Past Talks

In this talk I will present a joint work with Tristan Rivière in which we prove some new results about uniqueness of tangent maps for general pseudo-holomorphic and strongly approximable maps from an arbitrary almost complex manifold into projective algebraic varieties. As a byproduct of the approach and the techniques developed, in the same work we also show how to obtain the unique tangent cone property for a special class of non-rectifiable positive pseudo-holomorphic cycles.

Many interesting physical systems can be described at the microscopic level as particle dynamics and at the mesoscopic level with kinetic equations. In the wide field of regular two-body interactions, the link between these two regimes is mathematically well understood in the case of the mean-field limit (MFL).

In this talk, I will focus on the classical proof of the MFL for regular two-body potentials and I will illustrate the main ideas behind the MFL for particle systems with "topological" interactions, a model introduced in some biophysics works of the last decade

Spectral theory rules! This is particularly true in quantum mechanics. We will consider a system of Bosons (some class of particles) and analyze the spectrum of the operators (aka the Hamiltonian) that models that system. The main idea is that we can obtain effective equations for the eigenvalues of our Hamiltonians (one functional to rule them all).

In the last decades, there has been increasing interest in the isoperimetric problem with density, which consists in looking for minimizers of weighted perimeter among sets with fixed weighted volume, where the densities involved can possibly depend on the orientation of the set we are measuring.

In this talk, we first introduce the basic definitions and known results on the subject in the "single set case"; next, we are going to describe a recent generalization of the problem to the "cluster case", where two or more sets are considered, possibly having parts of their boundaries in common. In particular, we will describe the so called "\epsilon-\epsilon^\beta" property, which is fundamental to explore the main questions on existence, boundedness, and regularity of optimal clusters.

I present the "large data problem'' for diffusion equations in R^N - that is, how large can the initial data u_0 \in L^1_{loc}(R^N) be while still guaranteeing local-in-time existence and uniqueness of solutions. Attention is drawn to the role of explicit source solutions and the comparison between behavior of solutions to the linear heat equation and nonlinear porous medium type equations. For these large initial data and nonlinear problems, L^1 theory and/or convolution with a source solution do not suffice. I explain a few powerful nonlinear methods used to obtain well-posedness results - many going back to the 1970s and 80s. I will also mention the question of asymptotics and many open problems.

Thermal convection is a physical mechanism of heat propagation which, unlike thermal conduction, involves the motion of matter. When considering a porous layer heated from below and saturated by a fluid initially at rest, convection occurs due to a difference in density within the fluid, caused by the temperature gradient imposed across the layer. In such a case, we speak of “natural” thermal convection.

Mathematically, convection can be seen as the instability motion of the steady solution describing a situation with fluid at rest and heat spreading by conduction. In this talk, I’m going to show how to approach such a stability problem, by performing both linear and nonlinear analysis. Moreover, I will focus on how to model the fluid motion in presence of the solid matrix and the heat propagation within the layer.

After that, I will show an optimal stability result for the onset of convection in a rotating anisotropic porous medium. Finally, in the last part of the talk, I will point out what would happen if instead of the Fourier’s law, one would employ the Cattaneo’s law in order to describe the heat propagation in the solid matrix.

The positivity conditions for vector bundles in Complex Algebraic Geometry are usefull tools which influence the geometry of the base complex manifold. For example and just to give an idea: a variety $X$ is projective if and only if it carries on a positive line bundle.

So a natural question is: how can one extends these notions in presence of a non zero Higgs field on a vector bundle? Bruzzo, Graña Otero and Hernández Ruipérez have attempted to furnish an answer to this problem, with success. They proved almost all the properties for H-nflat Higgs bundle; but in particular one is very hard to prove: the vanishing of Chern classes for H-nflat Higgs bundles over smooth complex projective varieties.

In this talk I will introduce the Higgs bundles, the H-nflat-ness condition, state the basic properties of these objects, the status of the art of the previous conjecture, with particular emphasis on the rank 2 case. This last part is a work in progress with Ugo Bruzzo.

