# 2011

• Wednesday 14 December 2011 h. 15:00, room 2BC30
"Regular biproduct decompositions of objects"

-Abstract
Every vector space over a field K is the direct sum of a number of copies of the one-dimensional K-vector space K. Allowing scalars to be elements of a ring R instead of a field, we obtain a more general object called right (or left, depending on which side we write the scalars) R-module. Contrary to the trivial case of K-vector spaces, modules over R may or may not decompose into indecomposable submodules, and when they do, it is interesting to know whether their decompositions are unique in some sense or at least satisfy some sort of constraint. Beginning with the basics and with the classical results of the field we will end up giving some examples where modules have decompositions that satisfy a nice combinatorial condition. As a last step, we hint to a generalisation to the setting of biproduct decompositions in preadditive categories.
• Wednesday 30 November 2011 h. 15:30, room 1BC45
"What does "Inverse Problems" mean?"

-Abstract
Inverse Problems (IP) are described as situations where "the answer is known, but not the question, or where the results, or consequences are known, but not the cause" (Isakov, 2006). To better understand what this means, after an introductory overview, the talk will focus on two particular IP. The first one consists in the estimation of the quantity of pollutant released in a river, while the second one is about the estimation of the corrosion of an unobservable face of a metal slab. Both of them will be solved in a discrete context, using an adaptive parametrization.
• Wednesday 23 November 2011 h. 15:00, room 2BC30
"Boundedness and compactness of matrix operators in weighted spaces of sequences"
Zhanar TASPAGANBETOVA (University of Astana and Dip. Mat. Padova)

