Archivio 2016

"Due qubit si dicono entangled se la matrice che rappresenta lo stato del loro sistema ha rango due”. Questa affermazione vuole essere il punto di arrivo dell’incontro che mi propongo di animare. Non dimostrerò nulla di nuovo o di avanzato. Intendo banalmente presentare l’alfabeto di cui un matematico potrebbe sentire il bisogno per comprendere la frase di cui sopra.

Knotoid theory was created by V. Turaev in 2011. As classical knots, knotoids are represented by their diagrams on the 2-sphere. A knotoid diagram is a curve with self-intersections eqipped by over/undercrossing data. We define the equivalence relation on the set of knotoid diagrams and define some invariants of knotoids, as the Kauffman polinomial. The thickened torus is a direct product of the 2-torus and the interval I. A knot in the thickened torus is a simple closed curve. We define an equivalence relation on the set of knots in the thickened torus. We define the notion of knots of geometric degree 1 in the thickened torus and we construct the lifting map. This map gives a correspondence between knots of geometric degree 1 in the thickened torus and knotoids on the 2-sphere. We construct the switch-operation for knotoids and define prime knotoids. We prove the following theorem: the lifting map is injective for prime knotoids of complexity at least 2.

https://eventi.unibo.it/rassegna-scienza-cinema-pls2016/

We investigate the relations of two complexity notions in the zero entropy regime: mean equicontinuity and amorphic complexity. As it turns out, there is a close relationship in the minimal setting and we will present further results highlighting the interplay of these two concepts. Further, for mean equicontinuous subshifts we prove that amorphic complexity corresponds to the box dimension of the maximal equicontinuous factor and for certain Toeplitz subshifts we show how to calculate amorphic complexity using the theory of iterated function systems. If time permits, we will also elaborate on possible extensions of these notions, in particular with respect to more general group actions. This is work in progress with Gabriel Fuhrmann, Tobias Jäger and Dominik Kwietniak.

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We address the solution of PDE-constrained optimal control problems via semismooth Newton methods. Specifically, we consider problems with control constraints and with nonsmooth costs that are known to promote sparse optimal controls, i.e. controls which are identically zero on large parts of the control domain [HSW]. A typical example is the L$^1$ cost that has been used, e.g., for the optimal placement of control devices [S]. Following a discretize-then-optimize approach, we analyze the convergence properties of the Newton method applied to the discretization of optimal control problems with nonsmooth regularization terms. Moreover, we present the study of the impact of the control sparsity on the structure of the arising linear systems and propose preconditioners which exploit this information. Numerical experiments on 3D problems are presented

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We address the solution of PDE-constrained optimal control problems via semismooth Newton methods. Specifically, we consider problems with control constraints and with nonsmooth costs that are known to promote sparse optimal controls, i.e. controls which are identically zero on large parts of the control domain [HSW]. A typical example is the L$^1$ cost that has been used, e.g., for the optimal placement of control devices [S]. Following a discretize-then-optimize approach, we analyze the convergence properties of the Newton method applied to the discretization of optimal control problems with nonsmooth regularization terms. Moreover, we present the study of the impact of the control sparsity on the structure of the arising linear systems and propose preconditioners which exploit this information. Numerical experiments on 3D problems are presented.

La cronaca di ogni giorno ricorda a tutti noi come le scienze forensi siano sempre piu' protagoniste nella risoluzione di casi giudiziari. I laboratori forensi sono interdisciplinari per loro natura e offrono impiego alle professionalita' piu' diverse: biologi, fisici, chimici, ingegneri, matematici... Ogni settore del mondo scientifico puo' offrire il proprio contributo per fornire al Giudice elementi obiettivi di valutazione

I will begin with an introductory talk covering the basics of tractor calculus, focussing on the tight interaction between Einstein metrics and conformal geometry. In a second lecture, this machinery will then be applied to provide a general calculus of local and integrated hypersurface conformal invariants.

I will begin with an introductory talk covering the basics of tractor calculus, focussing on the tight interaction between Einstein metrics and conformal geometry. In a second lecture, this machinery will then be applied to provide a general calculus of local and integrated hypersurface conformal invariants.

We study the localisation and the existence of the eigenvalues of the generator of a contraction semigroup related to dissipative boundary problems for Maxwell system. The spectrum of the generator in the left half plane is formed by isolated eigenvalues with finite multiplicities and the corresponding solutions have an exponentially decreasing global energy. The localisation of such eigenvalues is important for the inverse scattering problems. We show that the eigenvalues are localisated in parabolic neighborhoods of the real axis or the imaginary one. For the ball we prove more precise results and we establish the existence of an infinite number of negative real eigenvalues.

