Visitors

Here is the list of Eccellenza visiting professors of the Department Guido Castelnuovo.

For an extended list of visiting scientists of the DMC please click here.

2023

During his stay at the DMC, G. Besson gave a PhD course on  "Concentration phenomenons in Geometry".

The first part of the course (prerequisites) was developed by A. Sambusetti and was about Ricci curvature, volume comparison theorems and the theory of the Laplacian in Riemannian geometry.

In the second part, G. Besson explained some concentration phenomena in geometry. Concentration was studied in three different contexts:

1. the law of large numbers after Émile Borel (which was interpreted by him has a concentration in the cube);

2. the concentration of the volume of the unit sphere on the equator (which is a geometric law of large numbers);

3. the Vitali Milmann's proof of  the Dvoretsky theorem on normed vector spaces. 

Besides the PhD course, G.Besson worked with A. Sambusetti on a joint project (in collaboration also with G. Courtois and S. Gallot) about differential rigidity of manifolds with upper curvature bounds which are close to each other with respect to the Gromov-Hausdorff distance. 

The conjecture is that the diffeomorphism type of CAT(1)-manifolds is locally constant. The method shall rely on comparison geometry for (local) barycenter maps, a tool which was developed by Besson, Courtois and Gallot and that has been intensely used by Sambusetti in different contexts.

During his one-month visit Prof. Goette carried on research activity in collaboration with Paolo Piazza. The core of their project can be summarised as follows:

give a rigorous definition of singular G2-structure, define the ν-invariant on it, find a relationship between this invariant and the analogous one defined on a resolution.

To attack these questions, Goette and Piazza have identified the degeneration of smooth G2-manifolds to G2-spaces with iterated incomplete edge singularities as one feasible approach.

In some simple examples, appearing in the list of Joyce, one can use the ν-invariant of a singular G2-space to compute the ν- invariant of the resolved smooth manifold, see the work of Fornasin and Scaduto. The hope is to pass to more complicated examples, typically of depth 2.

Besides this research activity, Prof. Goette also had fruitful discussions with Francesco Bei.

2024 / 2025

During his visit in March 2024 Prof. Massimo Ostilli taught a course on Complex networks. The course, addressed to PhD students but open to any interested researcher, was arranged in four two-hour lessons as illustrated in the following abstract:

Abstract

• Day 1. Basic Topology and Metrics of Equilibrium Networks (adjacency matrix, connectivity, loops, k-cliques, degree distribution and excess degree distribution); Lattices, trees and Bethe lattices; The Random Graph (Erdös-Rényi and Gilbert Models); The Configuration Model.

• Day 2. Shortest Path and the Small-World Effect in the Random Graph; Complex Networks and the Scale-Free concept; Ultra Small-World; Connected Components; Percolation and other Critical Phenomena running on Complex Networks.

• Day 3. Growing Networks (The Barabási-Albert model and its variants); Hidden-variable models; Distribution of the Maximal degree, Motifs, and Fluctuations in Complex Networks; Examples of Real World Networks.

• Day 4. According to audience’s interest, an advanced topic to be chosen among the following: Critical phenomena on Complex Networks with loops; Random Geometric Graphs; A “no-go” theorem for Quantum Mechanics on random graphs.

During his stay, Prof. Macrì will give the following series of lectures:

Title: Stable bundles on Fano and hyper-Kähler manifolds.

Program: The aim of this course is to present some recent advances in the theory of stable sheaves on higher dimensional varieties, in particular Fano and hyper-Kähler manifolds.

1. 1. We will start by reviewing the case of curves and surfaces.

   2. We will then turn to Fano threefolds, where the theory is in a certain sense a generalization of the case of curves, by studying in detail the case of cubic threefolds.

   3. Finally, we will consider hyper-Kähler fourfolds, where the theory gives a generalization of Mukai's theory on K3 surfaces.

During her stay, Prof. Moreau will give the following series of lectures:

Title: W-algebras and applications.

Program: W-algebras are certain vertex algebras associated with nilpotent elements of a simple Lie algebra. They generalize both Virasoro and Kac-Moody algebras. The apparence of the AGT conjecture in physics led many researchers toward to these algebraic structures. W-algebras were introduced and developed by Zamolodchikov, Fateev-Lukyanov, Feigin-Frenkel and Kac-Roan-Wakimoto, while their finite-dimensional analogues, the finite W-algebras introduced by Premet, are important in classical problems in representation theory. W-algebras are related to integrable systems, two-dimensional conformal field theory and the geometric Langlands program.

