Searcharxiv⌕ Search

arXiv subjects

Simon Pepin Lehalleur

Publications and source records attributed to Simon Pepin Lehalleur.

14 recordsLinked to original sources
arXiv2025

You Are What You Eat -- AI Alignment Requires Understanding How Data Shapes Structure and Generalisation

In this position paper, we argue that understanding the relation between structure in the data distribution and structure in trained models is central to AI alignment. First, we discuss how two neural networks can have equivalent performance on the training set but compute their outputs in essentially different ways and thus generalise differently. For this reason, standard testing and evaluation are insufficient for obtaining assurances of safety for widely deployed generally intelligent systems. We argue that to progress beyond evaluation to a robust mathematical science of AI alignment, we need to develop statistical foundations for an understanding of the relation between structure in the data distribution, internal structure in models, and how these structures underlie generalisation.

cs.LG↗
arXiv2024

Geometry of fibers of the multiplication map of deep linear neural networks

We study the geometry of the algebraic set of tuples of composable matrices which multiply to a fixed matrix, using tools from the theory of quiver representations. In particular, we determine its codimension $C$ and the number $θ$ of its top-dimensional irreducible components. Our solution is presented in three forms: a Poincaré series in equivariant cohomology, a quadratic integer program, and an explicit formula. In the course of the proof, we establish a surprising property: $C$ and $θ$ are invariant under arbitrary permutations of the dimension vector. We also show that the real log-canonical threshold of the function taking a tuple to the square Frobenius norm of its product is $C/2$. These results are motivated by the study of deep linear neural networks in machine learning and Bayesian statistics (singular learning theory) and show that deep linear networks are in a certain sense ``mildly singular".

math.AG↗
arXiv2024

Motivic mirror symmetry and $χ$-independence for Higgs bundles in arbitrary characteristic

We prove that the (twisted orbifold) motives of the moduli spaces of $\mathrm{SL}_n$ and $\mathrm{PGL}_n$-Higgs bundles of coprime rank and degree on a smooth projective curve over an algebraically closed field in which the rank is invertible are isomorphic in Voevodsky's triangulated category of motives. The equality of twisted orbifold Hodge numbers of these moduli spaces was conjectured by Hausel and Thaddeus and recently proven by Groechenig, Ziegler and Wyss via $p$-adic integration and then by Maulik and Shen using the decomposition theorem, an analysis of the supports of $D$-twisted Hitchin fibrations and vanishing cycles. Our proof in characteristic zero combines the geometric ideas of Maulik and Shen with the conservativity of the Betti realisation on abelian motives; to apply the latter, we prove that the relevant motives are abelian. In particular, we prove that the motive of the $\mathrm{SL}_n$-Higgs moduli space is abelian, building on our previous work in the $\mathrm{GL}_n$-case. We then use motivic nearby cycles to deduce the result in positive characteristic from that in characteristic zero. Using the same ideas, we prove motivic $χ$-independence for $\mathrm{GL}_n$-Higgs bundles.

math.AG↗
arXiv2022

Exponentiation of coefficient systems and exponential motives

We construct new six-functor formalisms capturing cohomological invariants of varieties with potentials. Starting from any six-functor formalism $C$, encoded as a coefficient system, we associate a new six-functor formalism $C_{\text{exp}}$. This requires in particular constructing the convolution product symmetric monoidal structure at the $\infty$-categorical level. We study $C_{\text{exp}}$ and how it relates to $C$. We also define motives in $C_{\text{exp}}$ attached to varieties with potential and study their properties.

math.AG↗
arXiv2022

Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces

Let $k$ be a field and let $\text{GW}(k)$ be the Grothendieck-Witt ring of virtual non-degenerate symmetric bilinear forms over $k$. We develop methods for computing the quadratic Euler characteristic $χ(X/k)\in \text{GW}(k)$ for $X$ a smooth hypersurface in a projective space or a weighted projective space. We raise the question of a quadratic refinement of classical conductor formulas and find such a formula for the degeneration of a smooth hypersurface $X$ in $\mathbb{P}^{n+1}$ to the cone over a smooth hyperplane section of $X$; we also find a similar formula in the weighted homogeneous case. We formulate a conjecture that generalizes these computations to similar types of degenerations. Finally, we give an interpretation of the quadratic conductor formulas in terms of Ayoub's nearby cycles functor.

math.AG↗
arXiv2021

Motives of moduli spaces of rank 3 vector bundles and Higgs bundles on a curve

We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank 3 and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a completion of) the Grothendieck group of effective Chow motives to isomorphisms in the category of Chow motives. For the Higgs moduli space, we use motivic Bialynicki-Birula decompositions associated to a scaling action with variation of stability and wall-crossing for moduli spaces of rank 2 pairs, which occur in the fixed locus of this action.

math.AG↗
arXiv2020

Motives of moduli spaces of bundles on curves via variation of stability and flips

We study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall-crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall-crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulas for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant).

math.AG↗
arXiv2019

On the Voevodsky motive of the moduli stack of vector bundles on a curve

We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky's category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localising tensor subcategory generated by the motive of the curve, using Bialynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.

math.AG↗
arXiv2019

On the Voevodsky motive of the moduli space of Higgs bundles on a curve

We study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky's triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

math.AG↗
arXiv2019

Triangulated categories of relative 1-motives

We construct and study a candidate for the standard motivic t-structure on the triangulated category of relative cohomological 1-motives with rational coefficients over a noetherian finite dimensional scheme S. This t-structure is defined as a generated t-structure, and we show it is non-degenerate. We relate its heart MM^1(S) with Deligne 1-motives over S; in particular, when S is regular, the category of Deligne 1-motives embeds in MM^1(S) fully faithfully. We also study the inclusion of DA^1(S) into the larger category DA^{coh}(S) of relative cohomological motives on S, and prove that its right adjoint, the motivic Picard functor, preserves compact objects.

math.AG↗
arXiv2017

Constructible 1-motives and exactness

We prove that arbitrary pullbacks, as well as Betti and étale realisation functors, are t-exact for the constructible motivic t-structure on the category of cohomological 1-motives over a base scheme.

math.AG↗
arXiv2016

On the relative motive of a commutative group scheme

We prove a canonical Kuenneth decomposition of the relative motive with rational coefficients of a smooth commutative group scheme over a noetherian finite dimensional base. This paper is a follow-up of "On the motive of a commutative algebraic group" arXiv:1312.4171

math.AG↗