The Graduate Seminar ("Seminario Dottorato" in Italian) started in 2006. It runs about twice per month, usually on Wednesday afternoon, except in Summer. Seminars are usually given by PhD students and PostDocs of the Department of Mathematics, but occasionally also by Senior Researchers. It is assumed that each Student of the Doctoral School will give a talk in the Seminar during his/her doctoral studies.

The Graduate Seminar is a double-aimed activity. On the one hand, speakers have the opportunity to think how to communicate their researches to a public of mathematically well-educated but not specialist people, by preserving both understandability and the flavour of a research report. On the other hand, people in the audience enjoy a rare opportunity to get an accessible-but-precise idea of what's going on in areas of mathematics that they might not know very well.

All speakers are required to prepare a short report on the the topic of their talk, which are collected in a booklet at the end of the  year.

List of 2019-2020 Seminars
(Click on title for abstract)

##### Federico Campanini, An overview on non-unique factorization

Abstract. In this seminar we present some basic facts about non-unique factorizations we hope will be interesting also for the non-algebraic audience. We start with the Fundamental Theorem of Arithmetic and we recall some elementary notions and examples on unique and non-unique factorization monoids. Then we move to the classical Krull-Schmidt Theorem for modules, which states that any module of finite length decomposes as a direct sum of indecomposable modules in an essentially unique way. We briefly discuss about some classes of modules in which direct-sum decompositions are not unique. Finally, we present some properties on factorizations in the monoid of polynomials with non-zero integral coefficients.

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##### Andrei-Florin Albisoru, Potential Theory and Boundary Element Method for the Laplace equation. An introduction

Abstract. We aim to give an overview of Potential theory for Laplace's equation. We introduce the fundamental solution of this equation. Next, we define the layer potentials and we state their properties. Using the layer potentials we will construct a solution of the interior Dirichlet problem for the Laplacian. We also describe a numerical method of solving Laplace's equation, namely the Boundary Element Method. Finally, we present some numerical results.

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##### Enrico Picari, An introduction to sheets of conjugacy classes in reductive groups

Abstract.It is very well-known today that Quantum Mechanics plays a crucial role in describing and understanding nature, at least (but not only) at the smallest scales, i.e. the ones of atoms and subatomic particles. The problem of connecting physical and mathematical features of Classical and Quantum Mechanics dates back to the early days of the theory and although reduction of one to the other is understood heuristically in terms of a limit process in which the Planck constant goes to zero, the mathematical ground of such procedure, known as Semi-classical Analysis, is still an active field of research which includes techniques of apparently distant topics like Harmonic Analysis and Deformations of Poisson Algebras. The aim of this seminar is to introduce the main features of elementary Quantum Mechanics, with a brief historical note, and to give an insight into the state-of-the-art of the semi-classical limit problem.

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##### Filippo Ambrosio, A smooth introduction to the semi-classical problem in Quantum Mechanics

Abstract.Linear algebraic groups arose as a generalization of Lie groups, introduced in the late 1800s to study continuous symmetries of differential equations. The development of the modern theory of algebraic groups with the use of algebraic geometry is mostly due to Borel: in the 1950s, his work led to the definition of Chevalley groups, an important family of finite simple groups. This suggests that algebraic groups can be approached from different perspectives (Group Theory, Algebraic Geometry, Combinatorics) and have applications in several directions (Invariant Theory, Physics). In the first part of the talk we will introduce basic notions and examples of linear algebraic groups. The last part of the seminar aims at describing some of the geometric structure of these groups.

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##### Alberto Righini, Stable hypersurfaces in the complex projective space

Abstract. The classification of complete oriented stable hypersurfaces in the complex projective space could be an important step for the classification of isoperimetric sets. Indeed, the boundary of an isoperimetric set, if smooth, is a hypersurface with constant mean curvature which is stable for variations fixing the volume. In this talk we give an introductory overview of the problem and present some new results, in particular we will characterize the geodesic spheres as the unique stable connected and complete hypersurfaces subject to a certain bound on the curvatures.

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List of 2018-2019 Seminars
(Click on title for abstract)>

Abstracts of 2018-2019 Seminars

##### Yan Hu, Congruent numbers, Heegner method and BSD conjecture

Abstract. The “Congruent number problem” is an old unsolved major problem in number theory. In this seminar we provide a brief introduction to it. We will start from the original version of the problem, and lots of objects will be introduced during the talk. If time permits, some current progresses relateted to the BSD conjecture will also be described.