These lectures will start with a short introduction to vertex algebras and basic examples. Then we will explain the construction of W-algebras that are obtained from affine vertex algebras by the BRST construction, and describe their properties. We will also discuss some applications of equivariant W-algebras that play a key role in the quantum geometric Langlands correspondence.

Prof. Hanke will hold a 6/8-hour mini-course on existence problems for positive scalar curvature metrics within the intensive research period focussed on geometric analysis.

The audience will be made of researchers, post-docs and phd students from our department and from other institutions.

During the first week of activities, 11th-15th November 2024, Christian Bär will hold a mini-course on spin manifold, Dirac operators, Atiyah-Singer index theorem and positive scalar curvature. The course aims to introduce some of the fundamental ideas and techniques involving Dirac operators, index theory and Riemannian metrics with positive scalar curvature.

During the second week, 18th-22nd November 2024, he will give a talk within the conference "Curvature and Geometric Analysis in Rome". In his talk he will explain  his recent contributions on various problems concerning the existence of positive scalar curvature.

Professor Tristan Rivière will give a course in the field of Calculus of Variations applied to Geometric Analysis (min/max methods, minimal surfaces and related questions). This will be one of the courses of the Winter school held during the week January 20-24, 2025.

Professor Rivière will also give a seminar in the workshop held in the week January 27-31, 2025. Both events are part of the trimester in geometric analysis held in Winter 2024-2025.

Professor Chambolle will hold a series of seminars during the winter school and the conference scheduled as part of the activities for the trimester, within the framework of the excellence project, between January and February 2025.

These seminars will focus on variational methods in the study of singularities and geometric flows, and will be specifically addressed to young PhD students.

Prof. Otto is one of the main speakers invited to the winter school in Geometric Analysis in January/February 2025.

Prof. Otto is one of the world's most influential mathematicians in mathematical analysis and applied mathematics. He is Leading Director of the Max Planck Institute for Mathematics in the Sciences in Leipzig, former Director of the Hausdorff Center for Mathematics (Bonn), and member of the Deutsche Akademie der Naturforscher Leopoldina. 

Prof. Otto's results on the variational formulations of geometric and gradient flows are among the most relevant in the field and fit perfectly the aim of the Trimester in Geometric Analysis.

During his visit to DMC, Professor Vojkan Jakšić will give a Ph.D. course entitled "Modern Non Equilibrium Statistical Mechanics" in which he will present a recent C*-algebraic approach to Statistical Mechanics which encompasses both the Classical and the Quantum case.

He will collaborate with several members of the Department, on research topics at the crossroads of Mathematical Physics, Operator Algebras, and Probability.

Minicourse "Nonequilibrium fluctuations".

In these lectures we review a recent result on a scaling limit of the space-time mass fluctuation field of Glauber + Kawasaki particle dynamics around its hydrodynamic mean curvature interface limit.

Here, the Glauber rates are scaled by $K=K_N$, the Kawasaki rates by $N^2$ and space by $1/N$.

"Chaos and integrability in open and driven quantum many-body systems".

The course will be on recent developments in open quantum many-body systems and dynamics of driven (Floquet) lattice systems. It will be divided into four lectures:

1. In the first lecture, I will discuss empirical diagnostics of quantum chaos of open quantum systems and present tenfold way of symmetry classification of the corresponding Liouvillians

2. In the second lecture, I will discuss Yang-Baxter integrability of open quantum systems and present some simple examples

3. In the third lecture, I will focus on exactly solvable steady states of boundary driven open spin chains and algebraic aspects of the underlying matrix product states

4. In the fourth lecture, I will introduce integrable spin lattice models in discrete space-time (integrable quantum circuits) with the corresponding unitary integrability structures in discrete time domain, and present some interesting conjectures on anomalous transport and anomalous fluctuations in these systems.

During his visit in September 2025, Prof. Jean Barbier will deliver a course dedicated to "Information theory and mean-field theory of high-dimensional Bayesian inference" within the summer school "Mathematical methods for high-dimensional data", which is one of the activities organized during the Data Science thematic period.

The course will cover the following topics:

• Models in high dimensional inference

• Information-theoretic versus algorithmic limits

• Spectral phase transition in the spike Wigner model: the BenArous-Baik-Péché transition.

• Concentration-of-measure in Bayesian optimal inference. 

• Derivation of the information-theoretic limits in the spike Wigner model by the replica method. 

• Introduction to the generalised approximate message-passing algorithm

The course will also include tutorials covering:

• Simulation of the spectral PCA estimator and comparison against the Bayesian estimator obtained by Monte Carlo methods.

