Archivio 2013

Abstract We provide some explicit examples of nonlocal s-minimal surfaces for 0<s<1, namely stationary for the s-perimeter functional, introduced first in the form of a singular integral, related to fractional Laplacian, by Caffarelli, Roquejoffre and Savin, which recovers the classical concept in the limit s-->1. We find and analyze stability of a class of Lawson cones which shows important differences with the classical case. In addition we construct an axially symmetric analog of the catenoid in R3 in the limiting case when s approaches 1.

Discretization Program of the famous Completely Integrable Systems and associated Linear Operators was developed in 1990s. In particular, specific properties of the second order difference operators on the triangulated manifolds were studied in the works of S.Novikov and I.Dynnikov since 1996. They involve factorization of operators, the so-called Laplace Transformations, New Discretization of Complex Analysis and New Discretization of $GL_n$ Connections on the triangulated $n$-manifolds. The general theory of the new type discrete $GL_n$ connections was developed. However, the special case of $SL_n$-connections was not selected properly. Indeed, it appears in the theory of important self-adjoint operators. In the present work we construct a Theory of SL_2 discrete connections on the triangulated 2-manifolds. They are deeply associated with real self-adjoint difference operators similar to complex line bundles (magnetic fields) in the 2nd order Schrodinger operators on the plane in the standard continuous quantum mechanics.

We study eigenfunction decomposition for periodic 1-dimensional Schroedinger with real singular finite-gap potentials. Many ''real'' inverse spectral data for periodic finite-gap operators(consisting of Riemann Surface with marked ''infinite point'', local parameter and divisors of poles) lead to operators with real but singular coefficients. These operators cannot be considered as self-adjoint in the ordinary (positive) Hilbert spaces of functions of x. In particular, it is true for the special case of Lame' operators with elliptic potential n(n+1)P(x) where eigenfunctions were found in XIX Century by Hermit. However, such Baker-Akhiezer (BA) functions present, right analog of the Discrete and Continuous Fourier Bases on Riemann Surfaces. It turns out that these operators for the nonzero genus are symmetric in the indefinite inner product, described in this work. The analog of Continuous Fourier Transform is an isometry in this inner product. Its image in the space of functions of the real variable x is described. The results presented have been obtained in collaboration with S.P. Novikov.

Direct channeling of intermediates via a physical tunnel between enzyme active sites is an established mechanism to improve metabolic efficiency. We present a theoretical model that demonstrates that coclustering multiple enzymes into proximity can yield the full efficiency benefits of direct channeling. The model predicts the separation and size of coclusters that maximize metabolic efficiency, and this prediction is in agreement with the spacing between coclusters in yeast and mammalian cells. The model also predicts that enzyme agglomerates can regulate steady–state flux division at metabolic branch points. We experimentally test this prediction for a fundamental branch point in Escherichia coli: The results confirm that enzyme colocalization within an agglomerate can accelerate the processing of a shared intermediate by one branch. Our studies establish a quantitative framework to understand coclustering-mediated metabolic channeling and its application to both efficiency improvement and metabolic regulation

Nella prima parte illustreremo la costruzione delle W-algebre classiche affini. Quindi descriveremo la loro applicazione nella teoria dei sistemi integrabili, ottenendo le cosiddette gerarchie di Drinfeld-Sokolov generalizzate.

Dynamical zeta functions of Selberg, Smale and Ruelle are analogous to the Riemann zeta function with the product over primes replaced by products over closed orbits of Anosov flows. In 1967 Smale conjectured that these zeta functions should be meromorphic but admitted "that a positive answer would be a little shocking". Nevertheless the continuation was proved in 2012 by Giulietti-Liverani-Pollicott. In my lecture I will present a proof of this result obtained by Dyatlov and myself and inspired by a trace formula of Guillemin and by recent work of Faure-Sjoestrand. It is based on a simple idea involving wave front sets and propagation of singularities: we apply methods of microlocal analysis to the generator of the flow, an in particular propagation results due to Duistermaat-Hoermander, Melrose and Vasy.

Le equazioni differenziali (ordinarie o alle derivate parziali) in presenza di perturbazioni aleatorie presentano delle difficoltà di analisi dovute alla natura singolare delle traiettorie tipiche delle perturbazioni. Il calcolo stocastico è stato introdotto da Itô negli anni '40 come quadro teorico all'interno del quale formulare rigorosamente tali equazioni (interpretandole come equazioni differenziali stochastiche). Negli anni '90 T. Lyons ha introdotto la teoria dei "rough paths", una teoria alternativa che presenta dei vantaggi rispetto al calcolo stocastico e permette di trattare alcune equazioni che non sono abbordabili con la teoria di Itô. Recentemente si sono trovati degli approcci teorici che permettono di trattare una classe ancora più ampia di equazioni e in particolare varie EDP. In questo seminario introdurrò uno di questi approcci, quello delle distribuzioni para-controllate, e mostrerò come lo si possa usare per risolvere alcune EDP aleatorie singolari. Dal punto di vista astratto le distribuzioni para-controllate forniscono un modo non-ambiguo di estendere certe operazioni non-lineari alle distribuzioni. Il punto chiave dell'approccio è l'uso dei paraprodotti, introdotti da J.M. Bony negli anni '80 nell'ambito dell'analisi microlocale delle PDE nonlineari. Il seminario non richiede alcuna familiarità con il calcolo stocastico ed è pensato per essere fruibile da analisti e probabilisti.

