Schedule

In convection-dominated problems, standard numerical schemes produce unsatisfactory solutions with spurious oscillations near the element boundaries. This short talk will present a numerical method for convection-diffusion problems. In particular, we have devised a Continuous Interior Penalty (CIP) stabilized Virtual Element Method (VEM). The idea of the CIP-VEM scheme is to penalize the jumps of the gradients of the functions across the edges of the polygons in the mesh. To achieve our goal, we insert an appropriate bilinear form J(·, ·) called jump operator in the weak formulation of the problem. We have devised three different versions for the jump operator that return similar results. After the presentation of the method, we will show some numerical experiments where it is possible to appreciate the benefits of the jump operator. We will also investigate the rate of convergence of the numerical solutions to the analytic solution in the L^2−norm and broken H^1−seminorm for problems with known solution.

Recently the topic of explainability of Artificial Intelligence has been gaining increasing interest from a large part of the scientific community. In first place there is the desire to investigate the mechanisms of AI more thoroughly in order to better understanding its core functioning, so that anomalies, unexpected behaviors and even counterfeits can be prevented. In second place the interest is strongly pushed forward by the demands of policymakers, who request techniques that can be understood by humans and relied upon for public decision-making, not only for the excellent results but also for an inherent property of transparency. In this context, Group Equivariant Non-Expansive Operators (GENEOs) have been developed to establish a mathematical theory of information processing agents. Equivariance is indeed the key property of GENEOs, it guarantees that they commute with a group of geometrical transformations of the data domain. This property translates to the fact that some transformations play a less important role in determining the outcome of the agent’s action on the data. In spite of this, Equivariance is not a common feature of the majority of Artificial Intelligence methods yet, in particular there are examples of Deep Neural Networks in which different configurations of the the same data lead to unwanted and counter-intuitive results. On the other hand, GENEO-based networks are defined to be equivariant by design and the choice of the architecture allows to encapsulate into the network prior knowledge about the problem. This knowledge, if available, allows the model to be simpler, more transparent and more interpretable.

Deep Neural Networks (DNNs) have become extremely popular after their successes in areas such as language processing and computer vision. Nowadays, DNNs have also found application in scientific areas, such as Statistics and Numerical Analysis, which has gained them the attention of both applied and theoretical mathematicians. In particular, a very active area of research concerns the use of DNNs for learning operators in high (infinite) dimensions, with applications ranging from optimal control to inverse problems.

Within this context, we develop several approaches and derive theoretical results that further unveil the ability of DNNs in learning mathematical operators. First, inspired by the more classical linear methods, we focus on autoencoder architectures, a class of DNN models used for nonlinear dimensionality reduction. There, we introduce the concept of minimal latent dimension, which we define through the so-called manifold n-width. We then proceed to bound this minimal dimension, revealing its intimate connection with the topological properties of the operator image.

Parallel to this, we discuss on how to deal with the high dimensions generated by the discretization of functional spaces. For hypercubic domains, a popular approach is to exploit the so-called Convolutional Neural Networks (CNNs), but an underlying theory is lacking. In light of this, we derive novel upper-bounds for the approximation error of CNNs, highlighting how the mathematical properties of the operator affect the design of the network architecture.

The importance of causality and the notion of intervention for ML applications is now well known in the scientific community. For example, in the context of Explainable AI, the role of counterfactual explanations and their use in automatic decision systems, including the possibility of user recourse (e.g., granting a loan). However, the application of these techniques in artificial intelligence still appears limited, also because of the strong assumptions typically needed to apply the most powerful causal theories (Pearl), such as the complete knowledge of the SCM describing the relationships between the variables of a dataset, rarely obtainable in practice. Often, starting from a specific dataset, one has available only the joint probability distribution and a PDAG (or, equivalently, a Markov Equivalence class).

We propose a sampling technique in the space of DAGs (Directed Acyclic Graphs) in order to estimate, using ensemble methods, the causal effects of a system of which we know the observational data (e.g., the joint distribution of its relevant variables), but have no a-priori causal information. This proposal aims to extend the application of causal inference techniques, which are essential for all problems of interventional nature i.e., policy evaluation, actionable recourse recommendation, and counterfactual explanations, weakening the assumptions typically required in the causal framework, almost always not achieved in real-world problems.

This paper studies a semiparametric quantile regression model with endogenous variables and random right censoring. The endogeneity issue is solved using instrumental variables. It is assumed that the structural quantile of the logarithm of the outcome variable is linear in the covariates and censoring is independent. The regressors and instruments can be either continuous or discrete. The specification generates a continuum of equations of which the quantile regression coefficients are a solution. Identification is obtained when this system of equations has a unique solution. Our estimation procedure solves an empirical analog of the system of equations. We derive conditions under which the estimator is asymptotically normal and prove the validity of a bootstrap procedure for inference. The finite sample performance of the approach is evaluated through numerical simulations. The method is illustrated by an application to the national Job Training Partnership Act study.

