arXiv subjects

Clément Dupont

Publications and source records attributed to Clément Dupont.

At least 19 recordsLinked to original sources

arXiv2026

Parametrically driven Kerr temporal soliton crystals

We theoretically investigate the dynamics of parametrically driven soliton crystals (PDSC) and their associated frequency combs in doubly resonant cavities with quadratic and cubic nonlinearities. We show that, in a regime with strong pump--signal walk-off where the homogeneous state is unstable, noise-seeded dynamics relaxes toward stable, equally spaced multi-soliton states. At fixed detuning, the driving strength acts as the primary control parameter for the soliton number. Pump signal walkoff extends pump depletion from a local perturbation to a global constraint, enabling long-range soliton interactions; in combination with the pump phase, this mechanism stabilizes the crystal's equal spacing and preserves comb coherence. Additionally, we identify a novel nonlinear state in parametric soliton crystals in which circulating solitons periodically alternate their intensities and group velocities, a phenomenon we term the {\it soliton-pursuing} state. Furthermore, due to the phase-selective nature of the optical parametric process, we show how different configurations of soliton phases determine the optical frequency combs, enabling odd-harmonic and subharmonic-like combs.

physics.optics↗

arXiv2026

Regularized integrals and manifolds with log corners

We introduce a natural geometric framework for the study of logarithmically divergent integrals on manifolds with corners and algebraic varieties, using the techniques of logarithmic geometry. Key to the construction is a new notion of morphism in logarithmic geometry itself, introduced by Howell, which allows us to interpret the ubiquitous rule of thumb ''$\lim_{ε\to 0} \log ε:= 0$'' as the restriction to a submanifold. Via a version of de Rham's theorem with logarithmic divergences, we obtain a functorial characterization of the classical theory of ``regularized integration'': it is the unique way to extend the ordinary integral to the logarithmically divergent context while respecting the basic laws of calculus (change of variables, Fubini's theorem, and Stokes' formula.)

math.DG↗

arXiv2026

Logarithmic morphisms, tangential basepoints, and little disks

We develop the theory of "virtual morphisms" in logarithmic algebraic geometry, introduced by Howell. It allows one to give algebro-geometric meaning to various useful maps of topological spaces that do not correspond to morphisms of (log) schemes in the classical sense, while retaining functoriality of key constructions. In particular, we explain how virtual morphisms provide a natural categorical home for Deligne's theory of tangential basepoints: the latter are simply the virtual morphisms from a point. We also extend Howell's results on the functoriality of Betti and de Rham cohomology. Using this framework, we lift the topological operad of little $2$-disks to an operad in log schemes over the integers, whose virtual points are isomorphism classes of stable marked curves of genus zero equipped with a tangential basepoint. The gluing of such curves along marked points is performed using virtual morphisms that transport tangential basepoints around the curves. This builds on Vaintrob's analogous construction for framed little disks, for which the classical notion of morphism in logarithmic geometry sufficed. In this way, we obtain a direct algebro-geometric proof of the formality of the little disks operad, following the strategy envisioned by Beilinson. Furthermore, Bar-Natan's parenthesized braids naturally appear as the fundamental groupoids of our moduli spaces, with all virtual basepoints defined over the integers.

math.AG↗

arXiv2025

Lie-operads and operadic modules from poset cohomology

As observed by Joyal, the cohomology groups of the partition posets are naturally identified with the components of the operad encoding Lie algebras. This connection was explained in terms of operadic Koszul duality by Fresse, and later generalized by Vallette to the setting of decorated partitions. In this article, we set up and study a general formalism which produces a priori operadic structures (operads and operadic modules) on the cohomology of families of posets equipped with some natural recursive structure, that we call "operadic poset species". This framework goes beyond decorated partitions and operadic Koszul duality, and contains the metabelian Lie operad and Kontsevich's operad of trees as two simple instances. In forthcoming work, we will apply our results to the hypertree posets and their connections to post-Lie and pre-Lie algebras.

math.CO↗

arXiv2025

Positive geometries and canonical forms via mixed Hodge theory

''Positive geometries'' are a class of semi-algebraic domains which admit a unique ''canonical form'': a logarithmic form whose residues match the boundary structure of the domain. The study of such geometries is motivated by recent progress in particle physics, where the corresponding canonical forms are interpreted as the integrands of scattering amplitudes. We recast these concepts in the language of mixed Hodge theory, and identify ''genus zero pairs'' of complex algebraic varieties as a natural and general framework for the study of positive geometries and their canonical forms. In this framework, we prove some basic properties of canonical forms which have previously been proved or conjectured in the literature. We give many examples and study in detail the case of arrangements of hyperplanes and convex polytopes.

