Euler characteristics in relative K-groups

Suppose that M is a finite module under the Galois group of a local or global field. Ever since Tate's papers [17, 18], we have had a simple and explicit formula for the Euler–Poincaré characteristic of the cohomology of M. In this note we are interested in a refinement of this formula when M also carries an action of some algebra [script A], commuting with the Galois action (see Proposition 5.2 and Theorem 5.1 below). This refinement naturally takes the shape of an identity in a relative K-group attached to [script A] (see Section 2). We shall deduce such an identity whenever we have a formula for the ordinary Euler characteristic, the key step in the proof being the representability of certain functors by perfect complexes (see Section 3). This representability may be of independent interest in other contexts. Our formula for the equivariant Euler characteristic over [script A] implies the ‘isogeny invariance’ of the equivariant conjectures on special values of the L-function put forward in [3], and this was our motivation to write this note. Incidentally, isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer) was also a motivation for Tate's original paper [18]. I am very grateful to J-P. Serre for illuminating discussions on the subject of this note, in particular for suggesting that I consider representability. I should also like to thank D. Burns for insisting on a most general version of the results in this paper

Donaldson-Thomas invariants and wall-crossing formulas

Notes from the report at the Fields institute in Toronto. We introduce the Donaldson-Thomas invariants and describe the wall-crossing formulas for numerical Donaldson-Thomas invariants.Comment: 18 pages. To appear in the Fields Institute Monograph Serie

The SnS_n-equivariant top weight Euler characteristic of Mg,nM_{g,n}

We prove a formula, conjectured by Zagier, for the $S_n$-equivariant Euler characteristic of the top weight cohomology of $M_{g,n}$.Comment: 26 pages, 5 figures; v2: added references to related work of Tsopm\'en\'e and Turchin and of Willwacher and \v{Z}ivkovi\'c, along with Remark 1.8, which explains the connection

Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays

The goal of this paper is to introduce a new method in computer-aided geometry of solid modeling. We put forth a novel algebraic technique to evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with regularized operators of union, intersection, and difference, i.e., any CSG tree. The result is obtained in three steps: first, by computing an independent set of generators for the d-space partition induced by the input; then, by reducing the solid expression to an equivalent logical formula between Boolean terms made by zeros and ones; and, finally, by evaluating this expression using bitwise operators. This method is implemented in Julia using sparse arrays. The computational evaluation of every possible solid expression, usually denoted as CSG (Constructive Solid Geometry), is reduced to an equivalent logical expression of a finite set algebra over the cells of a space partition, and solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig

Multiplicative Invariants and the Finite Co-Hopfian Property

A group is said to be, finitely co-Hopfian when it contains no proper subgroup of finite index isomorphic to itself. It is known that irreducible lattices in semisimple Lie groups are finitely co-Hopfian. However, it is not clear, and does not appear to be known, whether this property is preserved under direct product. We consider a strengthening of the finite co-Hopfian condition, namely the existence of a non-zero multiplicative invariant, and show that, under mild restrictions, this property is closed with respect to finite direct products. Since it is also closed with respect to commensurability, it follows that lattices in linear semisimple groups of general type are finitely co-Hopfian

The Absolute Relativity Theory

This paper is a first presentation of a new approach of physics that we propose to refer as the Absolute Relativity Theory (ART) since it refutes the idea of a pre-existing space-time. It includes an algebraic definition of particles, interactions and Lagrangians. It proposed also a purely algebraic explanation of the passing of time phenomenon that leads to see usual Euler-Lagrange equations as the continuous version of the Knizhnik-Zamolodchikov monodromy. The identification of this monodromy with the local ones of the Lorentzian manifolds gives the Einstein equation algebraically explained in a quantized context. A fact that could lead to the unification of physics. By giving an algebraic classification of particles and interactions, the ART also proposes a new branch of physics, namely the Mass Quantification Theory, that provides a general method to calculate the characteristics of particles and interactions. Some examples are provided. The MQT also predicts the existence of as of today not yet observed particles that could be part of the dark matter. By giving a new interpretation of the weak interaction, it also suggests an interpretation of the so-called dark energy