The geometric description of the Euler-Lagrange equations of a mechanical system determined by a Lagrangian function relies on the velocity phase space TM of a configuration manifold M. In discrete mechanics, the starting point is to replace TM by M x M, taking two nearby points as the discrete analog of a velocity vector. Discrete mechanics has been developed on Lie groups and Lie groupoids as well.

My talk is about the generalization of the discrete mechanics on Lie groups to non-associative objects, smooth loops, generalizing Lie groups. This shows that the associativity assumption is not crucial for mechanics and opens new perspectives. The motivating example is octonions, I will show how to construct the discrete Lagrangian and Hamiltonian mechanics on unitary octonions.

Symmetrization processes are a fundamental instrument in geometric analysis, allowing to prove many inequalities (Isoperimetric, Blaschke-Santalò, Brunn-Minkowski...) with very simple proofs. One of the main ingredients of such proofs is the asymptotic behavior of these processes, a topic which still presents many interesting open questions. In this talk we introduce a "well behaved" family of symmetrizations, which includes Steiner and Minkowski symmetrizations, and we prove that they all posses the same asymptotic behavior.

In this talk we will consider mathematical models evolving according to a stochastic dynamics in order to identify dynamical properties of real-life systems in the framework of non-equilibrium statistical mechanics. We will consider a specific problem in the general study of transitions from local minima to a global minimum, where the evolution is given by a Markov process. In particular, we analyze metastability in the context of a local version of the Kawasaki dynamics for the two-dimensional strongly anisotropic Ising lattice gas at very low temperature. Let $\Lambda\subset\mathbb{Z}^2$ be a finite box. Particles perform simple exclusion on $\Lambda$, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Along each bond touching the boundary of $\Lambda$ from the outside to the inside, particles are created with rate $\rho=e^{-\Delta\beta}$, while along each bond from the inside to the outside, particles are annihilated with rate $1$, where $\beta$ is the inverse temperature and $\Delta>0$ is an activity parameter. We consider the parameter regime $U_1>2U_2$ also known as the strongly anisotropic regime. We take $\Delta\in{(U_1,U_1+U_2)}$ and we prove that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We consider the asymptotic regime corresponding to finite volume in the limit of large inverse temperature $\beta$. We investigate how the transition from empty to full takes place. In particular, we estimate in probability, expectation and distribution the asymptotic transition time from the metastable configuration to the stable configuration. Moreover, we identify the size of the critical droplets, as well as some of their properties. We observe very different behavior in the weakly ($U_1<2U_2$) and strongly anisotropic regimes. We find that the Wulff shape, i.e., the shape minimizing the energy of a droplet at fixed volume, is not relevant for the critical configurations.

It is a well known result that the self-adjoint operators in Hilbert spaces, the more interesting for quantum mechanics, can be represented through the multiplication operator on a measure space. But what happens to non-self-adjoint operators?

In this short talk I will present how the de Branges spaces solve this problem: in fact, these entire functional spaces are also used as functional models for a particular family of non-self-adjoint operators.

On an arbitrary closed Riemannian manifold it is still an open question how many closed geodesics there are. To tackle this problem, one often studies the topology of the free loop space since Morse theory provides a link between the topology of the free loop space and the closed geodesics. In recent years, new algebraic structures - so-called string topology operations - were introduced on the homology of the free loop space, the most famous one being the Chas-Sullivan Product. One can ask now whether these string topology operations offer any new insight about the closed geodesics in a given manifold and vice versa. This talk gives an overview over the closed geodesics problem and over string topology and indicates relations between those two topics.

Magnitude is a recently introduced isometry-invariant of metric spaces which is obtained when the Euler characteristic of an enriched category is specialised to metric spaces. Despite its abstract origin in category theory, it is known to encode several interesting geometric invariants in integral geometry and geometric measure theory. In this talk, I will briefly survey recent developments in this field and point out a few open questions along the way. Maximum diversity of metric spaces, a closely-related invariant of information-theoretic flavour, will also be discussed.

