Here are

my nominees for the 2006 paper of the year award in mathematics &

mathematical physics : **in math.RA : math.RA/0606241**

: Notes on A-infinity

algebras, A-infinity categories and non-commutative geometry. I by

Maxim Kontsevich and

Yan Soibelman. Here is the abstract :

We develop

geometric approach to A-infinity algebras and A-infinity categories

based on the notion of formal scheme in the category of graded vector

spaces. Geometric approach clarifies several questions, e.g. the notion

of homological unit or A-infinity structure on A-infinity functors. We

discuss Hochschild complexes of A-infinity algebras from geometric point

of view. The paper contains homological versions of the notions of

properness and smoothness of projective varieties as well as the

non-commutative version of Hodge-to-de Rham degeneration conjecture. We

also discuss a generalization of Deligne’s conjecture which includes

both Hochschild chains and cochains. We conclude the paper with the

description of an action of the PROP of singular chains of the

topological PROP of 2-dimensional surfaces on the Hochschild chain

complex of an A-infinity algebra with the scalar product (this action is

more or less equivalent to the structure of 2-dimensional Topological

Field Theory associated with an “abstract” Calabi-Yau

manifold).

**why ?** : Because this paper

probably gives the correct geometric object associated to a

non-commutative algebra (a huge coalgebra) and consequently the right

definition of a map between noncommutative affine schemes. In a previous post (and its predecessors) I’ve

tried to explain how this links up with my own interpretation and since

then I’ve thought more about this, but that will have to wait for

another time. **in hep-th : hep-th/0611082** : Children’s Drawings From

Seiberg-Witten Curves by Sujay K. Ashok, Freddy Cachazo, Eleonora

Dell’Aquila. Here is the abstract :

We consider N=2

supersymmetric gauge theories perturbed by tree level superpotential

terms near isolated singular points in the Coulomb moduli space. We

identify the Seiberg-Witten curve at these points with polynomial

equations used to construct what Grothendieck called “dessins

d’enfants” or “children’s drawings” on the Riemann

sphere. From a mathematical point of view, the dessins are important

because the absolute Galois group Gal(\bar{Q}/Q) acts faithfully on

them. We argue that the relation between the dessins and Seiberg-Witten

theory is useful because gauge theory criteria used to distinguish

branches of N=1 vacua can lead to mathematical invariants that help to

distinguish dessins belonging to different Galois orbits. For instance,

we show that the confinement index defined in hep-th/0301006 is a Galois

invariant. We further make some conjectures on the relation between

Grothendieck’s programme of classifying dessins into Galois orbits and

the physics problem of classifying phases of N=1 gauge theories.

**why ?** : Because this paper gives the

best introduction I’ve seen to Grothendieck’s dessins d’enfants

(slightly overdoing it by giving a crash course on elementary Galois

theory in appendix A) and kept me thinking about dessins and their

Galois invariants ever since (again, I’ll come back to this later).

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