Stable cohomology is a Z-graded multiplicative cohomology theory generalizing Tate cohomology and first defined by Pierre Vogel. It is connected through a short exact sequence to absolute cohomology and bounded cohomology. In this talk we investigate the structure of the bounded cohomology as a graded bimodule using the Hopf algebra structure of the Ext algebra. We use the information on the bimodule structure of bounded cohomology to study the stable cohomology algebra as a trivial extension algebra.

We will discuss some results about higher symmetric powers of tautological bundles over Hilbert schemes of n points over a smooth complex algebraic surface X, associated to a line bundle L on X. The main point will be a structure theorem for (the push-forward of) triple and quadruple symmetric powers of tautological bundles over the symmetric variety, in terms of a filtration whose graded sheaves are invariants of exterior tensor products of L over the product X^n, with prescribed vanishing along diagonals. We believe that a similar picture should remain valid in general. Finally we will give some applications of the previous result; in particular we will present an effective vanishing theorem for (triple and quadruple) symmetric powers of tautological bundles twisted by a determinant, in presence of adequate positivity hypothesis.

Sunto: The game-theoretic or normalized $p$-laplacian finds applications in the study of tug-of-war games and the evolution of a surface by mean curvature and seems to get along well with balls. I will show two instances that justify this claim. The former is a natural version of the asymptotic mean value property on balls for $p$-harmonic and $p$-caloric functions, introduced by Manfredi, Parviainen and Rossi, and its relation to p-harmonious function, introduced by La Gruyer. The latter, that benefits from the fact that the game-theoretic $p$-laplacian becomes linear on one-dimensional and radial functions, concerns the construction of spherically symmetric barriers which are useful to control the short-time behavior of the solutions of certain initial-boundary value problems for the related evolutionary $p$-laplacian. This researches have been carried out in collaboration with M. Ishiwata (Osaka University) and H. Wadade (Kanazawa University), and D. Berti (Università di Firenze).

Una ricostruzione dettagliata del punto di partenza della formulazione originale dovuta a Tullio Levi-Civita della nozione di trasporto parallelo su una varietà.

Una ricostruzione dettagliata del punto di partenza della formulazione originale dovuta a Tullio Levi-Civita della nozione di trasporto parallelo su una varietà.

Una ricostruzione dettagliata del punto di partenza della formulazione originale dovuta a Tullio Levi-Civita della nozione di trasporto parallelo su una varietà.

Non-equilibrium statistical mechanics, as opposed to its equilibrium counterpart, remains rather poorly understood. While equilibrium measures are easily characterized in terms of specific ensembles, the measures that characterize non-equilibrium states, whether stationary or time-dependent, are peculiar in that they are singular and display fractal properties. This observation has far-reaching consequences, in particular in connection to thermodynamics and the production of entropy. In this talk, Prof. Gilbert will use simple models based on random walks to review some of these properties, emphasizing the role of dynamics on the macroscopic transport processes they underlie. The lecture is directed to a general audience.

Avvertenza. Ci scusiamo per un invio precedente con data errata. La data della terza conferenza è GIOVEDI' 16 GIUGNO ( e NON 15) ========================= Warning, The present talk is the third one of the seminar. The second talk will be held on Tuesday 14 June. Till the end of the 19th century, mathematicians had to work on numbers without any good definition of them. Instead they had to rely mostly on the theory of proportions as given in book V of Euclid's Elements, a theory usually assigned to Eudoxus, a mathematician of the time of Plato. Nevertheless, well before him, as soon as mathematicians had to use irrational magnitudes, they needed some theory of proportions more general than the arithmetical one. I will firstly comment on some solutions given by modern historians of mathematics. Then I will present another one which is consistent with the knowledge of early Greek mathematics, and I will study some of its consequences, in particular for Euclid's proof of Pythagoras' theorem (proposition 47 of the first book of the Elements).

Fifth and last lecture

The seminar consists of three two-hour talks. In this second part which may be also called "Archytas' solution revisited", I will try to show Archytas' solution as a natural one, well inside the human bounds of the Greek mathematics of his time. Then I will show how such reconstruction may solve some still open questions related to it.

We establish new lower bounds for the convergence radius of the Mayer series and the Virial series of a continuous particle system interacting via a stable and tempered pair potential. Our bounds considerably improve those given by Penrose and Ruelle in 1963 for the Mayer series and by Lebowitz and Penrose in 1964 for the Virial series. To get our results we exploit the tree-graph identity given by Penrose in 1967 using a new partition scheme based on minimum spanning trees.

We derive the (formal) Mayer series of the pressure of a system of classical particles interacting via a pair potential from the standard grand canonical partition function. We give two alternative representations of its coefficients and we discuss the combinatorial problem involved towards the question of its convergence. We conclude the talk by giving a brief review on results obtained and methods proposed , since 1963,to face this combinatorial problem.