If time permits, I will present an application of these ideas to number theory and share some results from my ongoing research in this direction.

In 1946, L.D.Landau proved the existence of damped solutions for the linearized Vlasov-Poisson equation near stable stationary states. The validity of this result for the nonlinear equation remained an open problem for many years, until the first work of scattering-type by E.Caglioti and C.Maffei in the 1D periodic case and the groundbreaking work by C.Villani and C.Mouhot for the general Cauchy problem.

In this talk, I will present some of the main ideas behind the Landau damping phenomenon in the linear and nonlinear setting. In particular I will focus on the backward scattering approach.

In this talk, we study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann’s theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which parametrizes all tau functions arising from such a sum. After introducing solitons solutions, we compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces.

This is joint work with Daniele Agostini, Yelena Mandelshtam and Bernd Sturmfels.

Goal of the talk is to study a nonlinear Schrödinger equation affected by a singular perturbation. Pointed out who is the corresponding limiting equation, we will see how to exploit the information on this last (simpler) problem to detect a set of expected solutions for the original problem. Some tricks will be employed in order to get compactness, an essential ingredient to gain existence of solutions in the framework of PDEs. Finally, a short introduction to the fractional nonlocal case will be presented.

Innovation is the driving force of human progress. Recent urn models reproduce well the dynamics through which the discovery of a novelty may trigger further ones, in an expanding space of opportunities, but neglect the effects of social interactions. In this talk I am going to focus on a model we have recently proposed, in which many urns, representing different explorers, are coupled through the links of a social network and exploit opportunities coming from their contacts. We will see that the pace of discovery of an explorer depends on its centrality in the social network. Our model sheds light on the role that social structures play in discovery processes and it's a novel approach in the modellization of team creativity and efficiency.

Schrödinger-type equations model a lot of natural phenomena and their solutions have interesting and important properties. This gives rise to the search for normalized solutions, i.e., when the L^2 norm is prescribed. We propose a simple and novel approach based on a minimization argument.

In this talk, we will present the strategy to tackle the so-called “singular perturbation problems” which arise from the mathematical models in fluid mechanics. In this context, we are interested in the inspection of fluids on large-scale (like the atmosphere and the oceans) where the dynamics is influenced by the Earth’s rotation effects.

The main goal is to describe the limit system when the rotation parameter tends to infinity, i.e. the Coriolis force becomes singular. To achieve this scope, we will explore the main steps of the analysis for a simplified barotropic Navier-Stokes model.

In the last part of the talk, I will show the main results in a more realistic case (the full Navier-Stokes-Fourier system), in which we take into account the temperature variations and the influence of external forces of physical relevance, like the centrifugal and gravitational ones.

If you pick two random integers, how likely is it that they are coprime? Starting from this basic question we will see what a natural density is (spoiler alert: some substitute for a probability measure) and how one can compute it. The philosophy is that computations over the integers are hard, but computations over the p-adic numbers are slightly easier. If we can deal with all p-adics, then we can patch this (local) data together to obtain the natural density (the global datum). We will also introduce a notion of expected value in this setting and see that we can play the same game as for the densities.

The coronavirus disease 2019 (COVID-19) pandemic is a worldwide threat with Italy being the first country in Europe to be heavily hit by the virus. The first studies concerning SARS-CoV-2 have been about studying the mechanism underlying the acute phase of this virus and the short-term outcomes. What is increasing now is the interest about long-term outcomes of this disease, the so called “post-COVID syndrome” of “chronic COVID syndrome”, the knowledge available on factors associated with serological response, duration of anti–SARS-CoV-2 antibody rise and the presence of humoral immunity.

Statistic can be used for a better understanding of the possible factors associated with persistence of the symptoms of COVID-19 and for a more accurate analysis of the progression of the clinical status of a patient that survived COVID-19.