-Abstract
One of the main problems in the theory of matrices is to find necessary and sufficient conditions for the elements of a matrix so that the corresponding matrix operator maps continuously one normed space of sequences into another space of sequences. Thus it is very important to find the norm of a matrix operator, or at least, an upper or lower bound for the norm. However, in several spaces, which are very important both theoretically and in the applications, such problems have not been solved yet in full generality for operators corresponding to arbitrary matrices. Therefore, in such spaces researchers have considered some specific classes of matrix operators and have established criteria of boundedness and compactness for operators of such classes.
We prove a new discrete Hardy type inequality involving a kernel which has a more general form than those known in the literature. We obtain necessary and sufficient conditions for the boundedness and compactness of a matrix operator from the weighted $l_{p,v}$ space into the weighted $l_{q,u}$ space defined by $(Af)_j:=\sum\limits_{i=j}^\infty a_{i,j}f_i$, for all $f=\{f_i\}_{i=1}^{\infty} \in l_{p,v}$ in case $1 • Wednesday 16 November 2011 h. 16:00, room 2BC30 "L^2 theory and global regularity for d-bar on pseudoconvex domains of C^n" Stefano PINTON (Dip. Mat.) -Abstract This seminar is divided into two parts. The first one is an introduction to the first order partial differential operator d-bar in a smooth bounded pseudoconvex domain D of C^n. Only preliminary definitions are given and the basic estimate, due to Morrey, Kohn and Hormander, is established. It yields the existence of the d-bar Neumann operator, that is, the inverse to the complex Laplacian and the construction of the canonical solution to the equation di-bar(u)=f, for f in Ker(di-bar), that is the solution orthogonal to the Kernel of d-bar. The second part is an introduction to the problem of the global regularity up to the boundary for the canonical solution of the d-bar equation with data regular up to the boundary. In particular it is shown how compactness estimates are sufficient for global regularity as well as the existence of "good defining functions". • Wednesday 2 November 2011 h. 15:00, room 2BC30 "The explicit Laplace transform for the Wishart process" Alessandro GNOATTO (Dip. Mat.) -Abstract The first part of the talk will provide an introduction to mathematical finance. We start with a brief historical and philosophical perspective on the field and briefly review the main problems one tries to solve. We concentrate on the valuation of simple derivatives and review the famous Black-Scholes formula. After that we recall how the weaknesses of this standard approach motivated the introduction of more advanced models, in particular stochastic volatility (SV) models. In the context of SV models we review the role played by characteristic functions and the fast Fourier transform. This last point will serve as an introduction to the results of the paper, where we derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral which extends the original approach of Bru (1991). We compare our methodology with the alternative results given by the variation of constants method, the linearization of the Matrix Riccati ODE's and the Runge-Kutta algorithm. The new formula turns out to be fast, accurate and very useful for applications when dealing with stochastic volatility and stochastic correlation modelling. • Wednesday 19 October 2011 h. 14:30, room 2BC30 "Hopf Algebras. An introduction" Agustin GARCIA (Univ. Cordoba - Argentina and Dip. Mat. - Padova) -Abstract In this introductory talk we will motivate and present the definition of a Hopf algebra and we will review some of its main properties. The talk will be illustrated with several examples. The concept of a Hopf algebra, in a slightly different version as we know it today, was introduced in the 50's and got its final shape towards the late 60's. They became popular in the 80's, with the appearance of quantum groups and their relation with phenomena of algebraic groups in positive characteristic. Soon after that, they spread over various fields of mathematics and mathematical physics. • Wednesday 15 June 2011 h. 14:30, room 2BC30 "On the essential dimension of groups" Dajano TOSSICI (Univ. Milano-Bicocca) -Abstract At the end of the nineteenth century many authors (Klein, Hermite, Hilbert for instance) studied the problem of reducing the number of parameters of a generic polynomial of fixed degree. This problem was motivated by the problem of finding a formula, in terms of the usual algebraic operations and of radicals, for the roots of polynomial equations. It is very well known that, later, Galois proved that for polynomial equations of degree greater than 5 this formula does not exists. In 1997 Buhler and Reichstein rewrote and generalized this problem in a more modern context. They introduced the notion of essential dimension of a finite group G, which, very roughly speaking, computes the number of parameters needed to describe all Galois extensions with Galois group G. If we consider the symmetric group S_n then one obtains the number of parameters needed to write a generic polynomial of degree n. In the talk, after recalling the classical problem described above and the precise definition of essential dimension of a group, we illustrate several examples and open problems. At the very end of the talk, if time is left, we quickly give an overview of results we obtained in collaboration with Angelo Vistoli about essential dimension of a group scheme, which is a generalization of the concept of group in the context of algebraic geometry. • Wednesday 8 June 2011 h. 14:30, room 2BC60 "Identification of Reciprocal Processes and related Matrix Extension Problem" Francesca CARLI (Padova - D.E.I.) -Abstract Stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of Markov random fields restricted to one dimension. This kind of processes are potentially useful for describing signals which naturally live in a finite region of the time (or space) line. Non-stationary reciprocal processes have been extensively studied in the past especially by Jamison, Krener, Levy and co-workers. The specialization of the non-stationary theory to the stationary case, however, does not seem to have been pursued in sufficient depth in the literature. Moreover, estimation and identification of reciprocal stochastic models starting from observed data seems still to be an open problem. This talk addresses these problems showing that maximum likelihood identification of stationary reciprocal processes on the discrete circle leads to a covariance extension problem for block-circulant covariance matrices. This generalizes the famous covariance band extension problem for stationary processes on the integer line. We show that the maximum entropy principle leads to a complete solution of the problem. An efficient algorithm for the computation of the maximum likelihood estimates is also provided. • Wednesday 18 May 2011 h. 14:30, room 2BC30 "A Viscosity approach to Monge-Ampere type PDEs" Marco CIRANT (Dip. Mat.) -Abstract In this introductory talk we present some results of existence and uniqueness of solutions to the Dirichlet problem for the prescribed gaussian curvature equation, a Monge-Ampere type equation arising in differential geometry. We implement the modern tools of viscosity theory, combined with new ideas of Harvey and Lawson; our point of view is also based upon Krylov's language of elliptic branches. Our case study will be the homogeneous equation, i.e. when the curvature is identically zero, for which we outline the proof of existence and uniqueness of a convex solution (in a weak sense). Then, we sketch how to generalize these kind of results to the non-homogeneous equation and show some open problems related to curvature equations of more general form. • Wednesday 4 May 2011 h. 14:30, room 2BC30 "Large Deviations in Probability Theory" Markus FISCHER (Dip. Mat.) -Abstract In probability theory, the term large deviations refers to an asymptotic property of the laws of families of random variables depending on a large deviations parameter. A classical example is derived from coin flipping. For each number n, consider the random experiment of tossing n coins. Let S(n) denote the number of coins that land heads up. The quantities S(n) and S(n)/n are random variables, S(n)/n being the empirical mean, here equal to the empirical probability of getting heads. If the coins are fair and tossed independently, then by the law of large numbers S(n)/n will converge to 1/2 as n tends to infinity. Consequently, given any strictly positive c, the probability that S(n)/n is greater than 1/2+c (or less than 1/2-c) goes to zero as n tends to infinity. But one can say more about the convergence of those probabilities of deviation from the law of large numbers limit. Indeed, the decay to zero is exponentially fast (in the large deviations parameter n) with rates that can be determined exactly. The exponential decay of deviation probabilities is a common property of families of random objects arising in many different contexts. The aim of this talk is to give an introduction to the theory of large deviations, illustrating it by elementary examples as well as in the context of a class of mean field models. • Wednesday 20 April 2011 h. 14:30, room 2BC30 "The Liouville Theorem for conformal maps: old and new" Alessandro OTTAZZI (Milano-Bicocca) -Abstract In this seminar we discuss a classical result of Liouville for conformal maps in euclidean spaces of dimension at least three. Most of the time will be devoted to present a proof which is not present in this form in the classical literature. This proof works in more generality and in fact we (A. Ottazzi and B. Warhurst) can prove a Liouville theorem for all nilpotent and stratified Lie groups endowed with a sub-Riemannian distance. In the last part of the seminar we shall describe the setting of such groups, and possibly discuss some open problems. • Wednesday 6 April 2011 h. 14:30, room 2BC30 "Robustness for path-dependent volatility models" Mauro ROSESTOLATO (Pisa - SNS) -Abstract In this introductory talk, we present a 2-dimensional market model in which only one component (say S) is observable in the market, while the other one (say P) is not observable, thus the choice of the starting point (S(0), P(0)) is a-priori subject to an error. This is the reason why we are interested firstly in investigating the dependence of the (S,P)-dynamics with respect to the initial condition, secondly in choosing an initial condition which minimizes the error. The first issue is classical even in the deterministic case. The most intuitive and general approach is to recur to estimates based on Gromwall lemma, while if one instead uses the differentiability with respect to the initial conditions such estimates can be significantly improved. We will review this in the case of ordinary differential equations and see how the results generalize to the current case and how this improvement makes the model applicable in reality. The second issue is treated by techniques based on invariant measures and the clever use of past observations: we will show that, by using the estimates based on the differentiability with respect to the initial conditions, the width of the past window required is linear with respect to the future time horizon. Finally, we present some numerical results. • Wednesday 23 March 2011 h. 14:30, room 2BC30 "Approximating Goldbach conjecture" Valentina SETTIMI (Dip. Mat.) -Abstract The Goldbach conjecture is one of the oldest unsolved problems in the entire mathematics and, since its appearance in 1742 to nowadays, a lot of mathematicians dealt with it. In my talk I will give an introduction to the origin of the Goldbach conjecture and then I will describe the most important developments in some problems related to it. In particular I will talk about the ternary Goldbach conjecture, the exceptional set in Goldbach's problem and the Goldbach-Linnik problem. Finally, I will give a short overview of our results which can be seen as approximations to the Goldbach-Linnik problem. • Wednesday 9 March 2011 h. 14:30, room 2BC30 "Mean-variance Optimization Problems in Financial Mathematics" Claudio FONTANA (Dip. Mat.) -Abstract Quadratic optimization criteria are ubiquitous in applied mathematics. In particular, they have been successfully exploited in financial mathematics in the context of hedging and portfolio selection problems, beginning with the Nobel prize-winning work of Markowitz (1952). In this introductory talk, we will survey the main aspects of mean-variance optimization problems, both from a mathematical and a financial point of view. Furthermore, we shall present an abstract and unifying approach for the solution of mean-variance problems, together with the related issue of mean-variance indifference valuation. • Wednesday 23 February 2011 h. 14:30, room 2BC30 "From Shafarevich's conjecture to finite flat group schemes" Hendrik VERHOEK (Roma 2) -Abstract I will give an introduction to, and overview of, the work of Fontaine, Abrashkin, Schoof, Brumer-Kramer and Calegari that came forth from Shafarevich's conjecture or question about the non-existence of non-zero abelian varieties over Q with everywhere good reduction. First I briefly discuss what this conjecture is about. Then we will see the work of the pioneers Fontaine and Abrashkin passing by: they independently proved there does not exist such an abelian variety. After that we will consider the work of Schoof, Brumer and Kramer and Calegari related to abelian varieties over Q with so called semi-stable reduction at some places. Finally I will say something about generalizations and open problems. Speaker's web page: http://www.mat.uniroma2.it/~verhoek/ • Wednesday 16 February 2011 h. 14:30, room 1A150 "Numerical simulation of the electrical behavior of Carbon Nanotubes" Vittorio RISPOLI (Dip. Mat.) -Abstract We introduce in an elementary way the physical setting used to model the electrical behavior of metallic Carbon Nanotubes (CNTs); our aim is to compute the current induced in a CNT by an external electrical field. In the proposed setting, the temporal evolution of electrons and phonons (the last ones needed to take into account quantum mechanics effects) is described by a system of Boltzmann Equations, a system of hyperbolic equations with collision terms. We will give an overview of the general theory on the numerical treatment of such type of equations and present two schemes with some details. • Wednesday 31 January 2011 h. 10:30, room 1BC45 "Factorization in categories of modules" Marco PERONE (Dip. Mat.) -Abstract We study the regularity of the behaviour of the direct sum decomposition in categories of modules. It is well known that in some easy categories the direct sum decomposition is essentially unique. In the last 15 years some interesting examples were found of categories where the decomposition is not essentially unique but there still is an outstanding regularity. With this seminar we want to present in an elementary way these examples, introducing step by step all the necessary ingredients from monoid theory and module theory. • Wednesday 19 January 2011 h. 14:30, room 2BC30 "The maximum matching problem and one of its generalizations" Yuri FAENZA (Dip. Mat.) -Abstract Given a graph G(V,E), a matching M is a subset of E such that each vertex in V appears as the endpoint of at most one edge from M. The maximum matching problem and its weighted counterpart are among the most important and studied problems in combinatorial optimization. In this talk, we survey a number of classical results on the topic and present a new algorithm for a non-trivial generalization of the maximum weighted matching problem. The talk will be accessible to a general audience. # 2010 • Wednesday 15 December 2010 h. 15:00, room 2AB40 "An introduction to Coxeter group theory" Mario MARIETTI (Univ. Padova and Roma La Sapienza) -Abstract Coxeter groups arise in many parts of algebra, combinatorics and geometry, providing connections between different areas of mathematics. The purpose of this talk is to give an overview to Coxeter group theory from algebraic, combinatorial and geometrical viewpoints. Some classical and more recent results will be presented. • Wednesday 1 December 2010 h. 14:30, room 2AB/40 "A simple model for Financial Indexes with some application" Alessandro ANDREOLI (Univ. Padova, Dip. Mat.) -Abstract Mathematical finance is applied mathematics concerned with financial markets. Two of its major subjects are: 1. Mathematically modeling the prices of assets and indexes; 2. Option Pricing. We first recall the Black & Scholes model for asset prices, then present an easy model that overcomes some weak points of Black & Scholes and other models (in particular the absence of multiscaling effects and of volatility autocorrelation decay). Finally, we give an overview of the option-pricing problem. • Wednesday 17 November 2010 h. 14:30, room 2AB40 "Semiconcave type results of the minimum time function" NGUYEN Khai Tien (Univ. Padova, Dip. Mat.) -Abstract We will give an overview of "semiconcave type" results of the minimum time function in the case of nonlinear control systems under general controllability assumptions. Moreover, in this connection we will show some regularity results of a function whose hypograph satisfies an exterior sphere condition. • Wednesday 3 November 2010 h. 14:30, room 2AB40 "The concept of supermodularity in aggregation functions and copulas" Maddalena MANZI (Univ. Padova, Dip. Mat.) -Abstract In many domains we are faced with the problem of aggregating a collection of numerical readings to obtain a typical value, not only in mathematics or physics, but also in majority of engineering, economical, social and other sciences. So, aggregation functions are used to obtain a global score for each alternative taking into account the given criteria, even if the problems of aggregation are very broad and heterogeneous. For example, there is a lot of contributions about the aggregation of finite or infinite number of real inputs, topics treating of inputs from ordinal scales, or also the problem of aggregating complex inputs (such as probability distributions, fuzzy sets). In this talk I will discuss the way to construct, in particular, supermodular aggregation functions, which can be analyzed under various aspects: algebraic, analytical, probabilistic. So, in the first part I will introduce the general concept of supermodularity, which comes from lattice theory, and we will see several basic examples. Then, we will be able to apply this theory to aggregation functions and, in particular, to a subclass of aggregation functions, i.e. the family of copulas. In the last part, we will see some results obtained with a particular intersection with fuzzy set theory. So, a basic background will be given also in this direction. The talk