Fourth lecture

The seminar consists of three two-hour talks. 'In this first part, I will present the origins of what is known as the "Delian problem": - The myths around it - The so-called Hippocrates' reduction - The modern presentations of Archytas' amazing solution, inspired directly by the gods if we follow van der Waerden.

Relazioni primitive tra punti: stare fra (ternaria) e equidistanza (binaria tra coppie = quaternaria). Assiomi brevi e geometricamente visibili per le geometrie euclidea, iperbolica e assoluta di dimensione arbitraria. Analisi metamatematica dettagliata e elegante. Studio del frammento nel solo ``stare fra'', con sorprese.

Third lecture

I will present an overview of some of my recent papers on the probabilistic features of dynamical systems in deterministic and random settings. In particular I will focus on the rate of mixing for non-uniformly hyperbolic systems, on limit theorems for sequential systems and on extreme value theory (in the non-stationary setting and with some indications for a program on infinite-dimensional dynamical systems).

Affine e proiettivo (con assiomi di pura incidenza) e rappresentazione algebrica vettoriale. Ordine e ortogonalità sono strutture addizionali indipendentemente introdotte, geometricamente con perfetti equivalenti algebrici, coi vari gradi intermedi (per il corpo delle coordinate: da generico al campo reale). Insomma, l'armonica riconciliazione moderna delle polemiche ottocentesche.

Second lecture

First lecture in a series of five for the PhD program and everyone who is interested

We use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a σ-porous set. The second result states that irregular points of a Lipschitz function form a σ-porous set. We use these observations to give a new proof of Pansu's theorem for Lipschitz maps from a general Carnot group to a Euclidean space. Joint work with Gareth Speight (University of Cincinnati).

Come è ormai ben noto siamo nel pieno di una rivoluzione demografica, ovvero di un fenomeno planetario che interessa, anche se in maniera diversa, tutti i paesi e che nel giro di poco più di mezzo secolo ha visto quasi raddoppiare l’aspettativa di vita alla nascita che si situa oggi in Italia intorno a 80,1 anni per gli uomini e a 84,7 anni per le donne. Se le condizioni economico-sociali e sanitarie non peggioreranno questo fenomeno proseguirà anche nei prossimi decenni facendo prevedere una società dove le persone anziane (i 65+) rappresenteranno più di un terzo della popolazione. L’invecchiamento si accompagna ad un declino (più o meno rapido a seconda delle singole persone) di pressoché tutte le funzioni (fisiche, cognitive), all’insorgenza di tutta una serie di patologie croniche e alla fine, per le persone più sfortunate, alla perdita dell’autonomia. In questo scenario bisogna tenere presente che l’invecchiamento è il risultato dell’interazione tra i nostri geni, gli ambienti in cui siamo vissuti e gli stili di vita che abbiamo seguito. L’invecchiamento comincia dunque sin dai nove mesi che passiamo nell’utero materno e risente particolarmente degli eventi dei primi anni di vita. C’è però da sottolineare che esistono persone come i centenari (100+), i semi-supercentenari (105+) ed i super-centenari (110+), che hanno raggiunto i limiti estremi della vita umana e sono vissuti fino a tardissima età in buona salute evitando o ritardando di parecchi decenni l’insorgenza delle maggiori patologie età associate. Abbiamo dimostrato che i figli ed i familiari di queste persone eccezionali hanno una salute migliore di soggetti di pari età non figli o parenti di centenari. E’ questo il modello che stiamo indagando a fondo nel nostro laboratorio con le tecnologie più avanzate (genetica, epigenetica, trascrittomica, metabolomica, metagenomica, proteomica, glicomica) al fine di individuare combinazioni di marcatori biologici che ci consentano di valutare l’età biologica di una persona rispetto alla sua età cronologica e che ci permettano di effettuare e valutare l’effetto di strategie di prevenzione delle maggiori patologie e sindromi geriatriche (fragilità) attraverso interventi basati soprattutto sulla combinazione di attività fisica e nutrizione, ovvero sulla ottimizzazione dello stile di vita. E’ stato dimostrato che adeguati stili di vita sani (es. dieta mediterranea) seguiti scrupolosamente possono neutralizzare fattori di rischio genetico che predispongono verso le patologie etàassociate. E’ dunque chiaro che siamo noi stessi i padroni della nostra salute e della qualità del nostro invecchiamento. Questo nuovo tipo di ricerche largamente interdisciplinari, produce dati ad alta dimensionalità con milioni di variabili per singolo soggetto e vede la collaborazione di medici (geriatri, pediatri, etc.), biologi, biotecnologici, bioinformatici, demografi, nanotecnologi, ma anche ed in misura sempre maggiore di statistici, fisici, matematici, chimici essenziali per la messa a punto delle metodologie (sequenziamento di genomi, spettrometria di massa, NMR, per non citarne che alcune) e soprattutto l’analisi e l’interpretazione dei BIG datA. Inoltre, poiché gli anziani non vivono nel vuoto ma in precise società e comunità che variano da nazione a nazione e che in Italia sono diverse tra Nord e Sud e tra città e campagna, e poiché la durata e la qualità della vita sono legate, tra l’altro, anche allo stato socio-economico, all’educazione, alla personalità, al sonno e alla capacità di neutralizzare gli stress della vita e di controllare le emozioni, questi studi hanno sempre di più bisogno dell’expertise di sociologi, economisti, neurologi e psicologi. Come si vede, una strategia complessa nella quale le tradizionali distinzioni tra le discipline si attenuano al fine di affrontare un problema estremamente complesso come l’invecchiamento e la qualità della vita che rappresenta una delle maggiori e più affascinanti sfide del nostro secolo

