Abstracts

I got my bachelor degree in mathematical engineering from the Politecnico di Milano. I then obtained my master degree and my PhD from the Università degli Studi di Milano. Currently, I am a Post-doc in Florence.

Title of the talk: Regularity results for elliptic PDEs

Abstract: In this seminar, I will talk about second-order regularity for elliptic partial differential equations. Starting from the classical Laplace operator (the Poisson equation), I will discuss various techniques to prove square integrability of the Hessian of solutions under suitable assumptions on the equation.
Time permitting, I will talk about quasilinear equations as well.

I have obtained a Master's degree in mathematics at the University of Milano-Bicocca. I am currently a PhD student at the University of Milano. My research interests are in the field of perturbative techniques applied to Hamiltonian PDEs.

Title of the talk: A KAM approach to the non-relativistic limit of the Klein-Gordon equation.

Abstract: It is heuristically well known that the non-relativistic limit of the nonlinear Klein-Gordon equation is the nonlinear Schrödinger equation. Several authors have proved rigorous results ensuring that solutions of the nonlinear Klein-Gordon equation, after a Gauge transformation, converge to solutions of the nonlinear Schrödinger equation uniformly on compact intervals of time.
I will present a result proving the existence of quasiperiodic solutions of the nonlinear Klein Gordon equation on the one-dimensional torus uniformly as c tends to infinity. I will also prove that, after a Gauge transformation, such solutions converge uniformly with respect to t in ℝ to solutions of the nonlinear Schrödinger equation.
This result is obtained by the application of KAM theory and NormalForm techniques, which I’m going to recall during the presentation.

I obtained my bachelor’s in Economics from University of Milan – Bicocca and my Master’s in Quantitative Finance from the University of Milan. At the moment I am a PhD student at Università Cattolica working on topics in Mathematical Finance. Specifically, I am investigating the solution to issues of consistency in intertemporal portfolio choices.

Title of the talk: Continuity results on state-dependent utilities and applications to non-linear expectations

Abstract: State-dependent preferences for a general state space were shown in Wakker and Zank (1999) to admit a numerical representation in the form of the integral of a state-dependent utility, as soon as pointwise continuity of the preference ordering  is assumed. In this work we prove that such a state-dependent function inherits pointwise continuity from the preference ordering, providing in this way a positive answer to a conjecture posed in the aforementioned seminal work. We further apply this result to obtain an explicit representation of conditional Chisini means in the form of a conditional certainty equivalent.

I am a PhD student in the university of Milan. I reside in the tangible realm, yet my study is situated in the p-adic world, a world closely aligned with the real world. More precisely, I am interested in p-adic Hodge theory of analytic varieties.

Title of the talk: An Introduction to Non-Archimedean Geometry

Abstract: As we know, real numbers are obtained by taking the completion of rational numbers with respect to the usual absolute value. If we do the same thing with respect to a new absolute value which reflects the "p-ness" of ℚ, then we get p-adic numbers. In this talk, we will give a brief overview of the (analytic) geometry over p-adic numbers, which allows us to use methods of analysis, topology and algebra.

I am a Phd student in Mathematics at ‘Scuola Internazionale Superiore di Studi Avanzati’ (SISSA), in Trieste, where I am attending the first year in the sector of Geometry and Mathematical Physics. My interests cover topics in Mathematical Physics related to the study of many-body quantum systems. More precisely, for my phd project I am focusing on the study of systems of many electrons on the lattice, in presence of two-body interaction or/and random disorder. About my education, I have studied Physics at Roma Tre university since September 2018 to September 2023; in my master thesis, supervised by prof. A. Giuliani, I have worked on the proof of a result of universality about the critical Hall conductivity, for a model of interacting electrons (the Haldane model).

