Archivio 2014

In the 1989 work "Quantum Field Theories and Jones Polynomial" E. Witten showed how the Quantum Field Theories formalism with the Chern-Simons action and compact gauge group can produce important invariants in low dimensional topology like the Jones polynomial. A combinatorial, formally correct, model for such theories was constructed right afterwards by Reshetikhin and Turaev. In order to be able to define quantum Chern-Simons theory with non compact gauge group, we will introduce a combinatorial model for a TQFT coming from the quantisation of the Teichmuller Space, as described by J.E. Andersen and R.M. Kashaev. This will permit us to define a new quantum invariant for triangulated 3-manifolds, conjecturally related to the hyperbolic volume. On the way we will need to introduce the Penner coordinates for the decorated Teichmuller space, and angle structures on triangulated pseudo 3-manifolds. If the time permits I will explain how this quantisation program can be extended to the case when these coordinates are complex.

What are they? In algebraic geometry they represent the intersection theory on moduli spaces of pseudo- holomorphic curves in almost Kahler manifolds. In mathematical physics they are symplectic invariants that can be read as coefficients in the multiplication table of the quantum cohomology ring of homogeneous varieties. In A-model string theory they represent well defined path integrals of the theory, and they play a fundamental role in the Mirror Symmetry for Calabi-Yau manifolds. Yes, but how can I compute them? They are usually hard to compute, we will go through some examples: (1) GW({pt}) and the Witten conjecture (2) GW(P^1) and the recently found quantum curve (3) GW(P^2) and the amazing Kontsevich-Manin Formula (4) Other intriguing examples, like the comparison between the quantum cohomologies of P^3 and a smooth quadric 3-fold that have very similar classical cohomology rings.

After a review of topological string theory and BPS invariants, I will discuss the type IIA physical realisation of the unoriented topological string introduced by Walcher, its M-theory lift, and I will show how it allows to compute open and unoriented topological amplitudes in terms of one-loop diagrams of BPS\M2-brane states.

I will review the new approach to F-theory compactifications pioneered in Arxiv:1410.4867. According to it, instead of resolving or deforming the singularities of the internal space, one can extract physically relevant information by adding more structure to the singular space itself. This new perspective also allows to treat more exotic brane configurations, such as T-branes. The mathematical framework in which the approach is formulated is known as Eisenbud '92s matrix factorizations. I will present an affine example of how this works, as well as a global, compact one.

n this talk we want to study the elliptic fibrations whose total space is a Calabi-Yau variety embedded as an anticanonical divisor in a suitable class of P^2-bundles over a surface. Once we fix the base surface, we will show that the number of such elliptic 3-folds is finite, using only intersection theoretic properties of the base. When the base surface is P^2, we will show that some of these fibrations admit in a very natural way some non-Kodaira fibres.

One of the long term goals of modern mathematical physics is to understand the quantisation of physical theories under a general and mathematically sound framework. Among physical theories, those presenting local symmetries of the action (gauge theories) are of fundamental relevance, and the related issues that appear when ensuing their quantisation are addressed under a variety of points of view. One of the most celebrated mechanism to quantise gauge symmetric theories is the BRST formalism, but there are many important examples where it does not apply or it is not optimal. The extension of BRST to more general symmetries that do not come from the action of a Lie group (or that do, up to the equations of motion) goes under the name of BV formalism. More recently it turned out that this formalism provides a natural way to treat gauge theories on manifolds with boundary, even when the theory can be cast in the BRST formalism. In this talk I will outline the generalities of the CMR (Cattaneo, Mnev, Reshetikhin) formalism for gauge theories on manifolds with boundary, focusing on examples both worked out and in progress.

This is a survey talk explaining the statement of the so called P=W conjecture (due to M. A. de Cataldo, T. Hausel and myself), its motivation, stemming from the determination, due to T. Hausel and F. Rodriguez-Villegas, of the Mixed Hodge polynomial of some character varieties, the few cases in which this conjecture was proved and the problems one finds to extend these proofs to the general case.

A generalized spline space is a space of piecewise functions, locally spanned by functions of polynomial and non-polynomial type, such as trigonometric or hyperbolic. Therefore generalized splines are a superset of polynomial splines. Compared to the latter, they are superior for their capacity of reproducing fundamental functions and for their effectiveness in handling shape preserving approximation problems. At the same time, generalized splines can be quite easily differentiated and integrated, which is their main advantage over Non-Uniform Rational B-Splines (NURBS). Despite being initially meant for Geometric Modeling application, in the latest years their interest has been mainly connected to Multiresolution Analysis and Isogeometric Analysis. A generalized spline space is suitable for applications when it has a local, computationally stable basis and when this feature is preserved under knot insertion. These requirements are equivalent to saying that the space must have an Optimal Normalized Totally Positive Basis. Unfortunately, not all generalized spline spaces admit such basis, and, even when this is the case, its computation may be very challenging. In this talk we present a simple and general approach for the construction and computation of the ONTP basis and we provide some insights on the problem of determining whether such basis exists in a given spline space.

Lens spaces are the simplest family of closed 3-manifolds; they may be defined in diffent ways. In this talk the focus will be on the genus one Heegaard splitting definition: a lens space is the glueing of the two solid tori through an homeomorphism of their boundaries that can be represented by an element of S L(2,Z). Lens spaces may be defined also by a quotient of the 3-sphere S^3 under the action of Z_p, by a glueing of the boundary of the ball B^3 and by integral or rational Dehn surgery on knots/links. The equivalence of all these definitions can be easily described through some pictures. At last, the classification results of lens spaces will be outlined, both up to diffeomorphism and up to homotopy type. If time will permit, it will be shown that every closed 3-manifold may be described by an Heegaard splitting, as well as Dehn surgery on knots/links.