will be based on some joint works with M. Cardin, M. Kalina, E. P. Klement and R. Mesiar. • Wednesday 20 October 2010 h. 14:30, room 2BC30 "The order complex of the coset poset of a finite group" Massimiliano PATASSINI (Univ. Padova, Dip. Mat.) -Abstract In this seminar we want to speak about a topological aspect of some objects in finite group theory. Let G be a finite group and let C(G) be the coset poset of G, i.e. C(G) = {Hg : H<G, g in G}. In order to give a topological interpretation to this object, we introduce the concept of order complex of C(G). The order complex was studied by Kennet Brown, who pointed out a connection between the Dirichlet polynomial of G and the reduced Euler characteristic of the order complex of C(G). In our talk, we first give an overview of the concepts of coset poset, order complex and Moebius function. Next we introduce the work of Kennet Brown concerning the order complex of the coset poset of a soluble group. Last we give an idea of our result about the non-contractibility of the order complex of the coset poset of a classical group. • Wednesday 23 June 2010 h. 14:30, room 1BC/50 "Topology of Kaehler and hyperkaehler manifolds" Julien GRIVAUX (Univ. Paris 6 - Pierre et Marie Curie) -Abstract This talk is an introduction to complex geometry. In the first part, we will introduce some of the basic objects in the field (such as complex manifolds, differential forms and cohomology groups) and see several examples. Then we will be able to state some fundamental results on the cohomology of compact Kaehler manifolds, which are a special class of complex manifolds. We will see how these results generate constraints on the topology of Kaehler manifolds. The last part will be devoted to the theory of hyperkaehler manifolds, wich is an active area of current research in complex algebraic geometry. The talk will be accessible for a general audience; but basic knowledge of differentiable manifolds will of course be helpful. • Wednesday 9 June 2010 h. 14:30, room 1BC/50 "Edge-connectivity augmentation" Roland GRAPPE (Univ. Padova - Dip. Mat.) -Abstract A graph is k-edge-connected if there exist k edge-disjoint paths between every pair of vertices. The problem of global edge-connectivity augmentation of a graph is as follows: given a graph and an integer k, add a minimum number of edges to the graph in order to make it k-edge-connected. We will comprehensively focus on this problem and the simple method of Frank that solves it. Then, we will see a few generalizations such as edge-connectivity augmentation of a graph with partition constraints (Bang-Jensen et al.), edge-connectivity augmentation of a hypergraph (Bang-Jensen and Jackson), and the unification of these two results (joint work with Bernath and Szigeti). Eventually, these problems can be formulated in an abstract form, leading to further generalizations. • Wednesday 26 May 2010 h. 14:30, room 1BC/50 "Holomorphic sectors and boundary behavior of holomorphic functions" Raffaele MARIGO (Univ. Padova - Dip. Mat.) -Abstract Forced extendibility of holomorphic functions is one of the most important problems in several complex variables: it is a well known fact that a function defined in an open set D of C^n extends across the boundary at a point where the Levi form of the boundary of D (i.e. the complex hessian of its defining function restricted to the complex tangent space) has at least one negative eigenvalue. A fundamental role in this result is played by analytic discs, i.e. holomorphic images of the standard disc. After describing the construction of discs attached to a hypersurface by solving a functional equation - Bishop equation - in the spaces of differentiable functions with fractional regularity, we will show how they induce the phenomenon described above, as well as the propagation of holomorphic extendibility along a disc tangent to the boundary of the domain. Finally, we will introduce a new family of discs, nonsmooth along the boundary, that will allow us to establish analogous results under various geometric conditions on the boundary of the domain. • Wednesday 12 May 2010 h. 15:00, room 1BC/50 "The l-primary torsion conjecture for abelian varieties and Mordell conjecture" Anna CADORET (Universite' de Bordeaux 1) -Abstract Let k be a field. An abelian variety A over k is a proper group scheme over k. It can be thought of as a functor (with extra properties) from the category of k-schemes to the category of abelian groups. One nice result about such a functor is: ~~~~ Theorem (Mordell-Weil): Assume that k is a finitely generated field of characteristic 0: then, for any finitely generated extension K of k, A(K) is a finitely generated group. In particular, the torsion subgroup A(K)_tors of A(K) is finite. ~~~~ For a prime l, the l-primary torsion conjecture for abelian varieties asserts that the order of the l-Sylow of A(K)_tors should be bounded uniformly only in terms of l, K and the dimension g of A. For g=1 (elliptic curves), this conjecture was proved by Y. Manin, in 1969. The main ingredient is a special version of Mordell conjecture for modular curves. The general Mordell conjecture was only proved in 1984, by G. Faltings. For g=2, the l-primary torsion conjecture remains entirely open. After reviewing the proof of Y. Manin, I would like to explain how the general version of Mordell conjecture can be used to prove - following basically Manin's argument - the l-primary torsion conjecture for 1-dimensional families of abelian varieties (of arbitrary dimension). This result was obtained jointly with Akio Tamagawa (R.I.M.S.), in 2008. • Wednesday 28 April 2010 h. 14:30, room 1BC/50 "Interest rate derivatives pricing when the short rate is a continuous time finite state Markov process" Valentina PREZIOSO (Padova - Dip. Mat. Pura e Appl.) -Abstract The purpose of this presentation is to price financial products called "interest rate derivatives", namely financial instruments in which the owner of the contract has the right to pay or receive an amount of money at a fixed interest rate in a specific future date. The pricing of these products is here obtained by assuming that the spot rate (i.e. the interest rate at which a person or an institution can borrow money for an infinitesimally short period of time) is considered as a stochastic process characterized by "absence of memory" (i.e. a time-continuous Markov chain). We develop a pricing model inspired by work of Filipovic'-Zabczyk which assumes the spot rate to be a discrete-time Markov chain: we extend their structure by considering, instead of deterministic time points, the random time points given by the jump times of the spot rate as they occur in the market. We are able to price with the same approach several interest rate derivatives and we present some numerical results for the pricing of these products. • Tuesday 13 April 2010 h. 14:30, room 1A/150 "Analytic and algebraic varieties: the classical and the non archimedean case" Alice CICCIONI (Padova - Dip. Mat. Pura e Appl.) -Abstract The complex line, as a set of points, can be endowed with an analytic structure, as well as with an algebraic one. The choice of the topology and the related natural definition of functions on the space determine different geometric behaviors: in the example of the line, there are differential equations admitting solutions in both cases, and some that can be solved only in the analytic setting. The first part of the talk will focus on the algebraic and analytic structures of a variety over the field of complex numbers, while in the second part we will give an overview of the analogous constructions for varieties defined over a non archimedean field, touching the theory of rigid analytic spaces and its relation to the study of varieties over a discrete valuation ring of mixed characteristic in the framework of syntomic cohomology. • Wednesday 24 March 2010 h. 15:00, room 2AB/45 "Finite and countable mixtures" Cecilia PROSDOCIMI (Universita di Padova - Dip. Mat.) -Abstract The present talk deals with finite and countable mixtures of independent identically distributed (i.i.d.) sequences and of Markov chains. After an easy introduction on mixture models and their main properties, we focus on binary exchangeable sequences. These are mixtures of i.i.d. sequences by de Finetti theorem. We present a necessary and sufficient condition for an exchangeable binary sequence to be a mixture of a finite number of i.i.d. sequences. If this is the case, we provide an algorithm which completely solves the stochastic realization problem. In the second part of the talk we focus on partially exchangeable sequences, that are known to be mixtures of Markov chains after the work of Diaconis and Freedman, and Fortini et al. later. We present a characterization theorem for partially exchangeable sequences that are mixtures just of a finite or countable number of Markov chains, finding a connection with Hidden Markov Models. Our result extends an old theorem by Dharmadhikari on finite and countable mixtures of i.i.d. sequences. • Wednesday 10 March 2010 h. 15:00, room 2BC/30 "A pseudometric for unbounded linear operators, extension to operators defined on different open sets and an application to spectral stability estimates for eigenfunctions" Ermal FELEQI (Universita di Padova - Dip. Mat.) -Abstract A distance on closed linear subspaces/operators has long been known. It was introduced under the name of "gap" or "opening in a Hilbert space context by Krein and coworkers in the 1940s. The first part of the talk will be of an introductory character and the main properties of the gap between subspaces/operators will be illustrated with the focus laid on spectral stability results. Next it will be shown how the notion of gap between operators can be adapted to study the spectral stability problem of a certain class of (partial) differential operators upon perturbation of the open set where they are defined on. An extension of the gap for operators defined on different open sets will be proposed and it will be estimated in terms of the geometrical vicinity or proximity of the open sets. Then, this will permit to estimate the deviation of the eigenfunctions of certain second order elliptic operators with homogeneous Dirichlet boundary conditions upon perturbation of the open set where the said operators are defined on. • Wednesday 24 February 2010 h. 15:00, room 2BC/60 "Moebius function and probabilistic zeta function associated to a group" Valentina COLOMBO (Padova, Dip. Mat.) -Abstract Many authors have studied the probabilistic zeta function associated to a finite group; in the last years the study has been extended to profinite groups. To understand how the probabilistic zeta function is defined, it is necessary to introduce another function associated to a group: the Moebius function. We will start considering finite groups: we will explain how these two functions are obtained and we will give some basic examples. Then we will define the profinite groups and proceed to investigate whether and how a probabilistic zeta function can be associated to them. This is not always possible: Mann has conjectured that for a particular class of profinite groups (PFG groups) the definition of this function makes sense. We will present some recent results which suggest that the conjecture is true. • Wednesday 10 February 2010 h. 14:30, room 2BC/30 "Examples of strategy designs in banking practice" Marco CORSI (Examples of strategy designs in banking practice) -Abstract While academic theory of financial mathematics emphasizes the concept of no-arbitrage models, in common practice the presence of arbitrage opportunities in the market in some cases can explicitly be taken into account. In this talk we will see how different kind of strategies (implied volatility strategy, volatility arbitrage strategy, etc.) can be practically implemented. • Wednesday 13 January 2010 h. 15:00, room 2BC/60 "Liouville-type results for linear elliptic operators" Luca ROSSI (Universita di Padova - Dip. Mat.) -Abstract This talk deals with some extensions of the classical Liouville theorem about bounded harmonic functions to solutions of more general partial differential equations. In the first part, I will introduce the only two technical tools needed to prove the Liouville-type result in the case of periodic elliptic operators: Schauder's a priori estimates and maximum principle. Next, I will discuss the role of the periodicity assumption, seeing what happens if one replaces it with almost periodicity. # 2009 • Wednesday 16 December 2009 h. 15:00, room 2BC/60 "On some aspects of McKay Correspondence and its applications" Luca SCALA (University of Chicago) -Abstract When we quotient C2 by a finite subgroup G of SL(2,C), and we take a minimal resolution Y of the kleinian singularity C2 /G, then Y is a crepant resolution and the exceptional locus consists of a bunch of curves, whose dual graph is a Dynkin diagram of the kind An, Dn, E6, E7, E8. In the eighties, McKay noticed that the Dynkin diagrams arising from resolutions of kleinian singularities are in tight connection with the representations of G. In the first and introductory part of the talk, we will explain the McKay correspondence and its key generalization by means of K-theory, due to Gonzalez-Sprinberg and Verdier. The latter point of view opens the way to the modern derived McKay correspondence, due to Bridgeland-King-Reid. We will then see some applications of the BKR theorem to the geometry of Hilbert schemes of points, due to Haiman, and some other consequences related to the cohomology of tautological bundles. • Wednesday 9 December 2009 h. 14:30, room 2AB/40 "Diffusion coefficient and the Speed of Propagation of traveling front solutions to KPP-type problems" Adrian VALDEZ (Ph.D., University of the Philippines) -Abstract In this talk, we shall concern ourselves with a general reaction-diffusion equation/system in a periodic setting concentrating on reaction terms of KPP-type. Our interest is focused on special solutions called traveling fronts. In particular, we look at how the minimal speed of propagation of such front solutions can be influenced by the different coefficients of the system. For this, an intensive discussion will be alloted specifically on the influence of the diffusion coefficient. • Wednesday 18 November 2009 h. 15:00, room 2AB/40 "Injective modules and Star operations" Gabriele FUSACCHIA (Univ. Padova, Dip. Mat.) -Abstract The problem of classifying injective modules in terms of direct decompositions does not admit, in general, a solution. Exhaustive results have been obtained, however, when restricting to special classes of domains, such as Prufer domains, valuation domains and Noetherian domains. After recalling some basic notions on injective modules and direct decompositions, we provide examples of domains in which the classification is not possible, and we give the classical results on valuation and Noetherian domains. Next we introduce the notion of star operation over a domain, a special kind of closure operator defined over the fractional ideals. Thanks to this concept, we show how the classification on Noetherian domains can be generalized, allowing to completely classify special subclasses of injective modules over domains which are not Noetherian. • Wednesday 28 October 2009 h. 14:30, room 2AB/60 "Typicality and Fluctuations: A different way to look at Quantum Statistical Mechanics" Barbara FRESCH (Univ. Padova, Dep. of Chemistry, Ph.D.) -Abstract Complex phenomena such as the characterization of the properties and the dynamics of many body systems can be approached from different perspectives, which lead to physical theories of completely different characters. A striking example of this is the duality, for a given physical system, between its thermodynamical characterization and the pure mechanical description. Finding a connection between these different approaches requires the introduction of suitable statistical tools. While classical statistical mechanics represents a conceptually clear framework, some problems arise if quantum mechanics is assumed as fundamental theory. In this talk, after a general introduction to the subject for non-experts, we shall discuss the emergence of thermodynamic properties from the underlying quantum dynamics. http://www.math.unipd.it/~maraston/SemDott/abstractFresch.pdf • Wednesday 14 October 2009 h. 15:00, room 2AB/60 "The d-bar-Neumann problem" TRAN Vu Khanh (Univ. Padova, Dip. Mat.) -Abstract The d-bar-Neumann problem is probably the most important and natural example of a non-elliptic boundary value problem, arising as it does from the Cauchy-Riemann system. The main tool to prove regularity of solutions in the study of this problem are L2-estimates: subelliptic estimates, superlogarithmic estimates, compactness estimates. In the first part of the talk we give motivation and classical results on this problem. In the second part, we introduce general estimates for "gain of regularity" of solutions of this problem and relate it to the existence of weights with large Levi-form at the boundary. (Keywords: q-pseudoconvex/concave domains, subelliptic estimates, superlogarithmic estimates, compactness estimates, finite type, infinite type. MSC: 32D10, 32U05, 32V25.) • Wednesday 17 June 2009 h. 15:00, room 2AB/40 "Finite p-groups" Eleonora CRESTANI (Ph.D. in Pure Math., Dip. Mat.) -Abstract A$p$-group is a group in which every element has order a power of$p$(where$p$is a prime). The first part of the seminar is an introduction to this area, and I will give same examples that, in particular, show why$p$-groups play such an important role in the theory of finite groups. In the second part of the seminar, some typical problems that arise in finite$p$-groups theory are presented, with a particular attention to the ones that I studied during my Ph.D. • Wednesday 3 June 2009 h. 14:30, room 2AB/40 "Social Interactions and heterogeneous agent models. Applications to Economics and Finance" Marco TOLOTTI (Università di Venezia) -Abstract Relying on my work in the field of contagion models, based on interacting particle systems, I will discuss some open issues concerning the applicability of complex systems in Economics and Finance. I will present some applications of a class of Markov models that are in line with recent research in Economic Theory. In particular I will highlight the importance of modeling social interactions, bounded rationality, heterogeneous agents and random utilities. [Keywords: heterogeneous agent models, intensity-based models, mean field interactions, non reversible Markov processes, phase transition, random utilities, social interactions, stochastic population processes, strategic complementarities.] • Wednesday 20 May 2009 h. 15:00, room 2AB/40 "Introduction to Moduli Spaces" Ernesto MISTRETTA (Univ. Padova, Dip. Mat.) -Abstract We will explain the meaning of moduli spaces as spaces parametrizing geometrical objects, giving some well known examaples as Grassmannians and Projective Spaces. We will focus our attention on the moduli space of triangles, constructing it and elucidating problems of symmetries and monodromy, that appear in more sophisticated cases. Time permitting we will illustrate open problems and recent progress in the theory of moduli spaces of curves. • Wednesday 13 May 2009 h. 15:00, room 2AB/40 "A global approach to multiobjective optimization" Alberto LOVISON (Univ. Padova, Dip. Mat.) -Abstract In real life situations, there are usually more than one objective to deal with in order to design a successful project. For instance, a good car should be fast, while being low consuming as possible, should protect occupants while keeping compact external dimensions, should be comfortable and maximize, or minimize, at the same time, many other performance indicators. Multiobjective optimization defines the mathematical framework for dealing with such problems. In this talk we propose a gentle introduction to this subject, strictly related to the standard single objective approach based on the study of critical points. We will recall the concept of Pareto critical set, introduced by Stephen Smale, and illustrate an effective algorithm for the global search of these critical sets. This topological approach allows the definition of a Morse theory for vector functions. On the other hand, severe restrictions derive from the curse of dimensionality and from the existence of structurally unstable singularities in higher dimensions. • Thursday 23 April 2009 h. 14:30, room 2AB/40 "A primer in Arakelov geometry" Vincent MAILLOT (professor at CNRS - Paris VI) [The seminar will be divided in two parts of 1h each, the first of which, of introductory type, will be suitable for a large public] -Abstract 1. In the first part of my talk, I'll introduce the basic notions and problems involved in the early developpements of Arakelov geometry. 