One of earliest goals of knot theory was the development of a knot tables. A knot table consist of a list of prime knots (knots that are not connected sums of non-trivial knots) that are sorted by their crossing number. Classically such tables were constructed combinatorially using knot invariants to distinguish inequivalent pairs (such as the Alexander and Jones polynomial). Moving apart from the classical knot theory in S^3, knots have been so far classified only in the projective space up to 6 crossings. We will present the computational methods used to construct knot tables of non-affine prime knots in the solid torus and the infinite family of lens spaces L(p,q). In addition, we will establish which of these knots are amphichiral. We will take a brief look at why classical knot invariants are very weak at distinguishing between these knots and will provide stronger invariants, namely the Kauffman bracket skein module and the HOMLFY skein module.

The notion of an orthogonal polynomial ensemble generalizes many important point processes arising in random matrix theory, probability and combinatorics. The most famous example perhaps is that of the eigenvalue distributions of unitary invariant ensembles (such as GUE) of random matrix theory. Remarkably, the study of fluctuations of these point processes is intimately connected to the study of one dimensional discrete Schroedinger operators. This talk will review recent work (partly joint with Maurice Duits) elucidating and exploiting this connection in the context of universality, laws of large numbers and central limit theorems.

Il Prof. Lucio Russo parlerà del suo libro "L'America dimenticata. I rapporti tra le civiltà e un errore di Tolomeo".

There is a wide class of quantum many body systems whose elementary excitations are well described in terms of massless Dirac particles; the most celebrated example is Graphene, but other examples include Spin Chains or Weyl semimetals. The methods developed in the last years in Constructive Quantum Field Theory and based on Renormalization Group are quite suitable to study such systems, and can provide mathematically rigorous results. In particular I will describe recent universality results obtained for the conductivity written in terms of Kubo formula, which in certain cases solved some controversy present in the physical literature.

Vengono presentati dei risultati preliminari sulla formazione di sub-shock in soluzioni di tipo struttura d'urto in miscele di gas multi-temperatura. Le miscele sono descritte da differenti modelli termodinamici. Sono considerate, infatti, sia miscele inerti di tre gas di Eulero, sia chiusure a livello dei 10 e 13 momenti delle equazioni di Boltzmann per miscele di quattro specie inerti o reagenti soggette a una reazione bimolecolare reversibile.

Nichols algebras are Hopf algebras in braided tensor categories with very particular properties; for instance, the positive parts of quantized enveloping algebras at a generic parameter, and their finite-dimensional counterparts when the parameter is a root of one, are Nichols algebras. The input datum to define a Nichols algebra is a braided vector space, that is a solution of the braid equation or equivalently of the quantum Yang-Baxter equation. Nichols algebras of diagonal type are by definition those corresponding to solutions of the braid equation given by a perturbation of the usual transposition given by a matrix of non-zero scalars. These Nichols algebras appear in the classification of pointed Hopf algebras with abelian group, within the method proposed by Hans-Jürgen Schneider and myself.

Il workshop si svolgera' da lunedi' 18 novembre a venerdi' 22 novembre. Ogni giorno in sala riviste a partire dalle 11.

In this talk we will discuss the existence and asymptotic behavior of solution to the Q-curvature problem in $\mathbb{R}^{n}$. The solutions satisfy a non-local differential equation with exponential non-linearity, generalizing the problem of Gaussian curvature in dimension 2. We first classify the solutions and give their asymptotics and then we prove the existence of non-spherical solutions to the problem. PS : this is a joint work with T.Jin, L.Martinazzi and J.Xiong.

Come ogni anno verranno presentati gli argomenti trattati nei Laboratori che si intendono organizzare nell'ambito del Piano Lauree Scientifiche.

A classical result of Kinchin says that for almost every real number x, the geometric mean of the first n digits in the continued fraction expansion of x converges to a number K=2.685... as n tends to infinity. On the other hand, for almost every x, the arithmetic mean of the first n digits tends to infinity. There is a sequence of refinements of the classical Arithmetic Mean - Geometric Mean inequality (called MacLaurin's inequalities) involving the k-th root of the k-th elementary symmetric mean, where k ranges from 1 (arithmetic mean) to n (geometric mean). We analyze what happens to these means for typical real numbers, when k is a function of n. We obtain sufficient conditions to ensure convergence / divergence of such means. Joint work with Steven J. Miller and Jake L. Wellens.

