# Graduate Seminar 2017/2018

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.

The Graduate Student is organized by Corrado Marastoni and Tiziano Vargiolu.

**List of 2017-2018 Seminars***(Click on title for abstract)*

- [27 June 2018] Torben Koch (Bielefeld University, Center for Mathematical Economics) “On a Particular Class of Singular Optimal Control Problems"
- [30 May 2018] Roberto Bramati (Padova, Dip. Mat. and KU Leuven) “Brascamp-Lieb inequalities and heat-flow monotonicity”
- [2 May 2018] Pietro Gatti (Padova, Dip. Mat. and KU Leuven) “Monodromy and Invariant Cycles”
- [16 May 2018] Marco Tarantino (Padova, Dip. Mat.) “Abelian Model Structures”
- [28 Mar 2018] Emma Perracchione (Padova, Dip. Mat.) “Kernel-based methods: a general overview”
- [14 Mar 2018] Yangyu Fan (Univ. Milano) “Tilting approach to the theorem of Fontaine-Wintenberger”
- [28 Feb 2018] Maria Flora (Padova, dSEA) “Optimal cross-border electricity trading”
- [14 Feb 2018] Roman Pukhtaievych (Padova, Dip. Mat.) “Effective conductivity of a composit material and its series expansion"
- [17 Jan 2018] Marija Soloviova (Padova, Dip. Mat.) “Norm attaining mappings”
- [20 Dec 2017] Marco Piccirilli (Padova, Dip. Mat.) “Stochastic models for energy forward markets”
- [6 Dec 2017] Simone Giovannini (Padova, Dip. Mat.) “Representation finite algebras and generalizations"
- [22 Nov 2017] Alekos Cecchin (Padova, Dip. Mat.) “Approximation and convergence in finite state Mean Field Games"
- [4 Ott 2017] Sebastiano Don (Padova, Dip. Mat.) “Fine properties of functions of bounded variation in Carnot-Caratheodory spaces"

**Torben Koch, On a Particular Class of Singular Optimal Control Problems **

Abstract. Optimal control problems have been attracted by many researchers in recent years. Its applications can be found in different fields of sciences such as physics or economics. In this talk, we study a particular class of singular optimal control problems, that is, the control variable is represented by an increasing and right-continuous process. Therefore the control measure is allowed to be singular with respect to the Lebesgue measure. The optimal control and value of the problem are typically characterized by a partial differential equation, the so-called Hamilton-Jacobi-Bellman (HJB) equation. In a first step we study a deterministic setting in which the controlled system is governed by an ordinary differential equation. We derive the associated HJB equation by employing standard techniques such as Taylor's theorem. In a next step, we extend the model and consider a specific stochastic singular control problem. The model we have in mind is that of a firm which aims to maximize its profits from selling energy in the market. Here, the control variable represents the firm's installation strategy of solar panels in order to produce energy. We assume that the energy price follows an Ornstein-Uhlenbeck process and is affected by the installation strategy of the firm. We find that the optimal installation strategy is triggered by a threshold, the so-called free boundary which separates the waiting region, in which it is not optimal to install additional panels, and the installation region where it is optimal to install additional panels. Finally, our study is complemented by an analysis of the dependency of the optimal installation strategy on the model's parameters.

**Roberto Bramati, Brascamp-Lieb inequalities and heat-flow monotonicity **

Abstract. Many well-known multilinear integral inequalities in Euclidean analysis, such as Hölder's and Young's inequalities, share a common feature: they are instances of the so called Brascamp-Lieb inequalities. In this introductory talk we will describe these inequalities and show how they can be derived as a consequence of a monotonicity property associated to the heat flow. We will also discuss to what extent some of the ideas and techniques can be adapted to non-Euclidean settings. In particular we will present a family of inequalities on real spheres involving functions that possess some kind of symmetry.

**Pietro Gatti, Monodromy and Invariant Cycles **

Abstract. We will define the monodromy operator for a 1-parameter family of curves. Focusing on a concrete example, a family of tori, we will compute explicitly the monodromy and show that its invariant part has a geometrical interpretation. This is an instance of the classical Invariant Cycles Theorem. After giving an intuitive introduction to log-geometry, we will briefly explain how this setup let us reprove the Invariant Cycles Theorem for a Semistable Family of Curves.

**Marco Tarantino, Abelian Model Structures **

Abstract.Model categories were introduced by Quillen in 1967 as an axiomatized setting in which it is possible to "do homotopy theory", by inverting a class of morphisms called weak equivalences. The construction involves the use of two more classes of morphisms, which, together with the weak equivalences, form what is called a model structure. In the case of abelian categories there are particular model structures, called abelian model structures, that can be constructed by means of objects rather than morphisms, using complete cotorsion pairs. We will present the theory of abelian model structures, showing how they can be applied to the particular case of R-modules to recover the derived category of the ring.

**Emma Perracchione, Kernel-based methods: a general overview **

Abstract. In this talk we present the general theory of Radial Basis Function (RBF) interpolation. In doing so, we follow the exposition line of the two books by G.E. Fasshauer (2007) and H. Wendland (2005). Such works provide a recent and extensive treatment about the theory of RBF-based meshless approximation methods. Thus, following their guidelines, we review the main theoretical features concerning positive definite functions and RBFs. Error bounds and error estimates for kernel-based interpolants will be presented as well. Moreover, all the results will be supported by basic numerical experiments carried out during the seminar with Matlab. To conclude, we also briefly review some recent research topics on RBF interpolation.