Title of the talk: Universality of the conductivity matrix in many-electron systems

Abstract: Universality is a central issue in statistical physics, consisting of the idea that some suitable macroscopic observables do not depend on the details of the microscopic system under analysis. I will discuss the study of universality for the conductivity matrix in the context of one of the most famous lattice 2D quantum models: the Haldane model. It describes the hopping dynamic of tight-binding electrons on a hexagonal lattice, in presence of a transversal and dipolar magnetic field, and possibly a density-density reciprocal interaction. When the interaction is zero, the model is exactly solvable and the values of the observables are explicit. In particular one is able to compute the conductivity matrix over the whole phase diagram. Unfortunately, as soon as the interaction is considered, the system is no longer integrable, and the goal of computing the conductivity becomes a hard challenge.
However, this problem has been solved in Mathematical Physics in the recent years, with methods based on the combination of rigorous renormalization group and Ward identities. It turns out that the conductivity matrix is actually universal (independent of the interaction), up to an analytic ‘deformation’ of the phase diagram of the non-interacting model.
The results I will present are based on a series of papers by A. Giuliani, I. Jauslin, V. Mastropietro, M. Porta, R. Reuvers and me.

I was born and raised in Milan; I studied Mathematics in UniMi for both my bachelor’s and master’s degree, graduating in February 2021 with the thesis “Approximation and Regularization for the Inverse Conductivity Problem” (advisor: prof. Luca Rondi). I am currently a PhD student at the University of Genoa; my advisor is prof. Giovanni S. Alberti.

Title of the talk: Compressed sensing for inverse problems

Abstract: Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional, up to logarithmic factors, to the sparsity of the unknown signal. The classical theory mostly considers either random linear measurements or subsampled isometries. In this talk, I will show how the theory of compressed sensing can also be rigorously applied to a variety of ill-posed inverse problems, including sparse X-ray and photoacoustic tomography. In the case when the unknown signal is s-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove stable recovery from a finite number of random samples under the condition m ≥ Cs, up to logarithmic factors, where C depends on the resolution of the signal to be recovered. This is joint work with Giovanni S. Alberti, Matteo Santacesaria and S. Ivan Trapasso.

I studied in Florence both for the Bachelor and the Master degree. Then, I moved to Torino for my PhD. My research interests are complex non-Kähler Geometry and the study of non-Kähler special metrics on complex manifolds with  special focus on Geometric flows and the study of elliptic equations connected to geometric problems.

Title of the talk: The use of PDEs for the search of special metrics in complex Geometry

Abstract: In modern Differential Geometry, the search for Riemannian  metrics which can be considered "special" is an active field of research. In the Riemannian setting,  Einstein metrics are the most well studied  special metrics.  An outstanding result by Yau in 1978 states that a compact  manifold  endowed with a Kähler structure satisfying further mild assumptions, we can always find Einstein metrics which are Kähler themselves.  The proof boils down to solve a complex Monge-Ampère equation. In this talk, after recalling the main aspects of non-Kähler Geometry and the physical motivations, I will  describe how one can construct special non-Kähler metrics  solving elliptic or parabolic equations.

I am a third-year PhD student in Mathematics Education at the University of Milan. I completed my bachelor's and master's degrees in Mathematics at Sapienza University of Rome. My research interests range from exploring teaching and learning challenges and opportunities for upper secondary school students in utilizing algebraic language as a thinking tool, to enhancing teachers' education through methodological innovations grounded in research for effective teaching practices.

Title of the talk: Mathematics Education Research for Mathematicians: why and how can we support the teaching and learning of mathematics?

Abstract: What does it mean to conduct research in mathematics education? During the talk, we will discuss some of the reasons why mathematics education research is of interest to mathematicians. We will delve into the topic of research in mathematics education by addressing some of the themes covered in my doctoral research, ranging from the primary difficulties and distorted views of algebra among students, to "innovative" teaching methodologies through which mathematics can be taught, such as mathematics laboratory and mathematical discussions.
The teaching of mathematics is a topic close to all of us for several reasons: we have been students for at least 13 years in school, we have studied advanced mathematics at university dealing with teachers we would describe as more or less skilled, we have tutoring assignments for university courses, and, who knows, most of us will teach mathematics in our future. One of the goals of this talk is to encourage reflection on multiple levels on what we can do as experienced mathematicians for the teaching and dissemination of mathematics.