Le relazioni di Dehn-Sommerville generalizzate sono un insieme di generatori per lo spazio delle relazioni tra i coefficienti degli f-vettori di bandiera dei politopi convessi e degli intervalli di Bruhat. Mostreremo esplicitamente le relazioni che si ottengono applicando una trasformazione di tipo inclusione-esclusione a queste relazioni e diverse applicazioni di questa formula per gli indici cd di poset euleriani e per i polinomi di Kazhdan-Lusztig. In particolare otteniamo una nuova formula non-ricorsiva per i polinomi di Kazhdan-Lusztig valida in completa generalità e mostriamo che questa formula non può essere ulteriormente semplificata. Questo lavoro è in collaborazione con F. Brenti (Roma)

In the 50's, the consensus was that all linear PDEs were solvable. Therefore it came as a surprise Hans Lewy in 1957 presented a non-vanishing complex vector field that is not locally solvable. Actually, the vector field is the tangential Cauchy-Riemann operator on the boundary of a strictly pseudoconvex domain. Hormander then proved in 1960 that almost all linear partial differential equations are not locally solvable. After a rapid development in the 60's Nirenberg and Treves formulated their famous conjecture in 1970: that condition (PSI) is necessary and sufficient for the local solvability of differential equations of principal type. Principal type essentially means simple characteristics, and condition (Psi) only involves the sign changes of the imaginary part of the highest order terms along the bicharacteristics of the real part. The Nirenberg-Treves conjecture was finally proved in 2006. We shall present the background, the main results, some examples and generalizations to systems of differential equations. We shall also study the propagation of singularities for operators of principal type satifying the more restrictive condition (P), for which we have complete results.

In the 50's, the consensus was that all linear PDEs were solvable. Therefore it came as a surprise Hans Lewy in 1957 presented a non-vanishing complex vector field that is not locally solvable. Actually, the vector field is the tangential Cauchy-Riemann operator on the boundary of a strictly pseudoconvex domain. Hormander then proved in 1960 that almost all linear partial differential equations are not locally solvable. After a rapid development in the 60's Nirenberg and Treves formulated their famous conjecture in 1970: that condition (PSI) is necessary and sufficient for the local solvability of differential equations of principal type. Principal type essentially means simple characteristics, and condition (Psi) only involves the sign changes of the imaginary part of the highest order terms along the bicharacteristics of the real part. The Nirenberg-Treves conjecture was finally proved in 2006. We shall present the background, the main results, some examples and generalizations to systems of differential equations. We shall also study the propagation of singularities for operators of principal type satifying the more restrictive condition (P), for which we have complete results.

Recently some attention has been attracted to the investigation of random fields on the sphere. This mostly because of the need of producing a reasonable model for the analysis and the statistical estimation of the data of the Cosmic Microwave Background, a cosmic radiation whose behavior gives important information on the Big-Bang. The most natural approach is to develop a random field into a Fourier series with respect to a natural orthonormal basis of the square integrable functions on the sphere (the spherical harmonics). One of the typical issues of the Big-Bang theory is that this cosmic radiation should be rotationally invariant (i.e. that its distribution is invariant by rotation). In this talk we first consider the problem of characterizing the joint distribution of the Fourier coefficients of a random field which is rotationally invariant. In particular we shall prove that if the Fourier coefficients are independent the they are necessarily Gaussian (and the random field is Gaussian too). Consequences, extensions and applications will follow.

In the 50's, the consensus was that all linear PDEs were solvable. Therefore it came as a surprise Hans Lewy in 1957 presented a non-vanishing complex vector field that is not locally solvable. Actually, the vector field is the tangential Cauchy-Riemann operator on the boundary of a strictly pseudoconvex domain. Hormander then proved in 1960 that almost all linear partial differential equations are not locally solvable. After a rapid development in the 60's Nirenberg and Treves formulated their famous conjecture in 1970: that condition (PSI) is necessary and sufficient for the local solvability of differential equations of principal type. Principal type essentially means simple characteristics, and condition (Psi) only involves the sign changes of the imaginary part of the highest order terms along the bicharacteristics of the real part. The Nirenberg-Treves conjecture was finally proved in 2006. We shall present the background, the main results, some examples and generalizations to systems of differential equations. We shall also study the propagation of singularities for operators of principal type satifying the more restrictive condition (P), for which we have complete results.

This paper investigates the effects of the ”uptick rule” (a short selling regulation formally known as rule 10a-1) by means of a simple stock market model, based on the ARED (adaptive rational equilibrium dynamics) modeling framework, where heterogeneous and adaptive beliefs on the future prices of a risky asset were first shown to be responsible for endogenous price fluctuations. The dynamics of stock prices generated by the model, with and without the uptick-rule restriction, are analyzed by pairing the classical fundamental prediction with beliefs based on both linear and nonlinear technical analyses. The comparison shows a reduction of downward price movements of undervalued shares when the short selling restriction is imposed. This gives evidence that the uptick rule meets its intended objective. However, the effects of the short selling regulation fade when the intensity of choice to switch trading strategies is high. The analysis suggests possible side effects of the regulation on price dynamics. JEL codes: C62, G12, G18 Keywords: Asset pricing model, Heterogeneous beliefs, Endogenous price fluctuations, Piecewise-smooth dynamical systems, Chaos, Uptick-rule.

