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C. Soulé

Publications and source records attributed to C. Soulé.

5 recordsLinked to original sources
arXiv2020

On the slopes of the lattice of sections of Hermitian line bundles

In this paper we study the distribution of successive minima of global sections of powers of a metrized ample line bundle on a variety over a number field. We develop criteria for there to exist a measure on the real line describing the limiting behavior of this distribution as one considers increasing powers of the bundle. When this measure exists, we develop methods for determining it explicitly. We present applications to the distribution of Petersson norms of cusp forms of increasing weight for SL_2(Z) and to the minimal sup norm of algebraic functions on adelic subsets of curves arising in capacity theory.

math.AG↗
arXiv2005

Mathematical approaches to differentiation and gene regulation

We consider some mathematical issues raised by the modelling of gene networks. The expression of genes is governed by a complex set of regulations, which is often described symbolically by interaction graphs. Once such a graph has been established, there remains the difficult task to decide which dynamical properties of the gene network can be inferred from it, in the absence of precise quantitative data about their regulation. In this paper we discuss a rule proposed by R.Thomas according to which the possibility for the network to have several stationary states implies the existence of a positive circuit in the corresponding interaction graph. We prove that, when properly formulated in rigorous terms, this rule becomes a theorem valid for several different types of formal models of gene networks. This result is already known for models of differential or boolean type. We show here that a stronger version of it holds in the differential setup when the decay of protein concentrations is taken into account. This allows us to verify also the validity of Thomas' rule in the context of piecewise-linear models and the corresponding discrete models. We discuss open problems as well.

q-bio.MN↗
arXiv1999

Direct images in non-archimedean Arakelov theory

We develop a formalism of direct images for metrized vector bundles in the context of the non-archimedean Arakelov theory introduced in our previous joint work with S. Bloch, and we prove a Riemann-Roch-Grothendieck theorem for this direct image.

math.AG↗
arXiv1997

Upper bounds for regularized determinants

Let $E$ be a holomorphic vector bundle on a compact Kähler manifold $X$. If we fix a metric $h$ on $E$, we get a Laplace operator $Δ$ acting upon smooth sections of $E$ over $X$. Using the zeta function of $Δ$, one defines its regularized determinant $det'(Δ)$. We conjectured elsewhere that, when $h$ varies, this determinant $det'(Δ)$ remains bounded from above. In this paper we prove this in two special cases. The first case is when $X$ is a Riemann surface, $E$ is a line bundle and $dim(H^0 (X,E)) + dim(H^1 (X,E)) \leq 2$, and the second case is when $X$ is the projective line, $E$ is a line bundle, and all metrics under consideration are invariant under rotation around a fixed axis.

dg-ga↗