We describe several results of maximal regularity for linear second order parabolic equations in spaces of Hoelder continuous functions: we shall consider Dirichlet, oblique derivative, Wentzell and dynamic boundary conditions

We discuss the modeling of shape memory alloys through a finite strain-version of the Souza-Auricchio model. By assuming an isotropic elastic response, a convenient formulation in terms of the Green-St-Venant transformation strain tensor can be established. A rate-independent inelastic constitutive relation is assumed, which is coupled to a quasi-static hyper-elastic energy with an additional regularizing interface-energy; the global existence of energetic solutions (i.e. a kind of variational evolution) to the corresponding boundary value problem is proven. In this framework, a rigorous linearization limit of the model at small strain can be established by means of the evolutive Γ-convergence for rate independent processes. A similar finite-strain approach is applied to provide a model for the magnetoelastic evolution in magnetic shape-memory materials.

In the recent years, Optimal transportation theory has been found to be very useful to describe geometric and analytic properties. We will focus on the geometric aspect, giving a short overview on the past theory and then discussing more recent applications, like the L ́evy-Gromov-Milman isoperimetric inequality for metric spaces.

In 1942 M. H. A. Newman formulated and proved a simple lemma of great importance for various fields of mathematics, including algebra and the theory of Gröbner–Shirshov bases. Later it was called the Diamond Lemma, since its key construction was illustrated by a diamond-shaped diagram. In 2005 the author suggested a new version of this lemma suitable for topological applications. In this talk we give a survey of results on the existence and uniqueness of prime decompositions of various topological objects: three-dimensional manifolds, knots in thickened surfaces, knotted graphs, three-dimensional orbifolds, and knotted theta-curves in three-dimensional manifolds. As it turned out, all these topological objects admit a prime decomposition, although it is not unique in some cases (for example, in the case of orbifolds). For theta-curves and knots of geometric degree 1 in a thickened torus, the algebraic structure of the corresponding semigroups can be completely described. In both cases the semigroups are quotients of free groups by explicit commutation relations.

ABSTRACT The ability of the modern graphics processing units (GPUs) to operate on large data can be exploited to perform computations in High Performance Computing (HPC) applications. Given the relevant number of libraries already available, little effort is required to parallelize numerical analysis approaches. We briefly describe NVIDIA GPU architecture, available software and programming tools. Fast Fourier Transform (FFT) speed-ups will be presented as an application example.

We study the dynamical properties of certain shift spaces. To help study these properties we introduce two new classes of shifts, namely boundedly supermultiplicative (BSM) shifts and balanced shifts. It turns out that any almost specified shift is both BSM and balanced, and any balanced shift is BSM. However, as we will demonstrate, there are examples of shifts which are BSM but not balanced. We also study the measure theoretic properties of balanced shifts. We show that a shift space admits a Gibbs state if and only if it is balanced. Restricting ourselves to S-gap shifts, we relate certain dynamical properties of an S-gap shift to combinatorial properties from expansions in non-integer bases. This identification allows us to use the machinery from expansions in non-integer bases to give straightforward constructions of S-gap shifts with certain desirable properties. We show that for any q∈(0,1) there is an S-gap shift which has the specification property and entropy q. We also use this identification to address the question, for a given q∈(0,1), how many S-gap shifts exist with entropy q? For certain exceptional values of q there is a unique S-gap shift with this entropy.

Language is a rich complex structure with the specific biological function of exchanging non-trivial information across human brains. Both the presence of common evolutionary history and cognitive constraints have lead to different languages sharing common features. In some cases this common patterns are so widespread that have become known as linguistic universals. While for languages that have diversified over the past few millennia traces due to common origin are beyond doubt, the question still remains of whether more distant language families still share linguistic structure. In the talk I will discuss a novel measure of relative entropy that can quantify the degree of order of words taking into account the full correlation structure of language. When this measure is applied to text corpora of languages from diverse linguistic families, an almost constant value emerges that can be interpreted as a quantitative statistical universal. Moreover, the analysis of simple models of language together with statistical evidence from real languages suggest that, during its evolution language has diversified under the constraint of keeping a precise balance between vocabulary diversity and the extent of long-range word correlations. It is also argued that the constancy of the relative entropy is probably a consequence of cognitive constraints of the human language faculty.