We propose new strategies to handle polygonal grids refinement based on Convolutional Neural Networks (CNNs). We show that CNNs can be successfully employed to identify correctly the “shape” of a polygonal element so as to design suitable refinement criteria to be possibly employed within adaptive refinement strategies. We propose two refinement strategies that exploit the use of CNNs to classify elements’ shape, at a low computational cost. We test the proposed idea considering two families of finite element methods that support arbitrarily shaped polygonal elements, namely Polygonal Discontinuous Galerkin (PolyDG) methods and Virtual Element Methods (VEMs). We demonstrate that the proposed algorithms can greatly improve the performance of the discretization schemes both in terms of accuracy and quality of the underlying grids. Moreover, since the training phase is performed off-line and is problem independent the overall computational costs are kept low.

The interaction between wind and suspension bridges is a broadly studied phenomenon from the experimental and numerical point of view. We aim at studying such phenomen from the analytical point of view, by exploiting the mathematical tools offered by the classical theory of the Navier-Stokes equations applied to fluid-structure-interaction problems. I will briefly illustrate you the last result that I obtained in this direction, trying to set it in a more general framework.

In particular, in an unbounded 2D channel, we consider the vertical displacement of a rectangular obstacle in a regime of small flux for the incoming flow field, modelling the interaction between the cross-section of the deck of a suspension bridge and the wind. I will show you the strategy to prove an existence and uniqueness result for a fluid-structure-interaction evolution problem set in this channel, where at infinity the velocity field of the fluid has a Poiseuille flow profile.

In many areas of science, such as physics, biology and engineering, phenomena are often modeled in terms of Partial Differential Equations (PDEs) that exhibit dependence on one or multiple parameters. As we change the values of the parameters, we move through the collection of all possible solutions, the so-called solution manifold. Here we address the problem of learning the structure of the solution manifold, which is of particular interest in contexts such as PDE-constrained Optimization and Uncertainty Quantification.

In general, finding an accurate yet computationally affordable approximation of the parameter-to-solution map is a truly challenging task. Motivated by the limitations of most state-of-the-art algorithms, we present a Deep Learning approach based on deep autoencoders and minimal representations. After establishing a few driving theoretical results, we report numerical experiments with particular emphasis on second order elliptic PDEs. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields, singular terms and stochastic coefficients.

Several types of organisms utilize an array of techniques to communicate with one another, either explicitly or implicitly. This communication often produces some form of collective behavior, whereby a group of these organisms form patterns, produce a consensus, and/or move synchronously with each other. There is great interest from scientists in finding and understanding the underlying mechanisms governing these types of collective behavior. These mechanisms usually take place in the microscale, on an individual level, and lead to macroscopic effects. In this talk, I will highlight our recent efforts in modeling the movement and pattern formation of phytoplankton and the curious social distancing behaviors seen in simple flocks.

The concept of group may be generalized to more general settings: in particular, one may define group objects in any category with finite products.

Similarly, also the notion of action of a group on a set may be generalized. In my talk, besides giving the main ideas of this, I will focus on the case of affine schemes and try to briefly explain how it is related to what I am working on.

Noncommutative geometry studies the property of a space X (topological, smooth manifolds, algebraic varieties, etc.) through the commutative algebras of (continuous, smooth, regular, etc.) functions over it. Thus, when one considers noncommutative algebras, one is thinking of noncommutative spaces.

In this talk, the focus is on principal fiber bundles. These objects are ubiquitous in both mathematics and physics. Their algebraic counterparts are co-module algebras over Hopf algebras.

The first part is devoted to explaining in detail what is the meaning of the last sentence and providing some examples which will guide us along through the entire talk. The rest turns the light on Galois objects of Hopf algebras which in this context are the algebraic version of principal fiber bundles over a singleton.

Automatic learning research focuses on the development of methods capable of extracting useful information from a given dataset.

