Hamiltonian group actions on symplectic Deligne–Mumford stacks and toric orbifolds

We develop differential and symplectic geometry of differentiable Deligne–Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks.     

The integral (orbifold) Chow ring of toric Deligne–Mumford stacks

In this paper we study the integral Chow ring of toric Deligne–Mumford stacks. We prove that the integral Chow ring of a semi-projective toric Deligne–Mumford stack is isomorphic to the Stanley–Reisner ring of the associated stacky fan. The integral orbifold Chow ring is also computed. Our results are illustrated with several examples.     

Castelnuovo–Mumford Regularity and Gorensteinness of Fiber Cone

In this article, we study the Castelnuovo–Mumford regularity and Gorenstein properties of the fiber cone. We obtain upper bounds for the Castelnuovo–Mumford regularity of the fiber cone and obtain sufficient conditions for the regularity of the fiber cone to be equal to that of the Rees algebra. We obtain a formula for the canonical module of the fiber cone and use it to study the Gorenstein property of the fiber cone.     

The Symplectic Deligne–Mumford Stack Associated to a Stacky Polytope

We discuss a symplectic counterpart of the theory of stacky fans. First, we define a stacky polytope and construct the symplectic Deligne–Mumford stack associated to the stacky polytope. Then we establish a relation between stacky polytopes and stacky fans: the stack associated to a stacky polytope is equivalent to the stack associated to a stacky fan if the stacky fan corresponds to the stacky polytope.     

Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford

We develop a cohomology theory for Deligne–Mumford stacks, adapted to Hirzebruch–Riemann–Roch formulas. For this, we define the cohomology with coefficients in the representations and a Chern character, and we prove a Grothendieck–Riemann–Roch formula for the associated Riemann–Roch transformation.     

Semisupervised data classification via the Mumford–Shah–Potts-type model

More and more high dimensional data are widely used in many real world applications. This kind of data are obtained from different feature extractors, which represent distinct perspectives of the data. How to classify such data efficiently is a challenge. Despite of existence of millions of unlabeled data samples, it is believed that labeling a handful of data such as the semisupervised scheme will remarkably improve the searching performance. However, the performance of semisupervised data classification highly relies on proposed models and related numerical methods. Following from the extension of the Mumford–Shah–Potts-type model in the spatially continuous setting, we propose some efficient data classification algorithms based on the alternating direction method of multipliers and the primal-dual method to efficiently deal with the nonsmoothing problem in the proposed model. The convergence of the proposed data classification algorithms is established under the framework of variational inequalities. Some balanced and unbalanced classification problems are tested, which demonstrate the efficiency of the proposed algorithms.     

The Hyers theorem via the Markov–Kakutani fixed point theorem

We present an application of the Markov–Kakutani common fixed point theorem to the theory of stability of functional equation by proving some version of the Hyers theorem concerning approximate homomorphisms.     

The Rohde–Schramm theorem via the Gaussian free field

The Rohde–Schramm theorem states that Schramm–Loewner Evolution with parameter κ (or SLE_κ for short) exists as a random curve, almost surely, if κ ≠ 8. Here we give a new and concise proof of the result, based on the Liouville quantum gravity coupling (or reverse coupling) with a Gaussian free field. This transforms the problem of estimating the derivative of the Loewner flow into estimating certain correlated Gaussian free fields. While the correlation between these fields is not easy to understand, a surprisingly simple argument allows us to recover a derivative exponent first obtained by Rohde and Schramm [14], subsequently shown to be optimal by Lawler and Viklund [17], which then implies the Rohde–Schramm theorem.     

The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3

In the moduli space M \mathcal{M} _g of genus-g Riemann surfaces, consider the locus RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} of Riemann surfaces whose Jacobians have real multiplication by the order O \mathcal{O} in a totally real number field F of degree g. If g = 3, we compute the closure of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} in the Deligne–Mumford compactification of M \mathcal{M} _g and the closure of the locus of eigenforms over RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} . Boundary strata of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} are parameterized by configurations of elements of the field F satisfying a strong geometry of numbers type restriction.     

Un isomorphisme de type Deligne–Riemann–Roch

We prove an Adams–Riemann–Roch theorem for projective morphisms between regular schemes, in the sense of the program of P. Deligne on the functorial Riemann–Roch theorem and we deduce some geometric consequences. To cite this article: D. Eriksson, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

The Sandwich theorem via Pataraia’s fixed point theorem

Positivity - The Sandwich theorem (König in Archiv der Mathematik 23:500–1972, 1972) yields the existence of a linear functional sandwiched in between a given superlinear functional and...     

The bang–bang theorem via Baire category: a dual approach

The paper develops a new approach to the classical bang–bang theorem in linear control theory, based on Baire category. Among all controls which steer the system from the origin to a given point \({\bar x}\), we consider those which minimize an auxiliary linear functional \({\phi}\). For all \({\phi}\) in a residual set, we show that the minimizing control is unique and takes values within a set of extreme points.     

Castelnuovo–Mumford Regularity of Some Modules

We give bounds for the Castelnuovo–Mumford regularity of the so-called sequentially κ-Buchsbaum modules and of the canonical modules of certain rings.