We provide and analyze analytical approximations of BSDEs in the limit of small non-linearity and short time, in the case of non-smooth drivers. We identify the first and the second order approximations within this asymptotics and consider two topical financial applications: the two interest rates problem and the Funding Value Adjustment. In high dimensional diffusion setting, we present numerical tests to illustrate the efficiency of the numerical schemes. Finally, we discuss the limit of this approach by assessing the possibility of higher order expansions.

In recent years, due to the method of supersymmetric localization, many exact results have been achieved in the study of supersymmetric gauge theories on compact spaces of various dimension and topology, discovering surprising structures. For example, it is becoming clear that partition functions of a large variety of such theories can be described in terms of fundamental "holomorphic blocks" and gluing rules. In this talk, I will introduce the subject by reviewing one of the best understood example, namely 3d N=2 theories defined on manifolds admitting a Heegaard decomposition in solid tori, where partition functions can be built by fusing 3d holomorphic blocks by elements in SL(2,Z). This was originally checked for S2xS1 and S3, and recently for any lens space L(r,1). Moreover, this picture can be lifted to 4d N=1 theories defined on L(r,1)xS1. Holomorphic blocks have been also proposed for 5d N=1 theories, and I will discuss how partition functions on S4xS1 and any toric Sasaki-Einstein manifold can be built out of them.

In this work, we discuss several methods that can solve topological changes during the Lagrangian surface evolution. The Lagrangian approach, contrarily to level-set methods, solves the surface evolution problem in a two-dimensional setting, but the processes of cutting and gluing the evolving surfaces are non-trivial. We present examples of mean curvature flow and an evolution by a constant normal speed. We made approximation of the mean curvature flow by a finite volume method -- the so called cotangent scheme. Then we suggest two methods that solve the most common topological changes."

Una breve panoramica su alcuni teoremi di esistenza e non esistenza per equazioni ellittiche nonlineari ad andamento critico.

In this talk we a introduce local dissipative mechanism able to produce exponential sta- bility. We will show that under some conditions no matter the quantity of dissipative mechanis we can introduce, the corresponding system is never exponentially stable. We will show situation that even there is a lack of polynomial stability. On the other hand, de- pending on the position, the material, and the dissipative mechanism, we will show some condition for which the local dissipative mechanism is effective and produce exponential stability of the system. We show some examples and applications.

The ``Lambert W'' is a not-so-widely-known special function implicitly defined by the equation z = W(z) exp (W(z)). Although its simple definition, which recalls the logarithm, its computation is made non-trivial by the lack of regularity and logarithm-like properties. Thus, iterative methods are a natural choice, which works very well in practice and can be easily extended to the matrix case. Nevertheless, turning to matrices poses some new challenges which require a non-trivial extension of the original definition and the development of a quite involved algorithm.

We are going to introduce Grid Homology, a purely combinatorial version of Knot Floer homology, for knots in the three sphere/lens spaces; after dealing with some of its properties, we are going to show Gallais' approach to lift coefficients from Z_2 to Z.

A seguito del Progetto Genoma siamo ormai in grado di leggere l'intero patrimonio genetico di qualunque specie. Per esempio conosciamo la sequenza dei 3,5 miliardi di basi che formano il DNA contenuto in una cellula umana. Ma sappiamo ancora molto poco sul significato biologico funzionale di questa sequenza. Più o meno il 2% del nostro DNA è una codifica chimico-algebrica, organizzata in circa 20.000 geni, della conformazione chimico-geometrica delle proteine e delle funzioni biologiche che esse possono svolgere proprio grazie a tale conformazione. I meccanismi molecolari alla base dei processi biologici di funzionamento del DNA e di espressione dei geni presentano incredibili analogie con le tecniche messe a punto dai matematici per lo studio della topologia dei nodi. Ecco perché la teoria dei nodi, e in particolare delle "matasse razionali", può essere usata come un potente microscopio per "vedere" anche ciò che neppure i microscopi elettronici possono mostrarci. Del restante 98% si sa molto poco. Fino a qualche tempo fa si pensava fosse "DNA spazzatura", residuo dell'evoluzione privo di un effettivo ruolo biologico. Oggi cominciamo capire che in realtà potrebbe giocare un importante ruolo epigenetico, cioè nell'attivazione o disattivazione dei geni anche in risposta agli stimoli ambientali. Lo studio delle proprietà geometriche e topologiche globali del DNA potrà dare un grande contributo alla comprensione delle dinamiche che regolano questi processi biologici fondamentali. Anche se, per dirla con le parole del famoso matematico Gromov, si tratta molto verosimilmente di uno di quei problemi "in bilico tra simmetria e caos, che attendono la nascita di una nuova matematica".

This talk aims at showing on an example how one can deal with nonlinearities and resonances with resurgent methods. We will pay special attention to the first Painlevé equation, which plays a significant role in many physical models : for instance, this equation and its resurgent structure were particularly adressed recently in string theory thanks of its interpretation in the context of 2D quantum gravity in the so-called double scaling limit. We will show how to get the Painlevé transcendents by Borel-Laplace summation, and we will interpret the formal integral in terms of holomorphic conjugations. We will describe the nonlinear Stokes phenomena and the Ecalle's holomorphic invariants.

The problem of finding sparse solutions to underdetermined systems of linear equations arises in several real-world problems (e.g. signal and image processing, compressive sensing, statistical inference). A standard tool for dealing with sparse recovery is the l1-regularized least squares approach that has been recently attracting the attention of many researchers. In this talk, we focus on variable fixing and active set approaches. We describe two different methods and analyze their convergence properties. Finally, we report numerical results on some test problems showing the effectiveness of the approaches.

We first present an overview on random motions at finite velocuty on the line an in the space. We study different forms of random motions, the probability distributions of which are solutions of the famous Euler-Poisson-Darboux equations which emerges also in the accelerated telegraph process. Finally, we present some fractional extensions of the finite-velocity random motions. References: 1) Fractional Klein-Gordon equations and related stochastic processes (2014) (with R. Garra and F. Polito) Journal of Statistical Physics, Vol. 155, 777-809. 2) Random flights governed by Klein-Gordon.type partial differential equations (2014) (with R. Garra). Stochastic Processes and their Appllications, Vol. 124, 2171-2187. 3) Flying randomly in R^d with Dirichlet displacements (2012) (with A. De Gregorio). Stochastic Processes and their Appllications, Vol. 122, 676-713. 4) Random flights in higher spaces (with A. De Gregorio) (2007). Journal of Theoretical Probability, Vol. 20, 769-806.