A large variety of learning methods exists, ranging from biologically inspired neural networks to statistical methods. A common trait in these methods is that they are parameterized by a set of hyperparameters, which must be set appropriately by the user to maximize the usefulness of the learning approach. In this talk I review hyperparameter tuning and discuss its main challenges from an optimization point of view. I provide an overview on the most important approaches for hyperparameter optimization problem, comparing them in terms of advantages and disadvantages, focusing on Gradient-based Optimization.

Since Kontsevich raised the homological mirror symmetry conjecture (HMS), the derived category of coherent sheaves has attracted greater attention from both mathematicians and physicists. Originally, HMS was stated for Calabi--Yau manifold with a mirror partner, and twisting Calabi--Yau manifolds by turning on a Brauer class becomes inevitable in associated superconformal field theories. Therefore, it is natural to wonder a twisted version of HMS.

In this talk, we explain two ways of twisting the derived category of projective spaces by a Brauer class. One is twisting projective spaces to obtain a Brauer--Severi variety and taking its derived category. The other is twisting the structure sheaf and taking its derived category of coherent modules. The latter leads us to the world of noncommutative schemes, as we obtain schemes whose structure sheaves are Azumaya algebras.

For a toric variety, HMS is reduced to the nonequivariant coherent--constructible correspondence (NCCC). NCCC is a relation between the derived category of coherent sheaves and the derived category of constructible sheaves on a torus. If time allows, we will introduce our current result on NCCC for Brauer--Severi varieties, which can be considered as a twisted version of HMS.

My research in probability has had interplay with the fields of analysis, combinatorics, and algebra. In this talk, I give examples of each of these connections and discuss how they were useful in my work. Notable appearances include generators of operator semigroups, integer compositions, and quasisymmetric functions.

The presentation will be non-technical and full of interesting exercises.

Evolving interfaces is a key feature of many practical applications as e.g. pattern dynamics, multi-phase flows, material design, and their approximation raise tremendous interesting numerical issues.

In particular, incompressible two-phase flows require an accurate description of the interface in order to select the proper phase properties and dynamic to consider on each subdomain. Indeed, while the volume of each phase remains constant in time, the interface may develop heavy distortion and change topology across time. There, a small error in the interface location may have dreadful repercussions on the state dynamic (e.g. when vortices are encountered).

Therefore, the designed numerical scheme has to pay a particular attention to the representation of the interface(s) and the description of its motion.

Using an implicit representation of the interface, the Level Set method is a popular technique that allows a complex interface shape and is suited to topology changes in time. However, it is not natively conservative, that is the quantity of each fluid is not preserved as the interface is evolved.

In this talk, we propose a correction technique to the Level Set method that enforces mass conservation while preserving the continuity of each connected component of the interface. Flexible, it can be used in combination with any meshed scheme updating the non-corrected Level Set field.

Mathematicians have long been interted in rigorously understanding the conductivity properties of disordered materials at the quantum level, in particular after the work of the Nobel Prize winning American physicist Philip W. Anderson (1923-2020)

In 1990, Klein, Lacroix and Speis analyzed a well studied random operator model for an electron moving on a portion of lattice of the form Z × [0, W], W ∈ N and subject to a random potential, called Anderson model on the strip. They showed, in particular, that such a model boasts spectral localization on all of its energy spectrum, a well defined mathematical property that is a very powerful signature of the electron getting trapped in a region by the potential.

In thie present work, we focus on a more general model of a quantum particle with internal degrees of freedom moving in a quasi 1D random medium (disordered quantum wire), that we call "generalized Wegner Orbital Model".

In particular, we prove spectral and dynamical localization at all energies for such a model suggesting that the disordered materials belonging to the wide class described by this model are all perfect insulators.

In this talk, I will start by introducing basic concepts related to Anderson Localization in general, then move to the specific model considered in this work, and outline our proof of its spectral localization. The proof combines techniques from probability theory, spectral theory of selfadjoint operator and ergodic theory, with an unexpected algebraic twist...