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##### Nicola Gastaldon, Exact and Meta-Heuristic Approach for Vehicle Routing Problems

Abstract. The Vehicle Routing Problem (VRP) includes a wide class of problems studied in Operations Research and relevant from both theoretical and practical perspectives. In its basic formulation, the problem is to find a set of routes for a given fleet of vehicles through a set of locations, so that each location is visited by exactly one vehicle and the total travel cost is minimized. Such problem is often enriched with many attributes rising from real-world applications, such as capacity constraints, pickup and delivery operations, time windows, etc. VRP belongs to the class of combinatorial optimization problems, and it is very hard to solve efficiently and researchers have developed many exact and (meta-)heuristic algorithms. The former takes advantage of the structure of the mathematical model to obtain a speedup through decomposition methods. The latter exploits heuristic techniques to obtain solutions that trade off quality and computational burden, such as evolutionary algorithms and neighborhood search routines. In our research, we consider the VRP arising at Trans-Cel, a freight transportation company based in Padova. We devised a Tabu Search heuristic implementing different neighborhood search policies, and now embedded in the tool supporting the operation manager at Trans-Cel. The algorithm runs in an acceptable amount of time both in static and dynamic settings, and the quality of the solutions is assessed through comparison with results obtained by a Column Generation algorithm that solves a mathematical programming formulation of the problem. Current research aims at developing data-driven techniques that exploit the information available from the company's repositories to support stochastic transportation demand arising in real time.

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##### Paolo Luzzini, Regular domain perturbation problems

Abstract. The study of the dependence of functionals related to partial differential equations and of quantities of physical relevance upon smooth domain perturbations is a classical topic and has been carried out by several authors. In this talk we will give an introductory overview about regular domain perturbation problems. We will provide concrete examples, highlight the motivations and the possible applications, and present an outline of some new results obtained in collaboration with P. Musolino and R. Pukhtaievych.

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##### Dimitrios Zormpas, Real Options: An overview

Abstract. Financial options are contracts that derive their value from the performance of an underlying asset. They give to their holder the right, but not the obligation, to buy/sell an asset at a predetermined price and time. Contracts similar to options have been used since ancient times. However, the most basic model for their pricing was proposed in the early 1970’s leading to a Nobel prize in 1997. In the late 1970's the term Real Options is coined by Stewart Myers. According to the real options approach an investment characterized by uncertainty and irreversibility is like a financial option on a real asset. For instance, a potential investor has the right but not the obligation to pay a given amount of money in order to make an investment and gain access to the corresponding profit flow. Using standard option pricing tools one can also study the option to leave a market, outsource production, mothball a production plant etc. In this seminar, I will refer to the correspondence between financial and real options and then present the simplest model in the real options literature that has to do with a potential investor who is considering undertaking an uncertain and irreversible investment. Then I will present a number of applications of the real options approach from the broad literature of operations management and finally make a reference to applications of the real options approach in energy economics.

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##### Maria Teresa Chiri, Conservation law models for supply chains

Abstract. Many real situations are modelled by nonlinear hyperbolic first order partial differential equations (PDEs) in the form of conservation or balance laws. Beside the classical case of Euler equations of gas dynamics, such PDEs arise for instance in traffic flow, gas pipelines, telecommunication networks, blood flow in arteries. In this talk, after a short review on the basic theory of scalar conservation laws, we introduce a new model for supply chains. Here, we are considering large volume production that allows a continuous description of the product flow in terms of conservation laws, accompanied by ordinary differential equations describing the processing capacities. A key feature of this model is the behaviour of solutions in presence of a discontinuous dynamics with respect to the unknown conserved quantity (number of parts being processed). This is a joint work with Prof. Fabio Ancona from University of Padova.

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##### Pelino Guglielmo, Mean field interacting particle systems and games

Abstract. Mean field theory studies the behaviour of stochastic systems with a large number of interacting microscopic units. Under the mean-field hypothesis, it is often possible to give a macroscopic easier description of the phenomena, which still allows to catch the main characteristics of the complex pre-limit model. The main purpose of the talk is to motivate a system of two coupled forward-backward partial differential equations, known as the mean field game system, which serves as a limit model for a particular class of stochastic differential games with N players. For reaching this goal, an introductive overview on macroscopic limits for mean field interacting particle systems and games under diffusive dynamics will be presented. In the last part of the talk I will briefly review my contributions in the context of finite state mean field games.

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##### Venturelli Federico, On the Alexander polynomial of line arrangements in P^2

Abstract. The Alexander polynomial was first introduced in the context of knot theory, and it was used to study the local topology of plane curve singularities; this notion was later extended to projective hypersurfaces (zero loci of a single polynomial equation in a projective space), which is the case that will be discussed in this talk. The Alexander polynomial of a hypersurface V encodes information on the monodromy eigenspaces of H^1(F,C), where F is the Milnor fibre of V; while these eigenspaces are well understood for smooth hypersurfaces, they are significantly harder to compute if the hypersurface is singular, even in the simplest cases i.e. hyperplane arrangements. In my talk I will try to give a basic introduction to this problem, explaining how the combinatorics of a hyperplane arrangement can help in determining its Alexander polynomial and presenting some known results; throughout the exposition some detours will be made, in order to discuss explicit examples and to introduce (or clarify) concepts that could be unfamiliar to non-specialists.