Abstract: Irreducible holomorphic symplectic manifolds are — together with complex tori and Calabi-Yau manifolds — building blocks for Ricci flat compact complex manifolds. Their geometry is very rich but very few examples are known and they are all somehow related to moduli spaces of sheaves on symplectic surfaces. Indeed, it is still an open problem to determine all deformation classes in a given dimension. I will present the known examples and hint at some ideas about how to possibly get some new ones.

Ample divisors play a fundamental role in algebraic geometry. They display beautiful geometric, cohomological and numerical properties, and any algebraic geometer says "ample divisor" at least once every day. On the other hand, also "ample enough" divisors which are not ample (namely big divisors) are interesting ones, and, for instance, one could try to understand how the locus where these fail to be ample, namely their "augmented base locus", is made.

The augmented base locus of a divisor is an "asymptotic base locus", and the purpose of this talk is to give an introduction to asymptotic base loci, with examples coming from hyper-Kähler geometry.

Calabi-Yau manifolds are a central object for both mathematics and physics. During the years, mathematicians and physicists have constructed an half bilion of examples of these manifolds, this makes the question of their classification and the study of their geometry tremendously interesting.

A way to construct a Calabi-Yau manifolds is to consider certain actions of finite group on complex tori and produce the quotients.

In this talk, I will introduce Calabi-Yau manifolds, presenting in particular those constructed as quotient of complex tori. Then, we will discuss the relations between the geometry of Calabi-Yau manifolds and complex tori.

Integral group rings are mathematical object that arises naturally studing representation theory but can be applied in other field of mathematics (i.e. correction codes, number theory).

In spite of their natural definition, we are not able to completly characterize their units, essentials to understand whathever we can recover the group from the ring structure. In fact there are plenty of conjecture concerning those units that are still open.

In this talk I’ll focus on the Prime Graph Problem, i.e. is true that for a finite group G then the GK (Gruenberg-Kegel) graph Γ(G) is the same as the GK graph Γ(V (ZG))? These graphs are (mostly) completely classified for cut (central units trivial) groups, which are groups defined by a “ring condition”. It has been proven that these groups are inverse semirational groups, particular case of the semirational ones, characterized by a “group condition.

Fusion systems first appear in Puig’s work in the 70s as a tool for studying local-global conjectures in Modular Representation Theory. They are categories providing an abstract model of the conjugation homomorphisms among subgroups of a p-subgroup of a (finite) group G. Nowadays fusion systems are of great interest also in Group Theory and Algebraic Topology. We will discuss about the main results that have already been achieved, such as a proof of the Martino-Priddy conjecture independent on the Classification of the Finite Simple Groups (CFSG), as well as ongoing research, in particular in connection with the revision work of the CFSG (Aschbacher’s and MSS’s programmes). We will also introduce more recent tools which were developed in the context of the Martino-Priddy conjecture, with a focus on how they may be exploited when studying fusion systems.

It is a well-known fact that the category of compact Hausdorff spaces and the category of algebras of the ultrafilter monad are equivalent. Is it possible to alter this equivalence with the aim of extend the notion of compact Hausdorff spaces?

Let Σ ⊆ Rⁿ be a cone, with Σ = R⁺×Ω and Ω ⊆ Sⁿ⁻¹ , n ≥ 2. We consider the problem of classifying positive solutions to:

( ∆u + u^n+2/n−2 = 0 in Σ u_ν = 0 on ∂Σ (1)

where the condition on ∂Σ is assumed whenever ∂Σ ≠ ∅. This problem arises from the study of extremals of Sobolev inequality as well as from Yamabe problem.

If Ω = Sⁿ⁻¹ then positive solutions of (1) are completely classified [Caffarelli-Gidas-Spruck CPAM 1989] and they are radially symmetric (the so-called bubbles). This classification result can be extended to convex cones and more general operators [Lions-Pacella-Tricarico IUMJ 1988, Ciraolo-Figalli-Roncoroni GAFA 2020], under suitable assumptions on the energy of the solution. Our goal is to understand what happens when the cone is not convex, in particular we aim at providing optimal conditions on Ω such that bubbles are not the only positive solutions of (1).

I start by introducing the Hardy space on the unitary disk H^2, I discuss some basic properties of functions belonging to this space, I speak about its multipliers algebra and finally I define some relevant subspaces, namely the sub-Hardy spaces H(b), which have interesting applications to functional analysis, in particolar to the study of contractive operators on Hilbert spaces. Then, I introduce another space of analytic functions on the disk, the Dirichlet space, which was first defined by A. Beurling in the ‘30s, and I discuss the problems that arise when studying its multipliers algebra. Finally, I show that mimicking the construction of the sub-Hardy spaces in the Dirichlet case comes with some challenges, related to the multipliers algebra.

Many physical phenomena are well described as the propagation of waves: the motion of the sea, the transmission of sound, electromagnetic waves (light, radio waves…). Their mathematical description is often extremely complicated and characterized by the coexistence of stable and chaotic behaviours. I will discuss some models of wave propagation described by nonlinear Partial Differential Equations briefly illustrating the main difficulties as well as some mathematical methods used to study them.