Although hunting for chaos is not as popular as it used to be, we dedicate this paper to the presentation of a new strange attractor. It concerns the most important dynamic process in our life, that is the emergence and evolution of love in interpresonal relationships. More precisely, we show that a romantic relationship can be unpredictable - chaotic - on the sole basis of the interplay of the characters involved. To do this, we analyse the celebrated triangular love story described in Jules et Jim, a novel by Henri-Pierre Roché, using a mathematical model. As in physics and biology, the theory of coupled oscillators proves to be a powerful tool in identifying minor but strategically important factors - here jealousy and complaisance - as sources of unpredictability. Our results also highlight the genius of François Truffaut, whose film - one of the masterpieces on love and friendship - is based on the novel. Truffaut successfully reproduced the turbulence of the feelings and the uncertainty over their future by systematically exploiting the metaphor of stretching and folding, both visually and in the scripts, and especially with the song "Le tourbillon de la vie" - a beautiful hymn to chaos. And all this was done in 1961, two years before the publication of Lorenz's paper!

In this talk we will discuss some results obtained by Gerard Le Caer (J Stat Phys 2010, 2011) concerning Dirichlet random walks, namely convex combinations of points on a sphere with weights having Dirichlet distribution. The proofs are based on a kind of Stieltjes transform, suited for distributions on a ball, which has special properties with regard to the Dirichlet distribution. A limiting regime is considered, from which the means of Dirichlet processes come out.

While the empirical and theoretical industrial organization literature has traditionally focussed on the growth rate of firm size, we take the position that profit rates, measured as the ratio of operating income to total assets, are economically more fundamental than logarithmic time differences in firm size, and hence should attain more attention in economic inquiry. The reason why we consider the profit rate so important from an economic viewpoint is that the economy is ultimately driven by the reallocation of capital, and investment decisions are governed by the rate of return per unit of invested capital. Additionally, profit rates exhibit peculiar time series and cross-sectional properties that make them appear an adequate quantity to study the complex interactions of competitive firms in a statistical equilibrium framework. Considering a sample of more than 500 long-lived publicly-traded U.S. companies, the purpose of this contribution is to explore the statistical features of profit rates, and to compare our findings to the properties of another very prominent degree of freedom for this problem: the growth rate of firm size. We find that the empirical densities of both quantities can be reasonably well described by the family of Subbotin distributions, but there are characteristic differences in their autocorrelation structures. Moreover, while there is a negative effect of size on the mean absolute deviation of growth rates, it turns out that neither location nor the dispersion of profit rates depends of firm size. In the second part of this paper, we employ the statistical equilibrium model of competitive firms proposed by Alfarano et al. (2012) to model the dynamic evolution of profit rates and show that this particular stochastic process reproduces the statistical characteristics of the profit rate time series. Moreover, we use the solution of the transient density of this process to estimate the diffusion coefficient, which allows us to measure the characteristic time scale for the dissipation of excess profits.

I will present some results regarding minimizers and interfaces of local and nonlocal Ginzburg-Landau functionals perturbed by an additive external random field. ( Results obtained in collaboration with Nicolas Dirr)

Let $(S,d)$ be a metric space, $\mathcal{G}$ a $\sigma$-field on $S$ and $(\mu_n:n\geq 0)$ a sequence of probabilities on $\mathcal{G}$. Suppose $\mathcal{G}$ countably generated, the map $(x,y)\mapsto d(x,y)$ measurable with respect to $\mathcal{G}\otimes\mathcal{G}$, and $\mu_n$ perfect for $n>0$. Say that $(\mu_n)$ has a Skorohod representation if, on some probability space, there are random variables $X_n$ such that \begin{equation*} X_n\sim\mu_n\text{ for all }n\geq 0\quad\text{and}\quad d(X_n,X_0)\overset{P}\longrightarrow 0. \end{equation*} It is shown that $(\mu_n)$ has a Skorohod representation if and only if \begin{equation*} \lim_n\,\sup_f\,\abs{\mu_n(f)-\mu_0(f)}=0, \end{equation*} where $\sup$ is over those $f:S\rightarrow [-1,1]$ which are $\mathcal{G}$-universally measurable and satisfy $\abs{f(x)-f(y)}\leq 1\wedge d(x,y)$. An useful consequence is that Skorohod representations are preserved under mixtures. The result applies even if $\mu_0$ fails to be $d$-separable. Various possible applications of such result are discussed.

Le soluzioni di PDE stocastiche tipicamente dipendono da tempo, spazio e caso. L'esistenza di densità per la distribuzione è una forma probabilistica di regolarità. Nel corso del seminario sarà presentata una tecnica per l'esistenza di densità rispetto alla misura di Lebesgue per le marginali finito-dimensionali della legge della soluzione delle equazioni di Navier-Stokes in dimensione 3 a tempo fissato. Si vedrà inoltre che tali densità posseggono una regolarità (minimale) in spazi di Besov. Metodi ormai classici come il calcolo di Malliavin non sembrano funzionare per questo problema, e questo è una caratteristica intrinseca del problema in esame. In effetti l'esistenza di densità deriva da un metodo probabilistico ad-hoc che si inserisce tra i recenti risultati (Kohatsu-Higa, Bally-Caramellino tra gli altri) sviluppati per ovviare a problemi di non-derivabilità di Malliavin. Si discuterà infine una limitata (qualitativa) estensione al caso ipoellittico, ovvero di rumore (ad esempio) finito-dimensionale. (Lavoro in collaborazione con A. Debussche, ENS Rennes)

We discuss the problem of modeling the emergence of coordinated movements of a network made of a large number of self-propelled particles (agents) in terms of microscopic decisions of each individual member. The emergence of phase space patterns persistent in time described by a large connected cluster of coherently moving particles is called flocking or swarming. We introduce the basic models exhibiting such a behaviour and review the major results about proving its stability for finite size systems as well as, in the kinetic limit, we show that the particle density satisfies Vlasov type equations under suitable conditions on the interaction.

