相关论文: Ray class fields of global function fields with ma…
These notes form part of a joint research project on the logic of fields with many valuations, connected by a product formula. We define such structures and name them {\em globally valued fields} (GVFs). This text aims primarily at a proof…
Group field theories are a generalization of matrix models which provide both a second quantized reformulation of loop quantum gravity as well as generating functions for spin foam models. While states in canonical loop quantum gravity, in…
This is the text from a talk at the Arbeitstagung 2011, which can serve as an introduction to arxiv:1009.0736 and arXiv:1007.0907. I first discuss how a global field is determined by a certain dynamical system, and how this relates to…
Each graph and choice of a commutative ring gives rise to an associated graphical group. In this article, we introduce and investigate graph polynomials that enumerate conjugacy classes of graphical groups over finite fields according to…
We give a full classification, up to equivalence, of finite-dimensional graded division algebras over the field of real numbers. The grading group is any abelian group.
We review and extend a technique for recovering a smooth function from its averages over a wide class of curves in a general region of Euclidean space. The method is based on complexification of the underlying vector fields defining the…
We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic…
We work with semi-algebraic functions on arbitrary real closed fields. We generalize the notion of critical values and prove a Sard type theorem in our framework.
We consider the locus of irreducible nonsingular rational curves of degree d Pn, n>2, meeting a generic collection of linear subspaces. When this locus is 0 (resp 1)- dimensional, we compute (recursively) its degree (resp. geometric genus).…
Rational conformal field theories produce a tower of finite-dimensional representations of surface mapping class groups, acting on the conformal blocks of the theory. We review this formalism. We show that many recent mathematical…
We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the…
The geometry of algebraic curves over finite fields is a rich area of research. In previous work, the authors investigated a particular aspect of the geometry over finite fields of the classical unit circle, namely how the number of…
Let $X$ and $Y$ be rational functions of degree at least two with complex coefficients such that $\mathbb{C}(X,Y)=\mathbb{C}(z)$. We study the problem of determining when the field extension $[\mathbb{C}(z):\mathbb{C}(X)\cap\mathbb{C}(Y)]$…
We describe algebraic curves $ X : F(x, y) = 0 $ defined over $\overline{\mathbb{Q}}$ that satisfy the following property: there exist a number field $k$ and an infinite set $S \subset k$ such that, for every $y \in S$, the roots of the…
We present a table containing the maximal number of rational points on a genus 3 curve over a field of cardinality q, for all q<100. Also, some remarks on Frobenius non-classical quartics over finite fields are given.
The reductions of conformal field theories which lead to generalized abelian cosets are studied. Primary fields and correlation functions of arbitrary abelian coset conformal field theory are explicitly expressed in terms of those of the…
Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a…
We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of "tame" differential fields. We state several…
A generic theory of a single real scalar field is considered, and a simple method is presented for obtaining a class of solutions to the equation of motion. These solutions are obtained from a simpler equation of motion that is generated by…
Linear superposition of gravitational fields is shown to be possible for a large class of spacetimes, in some specific coordinates. Explicit examples are presented.