Let C and D two curves of the same degree without any common component. I will prove that, under some natural restrictions, the union of the singular members of the pencil generated by C and D is a free divisor in the sense defined by Saito.

Let (W,S) be a Coxeter system and let * be an involution of the corresponding Coxeter diagram, that is, a self-inverse automorphism of W which preserves the set of simple generators S. Many fundamental properties of elements in a Coxeter group, such as the notion of a reduced expression, the length function, and the exchange and deletion principles have interesting, nontrivial analogues for just the subset of *-twisted involutions in W, by which we mean the elements w in W with w^* = w^{-1}. The first part of this mini-course will give an introduction to such combinatorial properties of twisted involutions, as developed by Hultman, Richardson, Springer, and others. As motivation, we will review along the way some of the applications of this theory to the study of symmetric varieties. The remainder of the course will be a survey of recent progress and open questions related to some problems in combinatorics and representation theory in which the twisted involutions of a Coxeter group play a central role. Topics will include among the following, as time allows: Lusztig and Vogan's recent study of the "twisted" analogue of the regular representation of an Iwahori-Hecke algebra; Rains and Vazinari's theory of "quasiparabolic sets," of which conjugacy classes of twisted involutions serve as important motivating examples; connections between certain variants of the Poincaré series defined for twisted involutions and q-analogues of orthogonal polynomials as studied by Cigler and others; and reduced expression counting problems for twisted involutions. The course should be accessible to graduate students with some prior exposure to the study of Coxeter groups, or at least with a little knowledge of finite group theory.

The recently introduced M&m sequences and associated mean-median map are studied. These sequences are built by adding new points to a set of real numbers by balancing the mean of the new set with the median of the original. This process, although seemingly simple, gives rise to complicated dynamics. The main result is that two conjectures put forward by Chamberland and Martelli are shown to be true for a subset of possible starting conditions.

We briefly review the classical concept of Harish-Chandra representations then we give its generalization to supergeometry and we outline its geometric realization in super vector bundles.

We investigate weighted Sobolev spaces on metric measure spaces (X, d, m). Denoting by ρ the weight function, we compare the space W1,p(X,d,ρm) (which al- ways concides with the closure H1,p(X,d,ρm) of Lipschitz functions) with the weighted Sobolev spaces W1,p(X,d,m) and H1,p(X,d,m) defined as in the Euclidean theory of ρρ weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that W1,p(X,d,ρm) = H1,p(X,d,m). We also adapt the results proved by Muckenhoupt and the ones proved by Zhikov to the metric measure set- ting, considering appropriate conditions on ρ that ensure the equality W1,p(X,d,m) = H1,p(X,d,m)

We define a simple, explicit map sending a morphism f:M→N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of kerf and cokerf. As an immediate corollary, we obtain a new proof of the algebraic stability of persistence, a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules.

We show that being a general fibre of a Mori fibre space is a rather restrictive condition for a Fano variety. In order to detect this property, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with rational singularities to be realised as a fibre of a Mori fibre space, which turn into a characterisation in the rigid case. As an application, we apply our criteria to solve the classically known smooth two-dimensional case, give an almost exhaustive answer for smooth threefolds and flag varieties and a further characterisation on the polytope in the Gorenstein toric case. An interesting connection with K-semistability is also investigated. (This is a joint work with A.Fanelli, R. Svaldi and L. Tasin; it started during the school Pragmatic 2013.)

The Gromov-Wasserstein distance --a variant of the Gromov-Hausdorff distance based on ideas from mass transport-- provides an intrinsic metric on the collection of all metric measure spaces. I will review its construction, main properties, lower bounds, and computation.

We study constructible sheaves on a topological surface with singular support prescribed by a legendrian knot in the unit cotangent bundle. As we explain, moduli spaces of such include the wild character varieties which parameterize connections with irregular singularities, but also many more spaces. We explore connections to known constructions in legendrian knot theory, to Khovanov-Rozansky knot homology, and to the P=W conjecture in nonabelian hodge theory.

This is a gentle introduction to the problem of the classification of low-dimensional spaces and to the program of geometrization of two and 3-manifolds, both started more than a century ago with the Poincaré conjecture and Uniformization theorem, and completed some years ago by Thurston’s and Hamilton’s work and finally by the stunning results of Perelman. The beautiful results by Thurston showed the hitherto unexpected richness of geometry in three dimensions, and the great importance of the hyperbolic world. Perelman showed as the metric and the topology of 3-manifolds are profoundly related, and that the metric acts as a function of energy. In this process the Ricci flow play a key role. All these works offer a new landscape of mathematics and of the deep connections between topology, geometry and analysis. A few remarks will be devoted to sketch this new landscape and to highlight some applications especially in relativistic and quantum physics.

We prove Lp-parabolic a-priori estimates for $\partial_t u + \sum_{i,j=1}^d c_{ij}(t)\partial_{x_i x_j}^2 u = f $ on $\R^{d+1}$ when the coefficients $c_{ij}$ are locally bounded functions on $\R$. We slightly generalize the usual parabolicity assumption and show that still $L^p$-estimates hold for the second spatial derivatives of $u$. We also investigate the dependence of the constant appearing in such estimates from the parabolicity constant. We extend our estimates to parabolic equations involving non-degenerate Ornstein-Uhlenbeck type operators. Finally we present some open problems.

The Gromov-Wasserstein distance --a variant of the Gromov-Hausdorff distance based on ideas from mass transport-- provides an intrinsic metric on the collection of all metric measure spaces. I will review its construction, main properties, lower bounds, and computation.