The theory of representations in dimension 2 and 3 in Lie groups has a long history. In the case of surfaces, it is studied after the development of Teichmuller theory, for example to classify holomorphic vector fibers. In the case of 3-varieties, representations in Lie groups have been used to distinguish 3-varieties as complements of knots or for purposes of geometrization. We. will give an introduction of the theory of representations in dimension 2 or 3.

What do Risk players, neural networks, and thermostats have in common? And what makes them different?

Using the language of category theory we can pose and answer these questions, and more. The research program of categorical cybernetics aims to build an interdisciplinary framework for the study of interactive control systems. Its primary goal is to provide uniform tooling to approach game theory, gradient-based learning, reinforcement learning, control theory and other forms of 'cybernetic systems'. We ask what is their common structure, especially when it comes to composing simple parts in complex wholes. How do we describe these composites? How do we patch together their behaviours? What can we say about emergent effects?

In this lightly technical talk I'll introduce the basic ideas and tools of categorical cybernetics, focusing mostly on conceptual motivation and intuitive results.

The most fundamental concept of game theory is Nash equilibrium. However, in many cases of interest, the search for Nash equilibria may not be the best way of studying the behavior of the population: what if the payoff they reach is too low? or what if the number of the players is too high to make calculations feasible? In order to answer these questions, I will introduce the notions of coarse correlated equilibrium and mean field game. If time allows, such concepts will be brought together in a continuous time stochastic framework.

The problem of the degradation of marble stones with different porosity used as building materials for thousands of years is a very important issue that has been observed in the last century. The first cause is due to the atmospheric pollutants, in particular the reaction of sulphur dioxide with calcareous surfaces which forms gypsum and black crusts. Recently, some mathematical models have been used to study the evolution of degradation phenomena, either pure statistical models or deterministic partial differential equation. In this work a first attempt of introduction of randomness in the modelling starting from a deterministic PDE model is presented. Randomness is introduced via stochastic dynamical boundary conditions. We motivate our choice via an analysis of the sulphur dioxide time series in Milano, through a filtration procedure for the identification of the deterministic and stochastic component of the process. We discuss the possible choices of the dynamical boundary conditions and the consequences for the solution to the PDE model. In particular, a mean reverting process with bounded noise as dynamical boundary condition has been introduced. Then a comparison study of a system of PDE describing the evolution of the sulphur dioxide and the calcite both with deterministic and stochastic boundary condition has been performed via numerical experiments.

In the study of cancer evolution and radiotherapy treatments, a key dynamic lies in the tumor-abioticfactors interaction. In particular, oxygen concentration plays a central role in the determination of the phenotypic heterogeneity of the cancer cell population, both from a qualitative and geometric point of view. We present a continuous mathematical model to study the influence of hypoxia on the evolutionary dynamics of cancer cells. The model is settled in the mathematical framework of phenotype-structured population dynamics and it is formulated in terms of systems of coupled non-linear integro-differential equations. Numerical simulations are performed using Galerkin finite element methods with the aim to test and represent various biological situations. Then, the effects of radiotherapy treatment are included in the model to analyze the influence of the heterogeneity in oxygen concentration and phenotypic distribution of cancer cells on the treatment effectiveness. Various therapeutical protocols are considered. Simulations show that the geometric characterization of tumor niches differentiated by phenotypic characteristics determines a heterogeneous response to radiotherapy. The analysis of the study results provides suggestions about possible therapeutic strategies to optimize the radiotherapy protocol in light of the phenotypic and geometric inhomogeneities of the tumor. The study constitutes the first step in the development of a mathematical tool for the delineation of patient-specific protocols which, in the perspective of personalized medicine, aim not only at the eradication of the tumor mass, but also at the optimization, in case of relapse, of the phenotypic composition of the tumor so that resistance to subsequent treatments can be avoided as possible.

Twistor spaces were first introduced by Roger Penrose in the ‘60s, as an attempt to build a “bridge” between General Relativity and Quantum Mechanics. During the years, due to their rich mathematical structure and the huge variety of features they carry, twistor spaces have been widely studied by many mathematicians, both from an algebraic and geometric viewpoint. A very interesting case occurs when a twistor space, seen as a 3-dimensional almost complex manifold, is associated to a Riemannian four-dimensional manifold: indeed, the fourth dimension is “unique” in Riemannian Geometry, since, in this case, the Riemann curvature operator admits a block form decomposition which drastically changes the way the curvature of the manifold can be interpreted and gives rise to new “special metrics”. The relations between a four-dimensional manifold and its twistor space are so strict that weak metric conditions on the latter may have strong topological consequences on the underlying four-manifold. In this talk, I will give a brief introduction about the most important features of four-dimensional geometry and Riemannian twistor spaces: I will also discuss the most important results obtained in this research field and some new rigidity theorems, hinting at some possible future directions in this topic.

In 1968, John Milnor developed a theory to classify singular points of complex hypersurfaces by looking at the topology of their surroundings. He found that the multiplicity of a function at a singular point equals a certain degree of the gradient of the function at that point. But what happens over a general field? We will introduce the Grothendieck-Witt ring of quadratic forms and the motivic degree map to try to give an answer.