2. In the second part, I'll give a more formal and systematic introduction to the subject. Time permitting, I'll explain the statement of the arithmetic Riemann-Roch theorem and I'll give a recent application to number theory. • Wednesday 8 April 2009 h. 15:00, room 1A/150 "Constraint Programming Techniques for Mixed Integer Linear Programs" Domenico SALVAGNIN (Ph.D. in Applied Math., Dip. Mat.) -Abstract Two paradigms in the field of optimization have reached a high degree of sophistication from the point of view of both theory and implementation: Constraint Programmming (CP) and Mixed Integer Programming (MIP). The CP and MIP paradigms have strengths and weaknesses that complement each other: thus an integration of the two has the potential to yield important benefits. In this talk I will provide a brief introduction of the two paradigms and present two cases of application of CP techniques, namely nogoods and propagation, to enhance MIP resolution algorithms, namely dominance detection and primal heuristics. • Wednesday 25 March 2009 h. 15:00, room 1BC/45 "Computing with Affine Algebraic Groups" Andrea PAVAN (Ph.D. in Pure Math., Dip. Mat.) -Abstract How can one solve the Rubik's Cube? The question turns out to be equivalent to a problem about groups, whose solution is provided by Computational Group Theory. More generally, CGT is concerned with designing and analyzing algorithms to compute information about groups which can be described by a finite amount of data. Examples include finite permutation groups, finitely presented groups, finitely generated matrix groups and polycyclic groups, which have been at the center of the subject since the beginning of the last century. On the contrary, very little work has been done on affine algebraic groups. These are, roughly speaking, groups whose elements are solutions to some system of polynomial equations in finitely many indeterminates. Although their structure is well understood, they have been rarely studied from a computational point of view. Two pioneers in the field are Grunewald and Segal, who developed the basis for many useful algorithms. In the first part of the talk we will give an introductory overview of both Computational Group Theory and the theory of affine algebraic groups. Then we will describe the work of Grunewald and Segal, as well as some improvements of their methods. • Wednesday 11 March 2009 h. 15:00, room 2AB/40 "The Dynamics of a Spin-Flip System by an Example: the Curie-Weiss Model" Francesca COLLET (Ph.D. in Applied Math., Dip. Mat.) -Abstract A spin system is a system composed by N sites at which is associated randomly a +1 or -1 value, called spin. Each spin is influenced by all the others in the same way and this makes it flip with a certain probability. The dynamics just mentioned are completely described by the time evolution of the Magnetization (the sum of all the spin values divided by N), so it is sufficient to study the behavior of this last quantity. As N grows to infinity, its limiting dynamics are deterministic (driven by ODE) and exhibit a phase transition: multiple equilibrium solutions arise depending on the value of a parameter, which is the inverse of the temperature. The Curie-Weiss model is a basic example of it. After having recalled some notions of Probability, we try to explain in a simple and intuitive way, avoiding the most part of the technicalities, how the Curie-Weiss model evolves in time at different temperatures. • Wednesday 25 February 2009 h. 15:00, room 2AB/40 "An invitation to Frobenius manifolds" Luca Philippe MERTENS (Ph.D. in Pure Math., SISSA Trieste) -Abstract Frobenius manifolds are geometric structures encoding the dispersionless limit of a bihamiltonian integrable hierarchy. They were introduced by B. Dubrovin to study the remarkable connection between 1+1 integrable systems, 2D topological field theories and Gromov-Witten invariants of symplectic manifolds discovered by E. Witten and M. Kontsevich for the case of the KdV hierarchy. The first part of the talk will be an introduction to key ideas and definitions. We will define an appropriate class of integrable hierarchies and we will show how one can associate a Frobenius manifold to them, and vice versa. We will present in detail the example of the Toda Hierarchy, encoding the Gromov-Witten Invariants of the complex projective line. The second part of the talk will focus on recent developments of the theory related to 2+1 integrable systems. This class of hierarchies naturally lead to the notion of infinite dimensional Frobenius manifold. We will present the case of the 2D Toda hierarchy, highlighting main differences with respect to the finite dimensional case and pointing out future applications of the theory. • Wednesday 11 February 2009 h. 15:00, room 2AB/40 "Solving Mixed Integer Programs with Gomory Cutting Planes" Arrigo ZANETTE (Ph.D. in Applied Math., Dip. Mat.) -Abstract Gomory cutting planes were first introduced by Gomory in 1958 to solve Integer and Mixed Integer ('60) Programs (MIP). However they were soon abandoned in favor of enumeration tecniques, until, in 1996, they were revisited by Balas et al., becoming a fundamental tool for commercial MIP solver. Despite their long history and relative success, the lack of understading on their practical behaviour makes Gomory Cutting Planes an interesting research topic. In particular it is clear that they might perform much better than current implementations have managed to do, but nobody has found the right way of using them yet. In the talk we will review Gomory cutting planes, their typical usage in commercial MIP solvers and recent research findings that might eventually lead to a new performance breakthrough. • Wednesday 28 January 2009 h. 15:00, room 2AB/40 "An introduction to p-adic analysis" Valentina DI PROIETTO (Ph.D. in Pure Math., Dip. Mat.) -Abstract The p-adic numbers were discovered by Hensel at the end of nineteenth century and in the last century they came to a central role in number theory. In the first part of this talk we shall give the definition of the field of p-adic numbers and present some results in elementary p-adic analysis always comparing with results on classical analysis over the real numbers; in the second part we shall describe an example which explains a basic case of the result proven in our Ph.D thesis. • Wednesday 14 January 2009 h. 15:00, room 1A/150 "(De)-Localization of some (1+1)-dimensional models" Martin BORECKI (Ph.D. in Applied Math., Technische Univ., Berlin) -Abstract We consider a (1+1)-dimensional model, i.e. a directed model for a linear chain. The chain is randomly distributed in space and undergoes an interaction with the environment and itself. Thus, it can be seen as a random polymer and we want to study its spatial distribution as a function of its length and its interaction parameters. The self-interaction consists of a Gradient and Laplacian mixture type, whereas the interaction with the environment is reduced to a delta-pinning, i.e. the chain gets a reward each time it touches the x-axis. We discuss the localization behaviour of the model, which displays remarkable differences (phase transitions) as the parameters of the interaction vary. Furthermore we consider what changes, if we additionally introduce an impermeable wall. Motivation, introduction and explanations will hopefully make the talk accessible to a large audience. # 2008 • Wednesday 10 December 2008 h. 15:00, room 1BC/45 "Some problems from Geometric Measure Theory: Plateau, Bernstein, Dido" Davide VITTONE (researcher in Pure Math., Dip. Mat.) -Abstract The aim of Geometric Measure Theory (GMT) is to approach geometric problems by means of measure-theoretical tools. After a brief presentation of some GMT definitions of "surface measure", we will summarize the history and main results about three classical questions: minimal surfaces (or Plateau problem), the Bernstein problem and the isoperimetric (or Dido) problem. • Wednesday 26 November 2008 h. 15:00, room 1C/150 "On matrices with the Edmonds-Johnson property" Alberto DEL PIA (Ph.D. in Applied Math., Dip. Mat.) -Abstract Integer programming is the problem of optimizing a linear function over the integral points in a polyhedron P, expressed as a system of linear inequalities. It is known that it is equivalent to optimizing such linear function over the polyhedron P_I, that is the convex hull of the integral points in P. One of the main problems in mathematical programming is to developed tools to get P_I, and one of such methods is the Chvatal-Gomory procedure. This procedure, starting from P, gives a sequence of smaller polyhedra, P', P'', ..., that converges to P_I in a finite number of iterations. The number of iterations needed to get P_I gives an order of complexity to the problems. We survey some old and new results of classes of problems in which P'=P_I. • Wednesday 12 November 2008 h. 15:30, room 1BC/45 "Synchronization and homomorphisms" Pablo SPIGA (Ph.D. in Pure Math., Dip. Mat.) -Abstract An automaton is a machine which can be in any of a set of internal states which cannot be directly observed. A synchronizing automaton is an automaton admitting a sequence of transitions which take the automaton from any state into a known state. In this talk we present some recent connections between synchronizing automatons, permutation groups and graph homomorphisms. All relevant definitions would be given during the talk. • Wednesday 29 October 2008 h. 15:00, room 2BC/30 "Information Flow on Trees: the Reconstruction Problem and the Purity of the Free Gibbs Measure" Marco FORMENTIN (Ph.D. in Applied Math., Dip. Mat.) -Abstract The Reconstruction Problem on a tree can be stated as follows. We send a signal from the root to the boundary, making a prescribed error at every edge of the tree. Suppose you know what happens at distance N from the origin of the tree. What can you say about the original signal sent from the root when N goes to infinity? This problem, concerning the flow of information on trees is equivalent to the purity of the free Gibbs Measure for the Ising/Potts models on a tree. Purity can be regarded as a special kind of phase transition. We review this equivalence and give old and new thresholds for the transition to purity for the free Potts Gibbs measure on regular trees. • Wednesday 15 October 2008 h. 15:00, room 1C/150 "The Grothendieck fundamental groups: a basic introduction", Nicola MAZZARI (Ph.D. in Pure Math., Dip. Mat. "F. Enriques", Milano) -Abstract We give a basic introduction to the Grothendieck fundamental group. In topology there are (at least) two ways to define the fundamental group$\pi_1(S,s)$of a topological space S. Namely we can view it as the set of loops based on a point s up to homotopy, or as the group of automorphism of the universal cover. Only this second approach can be made algebraic and allows to define the Grothendieck (or etale) fundamental group$\pi_1^\et(S,s)$of a scheme S with respect to a geometric point s. We will consider only affine schemes and we don't assume the reader familiar with algebraic geometry. • 11 giugno 2008 in aula 1BC/45 alle ore 15:00 "An overview on low degree non-abelian cohomology", Pietro POLESELLO (researcher, Dip. Mat.) -Abstract In the first part of the seminar, which will be of introductory level, I will recall some basic facts about the first cohomology set$H1(X;\underline G)$, for a given (not necessarily abelian) topological group G, such as the classification of principal G-bundles, the Hurwitz formula and the classification of G-coverings. The second part of the seminar will be devoted to the generalization of some of these results to the "second non abelian cohomology" of G. • 28 maggio 2008 in aula 1BC/45 alle ore 15:00 "An Introduction to Stochastic Fluid Dynamic Models" David BARBATO (researcher at our Dip. Mat.) -Abstract The Navier-Stokes problem, still unsolved by more than 150 years, represents the starting point for lots of mathematical research topics. The aim of the talk is to present selected fluidodynamic models, in the deterministic and stochastic case, developed from Navier-Stokes equations. In particular the GOY shell model, a Fourier system simplified with respect to the Navier-Stokes one, will be described, and some recent rigorous results discussed. Finally open questions and conjectures on turbolence flows will be presented. • 14 maggio 2008, in aula 1BC/45 alle ore 15:00 "Curve lengths and surface areas" Roberto MONTI (researcher at our Dip. Mat.) -Abstract We discuss different definitions for the length of a curve and for the area of a hypersurface in the Euclidean space and in more general metric spaces. The talk has an expository character and is an introduction to Geometric Measure Theory. • 30 April 2008 room 1BC/45 , at 15:00 "Chaotic phenomena described by stochastic equations", Luigi MANCA (grant holder at our Dip. Mat.) -Abstract It is well known that many natural phenomena such as population dynamics, stock exchange, diffusion of particles, can be seen as 'chaotic'. To give a mathematical description of these chaotic' phenomena has been developed the theory of stochastic processes and of the related stochastic differential equations. Starting by the fundamental concept of Brownian motion, I shall introduce the main ideas and the basic tools in order to understand some easy models driven by stochastic equations. Moreover, I shall describe how stochastic equations can be used to study some deterministic model. • 29 April 2008, room 2BC/60 at 11:30 "Quiver mutation and derived equivalence", Bernhard KELLER (professor at Paris 7) [The seminar will be divided in two parts of 40' each, the first of which, of introductory type, will be suitable for a large public] -Abstract 1. In the first part, we will define and study quiver mutation. This is an elementary operation on quivers (=oriented graphs) which was introduced by Fomin and Zelevinsky in the definition of cluster algebras at the beginning of this decade. The combinatorics behind quiver mutation are rich and varied. We will illustrate them on numerous examples using computer animations. 2. In the second part of the talk, we will "categorify" quiver mutation using representation theory. More precisely, by combining recent work of Derksen-Weyman-Zelevinsky and Ginzburg, we will show how quiver mutations give rise to equivalences between derived categories of certain differential graded algebras. These derived categories are closely related to cluster categories and thus to cluster algebras. This is joint work with Dong Yang. • 16 aprile 2008, in aula 1BC/45 alle ore 14:30 "The Basic Picture on sets evaluated over an overlap algebra" Paola TOTO (dottoranda in Matematica Pura - Università del Salento) -Abstract In his forthcoming book, G. Sambin introduces a new topological theory, called "The Basic Picture". In this theory both the notion of topological space and its point-free version are generalized. The concept of overlap algebra is also introduced in order to put in algebraic form the properties needed to define the new topological structures. In this seminar we shall give a tutorial introduction to our work, whose ultimate goal is to generalize such topological notions in the context of many-valued sets. In many-valued set theory sets are built by using propositions evaluated in an algebraic structure. To reach our goal a key point is to check whether the original algebrization of Sambin's topological notions can be considered also as the algebrization of their many-valued version. We prove that this is the case if and only if we take an overlap algebra as the underlying structure of truth values. • 02 aprile 2008, in aula 1BC/45 alle ore 15:00 "Numerical modeling for convection-dominated problems" Manolo VENTURIN (Borsista in Matematica Computazionale) -Abstract During the last years, there has been a great interest in the development of sophisticated mathematical models for the simulation of real life applications which involves convection-dominated phenomena. For example, these problems concern the solution of scalar advection-diffusion equations, the Navier-Stokes equations and the Shallow Water equations. The main goal of this seminar is to review the most important difficulties that arise in the numerical approximation of this kind of problems when convection dominates the transport process. Moreover, we present a method for the treatment of this equations with the use of the finite element discretization on the domain. • 27 febbraio 2008, in aula 1BC/45 alle ore 15:00 "Computing VaR and CVaR for energy derivatives", Giorgia CALLEGARO (Matematica Computazionale, S.N.S. di Pisa) -Abstract The aim of the talk is to give an idea of the possible applications of mathematics to energy derivatives markets, when computing the risk related to an investment in such a market. First of all we will introduce the notion of derivative asset, starting with an analysis of the basic cases of Call and Put options and arriving to the more complicated swing option case, that are all financial products generally traded on option markets all over the world, with