We argue that topological deformations and biological functions are dynamically linked in living systems. In the last four decades it has became clear that, although the informational content of the genetic code was embodied in a linear array of bases, it was the three-dimensional spatial structure of the DNA double helix and its capability to change that ultimately would govern its physiological functions. This is very likely a crucial point. As an illustration of this point, in perhaps the most striking biological examples of “form dictates function”, the two complementary parental strands of DNA must separate during semi-conservative replication in order to act as the templates for each of the two newly synthesized daughter strands. This discovery led to recognize that the structure of DNA, while elegant, burdened the cell with previously unimagined topological problems. A huge of theoretical findings and experimental data strongly suggests that the secrets of life and what allows the biological growth of all organisms maybe lies in topology.

The theoretical investigation of the interplay between geometrical and electronic properties in graphene-based systems such as single-walled carbon nanotubes (SWNTs) and graphene nanoribbons (GNRs) is crucial both for the advance of condensed matter physics in low-dimensional systems and the development of new opto-electronic devices. In this talk I will first survey the basics concepts needed to understand the electronic properties of SWNTs and GNRs in the framework of the tight-binding model; electronic band structures and optical selection rules in GNRs for both longitudinally and transversely polarized photons will be discussed and compared to those known for SWNTs. Then I'll briefly review some experimental evidence of the failure of single-particle approximation revealing the presence of electronic correlation effects in SWNTs. The exact diagonalization (ED) method of the Hubbard model will be presented as a full many-body approach going beyond perturbative and mean-field techniques for elucidating the excitonic fine structure of these systems. I will show how the use of appropriate quantum lattice models with a well definite set of boundary conditions allows to capture the main features of the excited state properties of these systems and their size dependence in the low-intermediate correlation regime.

Understanding the physics of nonequilibrium systems remains as one of the major challenges of modern theoretical physics. We believe nowadays that this problem can be cracked in part by investigating the macroscopic fluctuations of the currents characterizing nonequilibrium behavior, their statistics and associated structures. This fundamental line of research has been severely hampered by the overwhelming complexity of this problem. However, during the last years two new powerful and general methods have appeared to investigate fluctuating behavior that are changing radically our understanding of nonequilibrium physics: a powerful macroscopic fluctuation theory (MFT) and a set of advanced computational techniques to measure rare events. I will talk about the statistics of current fluctuations in nonequilibrium diffusive systems, using the macroscopic fluctuation theory as theoretical framework, and advanced Monte Carlo simulations of several stochastic lattice gases as a laboratory to test the emerging picture. We will see how the discovery of novel isometric fluctuation relations opens an unexplored route toward a deeper understanding of nonequilibrium physics by bringing symmetry principles to the realm of fluctuations.

We will present joint work with Luchezar L. Avramov and Srikanth B. Iyengar. In the following (R,m) will denote a commutative local Noetherian ring, and L, M finitely generated R-modules. It is well-known that if R is a Gorenstein ring and L an R-module, then Ext_R^i(L,R) = 0 for i > dim R. The natural question that we wonder is: what if R is not Gorenstein? We'll prove that if there exists an M such that L is contained in M, mL is not equal to mM and mM is contained in L, then Ext_R^i(L,R) does not vanish for infinitely many i's if and only if R is not Gorenstein. The main tool used is stable cohomology, in particular a new decomposition theorem of stable cohomology.

Motivated by the so-called orbit method to construct unitary representations of a reductive Lie group, Adams Huang and Vogan studied the model orbit of the complex algebraic group E8, that is, the adjoint nilpotent orbit whose coordinate ring is the multiplicity-free direct sum of all the irreducible representations. They also studied a real analogue of it, an adjoint nilpotent orbit of the split real form of E8. Under the Kostant-Sekiguchi correspondence this is associated with an orbit of the complex algebraic group Spin(16), which we are able to study by making use of our results on the projective normality of wonderful varieties, and proving a couple of conjectures left open by Adams Huang and Vogan. This is a joint work with Jacopo Gandini and Andrea Maffei.

We review local regularity in Lebesgue spaces for solutions u of quasilinear elliptic systems. We discuss Meier's result in such a particular case and we show how to improve it, if off diagonal coefficients are small when |u| is large: the faster off diagonal coefficients decay, the higher the integrability of u becomes.

We start with the simple (to state) problem on counting special trajectories for billiards in polygons whose angles are multiples of right angles. From here, we go on a journey that involves ergodic theory, algebraic geometry, and enumerative combinatorics. Joint work with A. Eskin and A. Zorich.

Our world, at different scales, abounds in knotted and linked rings. Qualitative similarities may be observed between forms of the microscopic world, as well as between those of the macroscopic world; and by extracting the structures, it is possible to study their dynamics thanks to topological concepts and methods. The seminar is aimed at showing, first, that knots and links are scale-independent objects carrying an amount of precious information and signification on the emergence of new forms and structures in the mathematical and real world, and secondly, that a rich manifold of interesting surfaces and spaces can be constructed from knots and braids by applying diagrammatic and projection techniques. This will allow us to illustrate some striking relationships and common features between topological, natural and aesthetic objects and spaces. With the help of some relevant examples from knots and knotted spaces, we will propose a method for thinking by diagrams and transformation in order to grasp some processes underlying the generation of objects and events.

One of the simplest examples in quantum mechanics, a free particle on the torus, already exhibits complicated autocorrelation functions if the initial wave packet is highly localized. Such autocorrelations can be written in terms of classical number-theoretical sums (Gauss/Weyl sums and generalization thereof). The limiting distribution on the complex plane of Weyl sums can be studied using the theory of continued fractions or homogeneous dynamics. We will focus on the latter approach, and show that the partial sums of a quadratic Weyl sum converge in distribution to a random process on the complex plane. The properties of this process come from the geodesic flow on a hyperbolic manifold. In particular, the anomalous modulus of continuity of sample paths (different from that of a Brownian Motion) is derived from a logarithm law for geodesics by Kleinbock and Margulis. Joint work with J. Marklof.