We point out a close connection between the Moser-Tardos algorithmic version of the Lovász Local Lemma, a central tool in probabilistic combinatorics, and the cluster expansion of the hard core lattice gas in statistical mechanics. We show that the notion of witness trees given by Moser and Tardos is essentially coincident with that of Penrose trees in the Cluster expansion scheme of the hard core gas. Such an identification implies that the Moser Tardos algorithm is successful in a polynomial time if the cluster expansion converges.

  Motivated by models of vegetation patterns in the life-science literature, where a remarkable form of `robust critical behaviour' was suggested, we study percolation phenomena under the invariant measures of certain contact-like processes. We show that the percolation transition is sharp. (For one of the models with three states this is done under an assumption which will be explained in the talk). This sharpness seems to contradict the above mentioned robustness.  In Part (I) I will introduce the models, give some percolation background, and discuss weak-influence inequalities. Part (II) will present the above mentioned results, which were recently obtained with Jakob Bjornberg and Markus Heydenreich.

We describe the theory of admissible systems for semisimple Lie groups and we describe its relation with hermitian symmetric spaces.

Motivated by models of vegetation patterns in the life-science literature, where a remarkable form of `robust critical behaviour' was suggested, we study percolation phenomena under the invariant measures of certain contact-like processes. We show that the percolation transition is sharp. (For one of the models with three states this is done under an assumption which will be explained in the talk). This sharpness seems to contradict the above mentioned robustness.  In Part (I) I will introduce the models, give some percolation background, and discuss weak-influence inequalities. Part (II) will present the above mentioned results, which were recently obtained with Jakob Bjornberg and Markus Heydenreich.

We discuss the nature of aperiodically ordered systems, in terms of their dynamical spectrum  and their diffraction spectrum, and compare those descriptions. We discuss how aperiodic order can arise in ground states and Gibbs states of lattice models. Moreover we discuss how aperiodically ordered systems can provide examples of types of order expected in disordered systems, such as (spin) glasses.

We introduce spatially explicit stochastic processes to model multispecies host-symbiont interactions. The host environment is static, modeled by the infinite percolation cluster of site percolation. Symbionts evolve on the infinite cluster through contact or voter type interactions, where each host may be infected by a colony of symbionts. In the presence of a single symbiontspecies, the condition for invasion as a function of the density of the habitat of hosts and the maximal size of the colonies is investigated in details. In the presence of multiple symbiont species, it is proved that the community of symbionts clusters in two dimensions whereas symbiont species may coexist in higher dimensions. From a joint work with N.Lanchier and F.Zucca.

We consider Gibbs measures for Ising models on Cayley trees, subjected to an infinite-temperature Glauber dynamics (independent spin flips). We discuss similarities and differences with the situation on regular lattices. In particular we find that on trees there is the possibility of different Gibbs measures acquiring different Gibbsian properties, and the possibility of measures becoming "totally non-Gibbsian" in the sense that all spin configurations become points of discontinuity for some conditional probability. This is based on joint work with V.Ermolaev, G.Iacobelli and C.Kuelske

Bootstrap percolation models are Cellular Automata with probabilistic initial conditions. We discuss some results and open problems on  the influence of anisotropy on properties of bootstrap percolation models in two and three dimensions. In particular we discuss  finite-size scaling behaviour and  sharp thresholds. (Joint work with Tim Hulshof, Anne Fey, Hugo Duminil-Copin). 

Empirical evidence in financial markets suggests that investors and fund managers use combinations of fixed and switching strategies based on various fundamental and technical analysis when making complicated investment decisions. By estimating a simple asset pricing model of heterogeneous agents to characterize the power-law behavior the DAX 30 from 1975 to 2007, we show that the market is dominated by the adaptive investors who constantly switch between the fundamental and trend following strategies, though there are about 30% of investors who never change their strategies over the time. By conducting econometric analysis via Monte Carlo simulations, we show that the autocorrelation patterns, the estimates of the power-law decay indices, (FI)GARCH parameters, and tail index of the model match closely the corresponding estimates for the DAX 30. The results provide further evidence on the explanatory power of heterogeneous agent models.

Bootstrap percolation models are Cellular Automata with probabilistic initial conditions. We discuss some results and open problems on  the influence of anisotropy on properties of bootstrap percolation models in two and three dimensions. In particular we discuss  finite-size scaling behaviour and  sharp thresholds. (Joint work with Tim Hulshof, Anne Fey, Hugo Duminil-Copin). 