• Step-by-step derivation of the information-theoretic limit in the spike Wigner model using the replica method.

During his stay, R. Abgrall will give a PhD course on "High-Order Numerical Methods for Hyperbolic Problems and Compressible Fluid Mechanics."

The course will focus on residual distribution schemes and active flux schemes.

R. Abgrall will emphasize the convergence results obtained so far, as well as the invariant domain properties of these methods.

Mariel Sáez is an Associate Professor at the mathematics department of Pontificia Universidad Católica de Chile, and will be visiting our department during the first semester of the academic year 2025-2026.

She is an expert in geometric analysis and, in particular, in geometric flows, minimal surfaces and fractional equations arising from geometry.

She will deliver a minicourse aimed to PhD students about “Eigenvalues of operators in conformal geometry”.

The study of spectral properties of differential operators has an extensive history and draws connections with several fields of mathematics. In the mini-course we will explore spectral properties of certain operators that are related to conformal geometry and their  connections with topology.

We will specially focus on three examples:

• The Laplace-Beltrami operator in two dimensions

• the Dirichlet-to-Neumann map associated with it

• the Paneitz operator.

We will finish the discussion with some open problems.

Our first objective is the construction of a double Hopf algebra of packed matrices, generalizing the construction of QSym from one to two dimensions.

In this setting, the first coproduct should correspond to the ordinal sum of alphabets, whereas the second one should correspond to the product of alphabets, with the lexicographic order. These presentations should also induce morphisms between packed matrices and QSym, in natural ways. The study of this object should follow: freeness and cofreeness, primitive elements, generators, exotic structures (for example, splitting of associativity or coassociativity, etc.

The second objective is the construction of a double Hopf algebra of double quasi-posets.

This should be related to the double Hopf algebra of “single” quasi-posets, with a first coproduct given by separation of the objects into a lower and a upper part, whereas the second coproduct would be given by a contraction-extraction process.

The third objective is to study the relation between these two constructions, through the notion of linear extensions,  allowing to construct a double bialgebra morphism from double quasi-posets to packed matrices, as from “single” quasi-posets to QSym.

Finally, we expect to apply these constructions to obtain results or interpretations in the theory of representations of symmetric groups. This should be based on the combinatorics of Young diagrams and tableaux and related quasi-orders on their columns and rows. An aim would be the construction of a functor of “doubling” objects, which could be applied on other objects such as graphs, hypergraphs, noncrossing partitions or others.

-  Themes of the short courses/seminars

Hopf algebras in combinatorics, Hopf algebras of matrices, chromatic and Tutte polynomial of a graph and of an hypergraph.

Andrea Thomann is Chargée de recherche de classe normal (CRCN) Tenured Associate researcher at Inria Nancy Grand-Est, Strasbourg University, France.

Her scientific interests are in scientific computing and the development of numerical methods for the simulation of muti-physics, multi-material and multi-phase problems. Currently she works on the development of  higher order methods and low-cost structure preserving all-speed schemes with the focus on thermodynamical compatibility and the preservation of asymptotic processes. 

She obtained her PhD in Computational Mathematics at Insubria University, Italy, thanks to a scholarship as Marie Sklodwska-Curie fellow from the Istituto Nazionale di Alta Matematica. Later she has been Post-doctoral researcher in Numerical Analysis JGU Mainz, Germany. She is the PI of AAPG2025-JCJC Funding for individual research projects carried out by young researchers aimed at creating their own working group Funding agency: Agence national de la recherche (ANR), French national research agency.

Keywords: Hyperbolic systems, multi-scale equations, IMEX methods, FV and DG schemes, HTC schemes, TVD schemes and MOOD procedures, well-balancing, relaxation methods, high order schemes, asymptotic preserving schemes, scientific computing, dissipative weak solutions.

Applications: Diffusive and sharp interface models, multi-phase flows, nonlinear elasticity, heat conducting fluids, ideal magnetohydrodynamics, Euler equations, gravity.

During his visit in January 2026, Prof. Pierpaolo Vivo will deliver a blackboard 20-hours course on "Foundation and modern applications of Random Matrix Theory".