math.AG↗

arXiv2025

A construction of the polylogarithm motive

Classical polylogarithms give rise to a variation of mixed Hodge-Tate structures on the punctured projective line $S=\mathbb{P}^1\setminus \{0, 1, \infty\}$, which is an extension of the symmetric power of the Kummer variation by a trivial variation. By results of Beilinson-Deligne, Huber-Wildeshaus, and Ayoub, this polylogarithm variation has a lift to the category of mixed Tate motives over $S$, whose existence is proved by computing the corresponding space of extensions in both the motivic and the Hodge settings. In this paper, we construct the polylogarithm motive as an explicit relative cohomology motive, namely that of the complement of the hypersurface $\{1-zt_1\cdots t_n=0\}$ in affine space $\mathbb{A}^n_S$ relative to the union of the hyperplanes $\{t_i=0\}$ and $\{t_i=1\}$.

math.AG↗

arXiv2024

An introduction to mixed Tate motives

Mixed Tate motives are central objects in the study of cohomology groups of algebraic varieties and their arithmetic invariants. They also play a crucial role in a wide variety of questions related to multiple zeta values and polylogarithms, algebraic K-theory, hyperbolic geometry, and particle physics among others. This survey article is an introduction to mixed Tate motives and their many facets. It was written for the proceedings of the Summer School on Motives and Arithmetic Groups held in Strasbourg in June 2022.

math.AG↗

arXiv2024

Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions

The goal of this paper is to raise the possibility that there exists a meaningful theory of `motives' associated to certain hypergeometric integrals, viewed as functions of their parameters. It goes beyond the classical theory of motives, but should be compatible with it. Such a theory would explain a recent and surprising conjecture arising in the context of scattering amplitudes for a motivic Galois group action on Gauss' ${}_2F_1$ hypergeometric function, which we prove in this paper by direct means. More generally, we consider Lauricella hypergeometric functions and show on the one hand how the coefficients in their Taylor expansions can be promoted, via the theory of motivic fundamental groups, to motivic multiple polylogarithms. The latter are periods of ordinary motives and admit an action of the usual motivic Galois group, which we call the `local' action. On the other hand, we define lifts of the full Lauricella functions as matrix coefficients in a Tannakian category of twisted cohomology, which inherit an action of the corresponding Tannaka group. We call this the `global' action. We prove that these two actions, local and global, are compatible with each other, even though they are defined in completely different ways. The main technical tool is to prove that metabelian quotients of generalised Drinfeld associators on the punctured Riemann sphere are hypergeometric functions. We also study single-valued versions of these hypergeometric functions, which may be of independent interest.

math.AG↗

arXiv2024

The localization spectral sequence in the motivic setting

We construct and study a motivic lift of a spectral sequence associated to a stratified scheme, recently discovered by Petersen in the context of mixed Hodge theory and $\ell$-adic Galois representations. The original spectral sequence expresses the compactly supported cohomology of an open stratum in terms of the compactly supported cohomology of the closures of strata and the combinatorics of the poset underlying the stratification. Some of its special cases are classical tools in the study of arrangements of subvarieties and configuration spaces. Our motivic lift lives in the triangulated category of étale motives and takes the shape of a Postnikov system. We describe its connecting morphisms and study some of its functoriality properties.

math.AG↗

arXiv2024

Motifs: un tour d'horizon

This text was published in the Gazette de la Société Mathématique de France in October 2023. It is an introduction to the theory of motives, from its sources to its more modern developments.

math.AG↗

arXiv2022

Progrès récents sur la conjecture de Zagier et le programme de Goncharov [d'après Goncharov, Rudenko, Gangl, ...]

This survey article is the written version of a talk given at the Bourbaki seminar in April 2021. We give an introduction to Zagier's conjecture on special values of Dedekind zeta functions, and its relation to $K$-theory of fields and the theory of motives. We survey recent progress on the conjecture and in particular the proof of the $n=4$ case of the conjecture by Goncharov and Rudenko.

math.NT↗

arXiv2021

Valeurs zêta multiples

This survey article is the written version of two talks given at the Journées X-UPS 2019 "Périodes et transcendance" at École polytechnique. We give a gentle introduction to the study of multiple zeta values, from Euler's solution to the Basel problem to modern Galois theoretic tools. We emphasize the role of the theory of motives which conjecturally controls the structure of the algebra of multiple zeta values.