Several objects can be associated to a list of vectors with integer coordinates: a toric arrangement, a zonotope, a vector partition function. The linear algebra of the list is encoded by the notion of a matroid, but the topology of the toric arrangement, as well as several properties of the other objects mentioned above, depend also on the arithmetics of the list: this is retained by the notions of a "arithmetic matroid" and of a "matroid over Z". After introducing briefly these structures, we will focus on two of their invariants: the arithmetic Tutte polynomial and the Tutte quasi-polynomial. Among their applications, we will show one to colorings and flows on CW complexes, which can be seen as a higher-dimensional generalization of Tutte's theorem for graphs. Finally we will show that the set of arithmetic matroids on a given matroid is endowed by a natural product, which corresponds to a convolution product of the corresponding arithmetic Tutte polynomials.

Abstract We will discuss three one space dimensional time dependent linear parabolic equations: the heat equation, the Black-Scholes equation (describing stock options) and the Cox-Ingersoll-Ross equation (describing bond markets). New results will involve representation of the solution semigroups, chaotic properties of the semigroups, and a new kind of Feynman-Kac type representation of the solution for the CIR equation.

In this talk we take a look at iterated function systems and more general applications. We look at whether given an infinite system a subsystem of any dimension exists. Application to continued fraction expansions and symbolic dynamics will be given.

Il seminario affronterà il tema del continuo e dei numeri reali da un punto di vista storico in una prospettiva didattica. La storia della matematica ci mostra molte evoluzioni nelle immagini del continuo, grandi dibattiti che hanno accompagnato i cambi di prospettiva sul tema dell'infinitamente piccolo e sulla relazione tra numeri e oggetti della geometria. In questa lunga storia le costruzioni o assiomatizzazioni dei numeri reali non costituiscono un punto di arrivo ma costituiscono una risposta a problemi di natura più epistemologica che "tecnica", in un particolare periodo storico. Parlare di numeri reali nella scuola secondaria viene considerato indispensabile, ma spesso i problemi posti agli studenti e le definizioni dei libri di testo non generano apprendimento e le concezioni su punti di accumulazione, numeri irrazionali, completezza e continuità, densità sono inesatte o poco significative, con conseguenze sull'appredimento della matematica a livello formale. Una ricerca condotta con insegnanti in servizio di scuola secondaria, ha fatto emergere alcune convinzioni e alcune scelte ricorrenti che possono far riflettere sulle difficoltà e prospettare percorsi formativi per insegnanti e studenti in cui, al contrario, la complessità del tema diventi occasione per comprendere aspetti fondazionali rilevanti per l'apprendimento della matematica e il passaggio dalla scuola secondaria all'università.

Let n \ge 2 and let \Omega be a bounded open set in R^n. We provide higher differentiability results for local solutions of elliptic systems of the type div A(x, Du)=0 in \Omega, with coefficients in Sobolev spaces. We find that differentiability assumptions on A transfer to Du with no losses in the order of differentiation.

In this talk we will discuss the problem of existence of non-trivial Dirac-Geodesics. This problem comes from the super-symmetric non-linear Sigma model in physics. The problem is firstly stated for the case of surfaces then we reformulate it for loops. We sketch the basic idea of constructing the homology and its computation and how it can be used to prove existence of non-trivial Dirac-Geodesics.

Utilizzando metodi di analisi spettrale e di teoria modulare di algebra di operatori, studiamo gli scambi di energia tra un sistema finito e un reservoir infinitamente esteso nel processo del ritorno all'equilibrio. Pi\`u precisamente, consideriamo un sistema hamiltioniano microscopico che descrive un sitema finito $\mathcal{S}$ interagente con un reservoir termico a temperatura inversa $\beta$, dove la costante di interazione dipende da un parametro $\lambda$.\\ Consideriamo le misure di probabiltà ${\mathbb P}_{\mathcal{S}, \lambda, t}$, ${\mathbb P}_{\mathcal{R}, \lambda, t}$ ottenute attraverso un protocollo a due misure successive dell'energia al tempo $0$ e al tempo $t$ per $\lambda$ fisso. Supponendo che il sistema sia mixing rispetto allo stato termale iniziale, mostriamo che in un oppurtuno limite per $\lambda$ e $t$, le misure limite coincidono. Il risultato pu\`o essere visto come un estensione della legge di conservazione dell'energia, che riguardava solamente i valori medi dell'energia scambiata. (in collaborazione con {V. Jak\v{s}i\'c, J. Panangaden, C-A. Pillet)

