NOTE: All seminars are held in person at the Department of Mathematics in Via Carlo Alberto 10, Turin.
Second semester 2024/2025
Wed 9 April, 2025. Aula 1, Palazzo Campana, 14:30-15:30
Tamas Makai (Munich University)
Title: Dispersion on the complete graph
Abstract: In the synchronous dispersion process, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018), we move particles between the vertices of a graph according to the following rules. Initially every particle is placed on a fixed vertex. In every subsequent step of the process every particle, which is located on a vertex with at least one other particle present moves to one of its neighbours uniformly at random at the same time. The remaining particles, that is the ones which are the sole occupant of a vertex, keep their position. The process stops once every particle is alone on a vertex. When the underlying graph is the complete graph on n vertices, Cooper et al. identified a phase transition in the running time of the process, when the number of particles is n/2 going from a logarithmic number of steps to exponentially many. We analyse this transition more closely. Namely when the number of particles is (1+o(1))n/2, we prove bounds for the upper and lower tail of the running time. In addition, when the number of particles differs from n/2 by only O(n^(1/2)), we also provide the limiting distribution of the running time.
Joint work with Umberto De Ambroggio, Konstantinos Panagiotou and Annika Steibel.
Wed 14 May, 2025. Aula 1, Palazzo Campana, 14:30-15:30
Dietrich von Rosen (Linköping University)
Title: On the use of quadratic subspaces in statistics
Abstract: Under a multivariate normality assumption, quadratic subspaces are particularly valuable for estimating parameters in linearly structured covariance matrices. The quadratic subspace assumption implies that the inverse covariance matrix also follows the same linear structure. The theory is strongly connected to the exponential family and explicitly expressed estimators are available
Wed 21 May, 2025. Aula 1, Palazzo Campana, 14:30-15:30
Sara Mazzonetto (University of Lorainne)
Title: Skew-sticky diffusions: parametric estimation via local time approximation
Abstract: We consider one-dimensional diffusions whose dynamics are influenced by the presence of a point-barrier that is either partially reflective (skew) or sticky. The nature of this barrier is encoded in bias and stickiness parameters. First, we describe the process and its properties, and then we discuss local time approximation and parameter estimation based on a trajectory observed at discrete times. We will see why, in the specific case of skew Brownian motion, the estimators converge at a non-standard rate towards a mixed Gaussian and the sticky case is different.
This talk is partially based on works in collaboration with A. Anagnostakis (IECL Metz) and A. Lejay (IECL/Inria Nancy).
Wed 28 May, 2025. Aula 1, Palazzo Campana, double seminar
14:30-15:30
Rémi Catellier (Université Côte d'Azur)
Title: Mathematical modeling and analysis for the growth of a mycellium network
Abstract: In this talk, I will present several modeling approaches for the growth dynamics of filamentous fungi, focusing on key behaviors such as apical and lateral branching, as well as weak self-repulsion. We explore a microscopic model describing the development of the hyphal network, its connection to macroscopic growth patterns, and associated simulations. We also investigate whether this model satisfies an optimal control principle that could underlie the observed dynamics. Finally, we propose a mechanistic explanation for branching processes at an even finer scale.
15:30-16:30
Wolfgang Bock (Linnaeus University)
Title: Recent Results on generalized Grey Brownian motion
Abstract: tba
First semester 2024/2025
Thu 31 October, 2024. Aula C, Palazzo Campana, 14:30-15:30
Sandra Fortini (Bocconi)
Title: Large-width asymptotics for ReLU neural networks with alpha-Stable initializations
Abstract: There is a recent and growing literature on large-width asymptotic properties of Gaussian neural networks (NNs), namely NNs whose weights are initialized according to Gaussian distributions. In such a context, two popular problems are: i) the study of the large-width distributions of NNs, which characterizes the infinitely wide limit of a rescaled NN in terms of a Gaussian stochastic process; ii) the study of the large-width training dynamics of NNs, which characterizes the infinitely wide dynamics in terms of a deterministic kernel, referred to as the neural tangent kernel (NTK), and shows that, for a sufficiently large width, the gradient descent achieves zero training error at a linear rate. We consider these problems for alpha-Stable NNs, namely NNs whose weights are initialized according to alpha-Stable distributions, i.e. distributions with heavy-tails. First, for alpha-Stable NNs , we show that if the NN's width goes to infinity then a rescaled NN converges weakly to an alpha-Stable stochastic process. As a difference with respect to the Gaussian setting, the choice of the activation function affects the scaling of the NN. Then, we study the large-width training dynamics of alpha-Stable ReLU-NNs, characterizing the infinitely wide dynamics in terms of a random kernel, referred to as the alpha-Stable NTK, and showing that, for a sufficiently large width, the gradient descent achieves zero training error at a linear rate. The randomness of the alpha-Stable NTK is a further difference with respect to the Gaussian setting, that is: within the alpha-Stable setting, the randomness of the NN at initialization does not vanish in the large-width regime of the training.