math.NT↗

arXiv2021

Single-valued integration and superstring amplitudes in genus zero

We study open and closed string amplitudes at tree-level in string perturbation theory using the methods of single-valued integration which were developed in the prequel to this paper. Using dihedral coordinates on the moduli spaces of curves of genus zero with marked points, we define a canonical regularisation of both open and closed string perturbation amplitudes at tree level, and deduce that they admit a Laurent expansion in Mandelstam variables whose coefficients are multiple zeta values (resp. single-valued multiple zeta values). Furthermore, we prove the existence of a motivic Laurent expansion whose image under the period map is the open string expansion, and whose image under the single-valued period map is the closed string expansion. This proves the recent conjecture of Stieberger that closed string amplitudes are the single-valued projections of (motivic lifts of) open string amplitudes. Finally, applying a variant of the single-valued formalism for cohomology with coefficients yields the KLT formula expressing closed string amplitudes as quadratic expressions in open string amplitudes.

math.NT↗

arXiv2021

Single-valued integration and double copy

We study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a $p$-adic period pairing at the infinite prime. The single-valued integration pairing is defined by transporting the action of complex conjugation from singular to de Rham cohomology via the comparison isomorphism. We show how quite general families of period integrals admit canonical single-valued versions and prove some general formulas for them. This implies an elementary 'double copy' formula expressing certain singular volume integrals over the complex points of a smooth projective variety as a quadratic expression in ordinary period integrals of half the dimension. We provide several examples, including non-holomorphic modular forms, archimedean Néron-Tate heights on curves, single-valued multiple zeta values and polylogarithms. In a sequel to this paper we apply this formalism to the moduli space of curves of genus zero with marked points, to deduce a recent conjecture due to Stieberger in string perturbation theory, which states that closed string amplitudes are the single-valued projections of open string amplitudes.

math.NT↗

arXiv2019

Universal Tutte characters via combinatorial coalgebras

The Tutte polynomial is the most general invariant of matroids and graphs that can be computed recursively by deleting and contracting edges. We generalize this invariant to any class of combinatorial objects with deletion and contraction operations, associating to each such class a universal Tutte character by a functorial procedure. We show that these invariants satisfy a universal property and convolution formulae similar to the Tutte polynomial. With this machinery we recover classical invariants for delta-matroids, matroid perspectives, relative and colored matroids, generalized permutohedra, and arithmetic matroids, and produce some new convolution formulae. Our principal tools are combinatorial coalgebras and their convolution algebras. Our results generalize in an intrinsic way the recent results of Krajewski--Moffatt--Tanasa.

math.CO↗

arXiv2018

Relative cohomology of bi-arrangements

A bi-arrangement of hyperplanes in a complex affine space is the data of two sets of hyperplanes along with a coloring information on the strata. To such a bi-arrangement, one naturally associates a relative cohomology group, that we call its motive. The motivation for studying such relative cohomology groups comes from the notion of motivic period. More generally, we suggest the systematic study of the motive of a bi-arrangement of hypersurfaces in a complex manifold. We provide combinatorial and cohomological tools to compute the structure of these motives. Our main object is the Orlik-Solomon bi-complex of a bi-arrangement, which generalizes the Orlik-Solomon algebra of an arrangement. Loosely speaking, our main result states that "the motive of an exact bi-arrangement is computed by its Orlik-Solomon bi-complex", which generalizes classical facts involving the Orlik-Solomon algebra of an arrangement. We show how this formalism allows us to explicitly compute motives arising from the study of multiple zeta values and sketch a more general application to periods of mixed Tate motives.

math.AG↗

arXiv2018

Odd zeta motive and linear forms in odd zeta values

We study a family of mixed Tate motives over $\mathbb{Z}$ whose periods are linear forms in the zeta values $ζ(n)$. They naturally include the Beukers-Rhin-Viola integrals for $ζ(2)$ and the Ball-Rivoal linear forms in odd zeta values. We give a general integral formula for the coefficients of the linear forms and a geometric interpretation of the vanishing of the coefficients of a given parity. The main underlying result is a geometric construction of a minimal ind-object in the category of mixed Tate motives over $\mathbb{Z}$ which contains all the non-trivial extensions between simple objects. In a joint appendix with Don Zagier, we prove the compatibility between the structure of the motives considered here and the representations of their periods as sums of series.

math.AG↗

arXiv2018

Brown's moduli spaces of curves and the gravity operad

This paper is built on the following observation: the purity of the mixed Hodge structure on the cohomology of Brown's moduli spaces is essentially equivalent to the freeness of the dihedral operad underlying the gravity operad. We prove these two facts by relying on both the geometric and the algebraic aspects of the problem: the complete geometric description of the cohomology of Brown's moduli spaces and the coradical filtration of cofree cooperads. This gives a conceptual proof of an identity of Bergström-Brown which expresses the Betti numbers of Brown's moduli spaces via the inversion of a generating series. This also generalizes the Salvatore-Tauraso theorem on the nonsymmetric Lie operad.

math.AG↗