**Yangyu Fan, Tilting approach to the theorem of Fontaine-Wintenberger **

Abstract. In this basic notion talk, I will explain an approach to the Fontaine-Winterberger isomorphism between Galois groups in characteristic 0 and characteristic p using the tilting equivalence developed by Scholze and Kedlaya-Liu.

**Maria Flora, Optimal cross-border electricity trading **

Abstract. Using econometric tools, we show statistical evidence of cross-effects on the price of electricity in neighbouring countries, due to electricity flows across interconnected locations. We build on this result to set up an optimal trading strategy, based not only on the price spreads observed over time among the selected interconnected countries, but also on the market impacts caused by the flows of electricity among them. Using the previous econometric analysis findings, we model the joint dynamics of electricity prices as including both temporary and permanent impacts of electricity trades, as well as driven by a common co-integration factor. We then pose an optimal control problem, and solve the resulting dynamic programming equation up to a system of Riccati equations, which we solve numerically to evaluate the performance of the strategy. We show that including cross-border effects in the trading strategy specification significantly improves the performance.

**Roman Pukhtaievych, Effective conductivity of a composit material and its series expansion **

Abstract. In this talk we discuss the asymptotic behavior of the effective conductivity in a dilute two-phase composite with non-ideal contact conditions at the interface. The composite is obtained by introducing into an infinite homogeneous planar matrix a periodic set of inclusions of a different material and the diameter of each inclusion is proportional to a real positive parameter. After a brief introduction on composit materials, transmission boundary conditions, and the effective conductivity, we will discuss the way how to obtain the series expansion for the effective conductivity, and a fully constructive method to compute explicitly the coefficients of such series by solving recursive systems of integral equations. Also, we will solve some of them in case the inclusions are in the form of a disk. The talk will be of an introductory type and is intended for a general audience.

**Marija Soloviova, Norm attaining mappings **

Abstract. This talk is about approximation by norm attaining mappings. We start with some basic notions of Functional Analysis. In the first part we recall the classical results in this field, like Bishop-Phelps-Bollobas' theorem, James' weak compactness theorem. In the second part we present our joint with Vladimir Kadets and Miguel Martin results of the paper "Norm-attaining Lipschitz functionals" (2016), where we introduce a concept of norm attainment for Lipschitz functionals. The seminar will be of introductory type.

**Marco Piccirilli, Stochastic models for energy forward markets **

Abstract. I will present a probabilistic modeling framework for forward prices, specifically designed for energy markets. Most of the presentation will be kept at an intuitive level, as far as this is possible and sensible. I will start by explaining the general framework of the talk and then move to our contribution, of course describing the underlying mathematical theory as well. This talk is based on joint work with Fred Espen Benth, Luca Latini and Tiziano Vargiolu.

**Simone Giovannini, Representation finite algebras and generalizations **

Abstract. An algebra is called "representation finite" if it has a finite number of indecomposable modules. Finite dimensional hereditary representation finite algebras are classified by Gabriel's Theorem: they are the path algebras of Dynkin quivers of type ADE. Recently, with the development of higher dimensional Auslander-Reiten theory, some interest has been raised by a generalization in dimension n of these algebras, which are called n-representation finite algebras. In this seminar we will recall some basic definitions and results about representation theory of finite dimensional algebras. Then we will give a naive idea of how some classical notions can be generalized to higher dimension and, finally, we will show some examples of 2-representation finite algebras.

**Alekos Cecchin, Approximation and convergence in finite state Mean Field Games.**

Abstract. Mean Field Games represent limit models for symmetric non-zero sum non-cooperative dynamic games, when the number N of players tend to infinity. We focus on finite time horizon problems where the position of each agent belongs to a finite state space. Relying on a probabilistic representation of the dynamics in terms of Poisson random measures, we first show that a solution of the Mean Field Game provides an approximate symmetric Nash equilibrium for the N-player game. Then, under stronger assumptions for which uniqueness holds, we prove that the sequence on Nash equilibria converges to a Mean Field Game solution. We exploit the so-called Master Equation, which in this framework is a first order quasilinear PDE stated in the symplex of probability measures.

**Sebastiano Don, Fine properties of functions of bounded variation in Carnot-Caratheodory spaces **

Abstract. Functions of bounded variation (BV functions) can be seen as a generalization of Sobolev maps and arise naturally in many problems in Calculus of Variations. Carnot-Carath\'eodory spaces are particular metric spaces that arise from the study of hypoelliptic differential operators. In this seminar we will introduce the notions of approximate continuity point, approximate jump point and approximate differentiability point for a generic L^1 function and we will show the so-called fine properties of BV functions, first in the case of Euclidean spaces, and then in the case of Carnot-Carath\'eodory spaces. In particular, the rectifiability of the approximate jump set, the approximate differentiability almost everywhere and the decomposition formula for the measure derivative of a BV function will be shown.