The boundary of the complex hyperbolic space may be identified with the usual Heisenberg group; this is a particular example of a Kahler-Einstein manifold having a CR manifold as boundary. We will consider the construction of the CR fractional Laplacian through an extension problem, in the same spirit as the Caffarelli-Silvestre exension for the real case. This allows to set up a fractional CR Yamabe problem.

We deal with the asymptotic behavior of the s-perimeter of a set E inside a domain U as s->0. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of E and U. Moreover, we construct examples of sets for which the limit does not exist. This is a joint work with A. Figalli, G. Palatucci, E. Valdinoci.

I relatori discuteranno i testi: Giuseppe Longo, Maël Montévil Extended Criticality, Phase Spaces and Enablement in Biology.Special Issue of Chaos, Solitons and Fractals, 2013. http://www.di.ens.fr/users/longo/files/CIM/ExtCritEnable.pdf Giuseppe Longo, Maël Montévil The Inert vs. the Living State of Matter: Extended Criticality, Time Geometry, Anti-Entropy - an overview. special issue of Frontiers in Physiology, n. 3, 39, 2012. http://www.di.ens.fr/users/longo/files/CIM/inertVSlife.pdf Davide Barbieri, Giovanna Citti, Giacomo Cocci, Alessandro Sarti A cortical-inspired geometry for contour perception and motion integration http://arxiv.org/abs/1301.3433 A. Sarti G. Citti On the Origin and Nature of Neurogeometry La nuova critica 2011 http://www.dm.unibo.it/~citti/curri/neurogeometry.pdf

il seminario e' ristretto ai soli partecipanti al progetto

il seminario e' ristretto ai soli partecipanti al progetto.

We review the basic theory of super G-spaces. We prove a theorem relating the action of a super Harish-Chandra pair  on a supermanifold to the action of the corresponding super Lie group G. The   proposed formulation does not use super Frobenius’ theorem but relies on Koszul’s realization of the structure sheaf of a super Lie group.

I will give an overview talk demonstrating links between pure analysis, such as AAK-theory for Hankel operators, and very applied topics such as how to efficiently compress a piece of music or how to process large volumes of seismic measurements.

This is joint work with Bernhard Keller, Daniel Labardini-Fragoso and Pierre-Guy Plamondon. We consider a skew-symmetric cluster algebra A. Then A is associated with a quiver Q and its combinatorics is "categorified" by the generalized cluster category associated with Q. By using such categorification we show that the cluster monomials of A are linearly independent, proving an old conjecture of Fomin and Zelevinsky. We also show that the exchange graph of A and its cluster complex are independent on the choice of coefficients, confirming another conjecture of Fomin and Zelevinsky. I will first give an introduction to additive categorifications of cluster algebras and then I will give the proof of these results.

Giuseppe Longo (ENS, Paris) e Alessandro Sarti (EHESS, Paris) discuteranno i testi Giuseppe Longo, Maël Montévil. The Inert vs. the Living State of Matter: Extended Criticality, Time Geometry, Anti-Entropy - an overview. special issue of Frontiers in Physiology, n. 3, 39, 2012. http://www.di.ens.fr/users/longo/files/CIM/inertVSlife.pdf Giuseppe Longo, Maël Montévil. Extended Criticality, Phase Spaces and Enablement in Biology.Special Issue of Chaos, Solitons and Fractals, 2013. http://www.di.ens.fr/users/longo/files/CIM/ExtCritEnable.pdf Davide Barbieri, Giovanna Citti, Giacomo Cocci, Alessandro Sarti A cortical-inspired geometry for contour perception and motion integration http://arxiv.org/abs/1301.3433 Alessandro Sarti Giovanna Citti On the Origin and Nature of Neurogeometry La nuova critica 2011 A. Sarti G. Citti on the Origin and Nature of Neurogeometry preprint

The topological complexity of motion-planning algorithms can be viewed as a fibrewise Lusternik-Schnirelmann category of certain fibrewise pointed spaces. We are going to describe a general change-of-fibre construction on fibrewise pointed spaces that allows the extension of standard methods from LS category to the topological complexity.

This short course is offered within the Workshop in Quantitative Finance (timetable: 9 May, 9-13; 10 May, 14-18). It reviews recent option pricing literature while placing a special emphasis on non-parametric pricing methodologies. In addition,it provides novel insights into the accuracy of option pricing models during the recent 2008 credit crisis. A direct comparison of alternative models is also made in regards to the 1987 stock market crash.

In questo seminario parlerò di alcuni risultati ottenuti in collaborazione con Enrico Le Donne (Università di Jyvӓskylӓ) nell'ambito delle isometrie dei gruppi di Carnot, e più in generale di varietà subFinsler. Il problema a cui ci siamo interessati è stato quello di caratterizzare le isometrie nei suddetti contesti metrici. Dopo aver fissato la notazione, ricorderò alcuni risultati presenti in letteratura sull'argomento. Quindi enuncerò i teoremi che costituiscono il nostro contributo originale al problema. Infine, se ci sarà tempo, accennerò alle tecniche usate nella dimostrazione.