Fractal-like structures are common in nature. They have a property called self-similarity, which means that each potion looks like the whole. Some of the most famous mathematical fractals, such as the Cantor, Julia and Mandelbrot sets or the Sierpinski carpet, are produced by a deterministic process and contain identical, scaled-down copies of the themselves. On the contrary, natural fractal-like objects usually look random and are self-similar only in a statistical sense, which means that each portion of the object looks similar but not identical to the whole. This talk will explore how combining self-similarity and randomness produces extremely interesting objects, whose analysis requires a mixture of techniques from different areas of mathematics. Random fractals have many applications to various fields of mathematics, the natural sciences and economics. It will explain how they appear in such diverse contexts as the modeling of financial markets, the theory of phase transitions, and cosmology.

La varietà sei dimensionale di O'Grady è classicamente ottenuta come risoluzione minimale di uno spazio di moduli di fasci semistabili su una superficie abeliana. In questo seminario tale varietà è ottenuta come quoziente per una involuzione di una varietà liscia e in tal modo è possibile calcolare i suoi numeri di Hodge. I risultati oggetto del seminario sono stati ottenuti da Mongardi in collaborazione con A. Rapagnetta e G. Saccà. Verranno anche presentati altri risultati sulla varietà di O'Grady, ottenuti da Mongardi in collaborazione con M. Wandel, e loro possibili sviluppi futuri.

The Fulton-MacPherson conguration space is a natural compactication of the conguration space of n ordered points on a smooth projective variety . The Kontsevich moduli space parametrizing stable maps from n-pointed rational curves to a projective space is another widely studied algebraic variety, and plays a central role in algebraic geometry, string theory and Gromov-Witten theory. These two spaces are indeed closely related. In this seminar we will discuss properties of fibrations of these spaces, and compute their automorphism groups.

Le due lezioni tratteranno principalmente la storia del 1800 e del 1900 e saranno un complemento al corso di Storia della Matematica del Prof. Giorgio Bolondi. Tra i matematici e gli argomenti che verranno trattati compariranno sicuramente Riemann, Laplace, la corrente Bourbakista e la matematica e fisica durante le Guerre Mondiali.

Saranno due lezioni di Storia della Matematica (la seconda il giorno 21 aprile) Le due lezioni tratteranno principalmente la storia del 1800 e del 1900 e saranno un complemento al corso di Storia della Matematica del Prof.Giorgio Bolondi. Tra i matematici e gli argomenti che verranno trattati compariranno sicuramente Riemann, Laplace, la corrente Bourbakista e la matematica e fisica durante le Guerre Mondiali.

Let W be an infinite Coxeter group with a finite set S of generators. In this talk we will consider the set Red(W) of all reduced S-expressions of elements of W. Brink and Howlett showed that Red(W) is a "rational language" by constructing a finite state machine, or automaton, which accepts precisely the words of Red(W). Their construction, which we will recall, is based on properties of the generalized root system attached to W. We introduce a new family of automata which all recognize Red(W) and whose definition involves the weak order of W. We will also state two conjectures concerning the minimality of these automata. This is joint work with C. Hohlweg and N. Williams (LaCIM, UQàM, Montreal)

Seminario di ricerca riservato ad esperti del settore

Seminario di ricerca riservato ad esperti del settore.

During the seminar I will present an overview of recent works dealing with the analysis and modeling of financial time series. The goal of the presentation is to informally elaborate on topics of potential interest for different fields, from statistical mechanics to mathematical finance and econometrics.

We present new results about optimal decay of correlations (lower bounds), for invertible and non-invertible dynamical systems with weak hyperbolicity.

Serie di seminari agli studenti della laurea triennale in matematica

  Let G'<G be an embedding of semisimple complex Lie groups. One of the fundamental problems in representation theory is about the description of the space of G'-invariants in a simple G-module, and the variation the dimension of this space as function of the highest weight. The Borel-Weil theorem allows to realize all irreducible G-modules as spaces of section of line bundles on the flag variety X=G/B. Then one may apply Hilbert-Mumford theory for the G'-action on X to study the spaces of invariants.   In a joint work with H. Seppänen, we aim at an explicit description of the G'-unstable locus for any line bundle on X. We have achieved this for certain classes of subgroups. This yields alternative proofs of some know results, as well as a description the GIT-classes of line bundles and some properties of the GIT-quotients. We show that many GIT-quotients are Mori dream spaces. We deduce results on the asymptotic growth of multiplicities of invariants in terms of a global Okounkov body for a suitable quotient.

