On properties of (weakly) small groups

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central product of a definable divisible group with a definable one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-by-abelian subgroup. First corollary : a weakly small group with simple theory has an infinite definable finite-by-abelian subgoup. Secondly, in a group with simple theory, a normal solvable group A of derived length n is contained in an A-definable almost solvable group of class n

The omega-inequality problem for concatenation hierarchies of star-free languages

The problem considered in this paper is whether an inequality of omega-terms is valid in a given level of a concatenation hierarchy of star-free languages. The main result shows that this problem is decidable for all (integer and half) levels of the Straubing-Th\'erien hierarchy

Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence

Single-valuedness of the eigenfunctions of the quantised Hitchin Hamiltonians is proposed as a natural quantisation condition. Separation of Variables can be used to relate the classification of eigenstates to the classification of projective structures with real holonomy. Using complex Fenchel-Nielsen coordinates one may reformulate the quantisation conditions in terms of the generating function for the variety of opers. These results are interpreted as a variant of the geometric Langlands correspondence.Comment: 30 pages; v2: relevant corrections, close to fina

Propriétés algébriques des structures menues ou minces, rang de Cantor Bendixson, espaces topologiques généralisés

Abstract. Small structures appear in the '60s together with Vaught's conjecture. Weakly small structures include both minimal and small structures. Definable sets in a weakly small structure are ranked by Cantor-Bendixson rank. We show computational properties of this rank, which imply a local descending chain condition on acl(0)-definable subgroups, and introduce a notion of local almost stabiliser. We deduce algebraic properties of weakly small structures. Among them, a weakly small field of positive characteristic is locally finite dimensional over its centre, and an infinite weakly small group has an infinite abelian subgroup. We then turn to small type-definable structures, showing that finitary small type 0-de_nable groups are the intersection of definable groups. We extend the result to finitary small type 0- definable monoids, rings, fields, categories and groupoids. We give local definability results concerning groups and fields type definable over an arbitrary set of parameters in small and simple theories. Finally, we reintroduce the Cantor Bendixson rank in its topological context, and show that the Cantor derivative can be seen as a derivation in a semi-ring of topological spaces. In an attempt to find a global Cantor rank for stable structures, we try to eliminate the word denumerable, omnipresent when one does topology, by replacing it by a regular cardinal k. We develop the notions of k-metrisable space, k-topology, k-compactness etc. and show an analogue of Urysohn's metrisability lemma and Cantor-Bendixson theorem.Les structures menues apparaissent dans les années 60 en lien avec la conjecture de Vaught. Les structures minces englobent à la fois les structures minimales et menues. Les ensembles définissables d'une structure mince sont rangés par le rang de Cantor-Bendixson. Nous présentons des propriétés de calcul de ce rang, une condition de chaîne descendante locale sur les groupes acl(0)-définissables ainsi qu'une notion de presque stabilisateur local, et en déduisons des propriétés algébriques des structures minces : un corps mince de caractéristique positive est localement de dimension finie sur son centre, et un groupe mince infini a un sous groupe abélien infini. Nous nous intéressons ensuite aux structures menues infiniment définissables, et montrons que les groupes d'arité finie infiniment 0-définissable sont l'intersection de groupes définissables. Nous étendons le résultat aux demi-groupes, anneaux, corps, catégories et groupoïdes infiniment 0-définissables, et donnons des résultats de définissabilité locale pour les groupes et corps simples et menus, infiniment définissables sur des paramètres quelconques. Enfin, nous réintroduisons le rang de Cantor dans son contexte topologique et montrons que la dérivée de Cantor peut être vue comme un opérateur de dérivation dans un semi-anneau d'espaces topologiques. Dans l'idée de trouver un rang de Cantor global pour les théories stables, nous essayons de nous débarrasser du mot dénombrable omniprésent lorsque l'on fait de la topologie, en le remplaçant par un cardinal régulier k. Nous développons une notion d'espace k-métrique, de k-topologie, de k-compacité etc. et montrons un k-analogue du lemme de métrisabilité d'Urysohn, et du théorème de Cantor-Bendixson