Il seminario proseguirà nei giorni 2 e 3 maggio, ore 14, ed è parte del Workshop in Quantitative Finance (Laurea Magistrale Quantitative Finance)

The geometric theory of symmetric spaces has a close relation with the Harish-Chandra modules, that is the representations of the pair consisting of a real semisimple group G of classical type and its maximal compact subgroup. Such representations are realized in the Frechet space of the sections of certain line bundles over the quotient of the complexification of G modulo a borel subgroup. We are going to examine the general theory and to discuss some examples. At the end we mention some work in progress on the extension of all of these results to supergeometry (joint collaboration with C. Carmeli, University of Genova, and V. S. Varadarajan, UCLA).

This talk develops some aspects of stochastic calculus via regularization for processes with values in a general Banach space $B$. A new concept of quadratic variation which depends on a particular subspace is introduced. An It\^o formula and stability results for processes admitting this kind of quadratic variation are presented. Particular interest is devoted to the case when $B$ is the space of real continuous functions defined on $[-T,0]$, $T>0$ and the process is the window process $X(\cdot)$ associated with a continuous real process $X$ which, at time $t$, it takes into account the past of the process. If $X$ is a finite quadratic variation process (for instance Dirichlet, weak Dirichlet), it is possible to represent a large class of path-dependent random variable $h$ as a real number plus a real forward integral in a semiexplicite form. This representation result of $h$ makes use of a functional solving an infinite dimensional partial differential equation. This decomposition generalizes, in some cases, the Clark-Ocone formula which is true when $X$ is the standard Brownian motion $W$. Some stability results will be given explicitly.\\ This is a joint work with Francesco Russo (ENSTA ParisTech Paris)

We study two families of polynomials that play the same role in the Temperley– Lieb algebra of a Coxeter group as the Kazhdan–Lusztig and R–polynomials play in the Hecke algebra of the group. Our results include recursions, non– recursive formulas, symmetry properties and expressions for the constant term. We focus mainly on non–branching Coxeter graphs.

We study two families of polynomials that play the same role in the Temperley– Lieb algebra of a Coxeter group as the Kazhdan–Lusztig and R–polynomials play in the Hecke algebra of the group. Our results include recursions, non– recursive formulas, symmetry properties and expressions for the constant term. We focus mainly on non–branching Coxeter graphs.

Il gruppo del vertice tropicale (introdotto da Kontsevich e altri) è generato da una classe di simplettomorfismi formali di C* x C*. Esso gioca un ruolo importante in alcuni problemi di geometria algebrica e fisica matematica, e può essere studiato con varie tecniche. Esistono approcci basati sull'enumerazione di curve razionali, rispettivamente sulla topologia di certi spazi di rappresentazioni. Darò un'introduzione elementare a questi risultati, menzionando una particolare dualità ottenuta in collaborazione con M. Reineke e T. Weist. Nella seconda parte vorrei approfondire una applicazione del vertice tropicale ad una equazione di tipo "ansatz di Bethe termodinamica" (lavoro in corso con S. A. Filippini). Alcuni lavori in ambito fisico-matematico prevedono che questa sia strettamente legata alla geometria di una classe di fibrati di Higgs.

Inizialmente introdotto (25 anni fa) da M. Culler e K. Vogtmann per studiare la coomologia di Out(F_n), l'outer space di un gruppo libero puo' essere visto come l'analogo dello spazio di Teichmuller di una superficie. Il seminario sara' incentrato sulla definizione e le proprieta' basilari dell'outer space.

In this talk we address the solution of quasilinear elliptic problems of monotone type by using the Mimetic Finite Difference (MFD) method. Under a suitable approximation assumption, we prove that the MFD approximate solution converges to the exact solution in a mesh-dependent energy norm. The resulting nonlinear discrete problem is then solved iteratively via linearization by applying the Kacanov method, which is shown to be convergent. Several numerical experiments confirm the theoretical analysis. This is a joint work with P.F. Antonietti (PoliMi) and N. Bigoni (PoliMi).

The BK inequality says, roughly speaking, that the probability that two events `occur disjointly' is smaller than or equal to the product of the two individual probabilities. It has been used fruitfully in percolation and related topics. Until recently, no inequalities of this type were known for situations where the underlying random variables are dependent. However, in the last two years considerable progress has been made in this direction. This talk will start with an introduction and then highlight these new developments. Partly based on joint work with Johan Jonasson and joint work with Alberto Gandolfi.

It is well-known from work by Kesten in the mid-eighties and work by Borgs, Chayes, Kesten and Spencer around 2000 that, informally speaking, the largest, the second largest, the third largest etcetera cluster in an $n$ times $n$ box in 2D critical percolation typically have size of order $n^2 \pi(n)$. Here $\pi(n)$ is the probability that a given vertex v has an open path to vertices at distance at least $n$ from v. Motivated by so-called frozen percolation problems (which I will briefly explain), we obtained some new modifications/refinements of these results. I will also address our work on an open problem in this area posed in a paper by Jarai (2003). Part of this talk is based on joint work with R. Conijn and D. Kiss.