Joint work with Stefano Favaro (Università di Torino) and Stefano Peluchetti (Sakana AI)
Thu 14 November, 2024. Sala Orsi, Palazzo Campana, 14:00-15:00
Marco Fuhrman (Università di Milano)
Title: Partially observed controlled Markov chains and optimal control of the Wonham filter.
Abstract: After a general introduction to the optimal control problem for a time-continuous Markov chain we will address the case of control with partial observation, where the control actions are chosen dynamically based upon the observation of an auxiliary process perturbed by a Brownian noise. Using techniques of optimal filtering the original problem is recast as a control problem with full observation for a process with values in the space of probabilities on the state space of the Markov chain (the controlled Wonham filter). The latter problem with be studied with various techniques, in particular by the dynamic programming equations.
This is joint work with Fulvia Confortola (Politecnico di Milano).
Thu 5 December, 2024. Aula 5, Palazzo Campana, 14:30-15:30
Fabio Antonelli (Università degli Studi dell'Aquila)
Title: Differentiating a killed Brownian motion in a smooth domain
Abstract: In this work in progress, we try to extend the probabilistic representation formula for the derivative of the semigroup associated to a multidimensional killed diffusion process, that was obtained by one of the authors in the half space by employing the normally reflected process at the boundary. In particular, the authors showed that when the components are uncorrelated at the boundary, jumps in the derivative formula appear in the components unconcerned by the reflection. Here, we try to extend the formula in the case of a correlated Brownian motion killed at the boundary of a smooth multi-dimensional bounded domain. This makes the boundary curvature mixed with correlation appear in the jump-diffusion derivative process as well as in the obliquely reflected process. The main idea is to exploit the half space results, by constructing a modified Euler scheme for the reflected process, by mapping the boundary manifold to the boundary of the half plane and back and proving that, under appropriate conditions, tightness of the scheme’s derivative is ensured, so providing the limit formula.
Joint work with Arturo Kohatsu-Higa.
Mon 20 January, 2025. Aula 1, Palazzo Campana, 14:30-15:30
Enrico Scalas (Sapienza - Università di Roma)
Title: Contare e classificare: un programma di ricerca per la probabilità applicata e la statistica
Abstract: Nel 1960, il fisico Eugene Wigner ha pubblicato un saggio intitolato "The unreasonable effectiveness of mathematics in the natural sciences". Durante il seminario presenterò argomenti miranti a dimostrare che l'efficacia della matematica nella descrizione della natura non è né irragionevole né sorprendente. Si tratta infatti di una "naturale" estensione delle nostre capacità di contare e classificare (capacità che condividiamo con numerose specie su questo pianeta) e (di una parte) del nostro linguaggio. Quando descriviamo il mondo che ci circonda usiamo proposizioni relative a oggetti e classi o categorie alle quali gli oggetti appartengono. Tali proposizioni diventano fatti se vere. Poiché non tutto è noto, possiamo attribuire una probabilità alle proposizioni e introdurre descrizioni del cambiamento in termini di catene di Markov. Da queste descrizioni, con appropriati limiti di scala, emergono alcune equazioni della fisica matematica, per esempio l'equazione di diffusione e le equazioni cinetiche di Boltzmann. Illustrerò questo percorso con un modello di scambio stocastico per la distribuzione della ricchezza. Usando questo punto di vista si evita la cosiddetta "fallacia logica della proiezione mentale", sottolineata da Jaynes, che identifica gli oggetti delle nostre teorie con oggetti reali e conduce a bizzarrie quali il "collasso della funzione d'onda" in meccanica quantistica.
Concluderò con l'illustrazione di alcune difficoltà di questo programma di ricerca, problemi di natura diversa, quali, per esempio, la vaghezza nella classificazione o la violazione dell'invarianza di Lorentz in teorie "discrete" della gravità.
Per le idee riassunte sopra sono in debito almeno con de Finetti e Carnap. Spero quantomeno di convincere sulla centralità della probabilità e dei processi stocastici nella descrizione della natura.
Bibliografia
Ubaldo Garibaldi, Enrico Scalas, Finitary Probabilistic Methods in Econophysics, Cambridge University Press 2010.
Bertram Duering, Nicos Georgiou, Sara Merino-Aceituno, Enrico Scalas (2022).
Continuum and thermodynamic limits for a simple random-exchange model. Stochastic Processes and their Applications, 149. pp. 248-277.
Organized by
Bruno Toaldo: bruno.toaldo@unito.it
Elena Issoglio: elena.issoglio@unito.it
Sponsors
Department of Mathematics "Giuseppe Peano", University of Torino