an "underlying" that can be anything, from foreign currencies to stocks, oranges, gas or timber. We will explain how the underlying price dynamics are modeled in energy markets, in basic cases and we will present the problems of "pricing" a derivative and computing the risk related to an investment. In particular, focusing on the gas market, we will explain how the fair price of swing options can be (numerically) computed, by applying the Dynamic Programming Principle and the vectorial quantization. In the same setting, we will also obtain numerical estimates, by means of stochastic recursive algorithms of the Robbins-Monro type, for two different risk measures, namely the "Value at Risk" (VaR) and the "Conditional Value at Risk" (CVaR). (Keywords: swing option, dynamic programming, quantization, risk measure, Robbins-Monro algorithms.) • 13 febbraio 2008, in aula 1BC/45 alle ore 15:00 "Cluster Algebras: an overview", Giovanni CERULLI IRELLI (Dottorato in Matematica Pura) -Abstract Cluster algebras wer introduced in 2001 by S. Fomin and A. Zelevinsky with the aim of study total positivity and canonical basis in semi-simple algebraic groups. After their introduction, the theory has been developed in several unexpected fields of mathematics, e.g. quiver representations, Grassmannians and projective configurations, a new family of convex polytopes (generalized associahedra) including as special case Stasheff's associahedron, Al. Zamolodchikov's Y-systems in thermodynamic Bethe Ansatz, discrete dynamical system, Teichmuller spaces and Poisson geormetry,etc.. . In this talk I will recall the definition of such algebraic structure and I will give some motivating examples arising from algebraic geometry. I will also try to give an idea of the connection with the previous theory with particular attention to quiver representations. • 30 gennaio 2008, in aula 2BC/30 alle ore 15:00 "Algorithms for the computation of the joint spectral radius", Cristina VAGNONI (Dottorato in Matematica Computazionale) -Abstract The asymptotic behaviour of the solutions of a discrete linear dynamical system is related to the spectral radius R of its associated family F; in particular, a system is stable if R <= 1 and there exists an extremal norm for F. In the last decades some algorithms have been proposed in order to find real extremal norms of polytope type in the case of finite families. However, recently it has been observed that it is more useful to consider complex polytope norms. In this talk we show an approach based on the notion of "balanced complex polytopes"; due to the strong increase in complexity of the geometry of such objects, the exposition will be confined to the two-dimensional case. In particular, we give original theoretical results on the geometry of two-dimensional balanced complex polytopes in order to present two efficient algorithms, one for the geometric representation of a balanced complex polytope and the other the computation of the corresponding complex polytope norm of a vector. • 16 gennaio 2008, in aula 1A/150 alle ore 15:00 "Metodi di teoria del potenziale per l'analisi di problemi col dato al bordo singolarmente perturbati", Matteo DALLA RIVA (Dottorato in Matematica Pura) -Sunto Si considererà un problema con dato al bordo definito su un aperto limitato dello spazio Euclideo 3-dimensionale. Tale aperto avrà un buco al suo interno. Il nostro scopo e' di descrivere il comportamento della soluzione del problema con dato al bordo quando il buco collassa ad un punto. Problemi di questo genere sono stati lungamente studiati tramite le tecniche dell' "analisi asintotica" (si vedano ad esempio i lavori di Keller, Kozlov, Movchan, Maz'ya, Nazarov, Plamenewskii, Ozawa e Ward). Illustreremo in un facile esempio quale tipo di risultato possiamo attenderci applicando tali tecniche. Poi mostreremo il risultato che si ottiene tramite l'approccio alternativo proposto da Lanza de Cristoforis in alcuni lavori a partire dal 2001 e metteremo in luce le principali differenze tra i due risultati. # 2007 • 12 dicembre 2007, in aula 2BC/60 alle ore 15:00 "Un calcolo logico per la computazione quantistica", Paola ZIZZI (Dottorato in Matematica Pura) -Sunto Il calcolo dei sequenti (LK), un sistema di deduzione logica introdotto da Gentzen inizialmente per la logica Classica, e in seguito esteso alla logica Intuizionistica (LJ), esiste oggi anche per le logiche sub-strutturali, come la logica Lineare di Girard, e la logica di Base di Sambin. In questo seminario, dopo una prima parte introduttiva, ci proponiamo di introdurre un adeguato calcolo dei sequenti per la computazione quantistica (finora descritta solo in termini di reti di cancelli logici quantistici). Dai risultati finora ottenuti, sembra che il calcolo dei sequenti della logica di Base, possa, con le opportune modifiche, servire a tale scopo. • 28 novembre 2007, in aula 2BC/30 alle ore 15:00 "Sistemi dinamici e insiemi di Aubry-Mather", Olga BERNARDI (assegnista in Matematica Pura) -Sunto Un sistema dinamico consiste di uno spazio delle fasi che descrive gli stati permessi ad un sistema e di una legge che definisce l'evoluzione temporale di questi stati. L'evoluzione può essere continua, come per le equazioni differenziali, o discreta, come per le mappe. Nello studio dei sistemi dinamici un ruolo fondamentale è svolto dagli insiemi invarianti per la dinamica. Dopo una introduzione per non-esperti ai sistemi dinamici, si definiscono gli insiemi invarianti di Aubry-Mather per una classe di mappe quasi-integrabili in 2 dimensioni e si discute la loro localizzazione tramite tecniche di regolarizzazione ispirate alle teorie di viscosita'. • 14 novembre 2007, in aula 2BC/30 alle ore 15:00 "Sistemi disordinati: vetri di spin e polimeri diretti" Agnese CADEL (Dottorato in Matematica Computazionale) -Sunto Si parla di sistemi disordinati (o complessi) quando sono presenti eterogeneità a livello microscopico e per questo manifestano una ricca varietà di comportamenti. Dopo una breve introduzione alla meccanica statistica dei sistemi complessi, parleremo dei due più famosi esempi di questo tipo di sistemi: i vetri di spin e i polimeri diretti. • 31 ottobre 2007, aula 2BC/30, alle ore 15:00 "Metodi di viscosita' per la riduzione dell'ordine di sistemi di controllo singolarmente perturbati", dott. Gabriele TERRONE (Dottorato in Matematica Pura) -Sunto Si considera un sistema di due equazioni differenziali ordinarie per la coppia di variabili$(x(t),y(t))$. L'evoluzione di$x(t)$e$y(t)$avviene su due scale temporali differenti: la velocità delle variabili "veloci"$y(t)$e' proporzionale ad un parametro positivo$(\epsilon)^{-1}$. Si determina una dinamica "limite", per le sole variabili "lente"$x(t)$, che rappresenta il comportamento del sistema originario quando$\epsilon$tende a zero. Si mostra anche che il sistema limite e' in grado di fornire informazioni sulla stabilita' del sistema originario. • 17 ottobre 2007, aula 2BC/30, alle ore 15:00 "Formulazioni estese per problemi di programmazione intera mista", dott. Marco DI SUMMA (Dottorato in Matematica Computazionale) -Sunto In certi problemi di ottimizzazione, detti problemi di programmazione intera mista, è necessario studiare regioni dello spazio definite da disequazioni lineari, con la condizione aggiuntiva che alcune delle coordinate possono assumere solo valori interi. L'analisi di queste regioni nel loro spazio naturale di definizione è resa complessa proprio dai vincoli di interezza. Tuttavia in certi casi l'introduzione di variabili aggiuntive permette di descrivere in modo molto più semplice la regione in esame. Tali formulazioni, date in uno spazio di dimensione superiore, sono dette "formulazioni estese" e sono di fondamentale importanza per la soluzione di problemi di questo tipo. In questo seminario, dopo un'ampia panoramica introduttiva, illustrerò una tecnica che consente di ottenere semplici formulazioni estese per una vasta classe di problemi. Metterò in evidenza potenzialità e limiti di questo approccio. (Lavoro in collaborazione con M. Conforti, F. Eisenbrand e L. Wolsey) • 26 settembre 2007, Aula 1C/150, alle ore 11.30 "Algebra and Topology", Prof. DAN SEGAL (Oxford-All Souls College) -Abstract Actually the subject begins with number theory. In the 1930s Wolfgang Krull extended the Fundamental Theorem of Galois Theory from finite Galois extensions to infinite Galois extensions. In order to obtain a bijective correspondence between intermediate fields and subgroups of the Galois group, Krull realized that it is necessary to consider the latter as a topological group: each field corresponds to a subgroup and conversely. The topology is defined by taking as neighbourhoods of the identity the Galois groups of the big field over (larger and larger) finite sub-extensions of the small field. In this way, the Galois group appears as the inverse limit of a system of finite (Galois) groups. A group that is the inverse limit of an inverse system of finite groups is called a profinite group. It is in a natural way a compact, totally disconnected topological group (inheriting these properties from the finite groups considered as finite spaces). An infinite abstract group may have many different structures as a profinite group (i.e. different topologies) (or of course none). But it was discovered by J-P. Serre in the 1970s that for certain kinds of profinite group, the topology is uniquely determined by the underlying group. These are the so-called finitely generated pro-p groups. Serre wondered whether the same might be true for finitely generated profinite groups in general; after about 30 years of partial results by several mathematicians, we have recently shown that the answer is "yes". In fact, what the proof does is to show that many closed subgroups can be constructed in a purely algebraic way. In the talk I will try to sketch some of the mathematics involved in the proof, and mention other related results and open problems. • 6 giugno 2007, aula 1AD/30, alle ore 15.00 "Un modello dinamico di contagio: interpretazione finanziaria di un modello di particelle interagenti a campo medio", Elena SARTORI (dottorato di ricerca in Matematica Computazionale) -Abstract Un gruppo di aziende attive sul mercato può essere rappresentato da un sistema di N particelle con interazione a campo medio. Definiremo per esse una dinamica tale per cui il modello risultante sarà non reversibile: di questo modello sarà interessante studiare il comportamento per tempi "lunghi" trovandone le equazioni della dinamica a volume infinito e le soluzioni stazionarie, ed in seguito le approssimazioni a volume finito (N grande, ma finito). Quantificheremo infine le perdite sofferte da un'istituzione finanziaria che possiede un portafoglio le cui posizioni sono date dalle N aziende che affrontano il rischio di credito. Grazie a questo approccio, che tiene conto dell'interazione tra aziende (a livello microscopico), saremo in grado di spiegare il fenomeno delle crisi di credito, ossia di periodi in cui molte aziende si ritrovano improvvisamente in uno stato di forte stress finanziario. • 30 maggio 2007, aula 1BC/50, alle ore 15.00 "Landau's Problems on Primes" (Seminario sui numeri primi), Janos PINTZ (Hungarian Academy of Sciences) -Abstract At the 1912 Cambridge International Congress, Landau listed four basic problems about primes. These problems were characterised in his speech as "unattackable at the present state of science". The problems were the following. (1) The Goldbach conjecture, that every even number exceeding 2 can be written as the sum of two primes. (2) The Twin Prime Conjecture, the existence of infinitely many twin primes. (3) Does there exist always at least one prime between neighbouring squares? (4) Are there infinitely many primes p such that p-1 is a square? All these problems are still open. In the lecture a survey will be given about partial results on problems (1)-(3), with special emphasis on the recent results of D. Goldston, C. Yildirim and the speaker on small gaps between primes. • 23 maggio 2007, aula 1BC/45, alle ore 15.00 "Accoppiamenti completamente monotoni per processi di Markov", Ida Minelli (assegnista in Matematica Computazionale) -Sunto Verranno discussi i concetti di monotonia e monotonia completa per processi di Markov a valori in uno spazio degli stati finito e parzialmente ordinato. La monotonia è una proprietà della matrice di transizione (o del generatore infinitesimale, nel caso di tempi continui) del processo. La monotonia completa è un concetto più forte del precedente ed è utile in molte applicazioni, ad esempio quando si vogliono ottenere simulazioni dalla misura stazionaria di una catena di Markov utilizzando algoritmi di simulazione perfetta. Al contrario di quanto accade per la monotonia, non esiste un criterio semplice per verificare la monotonia completa. Per questo motivo è naturale domandarsi per quali insiemi parzialmente ordinati i due concetti sono equivalenti. Questo problema è stato completamente risolto nel caso di processi a tempi discreti, ma si è rivelato più complesso nel caso di processi a tempi continui. • 16 maggio 2007, in aula 1BC/50, alle ore 15.00 "Numeri p-adici e studio coomologico delle varietà algebriche sopra un anello di valutazione discreta" Daniele CHINELLATO (dottorato di ricerca in Matematica Pura) -Sunto Sia p un primo. I numeri p-adici possono essere pensati come completamento dei razionali Q rispetto ad una distanza che permetta di codificare proprietà aritmetiche relative al primo p. Ciò permette di introdurre tecniche analoghe all'analisi reale nello studio di problemi diofantei. D'altro canto tali problemi (ad esempio la congettura di Mordell, o il teorema di Fermat) trovano una loro naturale collocazione nella teoria degli schemi sopra un anello (in generale...). In tale teoria nel corso degli anni sono state sviluppate adeguate tecniche coomologiche nella speranza di ricondurre lo studio di problemi aritmetici (e di altro genere) nell'alveo dell'"algebra lineare" (o semilineare). In una prima parte del seminario verranno introdotti i numeri p-adici e alcune loro proprietà aritmetiche e "metriche". In seguito si passerà ad illustrare brevemente il significato dello studio di un argomento diofanteo dal punto di vista della coomologia di una varietà (funzione zeta...). Infine, nell'ottica cosí introdotta esporrò brevemente la ricerca compiuta nella tesi di dottorato. • 18 aprile 2007, in aula 1BC/50 (ex 1B/50) alle ore 15.00 "Fasci ed equazioni differenziali lineari", Giovanni MORANDO (dottore di ricerca in Matematica Pura) -Sunto Le equazioni differenziali lineari sono tra gli oggetti matematici più trasversali. I metodi per studiarle cosí come le loro applicazioni possono essere di natura fisica, geometrica, analitica, numerica, algebrica... In questo seminario presenteremo l'approccio algebrico sviluppato dalla scuola giapponese di M. Sato sin dagli anni '60, basato sulla nozione di "D-modulo" (la generalizzazione algebrica dei sistemi di equazioni differenziali lineari). Uno dei risultati più significativi di questo approccio è la "corrispondenza di Riemann-Hilbert", una generalizzazione del 21mo problema di Hilbert (esiste un'equazione differenziale lineare le cui soluzioni olomorfe abbiano un gruppo di monodromia prescritto?). Cercheremo di spiegare, attraverso numerosi semplici esempi, la corrispondenza di Riemann-Hilbert in quanto equivalenza tra le categorie delle equazioni differenziali lineari a singolarità regolari (oggetti di natura analitica) e degli spazi delle loro soluzioni olomorfe (oggetti di natura topologica). Tale corrispondenza sottolinea quindi come oggetti di natura e utilizzo diversi siano in realtà strettamente legati. • 2 aprile 2007, in aula Aula 1AD/50 (ex 1D/50), alle ore 16.30 "To what extent is Lie theory for groupoids like that for groups?", Ieke MOERDIJK (professore all'università di Utrecht) -Abstract Lie groupoids play an increasingly important role in foliation theory, symplectic and Poisson geometry, and non-commutative geometry. In this lecture, we explain how some basic properties of Lie groups extend to groupoids, and how some other properties don't. (Keywords: Lie theory, non-commutative geometry) • 28 marzo 2007, in aula 1AD/50 (ex 1D/50), alle ore 14.30 "Problemi con preferenze ed incertezza", Maria Silvia PINI (dottoranda in Matematica Computazionale) -Sunto Molti problemi della vita reale presentano dei vincoli, cioè delle richieste che devono essere soddisfatte totalmente. A volte però risulta più naturale esprimere questi vincoli in maniera meno stringente tramite delle preferenze. Oltre alle preferenze, molti problemi reali sono caratterizzati da incertezza, cioè dalla presenza di eventi incerti che non possono essere controllati dall'utente. In alcuni casi l'utente può avere un'informazione di tipo probabilistico o possibilistico riguardo al verificarsi di questi eventi incerti, altre volte può non avere alcuna informazione. In questo seminario presenteremo dei formalismi che modellano problemi con vari tipi di preferenze e l'incertezza, analizzeremo le proprietà di questi formalismi e considereremo il caso in cui le preferenze sono espresse da più utenti. In questo contesto considereremo l'aggregazione di preferenze e analizzeremo proprietà desiderabili come la fairness e la non-manipolabilita', estendendo risultati ben noti nella teoria dei voti. Infine esamineremo scenari in cui alcuni utenti decidono di non rivelare tutte le loro preferenze su un certo insieme di alternative, per esempio per ragioni di privacy, e analizzeremo la complessità computazionale di calcolare le alternative che sono comunque ottime. • 14 marzo 2007, in aula 1C/50, alle ore 14.30 "Funzioni separatamente olomorfe e CR", Raffaella MASCOLO (dottoranda in Matematica Pura - 20o ciclo) -Sunto Esistono varie caratterizzazioni equivalenti per funzioni olomorfe definite su aperti di$\C^n\$, la prima delle quali è a proprietà i essere rappresentate localmente come somme di serie di potenze convergenti. E ovvio che una funzione olomorfa in più ariabili è lomorfa separatamente in ciascuna variabile. E proprio separando le variabili che molte delle proprietà noi familiari delle funzioni olomorfe di una variabile complessa, come la formula integrale di Cauchy, hanno una versione corrispondente in più ariabili complesse. D'altra parte è n fatto notevole che una funzione separatamente olomorfa sia in realtà i classe C1, e quindi olomorfa nel complesso delle sue variabili (Teorema di Hartogs, 1906).
L'esposizione intende affrontare il problema della separata analiticità d estendere la discussione al caso di funzioni separatamente CR su varietà CR, attraverso una rivisitazione ed una generalizzazione di un risultato di Henkin e Tumanov del 1983.
(Parole chiave: Teorema di Hartogs, funzioni CR, separata analiticità)
• 7 marzo 2007, in aula 1C/50, alle ore 14.30
"Tecniche di accelerazione della convergenza per soluzioni di EDP per la modellistica di semiconduttori", Maria Rosaria RUSSO (assegnista in Matematica Computazionale)

