**Daniela Tonon**

"* Mean Field games planning problems *"

Mean Field Games (MFG) systems have been introduced simultaneously in 2006 by Lasry and Lions and by Huang, Caines and Malhamé to describe Nash equilibria in differential games with infinitely many players. MFG theory lies at the intersection of mean field theories, optimal control, stochastic analysis and partial differential equations: in the simplest cases, the value of each agent is found by solving a backward Hamilton-Jacobi equation whereas the distribution of the agents' states evolves according to a forward Fokker-Planck equation. The importance of MFG is reflected by the fact that they are used to model systems that belong to very different areas such as economics, finance, social sciences and engineering. After introducing the MFG model we present a particular case, the planning problem, in which a central planner would like to steer a population to a predetermined final configuration while still allowing agents to choose their own strategies.

**Sara Farinelli**

"* One dimensional localization for curvature dimension condition *"

In the setting of spaces with synthetic Ricci curvature lower bound, the localization principle allows to translate the information about the curvature into a one dimensional information on a suitable family of geodesics. This is due to Cavalletti and Mondino who extended to abstract metric measure spaces the results of Klartag linking curvature properties of smooth weighted Riemannian manifolds with properties of geodesics coming from L1 optimal transport. While the picture is clear in setting of metric measure spaces with synthetic Ricci curvature bound and upper bound on the dimension, the validity of the localization principle has not been proved when removing an upper bound on the dimension. In this seminar we will go through the localization principle in the finite dimensional case, we will see the difficulties that appear when passing to the infinite dimensional case and present some progress in the direction of L1 optimal transport in a particular class of CD(K,∞) spaces.

**Laura Meneghetti**

"*A Reduced Order Approach for Artificial Neural Networks Applied to Object Recognition*"

Computer Vision is a thriving ﬁeld increasingly exploited in several scientific and engineering contexts in order to solve complex tasks such as the recognition and detection of objects inside pictures. A possible approach to deal with image processing problems is represented by Convolutional Neural Networks (CNNs). Such architectures well perform on complex tasks such as object recognition but may require a high number of layers to extract all the features of the problem at hand, leading to more parameters to be calibrated during the training phase. This naturally opens several computational issues in the learning procedure, as well as in the memory and space required by the model itself, especially in the case these networks have to operate in vision devices with limited hardware. A possible solution for the aforementioned problem is represented by the development of a dimensionality reduction technique for CNNs by employing Proper Orthogonal Decomposition (POD), a method widely used in the context of Reduced Order Modeling, or Higher Order SVD (HOSVD), to keep into account the intrinsic tensorial structure. The reduced network is then obtained by splitting the original one in two different nets connected by the reduction technique: the first one obtained by retaining a certain number of layers of the original model and a second one that deals with the classification of the features extracted by the previous part. In our works we propose several version of reduced networks to tackle two different problems: image recognition and object detection. For the first case, we provide the numerical results obtained by applying such method to benchmark CNNs, such as VGG-16 and ResNet-110, using CIFAR-10, CIFAR-100 and a custom dataset. For the object detection case, we present a possible generalization of the method proposed for Artificial Neural Networks to object detectors and in particular to SSD-300 or neural networks with a similar architecture We then provide the results obtained by training our reduced model against the PASCALVOC dataset.

There will be a round-table after the talks.