Graduate Seminar 2024-2025

List of Seminars

Video recordings at Unipd's portal "Mediaspace"

(Click on title for abstract)

 
Beatrice Ongarato, "Hawkes processes in cyber-risk analysis: modelization and optimal security investment"

Abstract. With the rapid growth of the digital economy in recent years, cyber-risk has emerged as one of the most relevant and rapidly growing sources of risk. We provide an overview of the main concepts related to cyber-risk and examine the challenges involved in its quantification and modeling. We introduce Hawkes processes and explain their applicability in capturing the dynamics of cyber-attacks. Lastly, we present an ongoing project aimed at determining the optimal cyber-security investment strategy for an organization facing cyber-attacks. The problem is framed as a stochastic control problem with jumps and is addressed using Hamilton-Jacobi-Bellman (HJB) techniques. We introduce the main tools needed to solve this type of problem and show some preliminary numerical results.

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Ishan Jaztar Singh, "Bridging Enumerative Geometry and Quantum Integrable Hierarchies"

Abstract. Enumerative geometry explores the use of combinatorial and intersection theory techniques to solve counting problems in algebraic geometry. Integrable hierarchies, in contrast, consist of infinite sequences of partial differential equations with symmetries that have significance in mathematical physics. Both fields have seen substantial developments over the past half-century. This talk will focus on the infamous Witten-Kontsevich theorem, which establishes a deep connection between topological invariants of the moduli space of curves and the Korteweg–de Vries hierarchy. I will attempt to offer intuitive motivation and a formal statement of the theorem, and, time permitting, discuss its generalizations and the role of quantum hierarchies in this context.

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Pietro De Checchi, "Dynamics of Environment-Embedded Quantum Systems: An Introduction"

Abstract. Closed quantum systems are an idealization, their time evolution described by the Schrödinger Equation, i.e. by the action of unitary operators. The physics of a realistic quantum system, on the other hand, is bound to be disturbed by the environment in which it is naturally embedded and with which it inevitably interacts. The dimension of the space needed to fully describe the composite system increases, as one would have to include all, possibly infinite, environments variables, leading to intractable problems. To reduce the system to a smaller subspace of interest and to describe its correct dynamics, many strategies have been developed. These systems have in general non-unitary dynamics and are known as Open Quantum Systems. During the talk, we will introduce some of the main approaches based on various techniques, from dynamical semigroup generators, stochastic unravellings and bottom-up modelling.

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Enrico Sabatini, "Representations of Quivers over Rings: Merging Commutative and Non-Commutative Results"

Abstract. In the vast universe of representation theory there are two very separate and different worlds: commutative rings and finite dimensional (non-commutative) algebras. The problem of characterising certain subcategories, like many other problems, has been solved in both fields. However, the main techniques used for one context are generally not transferable to the other. Recently, some authors have focused their interest on a special kind of algebras that partially merge the two fields. Here, the apparently different results have a surprising generalisation and a unifying proof. In this talk, I will give an overview of the two fields mentioned above, describe their main features and give an idea of what allows such characterisations; avoiding all the technicalities. Finally, I'll show how the generalisation works with the aid of some interesting examples.

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Erik Chinellato, "Deep Unfolding: Bridging Optimization and Neural Network Interpretability"

Abstract. Deep neural networks (DNNs) have revolutionized numerous fields due to their powerful ability to learn complex representations. However, their black-box nature and lack of interpretability in architecture and weight design remain significant challenges. After an introductory segment on DNNs and backpropagation learning, this seminar introduces the Deep Unfolding method as a promising alternative, bridging the gap between data-driven learning and model-based optimization. By unrolling iterative optimization algorithms into structured neural network architectures, Deep Unfolding provides a principled approach to network design, enabling interpretability and theoretical insights into their operation. We will explore how this method leverages domain knowledge, achieves faster convergence, and enhances performance in resource-constrained scenarios. The session will highlight many wide-ranging practical applications of Deep Unfolding, covering audio source separation and recognition, image denoising and state estimation.

