Modular Invariant Regularization of String Determinants and the Serre GAGA Principle

Since any string theory involves a path integration on the world-sheet metric, their partition functions are volume forms on the moduli space of genus g Riemann surfaces M_g, or on its super analog. It is well known that modular invariance fixes strong constraints that in some cases appear only at higher genus. Here we classify all the Weyl and modular invariant partition functions given by the path integral on the world-sheet metric, together with space-time coordinates, b-c and/or beta-gamma systems, that correspond to volume forms on M_g. This was a long standing question, advocated by Belavin and Knizhnik, inspired by the Serre GAGA principle and based on the properties of the Mumford forms. The key observation is that the Bergman reproducing kernel provides a Weyl and modular invariant way to remove the point dependence that appears in the above string determinants, a property that should have its superanalog based on the super Bergman reproducing kernel. This is strictly related to the properties of the propagator associated to the space-time coordinates. Such partition functions Z[J] have well-defined asymptotic behavior and can be considered as a basis to represent a wide class of string theories. In particular, since non-critical bosonic string partition functions Z_D are volume forms on M_g, we suggest that there is a mapping, based on bosonization and degeneration techniques, from the Liouville sector to first order systems that may identify Z_D as a subclass of the Z[J]. The appearance of b-c and beta-gamma systems of any conformal weight shows that such theories are related to W algebras. The fact that in a large N 't Hooft-like limit 2D W_N minimal models CFTs are related to higher spin gravitational theories on AdS_3, suggests that the string partition functions introduced here may lead to a formulation of higher spin theories in a string context.Comment: 30 pp. New results on Quillen metric and its relations with the Mumford forms and the tautological classes. References adde

Kahler Geometry and Burgers' Vortices

We study the Navier-Stokes and Euler equations of incompressible hydrodynamics. Taking the divergence of the momentum equation leads, as usual, to a Poisson equation for the pressure: in this paper we study this equation in two spatial dimensions using Monge-Ampere structures. In two dimensional flows where the Laplacian of the pressure is positive, a Kahler geometry is described on the phase space of the fluid; in regions where the Laplacian of the pressure is negative, a product structure is described. These structures can be related to the ellipticity and hyperbolicity (respectively) of a Monge-Ampere equation. We then show how this structure can be extended to a class of canonical vortex structures in three dimensions

Multipoint correlators in the Abelian sandpile model

We revisit the calculation of height correlations in the two-dimensional Abelian sandpile model by taking advantage of a technique developed recently by Kenyon and Wilson. The formalism requires to equip the usual graph Laplacian, ubiquitous in the context of cycle-rooted spanning forests, with a complex connection. In the case at hand, the connection is constant and localized along a semi-infinite defect line (zipper). In the appropriate limit of a trivial connection, it allows one to count spanning forests whose components contain prescribed sites, which are of direct relevance for height correlations in the sandpile model. Using this technique, we first rederive known 1- and 2-site lattice correlators on the plane and upper half-plane, more efficiently than what has been done so far. We also compute explicitly the (new) next-to-leading order in the distances ($r^{-4}$ for 1-site on the upper half-plane, $r^{-6}$ for 2-site on the plane). We extend these results by computing new correlators involving one arbitrary height and a few heights 1 on the plane and upper half-plane, for the open and closed boundary conditions. We examine our lattice results from the conformal point of view, and confirm the full consistency with the specific features currently conjectured to be present in the associated logarithmic conformal field theory.Comment: 60 pages, 21 figures. v2: reformulation of the grove theorem, minor correction

