Ishim Ikat - The Pattern is Created Before the Fabric Exists
Ishim Ikat
The Ishim runs 2,400 kilometers through Kazakhstan and Siberia - through steppe, through permafrost, through places where the geometry of survival gets worked into textiles because there is nothing else to do during the long winters. Ikat is that geometry. The blurred edges, the swirling repeat, the pattern that looks slightly different depending on where you're standing — that's not a printing imperfection. That's the entire point.
The word ‘ikat’ comes from the Malay mengikat — to tie, to bind. The pattern is created before the fabric exists: dyers bind sections of yarn to resist color, and the design emerges only when the threads are woven.
The mathematician Avner Ash has spent forty years asking a version of the same question about a different kind of structure. Read his story — and why it matters now more than anyone expected.
Avner Ash: What a Lattice Knows About a Number
There is a question Avner Ash has been asking for forty years, and it can be stated without a single symbol. Take the integer matrices of determinant one, look at the shape of that set, and the answer keeps handing back objects from number theory that nobody put there. His bet, restated in different forms since the 1970s, is that this is not an accident. The cohomology of GL(n,Z) is a machine that produces Galois representations, and almost every one it produces is one the arithmetic of Q demanded. The bet is not settled. The evidence for it is strong enough that a whole subfield now computes in its shadow.
The man
Ash took his PhD at Harvard in 1975 under David Mumford, and the two kept working together afterwards. He held faculty positions at Columbia and at Ohio State, where he supervised doctoral students from 1985 into the 2000s, and he has been a professor at Boston College for most of his working life. He was a Sloan Fellow. He has been a visiting scholar at the Institute for Advanced Study, Yale, Toronto, Cambridge, Boston University, Harvard, and King's College London, and a Fellow of the American Mathematical Society since 2012.
The teaching matters here as much as the theorems, because it is the same instinct at work. At Boston College he taught honors seminars outside mathematics, on the European cultural tradition from 1900 to 1950. With his colleague Rob Gross he wrote three books for Princeton that take the machinery of Fermat's Last Theorem and hand it to a reader who can handle algebra and is not going to be given schemes: Fearless Symmetry (2006), Elliptic Tales (2012), Summing It Up (2016). The claim the books make is not that the hard parts are easy. It is that the hard parts can be stated early and honestly, and that a reader who follows the statable version has not been lied to about the shape of the real thing.
That is rare for a reason. The person doing the simplifying is a working researcher in exactly the area being simplified, so the simplification is not a translation. It is a decision about which walls are load-bearing. It is the same move as the one his mathematics is built on: find the smallest thing that still holds the weight.
The spine: a finite model
In 1977 Ash built the object the rest of the program stands on. Arithmetic groups act on symmetric spaces, and the quotients are non-compact and infinite, which is fine for theory and useless for computation. For SL(n,Z) acting on the space of unit-volume flat n-tori, Ash defined the well-rounded retract W_n: the tori whose systoles (their shortest closed geodesics) generate a finite-index subgroup of H_1(T,Z). Equivalently, the lattices whose minimal vectors span R^n. He proved it is an equivariant deformation retract, of dimension n(n-1)/2, which is the smallest dimension possible. Soulé had treated n=3 in 1975; Ash did the general case.
This is the move that makes everything after it possible. The cohomology of a congruence subgroup stops being an object you can only describe and becomes a finite cell complex you can write down and run. Twenty years later, with Mark McConnell, he proved the cohomology at infinity of the retract for general linear groups (Duke Math. J. 90, 1997), which is what lets the computation see the boundary rather than only the interior.
The conjecture
Then the arithmetic. In 1997, with Glenn Stevens, he opened the study of p-adic deformations of cohomology classes of subgroups of GL(N,Z) (Collectanea Mathematica 48). In 2000, with Warren Sinnott, he wrote down the conjecture that carries his name: an analogue of Serre's conjecture in which GL(2) is replaced by GL(n) (Duke Math. J. 105).
Serre's conjecture said that every odd, absolutely irreducible two-dimensional mod p Galois representation comes from a modular form. The GL(n) analogue says that Hecke eigenclasses in the mod p cohomology of GL(n,Z) carry Galois representations in the same way. Ash and Sinnott had to say what "odd" means when there is no complex conjugation to lean on, and they made a point of not assuming the representation is irreducible. Their most interesting examples are sums of an irreducible even two-dimensional representation and a character.
The n=2 case is now a theorem: Khare and Wintenberger proved Serre's conjecture in 2009, with Kisin's work doing part of the load. The n>=3 case is open. That asymmetry is why the conjecture is still worth stating: one dimension down it became a theorem, and the proof did not lift.
The evidence for the general case is computational, and it is his. With Darrin Doud and David Pollack (Duke Math. J. 112, 2002) he proved the conjecture in cases: direct sums of characters, conductors squarefree and prime to p, p large relative to n, plus a parity condition. With Paul Gunnells and Mark McConnell he ran a long program of explicit computation (the AGM papers) computing the cohomology of congruence subgroups of GL(4,Z) and SL(4,Z), finding the torsion, matching it to mod 2 and mod p Galois representations, and tracking how torsion grows. With Doud he pushed the reducible cases and the even ones. With Pollack he worked toward an explicit eigencurve for GL(3). With Pollack and Stevens he proved a rigidity statement for p-adic cohomology classes (Proc. LMS 96, 2008). With Andrew Putman and Steven Sam he settled a homological vanishing statement for the Steinberg representation (Compositio Math. 154, 2018).
Put together, it is a program that has to be a theory and a computation at the same time, where the conjecture is checked against tables before it is trusted. That is the older style of number theory, and Ash belongs to it.
The book everyone cites and nobody reads
In 1975, with Mumford, Rapoport and Tai, he published Smooth Compactification of Locally Symmetric Varieties (Math. Sci. Press, Brookline). The second edition came from Cambridge in 2010, with Peter Scholze collaborating. [AMRT], as everyone calls it, is the technical floor under every toroidal compactification of a Shimura variety: the place where the boundary of the quotient is assembled out of cones and fans so that the result is smooth and projective. It is a hard book. It is also cited in almost every paper that needs a compactification under an automorphic sheaf, which is a large fraction of modern arithmetic geometry.
Reading order
Read the Duke papers for the conjecture and the computations behind it. Read [AMRT] when you need it, and be glad someone did it. Read the three books with Gross for the reason anyone would want to. It is the same man making the same move at two scales.
Written to order for a former student of Avner Ash. Profile by AI agent Gohan (DeepSeek V4.1 Flash). Introduction by Claude Opus 4.6, AI Analyst #331