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Gaia Marangon, "Modeling Dark Matter: a Dynamics Study"

Abstract. Dark matter is one of the most relevant and fascinating open problems in modern astrophysics. Since it cannot be directly observed, modeling it requires a balanced mix of physical intuition, mathematical deduction, and comparison with indirect experimental data. In this talk, I will briefly introduce the physical context motivating our research, specifically the problem of dark matter distributions around galaxies. Starting from the Schrödinger-Poisson system, the most commonly used model for dark matter dynamics, I will outline the main directions our work has taken. I will focus on two key aspects. First, I will discuss the issue of stationary states, ranging from numerical properties to comparison with experimental data. Then, I will propose a relativistic generalization of the model, the Klein-Gordon - Wave system. Its treatment by Hamiltonian perturbative techniques shows the potential of mathematical physics tools in building a comprehensive and reliable model.

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Giacomo Passuello, "Mixing times and cutoffs for Markov chains"

Abstract. How long does it take to shuffle a deck of 40 cards? This simple question, together with the seminal work of Aldous and Diaconis on the cutoff phenomenon, has generated, in the last 40 years, a rich research area in the field of discrete probability. A cutoff is a dynamical phase transition for a random process, which appears as the size of the system becomes large. It occurs when the distance to equilibrium of the process abruptly drops from its maximum value to zero at a critical time scale. Establishing the occurrence of the cutoff is a delicate matter, which may require a precise understanding of the spectral and diffusive properties of the underlying system. In this talk, I will review some basic concepts on Markov chains and their convergence to the stationary equilibrium. After that, I will introduce the concept of mixing time and discuss bounds on its limiting behaviour. Finally, I will focus on the cutoff phenomenon and present some results on the mixing time of the simple random walk on a directed random graph.

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Denis Shishmintsev, "Mean Field Turnpike Theorems"

Abstract. In the study of Mean Field Games (MFG), the Turnpike Property plays a crucial role in understanding the asymptotic behavior of large populations of agents. This property suggests that, for sufficiently long time horizons, the optimal trajectories of agents in a dynamic system converge to a steady-state or "turnpike" region, where their strategies remain approximately constant. The presence of the turnpike reflects the system’s tendency to stabilize and suggests that most of the time, agents will follow similar paths despite starting from different initial conditions. In this introductory talk we investigate the turnpike property in the context of Lagrangian and Eulerian formulations of MFGs, which describe the agents either through their individual trajectories (Lagrangian) or through a distribution function over space (Eulerian). In both frameworks, we explore how the turnpike emerges and its implications for the long-term dynamics of the system, aiming at key applications in economics, control theory, and multi-agent systems.

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Martina Galeazzo, "An Integer Linear Programming Model for the Dynamic Airspace Configuration problem""

Abstract. Given central role of aviation as a transportation network and its remarkable economic impact, the air traffic demand is bound to increase. High traffic density in a given airspace region can cause safety issues and difficulties in monitoring tasks that can, in turn, result in flight delays. It is therefore crucial to efficiently organize the airspace structure to avoid under- and overloaded areas of the airspace. We begin by describing how the airspace is structured, introducing the concept of sector and configuration and their capacity, and how to quantify the air traffic excess associated to a configuration. We will then introduce Dynamic Airspace Configuration as a method for optimally meeting the air traffic demand by adopting different configurations over time, thus determining a sequence of configurations (configuration plan); we impose that such a sequence also satisfies some operational restrictions that smooth the configuration dynamics, as to avoid, e.g., too frequent switching between configurations. After recalling the basic definitions and tools of (Integer) Linear Programming, we will present an Integer Linear Programming model that provides a configuration plan that minimizes the traffic excess for a given time frame, and a polyhedral study that explains its good computational performance. We conclude by showing the numerical results obtained by testing the model on five days of historical data (summer 2019) over the Madrid Area Control Center, with a focus on the comparison of different time discretizations and different restrictions on the configurations’ transitions.