The Farey sequence F(Q) is the collection of fractions between [0,1] whose denominator (when written in lowest terms) is at most Q. As Q grows, these points become uniformly distributed in the interval, so in some sense, look `random’. However, when you look at the gaps between them, they do not behave like those for uniformly distributed random variables, but instead follow an unusual law known as Hall’s Distribution. We will explain a proof of this result that uses horocycle flow on the space of lattices SL(2,R)/SL(2, Z), and discuss how this picture can be generalized to explicitly computing the gap distribution between directions of saddle connections on Veech surfaces, in particular the double pentagon, which we had used computer experiments to approximate. This talk will include elements from joint work with Y. Cheung, joint work with J. Chaika, and joint work with J. Chaika and S. Lelievre.

È ben noto che le equazioni differenziali alle derivate parziali e soprattutto quelle non lineari sono alla base dei settori piu’ importanti sia della matematica pura che di quella applicata ed anche della fisica, della meccanica, della economia, ecc. Paradossalmente però, a dispetto della sua fondamentale importanza, la teoria generale di EDPN non e’ stata sviluppata, mentre solo singole EDPN di interesse particolare sono state studiate con metodi di natura del tutto specifici. Così, al momento, tutta l’area rappresenta una sorta di eccelente artigianato, inoltre e’ stata per molto tempo opinione comune che una teoria unificante, solida ed efficiente fosse impossibile. Ma, di recente, in seguito alla cristallizzazione di due nuove teorie unificanti, la situazione è drasticamente cambiata. La prima di queste teorie, chiamata “calcolo differenziale sulle algebre commutative”, può essere considerata la “logica” del calcolo differenziale e delle sue rappresentazioni su algebre commutative. In questo contesto la geometria differenziale standard e, più in generale, tutte le “matematiche differenziali” standard, sono la rappresentazione di questa logica su algebre di funzioni lisce su varietà differenziabili. In particolare, tutte le strutture studiate nella geometria differenziale moderna, nella meccanica dei mezzi continui, nella teoria di campo, nella teoria del controllo, in relatività, ecc., sono solo elementi particolari di questa “logica”. La seconda teoria, chiamata “calcolo secondario”, è la specializzazione del calcolo differenziale nel senso summenzionato alle diffiety. Una diffiety formalizza l’idea intuitiva di “spazio di tutte le soluzioni” di un sistema di EDPN. Esse formano una nuova classe di varietà infinito-dimensionali sulle quali il calcolo differenziale necessario può essere sviluppato solo sulla base del calcolo differenziale su algebre commutative. Il calcolo secondario rappresenta una sintesi naturale e originale di elementi di algebra commutativa e algebra omologica, geometria differenziale e geometria degli spazi di jet, topologia algebrica e altri strumenti matematici mai immaginati prima nel contesto delle EDPN. Lo scopo del seminario e’ di ofrire una introduzione informale a questa nuova area diella matematica moderna, dei suoi legami naturali con la fisica, la meccanica, ecc, le delle sue problematiche e prospettive. Al termine del seminario sara’ presentata la nostra “Diffiety School”, una scuola estiva sul tema.

Abstract We will give an introductory presentation of the research field of geometric modeling and its applications, with specific attention to the use of splines for the representation of curves and surfaces. In particular, we will start by introducing basic notions of geometric modeling leading up to the definition of splines, which are piecewise functions with prescribed smoothness at the locations where the pieces join. Splines will be exploited for the representation of parametric curves and surfaces, and we will present their application in the context of computer-aided geometric design for shape description by means of approximation and interpolation methods. Finally, we will discuss some open problems in this topic and we will sketch some recent approaches for addressing them.

Let P be a finite poset. We will show that for any P-persistent object X in the category of finite topological spaces there is a P- weighted graph, whose weighted clique complex has the same P-persistent homology as X. Joint work with A. Patania (ISI-PoliTO), M. Scolamiero (KTH) and G. Petri (ISI). Work done within the EU funded FET Open Project "TOPDRIM".

Conferenza del Prof. Ivan Martin, Professor of Tissue Engineering, Director, Laboratory of Tissue Engineering, University Hospital, Basel, Switzerland.

ABSTRACT: l’analisi computazionale (computable analysis) potrebbe essere considerata come un “oggetto estraneo” al corpus della matematica classica, e al contempo non soddisfare i criteri di effettività o di immediata applicabilità richiesti dai matematici costruttivi o dagli analisti numerici. Eppure ha fornito negli ultimi anni strumenti utili per la dimostrazione di nuovi risultati in topologia classica e teoria degli insiemi, in particolare per quanto riguarda lo studio della discontinuità delle funzioni Borel-misurabili. Inoltre, alcuni recenti sviluppi hanno consentito di correlare il concetto di numero reale “random”, così come definito in teoria della ricorsività, e quello di funzione reale differenziabile. Più precisamente, l’analisi computazionale ha riconsiderato alcuni classici teoremi di analisi reale e teoria della misura riguardanti specifiche proprietà di derivabilità e ne ha formulato nuove versioni costituenti, per certi aspetti, “miglioramenti” degli enunciati originali. A titolo di esempio citiamo il Teorema di Lebesgue: ogni funzione non decrescente f:[0,1]-->R è differenziabile su tutti i reali al di fuori di un insieme di misura nulla. Questo enunciato non specifica la natura dell’insieme di eccezioni, il quale oltretutto è dipendente dalla scelta di f. Recentemente V. Brattka, J. Miller e A. Nies, sviluppando alcune ricerche preliminari di O. Demuth, hanno ottenuto una versione “migliorata” del classico risultato di Lebesgue: un numero reale z in [0,1] è “computably random” se e solo se ogni funzione reale computabile non decrescente f:[0,1]-->R è differenziabile in z. Si noti come il teorema classico di Lebesque sia un enunciato condizionale, mentre quello di Brattka, Miller e Nies fornisca una precisa caratterizzazione equivalente del concetto di differenziazione per un particolare tipo di funzioni. Inoltre viene individuato un ben determinato insieme di misura nulla che contenga tutte le le eccezioni possibili (quello dei numeri non computably random), aspetto tralasciato dall’originario Teorema di Lebesgue. Questo insieme risulta per di più essere del tutto indipendente dalla scelta di f. Simili risultati sono stati ottenuti anche per altri enunciati tradizionali inerenti il concetto di derivazione, quali il Teorema di Denjoy-Young-Sacks e quello di Differenziazione di Lebesgue, da un lato, ed alternativi concetti di randomness, quali Martin-Loef e Schnorr, dall’altro. Anche questi risultati saranno oggetto di presentazione nel corso del seminario.