I am an ex-future terrible soccer player, who almost failed mathematics many times in high school but who learned to appreciate its beauty, and who grew a fascination about how deeply it allows us to understand nature. 
After graduating as a naval engineer and as a M.Sc. in Mathematics in Brazil, I went to Roma for my Ph.D. studies at Sapienza.
My research focuses on overdetermined elliptic problems. I mainly investigate the influence of the geometry of unbounded regions on the solvability of overdetermined problems. In particular, I develop ideas based on domain variations and shape optimization.

Title of the talk: Overdetermined problems: what differential equations tell us about shapes

Abstract: Consider the partial differential equation:

−∆u = f (u) in Ω with u = 0 on ∂Ω . (1)

There is a very well-established theory for the existence of positive solutions, under quite general assumptions on f and Ω. However, as was shown by James Serrin in 1971, we can only hope to find positive solutions to (1) satisfying the additional boundary condition ∂u:

∂u/∂ν = constant on ∂Ω (2)

if Ω is a ball! This discovery (and its proof) led our community to formulate many interesting questions, which were found to have extraordinarily beautiful answers and led to even more interesting questions. Among these, we will briefly discuss:

• What does (1) mean, and why do we bother with (2)?

• The interplay between analysis and differential geometry: how PDEs relate to the problem of prescribing the curvature of surfaces

• A dual problem: if Ω is symmetric, do all solutions of (1) have the same symmetry of Ω?

• A variational formulation for the overdetermined problem in the “space of shapes”

I was an ALGANT Master's student at the Universität Regensburg and UniMi between 2020-22. I'm spending my time as a PhD student at UniMi (with Paul Arne Østvær) and at UBFC/UB, France (with Adrien Dubouloz) . My area of research culminates the tools from algebraic topology into algebraic geometry; in particular I do Motivic homotopy theory with a focus on A¹-contractibility of low-dimensional smooth affine schemes and the (deformed) Koras-Russell 3-folds.

Title of the talk: On shrinking algebro-geometric entities…

Abstract: One of the most ancestral cravings for any area in Mathematics is that its entities are well classified. The all-time favourite invariant isomorphism is usually too rigid for practical purposes, and it turns out that the homotopy theory offers a relaxed replacement while still preserving the vital integrity of the entity concerned. One such entity we will discuss here is the category of varieties (or schemes) with its associated classification theory performed by a tool called the 'A¹-homotopy theory'. This talk promises to facilitate understanding of this theory at a grass-root level by suitable analogies with that of the topological world. In particular, we will carefully define the notion of A¹-contractibility (vista: when is a scheme ≃ a point?) and spell out an interesting result - Uniqueness of affine n-space over fields, n = 0,1,2. To this end, we will glimpse the current state of the art and our recent conjecture on the above result extended over Dedekind rings.

After obtaining my bachelor and master degree in UniMi, I am now a PhD student in the same university. My interests are in Riemannian and differential geometry, in particular geometric analysis and PDEs on manifolds.

Title of the talk: Brownian motion on Riemannian manifolds.

Abstract: We will review some classical results regarding the interconnection between Stochastic Analysis (e.g. Brownian motion) and Riemannian Geometry.
We will see how the behaviour of a particle moving of Brownian motion inside a manifold reveals a fascinating interplay between the geometry of the manifold and its analytical properties, that is, the study of Partial Differential Equations defined on it.
Special emphasis will put on the relevance that such a theory holds in the context of the celebrated Uniformization Theorem for Riemannian Surfaces.

I am a second-year PhD student in Mathematics at the University of Milan, where I also got my Master’s and Bachelor’s degrees. My research is in category theory and categorical algebra. I am particularly interested in the applications of category theory to algebra, topology and logic.

Title of the talk: Torsion theories in non-pointed contexts

Abstract: Torsion theories are the categorical generalisation of the following basic properties of abelian groups.
1. There are two special classes of abelian groups: torsion groups and torsion-free groups;
2. every group homomorphism from a torsion group to a torsion-free group is null;
3. every abelian group A has a torsion subgroup T such that the quotient A/T is torsion-free.
These three properties are easily translated into the language of category theory in a pointed context (that is, in categories with “0”).
However, trying to understand what a torsion theory is in a category with no true notion of “0” allows us to reinterpret a wide variety of different mathematical concepts as examples of torsion theories. This has been done in various ways, and is still the object of mathematical research.