-Sunto
Nel seminario verranno presentati alcuni metodi di estrapolazione per accelerare la convergenza di successioni vettoriali. I metodi presi in analisi, sebbene derivino da filosofie differenti, sono in grado di accelerare il processo iterativo di convergenza di successioni vettoriali, anche senza nessuna informazione sul modo in cui esse vengono generate; questo li rende applicabili a diverse tipologie di problemi, lineari e non lineari. Verranno mostrati alcuni risultati relativi alla applicazione delle tecniche di accelerazione per la risoluzione di sistemi lineari di grandi dimensioni, e per accelerare il processo di convergenza di algoritmi iterativi non lineari di tipo Gummel (mappa di Gummel), solitamente usati per disaccoppiare le equazioni del modello Drift Diffusion, nella modellistica per semiconduttore.
(Parole chiave: estrapolazione vettoriale, modellistica per semiconduttori.)
• 14 febbraio 2007, in aula 1C/50, alle ore 15.00
"Dualità di Tannaka per il gruppoide di Lie", Giorgio TRENTINAGLIA (dottorando in Matematica Pura - 19o ciclo

-Sunto
La teoria dei gruppoidi di Lie unifica concettualmente nozioni geometriche alquanto disparate: varieta' differenziabili, gruppi di Lie, azioni lisce di gruppi di Lie su varieta', foliazioni su varieta', orbifolds ... Nel corso degli ultimi anni gli studiosi si sono scontrati spesso con le difficolta' legate al concetto di rappresentazione, senza tuttavia fornire una spiegazione soddisfacente dei motivi e del carattere di tali difficolta' (non era in particolare chiaro se queste ultime fossero davvero insormontabili). Il mio lavoro fornisce una risposta chiara e definitiva a questo annoso problema e apre nuove prospettive d'indagine. Nel mio discorso limitero' la teorizzazione astratta al minimo necessario, cercando di procedere per mezzo di esempi chiave.
(Parole chiave: teoria della rappresentazione, gruppoidi di Lie.)