Per ogni algebra di Lie complessa che ammette una forma reale Hermitiana esiste una scelta del sistema positivo di radici tale per cui la rappresentazione aggiunta dell'algebra compatta massimale stabilizza gli spazi di radici non compatti positivi (e negativi). Vedremo come estendere questo risultato alle superalgebre di Lie controgradienti e un'applicazione relativa alla costruzione di una struttura complessa sul quoziente della superalgebra reale modulo la sua compatta massimale.

Incontro realizzato nell'ambito del Ciclo di Seminari PROFESSIONE MATEMATICO

In the second seminar, we will introduce some more exotic mathematical objects. We will use orbifolds to represent chords and voice leadings, giving few examples to prove their efficacy in music analysis. Finally we will show how the orbifold's model can even be a useful didactical tool, and we will give a brief overview on some new research directions, involving braids, dynamical systems and persistent homology.

This is the second of a series of two talks where I will present a gentle introduction to the motivating problems of geometric group theory as introduced and solved by Dehn in the case of hyperbolic surfaces; the word and conjugacy problems. I will introduce the modern viewpoint of discrete groups as geometric objects, after Gromov, and discuss Gromov hyperbolic groups.

Today Krylov methods are the most popular iterative methods for solving linear systems. In this talk we consider their residual norms and their decrease. We will recall what are the known results for GMRES (Generalized Minimum Residual method) and we will show how to obtain exact expressions for the residual norms for diagonalizable matrices. They involve the eigenvalues and eigenvectors of the matrix as well as the right-hand side. To a certain extent they allow to explain the good or bad convergence of the method. Then we will study how to extend some of these results to other Krylov methods like the biconjugate gradient or the QMR method.

I will present a gentle introduction to the motivating problems of geometric group theory as introduced and solved by Dehn in the case of hyperbolic surfaces; the word and conjugacy problems. I will introduce the modern viewpoint of discrete groups as geometric objects, after Gromov, and discuss Gromov hyperbolic groups.

Parameter Optimization for Rational Krylov Methods and Applications Some problems in scientific computing, like the forward simulation of electromagnetic waves in geophysical prospecting, can be solved via approximation of f(A)b, the action of a large matrix function f(A) onto a vector b. Rational Krylov methods are very popular for these computations, and the choice of parameters in these methods is an active area of research. We provide an overview of different approaches for obtaining (in some sense) optimal parameters, with an emphasis on the exponential and resolvent function, and the square root. If time permits, we also discuss a surprising new application of the rational Arnoldi method for generating near-optimal absorbing boundary layers for indefinite Helmholtz problems.

ABSTRACT: l’analisi computazionale (computable analysis) potrebbe essere considerata come un “oggetto estraneo” al corpus della matematica classica, e al contempo non soddisfare i criteri di effettività o di immediata applicabilità richiesti dai matematici costruttivi o dagli analisti numerici. Eppure ha fornito negli ultimi anni strumenti utili per la dimostrazione di nuovi risultati in topologia classica e teoria degli insiemi, in particolare per quanto riguarda lo studio della discontinuità delle funzioni Borel-misurabili. Inoltre, alcuni recenti sviluppi hanno consentito di correlare il concetto di numero reale “random”, così come definito in teoria della ricorsività, e quello di funzione reale differenziabile. Più precisamente, l’analisi computazionale ha riconsiderato alcuni classici teoremi di analisi reale e teoria della misura riguardanti specifiche proprietà di derivabilità e ne ha formulato nuove versioni costituenti, per certi aspetti, “miglioramenti” degli enunciati originali. A titolo di esempio citiamo il Teorema di Lebesgue: ogni funzione non decrescente f:[0,1]-->R è differenziabile su tutti i reali al di fuori di un insieme di misura nulla. Questo enunciato non specifica la natura dell’insieme di eccezioni, il quale oltretutto è dipendente dalla scelta di f. Recentemente V. Brattka, J. Miller e A. Nies, sviluppando alcune ricerche preliminari di O. Demuth, hanno ottenuto una versione “migliorata” del classico risultato di Lebesgue: un numero reale z in [0,1] è “computably random” se e solo se ogni funzione reale computabile non decrescente f:[0,1]-->R è differenziabile in z. Si noti come il teorema classico di Lebesque sia un enunciato condizionale, mentre quello di Brattka, Miller e Nies fornisca una precisa caratterizzazione equivalente del concetto di differenziazione per un particolare tipo di funzioni. Inoltre viene individuato un ben determinato insieme di misura nulla che contenga tutte le le eccezioni possibili (quello dei numeri non computably random), aspetto tralasciato dall’originario Teorema di Lebesgue. Questo insieme risulta per di più essere del tutto indipendente dalla scelta di f. Simili risultati sono stati ottenuti anche per altri enunciati tradizionali inerenti il concetto di derivazione, quali il Teorema di Denjoy-Young-Sacks e quello di Differenziazione di Lebesgue, da un lato, ed alternativi concetti di randomness, quali Martin-Loef e Schnorr, dall’altro. Anche questi risultati saranno oggetto di presentazione nel corso del seminario.