In 2018, I obtained a master’s degree in mathematics from the University of Genoa. During this time, I studied mathematics education. While writing my thesis and during my subsequent year of research fellowship at the University of Parma, I focused on the research area related to prospective teacher education and interdisciplinarity between mathematics and physics. Currently, I am in the second year of my PhD at the University of Rome “La Sapienza”, where I continue to pursue my interest in this area.

Title of the talk: Prospective secondary teacher education and interdisciplinarity between mathematics and physics

Abstract: Contemporary social challenges, such as artificial intelligence and climate change, prompt interdisciplinary thinking and dialogue. This increases interest in these topics in mathematics education research. It also invites reflection on what kind of education that teachers may need. If interdisciplinarity is more than a simple “additive juxtaposition” of different disciplines, then it is reasonable to assume that “expert mathematics and physics teachers with an interdisciplinary perspective” are not just the combination of an expert mathematics teacher and an expert physics teacher.
In a European project in which I was involved, two possible levels of interdisciplinarity were identified: a “curricular” one (considering themes in the national curriculum and already structured within disciplines) and an “emergent” one (considering new forms of collaboration between disciplines in contexts outside school, such as in the study of climate change or artificial intelligence).
In this seminar, I will focus on my PhD research (supervised by Prof. Laura Branchetti, University of Milan, and Prof. Francesca Morselli, University of Genoa), the follow-up of the project, whose research goal is to study the prospective teacher education of mathematics and physics teachers in secondary schools with an interdisciplinary perspective and to contribute to the design of prospective teacher education activities. The design considers an interdisciplinary curricular topic (the parabola and the motion of projectiles) and the crucial role of textbooks as tools for educating: the analysis and comparison of excerpts from several physics textbooks is in fact a thought-provoking activity for prospective teachers. The research setting and the first results obtained will be presented, also with the broader aim of giving insights into a complex area of research in mathematics education such as (prospective) teacher education.

I am a third-year Ph.D. student in Mathematics at the University of Milano-Bicocca and at the Université de Haute Alsace. My research interests are in quantization of Lie bialgebras and related topics, such as Lie theory, Hopf algebras, Category theory, and Representation Theory. Previously, I got both the B.Sc. and the M.Sc. in Mathematics at the University of Bologna.

Title of the talk: Introduction to monoidal categories

Abstract: The aim of this talk is to give an introduction to the concept of a (braided) monoidal category and show some basic examples. If there is time, we shall discuss the more sophisticated example of the monoidal category obtained by the deformation of an infinitesimally braided one through a Drinfeld associator.

I obtained my Master degree in Mathematics at University of Rome -Sapienza. Now I am a PhD candidate in Mathematical Models at University of Rome - Sapienza. My research interests are analysis of PDEs, elliptic PDEs, blowing-up solutions, Lyapunov-Schmidt reduction.

Title of the talk: On the Brezis-Nirenberg problem

Abstract: I will provide an overview on the Brezis–Nirenberg equation on bounded domains. It is well known that existence and multiplicity of positive and sign-changing solutions to this equation is strictly affected by the geometry of the domain and the dimension of the euclidean space where the domain lies. In particular I will discuss the existence of blowing-up solutions to this problem involving a critical Sobolev exponent.

I'm a PhD student at the Department of Mathematics in the University of Milan. My field of research is stochastic optimal control: I am currently focusing on McKean-Vlasov SDEs and viscosity solutions to the associated PDE on spaces of probability measures. I am also interested in stochastic calculus and applications of optimal control to deep neural networks.