• 31 gennaio 2007, in aula 1C/50, alle ore 15.00
"Soluzione numerica di EDP per la modellistica di semiconduttori", dott. Roberto BERTELLE (dottorando in Matematica Computazionale - 19o ciclo)

-Sunto
I calcolatori elettronici hanno una grande importanza nello sviluppo e nello studio di nuovi dispositivi elettronici. Ogni dispositivo elettronico è, semplicemente, una regione dello spazio sede di campi e di correnti elettriche la cui conoscenza è di capitale importanza per la comprensione del comportamento fisico del dispositivo.
Il seminario è una introduzione e può essere suddiviso in più parti. Una prima parte riguarda la scrittura dei modelli matematici idonei a rappresentare compiutamente un dispositivo elettronico. Nella seconda parte, le equazioni del modello sono risolte numericamente mediante discretizzazione con elementi finiti lineari. Infine, la terza parte illustra, mediante esempi numerici riferentesi a modelli mono-dimensionali, l'efficacia e i vantaggi delle tecniche di risoluzione numerica descritte.
(Parole chiave: modellistica per semiconduttori, giunzione p-n, elementi finiti.)
• 17 gennaio 2007, in aula 2C/30, alle ore 15.00
"Rappresentazioni integrali per il Propagatore di Schrödinger", dott. Lorenzo ZANELLI (dottorando in Matematica Pura - 19° ciclo)