In recent years, the interdisciplinary field of imaging science has been experiencing an explosive growth in active research and applications. In this talk I shall present some recent and new work of modeling the inverse problem of removing noise and blur in a given and observed image. Here we assume the Gaussian additive noise is present and the blur is defined by some linear filters. Inverting the filtering process does not lead to unique solutions without suitable regularization. There are several cases to discuss: Firstly I discuss the problem of how to select optimal coupling parameters, given an accurate estimate of the noise level, in a total variation (TV) optimisation model. Secondly I show a new algorithm for imposing the positivity constraint for the TV model for the case of a known blur. Finally I show how to generalise the new idea to the blind deconvolution where the blur operator is unknown and must be restored along with the image. Again the TV regularisers are used. However with the splitting idea, our work can be extended to include other high order regularizers such as the mean curvature. Once an observed image is improved, further tasks such as segmentation and co-registration become feasible. There will be potentially ample applications to follow up. Joint work with B. Williams, J. P. Zhang, Y.Zheng, S. Harding (Liverpool) and E. Piccolomini, F. Zama (Bologna). Other collaborators in imaging in general include T. F. Chan, R. H. Chan, B. Yu, N. Badshah, H. Ali, L. Rada, C. Brito, L. Sun, F. L. Yang, N. Chumchob, M. Hintermuller, Y. Q. Dong, X. C. Tai, etc.

Daremo un'introduzione paradigmatica alla combinatoria algebrica attraverso l'esempio citato nel titolo.

In recent years large data sets have been collected and analyzed, typically, by using some machine learning algorithms. However, many types of data demand a much more in-depth analysis that requires simulation, that is, solving partial differential equations, and optimization to estimate parameters. In this talk we discuss examples for such data sets in earth science. We describe the setting in which vast amounts of geophysical data is collected from the air. We present the large scale modeling that is required to simulate such a data set and the inverse problems that arise from these types of problems. We show that by using traditional techniques these problems cannot be solved in reasonable time on reasonable hardware. We then discuss a new set of algorithms that enable us to solve such problems. These algorithms are based on a concept we call domain of interest computation for the forward coupled with stochastic programming for the inverse. We show that by using this combination we are able to solve very large scale inverse problems, using a rather modest hardware in reasonable time.

In recent years, the interdisciplinary field of imaging science has been experiencing an explosive growth in active research and applications. In this talk I shall present some recent and new work of modeling the inverse problem of removing noise and blur in a given and observed image. Here we assume the Gaussian additive noise is present and the blur is defined by some linear filters. Inverting the filtering process does not lead to unique solutions without suitable regularization. There are several cases to discuss: Firstly I discuss the problem of how to select optimal coupling parameters, given an accurate estimate of the noise level, in a total variation (TV) optimisation model. Secondly I show a new algorithm for imposing the positivity constraint for the TV model for the case of a known blur. Finally I show how to generalise the new idea to the blind deconvolution where the blur operator is unknown and must be restored along with the image. Again the TV regularisers are used. However with the splitting idea, our work can be extended to include other high order regularizers such as the mean curvature. Once an observed image is improved, further tasks such as segmentation and co-registration become feasible. There will be potentially ample applications to follow up. Joint work with B. Williams, J. P. Zhang, Y.Zheng, S. Harding (Liverpool) and E. Piccolomini, F. Zama (Bologna). Other collaborators in imaging in general include T. F. Chan, R. H. Chan, B. Yu, N. Badshah, H. Ali, L. Rada, C. Brito, L. Sun, F. L. Yang, N. Chumchob, M. Hintermuller, Y. Q. Dong, X. C. Tai, etc.

We briefly review the concept of real structure of a supermanifold. Applications to the case of the super LIe group S^1|1 are given. A form of the Peter Weyl theorem for S^1|1 is discussed.

Math-for-Industry (MI) is a new research area that serves to create advanced technologies in response to industrial needs by innovating flexible and versatile methods in mathematics, statistics and computing. Prof. Kanzo Okada will share IMI’s efforts and activities to advance both mathematics and technologies through their mutual interactions as well as to nurture world-class navigators for MI.

The goal of this talk is to bring to light some recently discovered connections between problems about graph colourings and problems about the approximation of real numbers by rationals. The connections arise as a result of the fact that many statements about the quality of approximation of real numbers can be phrased as problems about the orbits of points in certain spaces (e.g. compact metric spaces) under the action of particular groups. Once these group actions are identified, there is a correspondence between questions about the approximations and questions about the Cayley graph of the given group. Information on either side of this correspondence gives information about the other. Several authors have used this machinery to transfer information from Diophantine approximation to give upper bounds for the chromatic number of Cayley graphs. Our main result shows that interesting information about Diophantine approximation can also be obtained by going in the other direction.We show how lower bounds for the chromatic number of certain Cayley graphs can be used to give a new proof of the p-adic Littlewood Conjecture for quadratic irrationals.

Abstract Negli ultimi anni la rivoluzione digitale ha permesso la raccolta di enormi quantita' di dati che permettono di quantificare la complessità dei sistemi socio-tecnici. Questi dati alimentano modelli computazionali che si propongono come strumenti adeguati per affrontare la studio di fenomeni come la diffusione delle epidemie su grande scala, la propagazione dell'informazione nella rete o la formazione del consenso e della conoscenza nei sistemi sociali. Questi modelli mirano a fornire un potere predittivo per i sistemi sociali che per lungo tempo e' stato ritenuto come irraggiungibile. Verra' presentata una rassegna dei risultati piu' recenti nel campo e delle sfide e che ci attendono nella predizione dei sistemi tecno-sociali. Short Bio Alessandro Vespignani insegna fisica e informatica come Distinguished Sternberg Professor alla Northeastern University di Boston dove dirige il laboratorio per la modellizzazione di sistemi biologici e tecno-sociali. Vespignani e' anche associato all'Istituto per le Scienze Sociali Quantitative alla Harvard University e direttore scientifico della Fondazione ISI di Torino. È noto per i suoi lavori sulle reti complesse, e in particolare per l'attività di ricerca sulle applicazioni della teoria delle reti alla diffusione della malattie infettive e lo studio delle proprietà topologiche di Internet. È autore di diverse monografie scientifiche e di più di 140 articoli scientifici nelle maggiori riviste internazionali.

