At 5:00 PM, the group will meet the guide in the courtyard of the Estense Castle, where the guided tour will begin.

The imposing structure of the Castle, a symbol of the city of Ferrara and of the power and wealth of the House of Este, will be admired. The history of this late 14th-century fortress, which later became the residence of the Este Dukes in the second half of the 15th century, will be told. The castle is still characterized by its large defensive moat, full of water.

The tour will then continue towards the medieval historic center to discuss the Cathedral, dedicated to St. George, the city's patron saint. If no services are being held, visitors will be able to explore the 18th-century interior. From the outside, the beautiful façade of the Este Ducal Palace, now the Town Hall, will be admired. This palace, dating back to the 13th century, was the first Este residence in the city.

We will then proceed towards the oldest part of the medieval center to enjoy the atmosphere of the splendid Via delle Volte with its characteristic arches and the picturesque and quiet alleys of the Jewish Ghetto, once home to important figures such as Isacco Lampronti and Giorgio Bassani.

Welcome address by the Rector Prof. Laura Ramaciotti and the President of INdAM Prof. Cristina Trombetti

Abstract: 

Why all the drums are round! This assertion is mathematically rephrased as follows: among all membranes with prescribed area, the one producing the lowest fundamental frequency has a circular shape. But what about the shapes of membranes with higher extremal frequencies? The general question in which are are interested concerns the relationship between the geometry of the domain and the spectrum of a differential operator defined on this domain. In this talk I will focus on the Neumann eigenvalues of the Laplace operator on domains of Euclidean space and on spheres. Together with a discussion about existence/non existence of optimal geometries and possible relaxation on densities, I will give some numerical approximations of the best geometries and densities. A particular attention will be given to the lowest two non trivial eigenvalues for which a full answer is given in any dimension of the Euclidean space. A surprising phenomenon occurs on spheres: while a complete answer can be given for the second eigenvalue, for the first one an unexplained phenomenon occurs. The results presented in this talk have been obtained in series of collaborations with R. Laugesen, A. Henrot, E. Martinet, M. Nahon and E. Oudet. 

Abstract: 

Given a compact Riemann surface, a classical theorem of of Narasimhan and Seshadri characterize vector bundles arising from unitary representations of the fundamental group as the polystable vector bundles of degree 0. Given a projective curve with good reduction over a local field of mixed charateristic 0-p, works of Faltings and Deninger-Werner associate semistable Higgs bundles of degree 0 to representations of the fundamental group. We explain some progress towards the question of characterizing which vector bundles arise in this way. 

Abstract: 

Modern applications such as machine learning involve the solution of huge scale nonconvex optimization problems, sometimes with special structure. Motivated by these challenges, we investigate more generally, dimensionality reduction techniques in the variable/parameter domain for local and global optimization that rely crucially on random projections.

We describe and use sketching results that allow efficient projections to low dimensions while preserving useful properties, as well as other tools from random matrix theory and conic integral geometry.  We focus on functions with low effective dimensionality, a common occurrence in applications involving overparameterized models and that can serve as an insightful proxy for the training landscape in neural networks. We obtain algorithms that scale well with problem dimension, allow inaccurate information and biased noise, have adaptive parameters and benefit from high-probability complexity guarantees and almost sure convergence. 

Abstract: 

Inflation of a membrane tube is a classical problem but has received renewed interest in recent years. It is well-known that when one end of the tube is closed and is allowed to extend freely during inflation, the pressure vs volume curve corresponding to uniform inflation has an up-down-up behaviour. This explains the phenomenon that when a tubular party balloon is inflated, it would quickly evolve into a "multi-phase" state: the coexistence of a thick section and a thin section. Another less understood loading scenario that may be more relevant in hi-tech applications is when the tube is first stretched axially to a specified length and then inflated with both ends fixed. We report our recent studies in which we treat localised bulging as a bifurcation problem and are able to give the entire inflation process a complete analytical description for both loading scenarios. We wish to promote this problem as a textbook example to illustrate a variety of important concepts and techniques in mathematics and mechanics such as strain localisation, snap-through, phase transition, and self-consistent asymptotic derivation of 1D models, etc. We also show how understanding the tube inflation problem has guided our studies of other more challenging elastic localisation problems such as the axisymmetric (thickness) necking of a radially stretched circular membrane.

Reference:  Y.B. FU, Elastic localizations. In "Electro-and magneto-mechanics of soft solids: Constitutive modelling, numerical implementation, and instabilities", CISM Courses and Lecture Notes (eds K. Danas and O. Lopez-Pamies), 2024.

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