula deserves further study and could potentially help us to better understand spin glasses, including those models for which the Parisi formula is already proved rigorously.
central problems in what may be called extremal design theory: minimum-degree thresholds for decompositions, and probabilistic thresholds for Steiner systems.
At present time, [Problem] 2.6 is open for all [math], and we believe it is a very challenging and interesting problem; plus: an ergodic-theoretic approach to higher correlations, an analytic proof of the CSP dichotomy theorem, quantitative density Hales-Jewett.
The study of the spectrum of such random matrices is largely open; the stated objectives are hardness of local search and of low-degree polynomial algorithms below the conjectured threshold.
the formula so obtained involves some geometric data that are currently not well understood, namely Euler characteristics of stalks of Iwahori-equivariant intersection cohomology complexes on the affine flag variety.
The original conjecture remains open, but there has been renewed interest and activity around it recently; the matroid inequality
has been conjectured, but any proof will require new ideas;
It would be very interesting to have a unified approach, but at present we do not.
is still open in general; the q-character of a general simple module
is not known.
No such positive proportion non-vanishing result is currently known by any method; the conditional results need
the Generalised Riemann Hypothesis and the Ratios Conjecture.
the optimal spectral gap result is not known for the Brooks-Makeover model; the conjecture: the optimal gap should always hold.
Is there a simple sharply 2-transitive group of positive characteristic in which every element is a product of a bounded number of translations?— among her
urgent questions that remain.
Is there a separated Deligne–Mumford stack that is not a global quotient stack?— with their local-structure conjecture for stacks with reductive stabilizers, and Θ-stratification questions for K-stability moduli.
provide evidence but not a proof that the GWW polygons are not Steklov-isospectral; computability of continuous/essential spectra is
even more challenging.
Applying any of these methods to the thirteen-sphere problem is still far beyond reach for current computers.
Honest coverage notes: the PDE sections yielded few verifiable open-problem quotations in this pass and deserve a targeted second reading; the education, history and modelling sections state agendas rather than problems; fourteen plenary texts exist only in the SIAM volumes and are not yet incorporated. The roster grows on the second pass — it does not shrink.
Twenty sections, 21 plenary lecturers, roughly 205 scientific lecture slots. The full speaker list is on the official congress site; the proceedings tables of contents are registered with SIAM. What is worth extracting here is the cluster structure — the same methods surfacing in different sections' lectures:
Slot arithmetic, for a sense of weight: the analysis-adjacent sections (8, 10, 11) total 41 scientific slots — the congress's largest thematic block; the algebraic block (2, 3, 4, 7) totals 50. Affiliations worth updating one's mental map by (all from the official database): Joshua Zahl is now at the Chern Institute of Mathematics, Nankai; Dennis Gaitsgory at MPIM Bonn; Yu Deng at Chicago; Tim Austin at Warwick; Moritz Hardt at MPI Tübingen.
Written as moves — when stuck on X, try Y, because Z — with the entry rule that two different fields must actually be using the technique, per a primary source.
When stuck bounding the mixing of a Markov chain on combinatorial objects, encode the distribution in a generating polynomial and prove log-concavity / spectral independence: local spectral control implies global mixing. The Abacus citation records the import chain — the log-concave polynomials papers “initially leveraged a breakthrough in Combinatorial Hodge Theory by Adiprasito, Huh, and Katz.”
When stuck proving a transcendental object algebraic, place it in an o-minimal structure and apply Pila–Wilkie or definable GAGA: tame topology turns transcendence into point-counting with controlled geometry. The Tsimerman citation calls this “the recasting of o-minimality as a fundamental method.”
When stuck on optimal spectral gaps for random covers or hyperbolic surfaces, prove strong convergence of random permutation representations: operator-norm convergence transfers directly to Laplacian spectra (van Handel, Monk–Naud).
When stuck on an oscillatory-integral or dimension bound, decompose into wave packets and induct on scales: orthogonality is only accessible scale-by-scale. The Wang citation is the compressed record — local smoothing, Falconer, Furstenberg, Kakeya.
When stuck on a degenerating family of Kähler metrics, formulate the limit on the Berkovich / test-configuration side, where Monge–Ampère theory extends, then pull back. Witt Nyström: a Calabi–Yau theorem “formulated in the language of non-Archimedean Kähler geometry” (arXiv:2510.03044).
When stuck on a closing lemma or structure theorem for area-preserving maps, use ECH/Floer spectral invariants and their Weyl laws — “quantities defined by Floer theory tend to have good continuity” (Cristofaro-Gardiner); the proofs of the strong closing lemmas “use the notion of spectral invariants from symplectic geometry” (Irie). The Simplicity Conjecture fell to the same transfer.