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##### Maren Diane Schmeck, An introduction to stochastic control in discrete time with an application to the securitization of systematic life insurance risk

Abstract. The basic idea behind insurance is to diversify risks. If a systematic risk is involved, this idea does not work well any more. So the idea arose to transfer the insurance risk to financial markets. Even though not perfectly linked to the own portfolio, these securitisation products work similarly to a reinsurance contract. For an investor, the products give a possibility to diversify an investment portfolio. Also insurers may act as investor and in this way diversify their own risk to regions where they have not underwritten contracts. The literature on securitisation products considers either the point of view of an investor, or the product is used to perform a Markovitz optimisation. From the point of view of an insurer, this only partially answers the question how to choose a securitisation portfolio. We will here use utility theory and stochastic control in discrete time to determine the optimal portfolio. In order to simplify the presentation we consider the case of a mortality catastrophe bond. Similar consideration would also apply for other securitisation products. The first part of the presentation will give an introduction to the methodology that we use in our research: stochastic control in discrete time. That is, we will look at the dynamic programming principle, also called Bellman’s equation and some results about the optimal strategy.

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##### Giovanna Giulia Le Gros, Covers and envelopes of modules

Abstract. Approximation theory of modules is the study of left or right approximations of modules, also known as covers or envelopes, with respect to certain classes of modules. For a class C of R-modules, the aim is to characterise the rings over which every module has a C-cover or a C-envelope and furthermore to characterise the class C itself. For example, if one considers the class of injective modules, then it is well-known that every module has an injective envelope (or injective hull). Instead, Bass proved that projective covers rarely exist and characterised the rings over which every module admits a projective cover, which are known as perfect rings. Moreover, precovers and preenvelopes are strongly related to the notion of a cotorsion pair, which is a pair of Ext-orthogonal classes in the category of R-modules. The aim of this talk is to give a basic introduction to the theory of covers and envelopes, and to describe them with respect to some well-known classes of R-modules, along with a review of concepts in homological algebra that will be useful in this exposition.

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##### Claudio Fontana, “Probability and Information in Finance"

Abstract. In mathematical finance, tools from stochastic analysis are applied to the study of investment and valuation problems arising in financial markets. In this talk, we introduce some basic and fundamental concepts and results, with a focus on no-arbitrage properties and optimal investment problems. After a general overview, we will discuss the role of information and explore the interplay between information, arbitrage, and optimal investment.

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##### Davide Barco, “An introduction to Riemann-Hilbert correspondence"

Abstract. The 21st Hilbert problem concerns the existence of a certain class of linear differential equations on the complex affine line with specified singular points and monodromic groups. Arising both as an answer and an extension to this issue, Riemann-Hilbert correspondence aims to establish a relation between systems of linear differential equations defined on a complex manifold and suitable algebraic objects encoding topological properties of the same systems. The goal was first achieved for systems with regular singularities, thanks to the works by Deligne, Kashiwara and Mebkhout. Moreover, Deligne and Malgrange established a generalized correspondence (called Riemann-Hilbert-Birkhoff correspondence) for systems with irregular singularities on complex curves, encoding and describing the Stokes phenomenon which arises in this case. In more recent years, the correspondence has been extended to take account of irregular points on complex manifolds of any dimension by D'Agnolo and Kashiwara. In this talk we give a basic introduction on the subject by providing concepts and classical example from the theory.

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##### Elena Bachini, “Including topographic effects in shallow water modeling"

Abstract. Shallow water equations are typically used to model fluid flows that develop predominantly along the horizontal (longitudinal and lateral) direction. Indeed, the so-called Shallow Water (SW) hypothesis assumes negligible vertical velocity components. The typical derivation of the SW equations is based on the integration of the Navier-Stokes equations over the fluid depth in combination with an asymptotic analysis enforcing the SW assumptions. In the presence of a general terrain, such as a mountain landscape, the model must be adapted to geometrical characteristics, since the bottom surface can be arbitrarily non-flat, with non-negligible slopes and curvatures. After an introduction on the standard SW model, we will present a new formulation of the two-dimensional SW equations in intrinsic coordinates adapted to general and complex terrains, with emphasis on the influence of the geometry of the bottom on the solution. The proposed model is then discretized with a first order upwind Godunov Finite Volume scheme. We will give an overview of the numerical method and then show some results. The results indicate that it is important to take into full consideration the bottom geometry and slope even for relatively mild and slowly varying curvatures.

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##### Giacomo Graziani, “Serre's p-adic modular forms and p-adic interpolation of the Riemann zeta function"

Abstract. The so-called zeta functions are among the most famous and discussed objects in mathematics, the simplest of which is the (in)famous Riemann zeta function. In order to work with them (and with the strictly related L-functions as well), mathematicians decided to isolate simpler pieces and hence ultimately to address the problem of their p-adic interpolation. In this seminar, after introducing the various objects involved, we will focus on easiest example of the Riemann zeta function and describe the surprising interpolation exploited by Serre using his notion of p-adic modular forms.

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