Serie di seminari agli studenti del corso di laurea in matematica

In this talk, based on total variation (TV), we propose a method for reconstructing blurred images corrupted by impulse noise. Following the idea of the two-phase method, we utilize a noise detector to identify image pixels contaminated by noise, and then we reconstruct the noisy pixels by solving an equality constraint total variation minimization problem that preserves the exact values of the noise-free pixels. The proposed method improves the computational efficiency and has the advantage of being parameter-free. Numerical results suggest that the method is competitive in terms of its restoration capabilities and speed.

In questo seminario presenteremo alcuni risultati recenti relativi al classico problema inverso di Shape-from-Shading. Nel modello classico si vuole ricostruire una superficie a partire da una singola immagine supponendo che le proprietà di riflessione della superficie siano uniformi (superficie Lambertiana) e che la luce venga da un'unica sorgente posta all'infinito. Queste ipotesi sono evidentemente poco realistiche e limitano l'utilizzazione pratica del modello, ma si traducono in una equazione eiconale abbastanza facile da risolvere. Presenteremo due modelli di riflessione per la risoluzione del problema di Shape-from-Shading nel caso di superfici non Lambertiane (il modello proposto da Oren e Nayar e quello introdotto da Phong) e li confronteremo con il modello classico Lambertiano, sotto l'ipotesi di proiezione ortografica. Questi modelli sono stati proposti da autori appartenenti ad ambiti diversi col fine di prendere in considerazione superfici più realistiche come quelle rugose o con caratteristiche speculari, ma mancava una formulazione matematica coerente ed unificata. Il vantaggio di utilizzare una formulazione matematica unica è la possibilità di adattare facilmente un singolo modello differenziale a diverse situazioni, modificando solo alcuni parametri. Vedremo nel corso del seminario come si derivano le equazioni di Hamilton-Jacobi associate a tali modelli nei diversi casi, dipendenti dalla posizione della sorgente di luce e/o dalla posizione dell'osservatore. Questi casi si riconducono tutti ad un problema generale scritto in forma di punto fisso. Illustreremo anche come l'approssimazione numerica semi-Lagrangiana che proponiamo sia valida per il modello generale e possa essere facilmente adattata ai casi particolari. Enunceremo e dimostreremo anche le proprietà dell'operatore discreto, proprietà che garantiscono la convergenza del modello discreto al modello continuo. Confronteremo le prestazioni numeriche dei vari modelli su una serie di immagini sintetiche e reali. Infine, tratteremo brevemente l'estensione di questi modelli al caso stereo fotometrico nel quale vengono usate più immagini della superficie prese in condizioni di luce diverse ed occorre risolvere un sistema di EDP non lineari. Lavori in collaborazione con R. Mecca e S. Tozza.

We will show a theorem studying the distribution of eigenvalues in clusters generated by introducing a perturbation V, with apropiated decay at infinity, of the Landau problem in the high energy limit. The result involves averages of V along straight lines, which can be restated as a suitable high energy limit of normalized averages of V along circles (classical orbits of the unperturbed problem). This is joint work with Alexander Pushnitski and Georgi Raikov.

A partire dalla pubblicazione, nel 1985, del Teorema di "non-squeezing" di Gromov, l'esistenza (o meno) di embeddings simplettici ha un ruolo chiave nello studio delle varietà simplettiche. Anche nei casi più semplici, come per esempio nel caso di embeddings simplettici di ellissoidi, ci sono ancora molte domande aperte. La descrizione del comportamento della "embedding capacity" di un ellissoide simplettico 4-dimensionale (McDuff e Schlenk, 2012) è uno dei risultati più sorprendenti e affascinanti in quest'ambito. In questo seminario introdurrò queste tematiche e presenterò alcuni nuovi risultati sulla "embedding capacity" di un ellissoide simplettico in varietà toriche di dimensione 4. Il seminario è basato su un lavoro (in corso) in collaborazione con Dan Cristofaro-Gardiner (Harvard), Tara Holm (Cornell) e Ana Rita Pires (Fordham).

Given a family of feasible subsets of a ground set, the packing problem is to find a largest subfamily of pairwise disjoint family members. Non-approximability renders heuristics attractive viable options, while efficient methods with worst-case guarantee are a key concern in computational complexity. This work proposes a novel near-Boolean optimization method relying on a polynomial multilinear form with variables ranging each in a high-dimensional unit simplex. These variables are the elements of the ground set, and distribute each a unit membership over those feasible subsets where they are included. The given problem is thus translated into a continuous version where the objective is to maximize a function taking values on collections of points in a unit hypercube. Maximizers are shown to always include collections of hypercube disjoint vertices, i.e. partitions of the ground set, which constitute feasible solutions for the original discrete version of the problem. A gradient-based local search in the expanded continuous domain is designed. Approximations with polynomials of bounded degree and near-Boolean coalition formation games are also finally discussed.