When stuck on almost-everywhere convergence along arithmetic sequences, run the number-theoretic circle method through modern multi-frequency harmonic analysis (Ionescu–Wainger). Bourgain's toolkit ranged “from harmonic analysis and number theory to probability and the theory of Banach spaces” (Mirek).
When stuck on a positive-scalar-curvature obstruction, probe with stable minimal hypersurfaces and μ-bubbles: stability converts curvature positivity into descent on dimension — the method that resolved Gromov's aspherical conjecture in dimensions 4 and 5 (Chodosh).
When stuck relating holomorphic dynamics to Kleinian groups, mate them through correspondences and Schwarz reflections, transferring bounds by surgery — “the Douady-Ghys qc surgery… transfers real a priori bounds from Blaschke maps to complex Siegel polynomials” (Dudko; Lomonaco–Mukherjee).
When stuck on an extremal packing configuration, use SDP hierarchies reduced by representation theory, then round to exact certificates: “conic optimization…, symmetry reduction via harmonic analysis, and techniques for rounding numerical approximations to exact and easily verifiable solutions” (Vallentin).
When stuck on positivity or unimodality for combinatorial data, build the intersection-cohomology package purely combinatorially: hard Lefschetz and Hodge–Riemann survive without any variety (Braden–Proudfoot; “Huh and Wang showed that the same argument applies to arrangement Schubert varieties”).
Recorded at the level of the programme rather than verified texts: Peyré, Optimal and Diffusion Transports in Machine Learning; Rigollet, The Mean-Field Dynamics of Transformers; Zuazua, Machine Learning and Control — three sections institutionalizing the same direction.
Separating three things that press coverage tends to blur: claims made in lectures by mathematicians; claims made by companies or press around the event; and inference. Only the first is evidence of where the field stands. The load-bearing text is Alex Kontorovich's special plenary, The Shape of Math To Come (arXiv:2510.15924), whose author dates his snapshot September 2025. His structural claims, in his own words:
“For mathematics, a non-zero error rate creates an unusable tool. If we cannot distinguish the correct results from the flawed ones without verification, then that defeats the whole purpose of automation.” — on the 100-AI-papers-a-day thought experiment; scaling accuracy does not fix it
“Compare this to a different scenario: an AI that produces only 10 papers per year, but each comes with a formally verified proof… That is a tool I would use enthusiastically.”
“As things currently stand, I cannot even state formally most theorems of interest to me, much less prove them. The existing formal libraries would need to grow by orders of magnitude…” — unless “formalization can be ‘supercharged’ by automation.” — the library gap, and the motivation for his proposed “quasi-autoformalization” architecture
He entertains the opposite outcome explicitly — “perhaps natural language AI will, after all, be sufficient for all of mathematics?” — as an open question, and discloses his position on the Lean FRO advisory board.
Around this one reflective plenary, the congress treated machine learning as ordinary mathematical subject matter: Bartlett's plenary on implicit bias and benign overfitting, Rigollet on transformer dynamics, Peyré on transport, Zuazua on control, Hardt on prediction — and the Gauss citation tying Nesterov's acceleration to modern AI training. There was a roundtable on mathematics education in the AI era, and public lectures by Terence Tao and Geordie Williamson (no public texts yet).
On the evidence of the 192 published lecture texts: in July 2026, machine assistance had not yet entered the congress's mathematical results — it entered its reflection about itself.
There was a political dimension — concerns about holding the congress in the United States — which this report records factually and briefly, with every statement attributed and no position taken.
Still to come from the congress itself: Proceedings Volume 1 (the prize laudationes), the remaining plenary texts, and the promised session recordings. This page will be revised when they appear.
Primary sources only, in tiers: the IMU award pages and citation PDFs (canonical for all prize texts); the official 305-entry speaker database; the SIAM Proceedings of the ICM 2026, Volumes 2, 3, 4, 5, 6, 7 (192 chapters, tables of contents via Crossref); and the 56 lecture texts their authors posted on arXiv. Journalism was used only to decide where to look.
Method: the lecture texts were read with AI assistance under a hard rule — every quotation used here was mechanically re-verified, character for character, against the source text; anything paraphrased was discarded. Where extraction lost a formula, the marker [math] is shown rather than a reconstruction, and the linked paper is the authority. Where the primary source does not exist yet (laudationes, recordings, fourteen plenary texts), the gap is stated instead of filled.
Compiled July 2026 by Yan He. Corrections welcome — contact.