Title of the talk: Optimal control of McKean-Vlasov equations

Abstract: In this talk we will present a theory of optimal control for McKean-Vlasov stochastic differential equations.
Optimal control studies controlled (stochastic) dynamical systems in order to find and describe optimal strategies which minimise / maximise a cost / gain associated with the dynamics. We will mainly consider the approach based on the dynamical programming principle, from which a PDE (Hamilton-Jacobi-Bellman equation) is derived. The value function,  which encodes the minimum/maximum value that the system can reach depending on the starting time and state, is then characterised as its unique solution (in a suitable viscosity sense). When the dynamical system is described by a McKean-Vlasov SDE, the coefficients of the state equation depend not only on the trajectories of the state process, but also on its law. We thus explain that in fact the value function in this setting depends on the initial condition only through its law. We then derive an associated PDE on a Wasserstein space of probability measures and discuss the main difficulties arising when we want to prove that the value function still solves it uniquely in a suitable sense.

I obtained my master degree in Mathematics at University of Milan. I am currently a PhD student in Mathematics at University of Milan, where I work in the area of Numerical Analysis. My research interests lie in the field of inverse problems, with applications to biomedical imaging.

Title of the talk: Hybrid knowledge and data-driven approaches for Diffuse Optical Tomography reconstruction

Abstract: Diffuse Optical Tomography (DOT) is an emerging medical imaging technique which employs Near Infrared (NIR) light to estimate the spatial distribution of optical coefficients in biological tissues for diagnostic purposes, in a non-invasive and non-ionizing manner. NIR light undergoes multiple scattering throughout the tissue, making DOT reconstruction a severely ill-conditioned problem [1].
In this contribution, we adopt a hybrid approach that combines model-based and deep learning techniques to solve PDE-constrained inverse problems of the form

θ* = arg minθ L(y,ỹ)     (1)

where L is a loss function which typically contains a discrepancy measure (or data fidelity) term as well as prior information on the solution. In the context of inverse problems like (1), one seeks the optimal set of physical parameters θ, given the set of observations y. Moreover, ỹ is the computable approximation of y, which may be both obtained from a neural network and in a classic way via the resolution of a PDE with given input coefficients. Our idea is to exploit Graph Neural Networks (GNNs) as a fast forward model that solves PDEs: after an appropriate construction of the graph on the spatial domain of the PDE, the message passing framework allows to directly learn the kernel of the network which approximates the PDE solution [2]. Due to the severe ill-conditioning of the reconstruction problem, we also learn a prior over the space of solutions using an autoencoder-type neural network which maps the latent code to the estimated physical parameter, that is passed to the GNN to obtain the prediction. The latent code is finally optimised to minimise the difference between the recorded and predicted data.

References
[1] A. Benfenati, G. Bisazza, P. Causin, A Learned SVD approach for Inverse Problem Regularization in Diffuse Optical Tomography, arXiv preprint arXiv:2111.13401, (2021)
[2] Q. Zhao, D.B. Lindell, G. Wetzstein, Learning to Solve PDE-constrained Inverse Problems with Graph Networks, arXiv preprint arXiv:2206.00711, (2022)

I am a PhD student from Politecnico di Milano. I am a member of the iHeart group that investigates and models the human cardiovascular system. I studied maths at Università Statale degli Studi di Milano and I got the master’s degree in 2020 in applied mathematics.

Title of the talk: LSTM neural networks and variational autoencoders for cardiovascular inverse problems

Abstract: The impact of neural networks and deep learning in recent years has been profound and unprecedented. The number and variety of applications of the methods have skyrocketed over just a few years. The healthcare field represents one of these applications: the approximation of the solutions of multiphysics cardiac models, that combine the 3D modelling of the cardiac function and the 0D modelling of the circulatory system, is characterised by a high computational cost. To make the mathematical model patient-specific, one of the main challenges of cardiac modelling is to solve inverse problems that require many simulations of the original model, thus increasing the computational cost.

In view of applying the techniques of this talk to a 3D-0D setting, we will focus on a fully 0D cardiovascular model that we will surrogate by means of an LSTM neural network, cutting down its computational cost. To solve the inverse problem, we will apply a variational autoencoder to the 0D cardiovascular model. In such a way, the uncertainty quantification of model parameters, that usually requires many model simulations, will be an output of the variational autoencoder. The results of the variational autoencoders will be compared to the ones of standard optimization methods both in terms of time cost and (empirical) accuracy.