On the Stability of Random Matrices

Η διδακτορική διατριβή αποσκοπεί σε μια πολυδιάστατη οικογένεια προβλημάτων στα μαθηματικά. Επιπλέον ένα από τα τρία θέματα που παρουσιάζονται στη διατριβή αυτή είναι η διαδικασία της διάχυσης σε δυναμικά χρηματοοικονομικά δίκτυα. Τα γραφήματα αυτά είναι συνεκτικά, κατευθυνόμενα και σταθμισμένα δίκτυα τραπεζών (κόμβοι), όπου τα αρχικά κεφάλαια και η ποσότητα δανείων που μεταφέρονται από την μία τράπεζα (κόμβο) στην άλλη θεωρούνται γνωστά. Με άλλα λόγια προσπαθούμε να θεραπέυσουμε προβλήματα στο χρηματοοικονομικό τομέα όπου αρχικά έχουμε τραπεζικά δυναμικά δίκτυα που χαρακτηρίζονται από διαφορετικά επίπεδα αρχικών μοχλεύσεων και καταλήγουμε να έχουμε ένα σταθερό τραπεζικό δίκτυο χάρη στη διαδικασία της διάχυσης. Επίσης, παρουσιάζουμε αρκετά παραδείγματα όπου επιβεβαιώνετε η θεωρία μας και μέσω πρακτικών παραδειγμάτων. Τέλος, έχουμε την πεποίθηση ότι η δουλειά αυτή μπορεί να επεκταθεί κάνοντας χρήση της μαθηματικής θεωρίας ελέγχου με σκοπό να κάνουμε αυτά τα δίκτυα πιο εύκολα ελέγξιμα.Επιπλέον, ένα άλλο πρόβλημα που παρουσιάζετε στη διατριβή αυτή είναι η έννοια των almost zeros πολυωνυμικών διανυσμάτων ή πινάκων. Στο θέμα αυτό το κίνητρό μας ήταν η κατανόηση αλλά και να παρουσιάσουμε πως η έννοια των almost zeros εξαρτάται από την τυχαιότητα. Πιο συγκεκριμένα μελετάμε και παρουσιάζουμε τα στατιστικά χαρακτηριστικά των almost zero και την τιμή τους για δοθέντες τυχαίους πολυωνυμικούς πίνακες. Τέλος, διάφορα παραδείγματα παρουσιάζονται για να είναι πιο κατονοητή η μελέτη αυτού του θέματος.Τέλος, το τελευταίο θέμα στο οποίο πραγματευόμαστε είναι η μελέτη σε γραμμικά, χρονικά-αμετάβλητα πολυμεταβλητά συστήματα που περιγράφονται από συστήματα εξισώσεων. Πιο συγκεκριμένα εστιάζουμε σε τυχαία συστήματα όπου οι παράμετροι έχουν αντικατασταθεί από τυχαίους πίνακες. Επιπροσθέτως, ορίζουμε την έννοια της ε-ελεγξιμότητας για τυχαία συστήματα για δοθέν θετικό αριθμό ε. Επίσης, θεωρούμε τη στοιχειώδη κατηγορία τυχαίων πινάκων Gaussian ortogonal ensemble (GOE) και υπολογίζουμε ότι το ε-μη-ελεγξιμότητα ενός τέτοιου τυχαίου συστήματος εξαρτάται από την κατανομή των στοιχείων του. Στην περίπτωση αυτή η ε-μη-ελεγξιμότητα υπολογίζεται.The aim of this thesis is multiple. Initially we study the diffusion process on dynamicalfinancial networks. To be more precise, we study the effect of diffusion method to interbanknetworks in relation to connected, directed and weighted networks. We consider networks ofn different banks which exchange funds (loans) and the main feature is how the leverage'sof banks can be chosen to improve the financial stability of the network. This is done byconsidering differential equations of diffusion type.Furthermore, we investigate the problem of almost zeros of polynomial matrices as usedin the system theory. It is related to the controllability and observability notion of systems aswell as the determination of Macmillan degree and complexity of systems. We also presentsome new results in this important invariant in the light of randomness and we prove anuncertainty type relation appearing in such ensembles of operators.In addition, we introduce the concept of ε-uncontrollability for random linear systems,i.e. a linear system in which the usual matrices have been replaced by random matrices. Wealso estimate the ε-uncontrollability in the case where the matrices come from the Gaussianorthogonal ensemble. Our proof utilizes tools from systems theory, probability theory andconvex geometry.This thesis is divided into three parts: Introduction and literature review material in PartI, methodological tools which were used in this thesis in Part II, and the rest consists of threeresearch papers among which the first one has been published in a collected volume underSpinger publications and the other two have been published in Institute of Mathematics andits Applications journal (IMA), in Part III.In Part I and especially in Chapter 1 we present an introduction and we provide information related to the motivation and the structure of this thesis. In addition in Chapter 2 we havea literature review about what other scientists have done so far on topics like mathematicalcontrol theory, diffusion process, graphs/networks and random matrix theory.The methodological tools needed for the derivation of the main research results presentedin Part III are given in Part II, Chapter 3. Furthermore, in this part, topics and notions frommathematical control theory, like controllability/observability, are presented. Moreover, wedescribe not only the meaning of RMT but also the the given examples of what a randommatrix is and what makes it so special. Another topic that is covered in this Chapter is graphtheory and examples about connected, directed and weighted graphs/networks and algebraicproperties of graphs are given. Finally, the topic of diffusion process is presented as well.Part III constitutes the most important part of this research study which is presented inthis thesis. Thus, in Chapter 4, we describe the diffusion process on connected, directed andweighted interbank loan networks. We prove that diffusion can drive the bank network to itssteady state where the leverages of all the banks of the network are equal. Diffusion in thebank network is modelled with a system of coupled ordinary linear differential equations. Thelast argument, we ended up with two different methods to the same results and a great success.In Chapter 5, we present the notion of almost zero of polynomial vectors and matrices in termsof a minimization problem. The purpose of this chapter is twofold: (a) Firstly, to explorefurther the algebraic properties and computations of almost zeroes of polynomial vectors ormatrices and present new results of higher order almost zeros; (b) Secondly, to relate thenotion of almost zero to that of randomness i.e. what are the statistical characteristics ofthe almost zero and its norm polynomial value in a given random ensemble of polynomials.In Chapter 6, we consider random systems where the parameters which are matrices havebeen replaced with random matrices. Moreover, we define the ε-uncontrollability for randomsystems and we also describe the GOE. In addition, we give the detailed calculation of theε-uncontrollability of a random system where the matrix A comes from the GOE