It is possible to recognize abstract structures lying behind music. Some music's features can be naturally described borrowing a largely used and well known set of tools in mathematics. In the first part of this "math-music" journey, after a preliminary introduction about music theory, we will show how the Tonnetz (an abstract graph which represents the interaction among notes) can be a useful, low-dimensional representation of some basic musical objects as triads, and how it is possible to achieve a generalization of the standard Tonnetz, inducing a geometrical and topological structure on the graph.

Le sheet per l'azione di un gruppo algebrico G su una varieta' X sono le componenti irriducibili dei sottoinsiemi X(n) di X dati da tutti i punti la cui orbita ha dimensione fissata n. Molte importanti proprieta' delle classi di coniugio sono preservate lungo le sheet. Nel caso dell'azione aggiunta di G (riduttivo) sulla sua algebra di Lie, le sheet sono state descritte e parametrizzate da Borho e Kraft intorno al 1980. Mostreremo una parametrizzazione ed una descrizione delle sheet per l'azione di coniugio di G su se stesso, analoga a quella di Borho e Kraft ed ottenuta in collaborazione con F. Esposito. Utilizzeremo questi risultati per mettere in relazione le sheet in G con una partizione di G recentemente introdotta da Lusztig nel contesto della corrispondenza di Springer.

In this talk I first discuss the existence of eigenfunctions for the transfer operator of the Farey map. The main problem is that these operators are not quasi-compact on spaces of C^k functions for any k. However, on a suitable space of holomorphic functions, it is possible to have a characterization of the eigenfunctions. Then in the second part I will show that these eigenfunctions are related to Maass cusp and non-cusp forms of the modular surface, that is eigenfunctions of the hyperbolic Laplacian on the hyperbolic half-plane quotiented by the action of full modular group PSL(2,Z). This is joint work with Stefano Isola.

We will look at embeddings and composition operators on coinvariant subspaces of the Hardy space, so called model spaces. In particular we investigate the case in which the defining inner function is one-component, developing an extension process to reduce to the study of an equivalent embedding operator on a Hardy-Smirnov space of a different domain. From there we deduce criteria for embeddings and composition operators to be p-Schatten, obtaining characterizations that are in the spirit of previous work of Baranov and Luecking, Zhu. Based on joint work with Alexandru Aleman, Yura Lyubarskii, and Eugenia Malinnikova.

Si considerano sistemi di equazioni differenziali lineari a commutazione. Si propone un nuovo metodo per l'approssimazione degli esponenti di Lyapunov, basato sulla costruzione iterativa di politopi invarianti per i sistemi discretizzati ottenuti imponendo che gli istanti di commutazione siano multipli di un dato numero positivo (tempo di switching). Questi politopi vengono successivamente usati per generare una funzione di Lyapunov congiunta lineare a tratti, che consente di ottenere stime bilaterali per gli esponenti di Lyapunov. In questo modo e' possibile determinare se un sistema e' uniformemente stabile oppure stabilizzabile. Si tratta di un lavoro in collaborazione con Linda Laglia (L'Aquila) e Vladimir Protasov (Mosca).

Si presenta un modello matematico del codice genetico basato su rappresentazioni non univoche dei numeri interi. In particolare, si dimostra che la degenerazione del codice genetico (sia nucleare sia mitocondriale) può essere descritta mediante sistemi di numerazione tipo non-potenza, tipicamente poco studiati dal punto di vista matematico. Il modello permette di interpretare in termini matematici tutte le simmetrie note del codice genetico; inoltre, evidenzia nuove simmetrie non precedentemente descritte. Un esempio importante di tali simmetrie è rappresentato dalla partizione del codice in classi dicotomiche. Oltre a essere definite come funzioni non lineari dell'informazione contenuta in un dinucleotide, le classi dicotomiche sono legate a specifiche interazioni biochimiche e possiedono simmetrie legate al gruppo V di Klein. La codifica dell'informazione genetica per mezzo delle classi dicotomiche permette di mettere in luce strutture universali di correlazione e di ottimizzazione dell'informazione genetica. Il modello evidenzia l'esistenza di proprietà fortemente conservate in termini evolutivi e permette di analizzare sulla base di primi principi sia l'origine della codifica delle proteine sia le funzioni biologiche associate a tali proprietà. Tra queste ultime spicca la presenza di meccanismi per la rilevazione e la correzione degli errori analoghi a quelli utilizzati nella trasmissione di dati digitali in applicazioni di tipo ingegneristico. A tale proposito si presentano anche alcuni risultati sui codici circolari, strutture matematiche coinvolte nei meccanismi di mantenimento del frame di lettura nella sintesi delle proteine.

Sia w una parola e G_w l'insieme dei valori che essa assume in un gruppo G. Tratteremo il seguente problema: se G_w è ricoperto da un numero finito o numerabile di sottogruppi di G con una certa proprietà, cosa possiamo dire del sottogruppo verbale w(G) generato da G_w? Ci occuperemo in particolare di gruppi profiniti.

Vector fields naturally arise in many branches of mathematics and physics. Recently it was discovered that Lax pairs for many important multidimensional integrable partial differential equations (PDEs) of hydrodynamic type (also known as dispersionless PDEs) consist of vector field equations. These vector fields have complex coefficients and their analytic, in the spectral parameter, eigenfunctions play an important role in the formulations of the direct and inverse spectral transforms. In this seminar we prove existence of eigenfunctions of the basic vector field associated with the celebrated dispersionless Kadomtsev-Petviashvili equation, which are holomorphic in the spectral parameter $\lambda$ in the strips $|\Im\lambda|> C_0$. This research has been done in collaboration with P.M. Santini (Università La Sapienza)