We test the ability of the Cyclically Adjusted Price Earnings (CAPE) ratio introduced by Robert Shiller to predict the future long run performances of international stock markets. We devote the first part of the seminar to the empirical analysis of numerous equity indices. We question if the striking ability of CAPE to predict returns of the index is a genuine effect or if it is a spurious consequence of the persistence and endogeneity of the regressor. The evidence rooted on a bootstrap analysis of the regression plays in favour of statistically significant predictability of long run yields for the US market (Standard & Poor and NYSE monthly data are available since 1870 and 1926, respectively). The same analysis extended to ten different countries is less conclusive. We believe that this is largely due to fact that the time series employed in the analysis are quite short (MSCI data are recorded since January 1970). In the second part, we provide a theoretical justification of the empirical observations from the US market. We propose a simple model of the price dynamics in which the return growth depends on three components: a) a momentum component, naturally justified in terms of agents' belief that expected returns are higher in bullish markets than in bearish ones; b) a fundamental component proportional to the log earnings over price ratio at time zero. The initial value of the ratio determines the reference growth level, from which the actual stock price may deviate as an effect of random external disturbances, and c) a driving component ensuring the diffusive behaviour of stock prices. Under these assumptions, we are able to prove that, if we consider a sufficiently large number of periods, the expected rate of return and the expected gross return are linear in the initial time value of the log earnings over price ratio, and their variance goes to zero with rate of convergence equal to minus one.

La ricerca di una definizione di funzione regolare sui quaternioni che estendesse la nozione di funzione olomorfa di una variabile complessa ha prodotto teorie differenti. L'esempio sicuramente più conosciuto è quello delle funzioni Fueter regolari, introdotte dal matematico svizzero Fueter negli anni 30 del secolo scorso. Più recentemente, nel 2006, Gentili e Struppa, ispirandosi ad un lavoro di Cullen del 1965, hanno dato la nuova definizione di funzione slice regolare, che ha dato vita ad una teoria piuttosto ricca, e tuttora in grande sviluppo. In questo seminario, dopo aver introdotto la classe delle funzioni slice regolari ed averne mostrato alcune proprietà fondamentali, ci concentreremo su risultati di teoria geometrica sulla palla unitaria dei quaternioni. In particolare vedremo come alcuni risulati classici si siano potuti estendere in questo nuovo ambito.

We present a unifying algebraic representation for orthogonal polynomial sequences both in the univariate and multivariate case. This representation gives new formulae for orthogonal polynomials arising from probability and alternative proof of the Favard’ theorem. Moreover orthogonal polynomials are connected via cumulants to invariant theory of binary forms.

In this talk we provide global subelliptic estimates for the linearized inhomogeneous Boltzmann equation without angular cutoff, and show that some global gain in the spatial direction is available although the corresponding operator is not elliptic in this direction. The proof is based on a multiplier method and the so-called Wick quantization, together with a careful analysis of the symbolic properties of the Weyl symbol of the Boltzmann collision operator. This is a joint work with R. Alexandre and F.Herau.

This Workshop is the 14th edition of a successful event which is held each year in an Italian city. It is an important meeting point for scholars and practitioners working in Mathematical Finance.This year 38 talks are scheduled.

The ordinary level-set method [1] used for tracking front contours is randomized to include the effects of turbulence.Turbulence randomly transports particle of the medium and therefore the contour gets a random character.The level-set method is randomized by considering a distribution of the contour according to theprobability density function of the turbulent displacement of the medium particles. In particular, this Lagrangian particle approach can be used to study the propagation of a reacting front in turbulent flows. A physical requirement connecting particle turbulent dispersion and the reacting front velocity is obtainedfrom equating the expansion rates of the front progression and of the reactant particles spread, so that the process follows to be fully determined by turbulence characteristics. The application of the method to turbulent premixed combustion is discussed [2]. The physical connection between particle dispersion and the front velocity compares favorably with experimental data. Moreover, in the case of a zero-curvature flame, with a non-Markovian parabolic model for turbulent dispersion, the formulation yields the classical Zimont equation extended to all elapsed times. The formulation extension to consider the acoustic noise is also addressed. The application of the method to model turbulence effects in wildland fire propagation [3] is also discussed. It emerges to be suitable, more than literature approaches based on the level-set method, to simulate the fire overcaming a fire breakzone because of the diffusion of the hot air behind it. [1] Sethian J.A. and Smereka P., Level set methods for fluid interfaces. Annu. Rev. Fluid Mech. 35, 341-372 (2003). [2] Pagnini G. and Bonomi E., Lagrangian formulation of turbulent premixed combustion. Phys. Rev. Lett. 107, 044503 (2011) [3] Pagnini G. and Massidda L., The randomized level-set method to model turbulence effects in woldland fire propagation. In D. Spano, V. Bacciu, M. Salis, C. Sirca (Eds.): Modelling Fire Behaviour and Risk, pp. 126-131. Proceedings of the International Conference on Fire Behaviour and Risk, Alghero, Italy, 4--6 October (2011)