-Abstract
Si descrive una classe di rappresentazioni per la soluzione dell'equazione di Schrödinger, che descrive il comportamento di una particella non relativistica in un campo elettromagnetico. A tal fine si evidenzia una diretta connessione fra il problema geometrico legato al flusso Hamiltoniano classico e il problema analitico della rappresentazione integrale del nucleo del Propagatore di Schrödinger.

# 2006

• 19 dicembre 2006, in aula 2B/40, alle ore 16.00
"Ramification"
prof. Ahmed Abbes (CNRS, Paris 13)

-Abstract

Starting from the concept of multiplicity of a fixed point in complex analytic geometry, we will introduce the notion of ramification and explain its rich structure in positive characteristic. Then we will give some geometric incarnations of theses phenomenaes.

• 30 novembre 2006, in aula 1B/50, alle ore 14.30
"Averaging and capture into resonance in low-dimensional dynamical systems"
prof. Anatoly Neishtadt, Space Research Institute, Mosca Russia

[Tutorial; 2 ore accademiche con breve intervallo]
-Abstract
Small perturbations imposed on an integrable nonlinear multifrequency oscillatory system cause a slow evolution. According to the classical averaging method, for an approximate description of this evlution one should average the equations of motion over the phases of fast oscillations of the unperturbed system. However, due to resonaces arising, during the evolution, between the frequencies of fast oscillations, the behavior of the system can differ in some essential way from the one predicted by the averaging method. The talk is devoted to an introductory description of such phenomena.