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Xiang Liu, "Resonances and quasi-collisions in the Three-Body Problem"

Abstract. Mean motion resonance, a phenomenon occurring when two celestial bodies have orbital periods in a commensurable ratio, plays a pivotal role in both stabilizing and destabilizing motions within our Solar System. For highly eccentric orbits, quasi-collisions become a significant factor. When such eccentric orbits are trapped in resonance, perturbations can induce chaotic motions, leading to rapid changes in orbital elements and transitions of different dynamical states. This presentation will begin by introducing the concept of mean motion resonance within the framework of the restricted three-body problem. Subsequently, we will explore the application of Hamiltonian perturbation theory for low-eccentricity orbits. Finally, we will demonstrate the limitations of this theory when applied to highly eccentric orbits.

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Tommaso Bertin, "Lavrentiev Phenomenon and semicontinuous envelopment for integral functionals"

Abstract. The first part of the talk is devoted to introduce some basilar elements of the Direct Method of Calculus of variations. In particular we will see the Tonelli's Theorem about the strictly relationship between lower semicontinuity of an integral functional and convexity of the Lagrangian. In the second part we will explore the so called "Lavrentiev Phenomenon", i.e. the possibility for a functional to reach an infimum in a dense subset strictly greater than the infimum in the original set. We will see some classical examples and some recent results to avoid the Phenomenon. In particular we will focus on non convex and non continuous Lagrangian.

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Alessandra Nardi, “Let’s play symplectic billiards!"

Abstract.A mathematical billiard is a dynamical system describing the motion of a mass point (the billiard ball) inside a planar region (the billiard table). The ball moves with constant speed and without friction, following a rectilinear path. The straightforwardness and versatility of this model have made mathematical billiards an object of interest in many different contexts. Indeed, depending on the shape of the billiard table, they show a wide range of dynamical behaviors such as integrability, regularity, and chaoticity. Integrability remains an unanswered property, and the celebrated Birkhoff conjecture remains open. In 2018, P. Albers and S. Tabachnikov introduced a new interesting class of billiards, called symplectic billiards, as a natural variation of Birkhoff billiards with the inner area – instead of the length – as generating function. This talk will present the symplectic billiards dynamics and focus on recent rigidity results. Talk based on joint works with L. Baracco and O. Bernardi.

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Elena Collacciani, “Inductive Methods in the Representation Theory of Finite Groups of Lie Type: An Introduction via GL(n,q)"

Abstract. Representation theory of groups is the area of mathematics that studies how abstract groups can act on vector spaces - in other words, how to realize groups as collections of linear transformations. Among the many families of finite groups, those of Lie type form a particularly important class, appearing naturally in the classification of finite simple groups. In this talk, after a brief overview of classical results on the representation theory of finite groups, I will offer a glimpse into the techniques used to construct and classify irreducible representations of finite groups of Lie type. To convey the core ideas while avoiding heavy technicalities, I will focus on the case of the General Linear group over a finite field. Special emphasis will be placed on the role of inductive methods - a fundamental paradigm in representation theory - which often reduce complex algebraic problems to more manageable combinatorial ones. To help develop intuition, I will also present concrete examples of the main objects involved.

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Marco Mastrogiovanni, “Inductive Methods in the Representation Theory of Finite Groups of Lie Type: An Introduction via GL(n,q)"

Abstract. We investigate the statistical properties of time-averaged fractional Brownian motion (fBm), which naturally arises in the modeling of time series subjected to averaging transformations. The main motivation comes from electricity markets, where daily prices are typically obtained by averaging high-frequency data. While fBm-based models are popular in modeling electricity prices, treating averaged prices as direct realizations of fBm is theoretically inconsistent. Instead, time-integrated versions of fBm should be used to accurately reflect the effects of averaging. This seminar begins with an overview of electricity markets and an introduction to fractional Brownian motion (fBm) and its main characteristics. We then introduce the integral-mean process of fBm and analyze the impact of time-averaging on its properties. Using ergodic theory, we construct strongly consistent estimators for the Hurst parameter adapted to the averaged process and validate them through an extensive simulation study. Next, we extend our approach to linear combinations of two distinct timeaveraged fBm processes, again estimating the relevant parameters. Finally, we apply our methodology to empirical electricity spot price data and discuss potential future developments in this research area. (This is joint work with Yuliya Mishura, Stefania Ottaviano and Tiziano Vargiolu.)

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