A reduction theorem for the generalised Rhodes' Type II Conjecture

One of the milestones in the theory of semigroups and automata is the Krohn-Rhodes Theorem. It states that every finite semigroup S divides a wreath product of finite simple groups, each of them divisor of S, and finite aperiodic semigroups, i. e. semigroups with trivial maximal subgroups. The smallest number of groups in any Kohn-Rhodes decomposition is called the group complexity of the semigroup. Since there is no obvious way to compute the complexity of a finite semigroup in general, the decidability of this number is one of the most important open problems in finite semigroup theory and the search for the solution has led to the development of many tools and ideas that are useful in finite semigroup theory and of independent interest. At this point, the notion of the kernel of a semigroup came on the scene. It was introduced by Rhodes and Tilson in their seminal paper of 1972 as the set of elements related to the indentity under every relational morphism between the semigroup and a group, and its computability would have important consequences in the solution of the complexity problem. In fact, one of the main results in Rhodes and Tilsons's paper is a description of the regular elements of the kernel of a semigroup. That result led to the Rhodes' conjecture, originally called the Type II Conjecture: the kernel of a semigroup is the smallest subsemigroup containing the idempotents and closed under weak conjugation. This conjecture attracted the attention of many semigroup theorists during about two decades before being solved. It was proved independently by Ash, and Ribes and Zalesskii using diferents methods. Like many problems with a no immediate solution, once they are solved, not only a great number of consequences spring from their solution, but also new questions related with them can be set out. A first natural step is to extend the definition of kernel of a semigroup to an arbitrary variety of finite groups F: the F-kernel of a finite semigroup S is the subsemigroup of S consisting of all elements of S such that relate to the identity under every relational morphism of S with a group in F. The generalised Rhode's Type II Conjecture asks if the F-kernel associated with a given variety of finite groups F is computable for every semigroup. Since generalised Rhodes' Type II Conjecture is completely general, it is natural to hope that some argument might exist that would prove it, but up to now no one seems to have an inkling of how such a proof might proceed. Failing that, one could try to reduce the problem to a question about some restricted class of finite semigroups. Indeed, in the important case where F is extension closed, a theorem of Ribes and Zalesskii allows us to conclude that the F-kernel is computable if its regular elements are computable. The main result of this thesis is meant to provide a decisive step towards verifying the generalised Rhodes' Type II Conjecture for extension closed varieties

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.Comment: Published in at http://dx.doi.org/10.1214/13-AOP890 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Representation Theory of Finite Semigroups, Semigroup Radicals and Formal Language Theory

In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; and \v{C}ern\'y's conjecture for an important class of automata

On the insertion of n-powers

In algebraic terms, the insertion of $n$-powers in words may be modelled at the language level by considering the pseudovariety of ordered monoids defined by the inequality $1\le x^n$. We compare this pseudovariety with several other natural pseudovarieties of ordered monoids and of monoids associated with the Burnside pseudovariety of groups defined by the identity $x^n=1$. In particular, we are interested in determining the pseudovariety of monoids that it generates, which can be viewed as the problem of determining the Boolean closure of the class of regular languages closed under $n$-power insertions. We exhibit a simple upper bound and show that it satisfies all pseudoidentities which are provable from $1\le x^n$ in which both sides are regular elements with respect to the upper bound

Complex Gaussian multiplicative chaos

In this article, we study complex Gaussian multiplicative chaos. More precisely, we study the renormalization theory and the limit of the exponential of a complex log-correlated Gaussian field in all dimensions (including Gaussian Free Fields in dimension 2). Our main working assumption is that the real part and the imaginary part are independent. We also discuss applications in 2D string theory; in particular we give a rigorous mathematical definition of the so-called Tachyon fields, the conformally invariant operators in critical Liouville Quantum Gravity with a c=1 central charge, and derive the original KPZ formula for these fields.Comment: 66 pages, 5 figures, contains open problems and application in 2 dimensional string theory. The new version contains the KPZ formula for the Tachyon fields and further discussions about application

Local description of phylogenetic group-based models

Motivated by phylogenetics, our aim is to obtain a system of equations that define a phylogenetic variety on an open set containing the biologically meaningful points. In this paper we consider phylogenetic varieties defined via group-based models. For any finite abelian group $G$, we provide an explicit construction of $codim X$ phylogenetic invariants (polynomial equations) of degree at most $|G|$ that define the variety $X$ on a Zariski open set $U$. The set $U$ contains all biologically meaningful points when $G$ is the group of the Kimura 3-parameter model. In particular, our main result confirms a conjecture by the third author and, on the set $U$, a couple of conjectures by Bernd Sturmfels and Seth Sullivant.Comment: 22 pages, 7 figure