The Voynich manuscript is a medieval text that has so far resisted all attempts of decipherment. It is often referred as the "most mysterious manuscript" and is considered a cryptographic and linguistic enigma. Its vellum has been dated to the second half of the 15th century, but apart from that, its origin, purpose, and meaning, still remain unknown. Here we overview some recent statistical evidence and present new findings supporting the hypothesis that the Voynich manuscript contains a genuine linguistic structure. We describe novel approaches to the problem of the Voynich manuscript using methods from information theory and statistics that quantify global patterns in the distribution of words. These methods allow the extraction of putative semantic networks between the most informative words. We also present recent evidence coming from the scaling properties of word frequencies and correlations that points towards an underlying structure compatible with natural languages. Finally, we discuss possible new research pathways that could be followed to shed some light on the mystery of the Voynich Manuscript.

Aim of this talk is to detect shared features in biological information processing networks, keeping the focus on neural and immune nets. We will proceed by steps: at first we will discuss simplest statistical mechanical models able to store one bit of information only (i.e. the paradigmatic Curie-Weiss and Mattis models) but we will address their thermodynamics using only variational principles stemmed from Lagrangian mechanics (rather than the classical statistical mechanical route). Then we will extend models and techniques toward the Hopfield scenario and we will use it to briefly revise neural networks and Hebbian learning. Next step will be to adapt this scaffold to lymphocyte networks and this will naturally give rise to the theory of autonomous parallel processing, that mathematically mirror the key ability of the immune system to defend its host from several pathogens at once. Finally, we will discuss why so different systems may share such an underlying conceptual description by showing that the transfer functions of ferromagnets (statistical mechanics), neurons (neurobiology), operational amplifiers (artificial intelligence) and lymphocytes (immunology) are identical.

Dale’s principle states that neurons in the nervous system have an exclusive physiological effect, each neuron either excites or inhibits all its synaptic targets. It is usually impossible for a neuron to excite some of its targets while inhibiting others. This principle has been known for more than fifty years, and yet there is no explanation for its function. Supported by theoretical analysis and experimental data, I propose that the function of Dale's principle is to maintain thermodynamic equilibrium in neural circuits. I study a nonlinear dynamical model characterized by a given synaptic matrix and input noise. Using the Fokker-Planck equation, I calculate the synaptic matrix consistent with thermodynamic equilibrium, and I show that it must satisfy Dale's principle under quite general assumptions. In order to test the theoretical predictions, I analyze the activity of neurons in the awake primate brain, and I show that collective neural dynamics displays temporal reversibility, which is a hallmark of thermodynamic equilibrium. I conclude by speculating on the significance of equilibrium and reversibility on brain function. While out-of-equilibrium dynamics may better support neural computation, equilibrium represents a desirable "idle" state.

In written language, the choice of specific words is constrained by both the particular semantic context consistent with the message to be transmitted, and grammatical requirements. To a significant degree, the semantic context is also affected by a larger cultural and historical environment, which in turn also influences matters of style and fashion. Over time, those environmental influences leave an imprint in the statistics of language use, leading to some words becoming more common while others are used less frequently. I will present a data-driven study of the statistics of language use over time based on the analysis of word frequencies extracted from more than 4.5 million books written over a period of 300 years (Google Ngram database). I will show evidence of systematic oscillatory patterns in word use that are highly consistent across different words. Moreover, while the periods of the oscillations are independent of the particular word, complex network analysis reveals that semantically related words show strong phase coherence. Ultimately, the origin of these previously unknown patterns in the statistics of language may be a consequence of the underlying broader cultural dynamics.

Data una ipersuperficie cubica liscia X di dimensione 3, la sua Jacobiana intermedia J(X) è una varietà abeliana principalmente polarizzata di dimensione 5, che fu usata da Clemens e Griffiths per dimostrare la non razionalità di X. Se Y è una ipersuperficie cubica liscia di dimensione 4, possiamo considerare la famiglia \mathcal X \to P^5 delle cubiche 3-dimensionali che sono sezione iperpiane di Y. Sull'aperto U di P^5 che parametrizza sezioni iperpiane lisce, si può considerare la Jacobiana intermedia relativa J_U \to U le cui fibre sono le Jacobiane intermedie delle fibre di \mathcal X_U \to U. Nel 1995 Donagi e Markman hanno costruito una forma simplettica olomorfa su J_U, rispetto a cui la fibrazione J_U \to U è Lagrangiana (o, equivalentemente, un sistema completamente integrabile algebrico). Da quel momento ci sono stati parecchi tentativi di trovare una compattificazione liscia J di questa fibrazione, con la proprietà ulteriore che la forma simplettica su J_U si estende ad una forma simplettica e olomorfa su tutta J. In un lavoro svolto in collaborazione con C. Voisin e R. Laza risolviamo questo problema usando le varietà di Prym relative e costruendo una compattificazione naturale di J_U che ammette una forma simplettica olomorfa.