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Benjamin Collas

Publications and source records attributed to Benjamin Collas.

9 recordsLinked to original sources
arXiv2026

Fundamental Mathematics in the Age of AI -- The Residue, the Journey, and the Ecology

Large language models have begun refuting long-standing conjectures and solving long-open problems. The introspection this has prompted about the future of mathematical discovery is well under way, and the anxiety accompanying it legitimate -- but both, we claim, are attached to the wrong loss. What machines now produce is the countable part of mathematics -- theorems, proofs, refutations -- which was always the work's residue, not its product. The distinction is old, and not economic: a result can be taken in its finished essence, or in the operations that engendered it. The product is human understanding: not a stock of results but a collective, hard-won way of deciphering the world and acting upon it. The two are arcs of a single loop: understanding tells us where to look; looking produces the residue; and taking it up again, one journey at a time, rebuilds shared understanding. Machines are strong on the countable arc, absent from the one that feeds it. The peril is to leave the loop open. AI did not create the confusion between residue and product; it has called a bluff long on the books, driving the cost of the residue towards zero and making the scarce thing visible at last. A new instrument makes a new way of working before it makes a new result. The pressing questions are therefore institutional: who can check an announced result, whoever announces it; what work and training become for the next generation of researchers; and whether the one thing that cannot be mass-produced -- the journey that nourishes a shared understanding -- continues to be funded. Mathematics, we argue, is uniquely placed among the sciences on the first -- a proof answers to no one's permission -- and uniquely exposed on the other two: teaching cannot go on as before, and no collective position yet exists; and the journey has never had a price our institutions knew how to pay. The decision is ours.

math.HO↗
arXiv2026

Anabelian geometry for Deligne-Mumford curves

We develop an anabelian framework for general Deligne-Mumford curves, showing that their stack and orbifold structures are encoded in the group-theoretic properties of their \'etale fundamental groups. After establishing the required properties for profinite F-groups, we prove that fundamental geometric features, including hyperbolicity, affineness, and inertia data, can already be detected from low-level solvable quotients of the associated profinite groups, namely at the optimal 3-step level. As a consequence, we obtain some anabelian reconstruction results for Deligne-Mumford curves, their rigidifications, and their coarsification. While the m-step Grothendieck conjecture doesn't hold for Deligne-Mumford curves, we establish a 5-step anabelian theorem for the rigidification of affine Deligne-Mumford curves, namely affine stacky curves. A certain emphasis is given to the role of stack inertia groups.

math.AG↗
arXiv2026

Anabelian perspectives in Galois-Teichm\"uller theory

By exploiting the arithmetic homotopy of the moduli spaces of curves, Galois-Teichm\"uller theory stands at the interface of braid-mapping class groups and of anabelian geometry. Starting from the classical braid-theoretic construction of the Grothendieck-Teichm\"uller group, we review how anabelian geometry -- beginning with the foundational work of Nakamura -- provides the arithmetic mechanisms underlying its definition. We then explain how the combinatorial anabelian geometry developed by Hoshi and Mochizuki recasts these constructions within a purely group-theoretic and algorithmic framework. In particular, we describe how the group GT emerges as an anabelian object and how, once freed from auxiliary or artificially imposed containers, the anabelian algorithms yield a combinatorial reconstruction of the absolute Galois group of rational numbers. The perspective developed here highlights a conceptual shift from explicit braid-theoretic computations to functorial and algorithmic forms of anabelian reconstruction.

math.AG↗
arXiv2025

Symmetries of spaces and numbers -- anabelian geometry

``Can number and geometric spaces be reconstructed from their symmetries?'' This question, which is at the heart of anabelian geometry, a theory built on the collaborative efforts of an international community in many variants and with the Japanese arithmetic school as a core, illustrates, in the case of a positive answer, the universality of the homotopic method in arithmetic geometry. Starting with elementary examples, we first introduce the motivations and guiding principles of the theory, then present its most structuring results and its contemporary trends. As a result, the reader is presented with a rich and diverse landscape of mathematics, which thrives on theoretical and explicit methods, and runs from number theory to topology.

math.NT↗
arXiv2023

On Oda's problem and special loci

Oda's problem, which deals with the fixed field of the universal monodromy representation of moduli spaces of curves and its independence with respect to the topological data, is a central question of anabelian arithmetic geometry. This paper emphasizes the stack nature of this problem by establishing the independence of monodromy fields with respect to finer special loci data of curves with symmetries, which we show provides a new proof of Oda's prediction.

math.AG↗
arXiv2018

Hurwitz Stacks of Groups Extensions and Irreducibility

We study the irreducible components of special loci of curves whose group of symmetries is given as certain group extension. We introduce some relative Hurwitz data, which we show by using mixed \'etale cohomology theory, identifies some irreducible components for rational and normal non-abelian special loci and Hurwitz spaces. A heuristic, that is supported by three classes of examples, provides an additional context for building further irreducible loci.

math.AG↗
arXiv2018

Monodromy of elliptic curve convolution, seven-point sheaves of $G_2$-type and motives of Beauville type

We study the Tannakian properties of the category of perverse sheaves on elliptic curves endowed with the convolution product. We establish that for certain sheaves with unipotent local monodromy over seven points the corresponding Tannaka group is isomorphic to $G_2$. This monodromy approach generalizes a result of Katz on the existence of $G_2$-motives in the middle cohomology of deformations of Beauville surfaces.

math.AG↗
arXiv2014

On Galois action on stack inertia of moduli spaces of curves

We establish that the geometric action of the absolute Galois group on the \'etale fundamental group of moduli spaces of curves induces a Galois action on its stack inertia subgroups, and that this action is given by cyclotomy conjugacy. This result extends the special case of inertia without \'etale factorisation previously established by the authors. It is here obtained in the general case by comparing deformations of Galois actions. Since the stack inertia corresponds to the first level of the stack stratification of the space, this results, by analogy with the arithmetic of the Deligne-Mumford stratification, opens the way to a systematic Galois study of the stack inertia through the corresponding stratification of the moduli stack

math.AG↗