Knots, Trees, and Fields: Common Ground Between Physics and Mathematics

One main theme of this thesis is a connection between mathematical physics (in particular, the three-dimensional topological quantum field theory known as Chern-Simons theory) and three-dimensional topology. This connection arises because the partition function of Chern-Simons theory provides an invariant of three-manifolds, and the Wilson-loop observables in the theory define invariants of knots. In the first chapter, we review this connection, as well as more recent work that studies the classical limit of quantum Chern-Simons theory, leading to relations to another knot invariant known as the A-polynomial. (Roughly speaking, this invariant can be thought of as the moduli space of flat SL(2,C) connections on the knot complement.) In fact, the connection can be deepened: through an embedding into string theory, categorifications of polynomial knot invariants can be understood as spaces of BPS states. We go on to study these homological knot invariants, and interpret spectral sequences that relate them to one another in terms of perturbations of supersymmetric theories. Our point is more general than the application to knots; in general, when one perturbs any modulus of a supersymmetric theory and breaks a symmetry, one should expect a spectral sequence to relate the BPS states of the unperturbed and perturbed theories. We consider several diverse instances of this general lesson. In another chapter, we consider connections between supersymmetric quantum mechanics and the de Rham version of homotopy theory developed by Sullivan; this leads to a new interpretation of Sullivan's minimal models, and of Massey products as vacuum states which are entangled between different degrees of freedom in these models. We then turn to consider a discrete model of holography: a Gaussian lattice model defined on an infinite tree of uniform valence. Despite being discrete, the matching of bulk isometries and boundary conformal symmetries takes place as usual; the relevant group is PGL(2,Qp), and all of the formulas developed for holography in the context of scalar fields on fixed backgrounds have natural analogues in this setting. The key observation underlying this generalization is that the geometry underlying AdS3/CFT2 can be understood algebraically, and the base field can therefore be changed while maintaining much of the structure. Finally, we give some analysis of A-polynomials under change of base (to finite fields), bringing things full circle.</p

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Inflation:Generic predictions and nilpotent superfields

Met de observaties van de Planck satelliet begint de kosmologie aan een nieuw tijdperk waarin het universum bestudeerd kan worden met hoge precisie. Dit levert interessante informatie op over het zeer vroege universum, inclusief de inflatieperiode. Deze kosmologische inflatie werd in dit doctoraat op twee manieren theoretisch bestudeerd. Met de eerste methode wordt een groot aantal inflatiemodellen met de data van de CMB, zoals gemeten door de Planck satelliet, met elkaar vergeleken. Hieruit kunnen generieke voorspellingen worden afgeleid voor de verschillende parametrisaties waarmee deze modellen zijn verkregen. Door verschillende parametrisaties van de potentiaal te vergelijken, wordt geconcludeerd dat modellen behorende tot de groep van de plateau inflatiemodellen beter overeenkomen met de CMB dan de zogeheten polynomische modellen. De tweede benadering bestudeert inflatiemodellen in supergravitatie, hetgeen een extensie is van zowel het standaardmodel van de deeltjesfysica als van de algemene relativiteitstheorie door middel van een nieuwe symmetrie genaamd supersymmetrie. In dit onderzoek bestuderen we een bepaalde inbedding van inflatie in supergravitatie en bestuderen we de theoretische consistentie. Daarnaast zijn we in staat om donkere materie, een ander probleem in de kosmologie, te bestuderen. In een deel van de parameterruimte van ons model wordt deze donkere materie verklaard door een deeltje in de supersymmetrische sector. Hierdoor zijn wij in staat om in ons model zowel supersymmetriebreking, inflatie als donkere materie te beschrijven

Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories

Seiberg-Witten geometry of mass deformed N=2 superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space M of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space of holomorphic G-bundles on a (possibly degenerate) elliptic curve defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group G. The integrable systems underlying, or, rather, overlooking the special geometry of M are identified. The moduli spaces of framed G-instantons on R^2xT^2, of G-monopoles with singularities on R^2xS^1, the Hitchin systems on curves with punctures, as well as various spin chains play an important role in our story. We also comment on the higher dimensional theories. In the companion paper the quantum integrable systems and their connections to the representation theory of quantum affine algebras will be discussedComment: 197 page