This course is one of the activities organized during the Data Science thematic period ( link ) and will cover the following topics:

• Simple classification of random matrix models. Gaussian and Wishart ensembles

• Level spacing statistics: Poisson vs Wigner–Dyson

• Coulomb gas method

• Orthogonal polynomial technique and numerical checks

• Largest eigenvalue of a random matrix.  Comparison with Extreme Value Statistics for i.i.d. random variables.  Tracy–Widom distribution and third-order phase transitions

• The Replica method.  Edwards–Jones formalism.  Applications to full and sparse matrices (random graphs).  Large deviation function of the largest eigenvalue using replicas.  Replica theory of random linear systems with quadratic constraints and associated phase transitions

• Free probability.  Sum of free random matrices.  Blue’s function.  R and S transforms

• RMT for high-dimensional data analysis (matrix denoising, spiked models with structured noise, dictionary learning)

In addition, Prof. Vivo will deliver two research seminars on his recent activity that combines Statistical Mechanics methods with real-life applications to economics, biology, and legal systems.

Dr Vivo will also make himself available to lead short master classes and tutorials on specific topics of interest for the postgraduate cohorts in the department, including hands-on and computational sessions, student-led journal clubs and seminars, collective critical analysis of a recent research paper from a peer-review perspective.

Professor Albin will hold a mini-course on pseudodifferential operators on non-compact spaces. In contrast to the situation on compact manifolds, in this setting elliptic operators do not always have a finite dimensional space of solutions.

The course will explain the supplemental conditions needed in settings including spaces with cylindrical ends and asymptotically hyperbolic manifolds.

Analysis on the former is central to the Atiyah-Patodi-Singer index theorem while the latter plays a key rôle in modern conformal geometry and the mathematical approach to the AdS/CFT correspondence in string theory.

The proposed mini-course will provide graduate students and young researchers with very valuable tools. The pseudodifferential calculi that will be introduced are examples of a general philosophy initiated by Richard Melrose and pursued by many mathematicians in the last 30 years.

Having access to these ideas and these tools will be important for a young researcher.

During his visit in January 2026, Prof. Alessandro Ingrosso will deliver a course on "Statistical Mechanics of Neural Networks and Learning Systems"  within the tematic period dedicated to Data Science ( link ).

The course will consist of 16 hours dedicated to the exploration of fundamental and advanced topics in the Statistical Mechanics of Neural Networks and Learning Systems, according to the following syllabus:

• Equilibrium theory of learning in neural networks:

1. the Gardner approach to learning in simple perceptrons

2. learning with structured dataset

3. beyond perceptrons: neural networks with one hidden layer and beyond

4. kernel renormalization perspective on learning in neural networks

5. a Random Matrix perspective on learning in large-width neural networks

• Dynamics and learning:

  1. online learning dynamics in narrow networks

  2. learning in the neural-tangent-kernel (NTK) regime in large-width networks

  3. recurrent networks: dynamical mean-field theory and dimensionality

There will be theoretical lectures and interactive sessions.

During his stay at the DMC, prof. Gautam will give a course on "Yang-Baxter equation and representation theory".

The aim of his course is to introduce the Yang–Baxter equation, and how it arises as a sufficient condition for solvability of the lattice models of statistical mechanics. The related algebras of symmetries (called quantum groups) and their representation theory will be studied.

Furthermore, he will discuss various interesting systems of difference-differential equations that are naturally associated with these representations. Time permitting, he will explain how the monodromy of these difference–differential equations ties various representation categories together.

Professor Gianluca Pacienza is an internationally renowned researcher in algebraic geometry.

His work has made significant contributions to

• birational geometry

• the study of deformations of rational curves

• and hyperkähler varieties

employing a wide range of modern techniques.

Professor Pacienza will deliver a series of lectures on the birational geometry of hyperkähler varieties, aimed at both early-stage PhD students and researchers in the field.

During his visit, Albert FATHI will give a mini-course (8 hours) entitled Weak KAM Theory on metric spaces.

It is a "gentle" introduction to weak KAM theory and its connection with the theory of viscosity solutions from a purely metric point of view.

The course is aimed at PhD students, Master's degree students in Mathematics and Mathematics for Applications, and researchers interested in these topics.

Title (8 hours): Weak KAM Theory on metric spaces.

Abstract.

We introduce a framework for the ``viscosity theory of the Hamilton-Jacobi equation’’ on more general metric spaces avoiding any recourse to tangent spaces or PDE’s. This allows for more geometric analysis aspects.

It also applies to the space of probability measures on a compact space, allowing for example a better analysis of the action of the``induced action of a geodesic flow'' (or an Euler-Lagrange flow) on a candidate for tangent space to the space of probability measures of a Riemannian manifolds, bridging with optimal transport and the recent formidable advances on the geometric and analytic aspects of these probability spaces.