MathPolo 2025
A Mathematical Bridge Between Nanjing and Rome
Rome, October 2-3, 2025
Following in the footsteps of Marco Polo, this workshop seeks to foster mathematical collaboration and exchange between China and Italy. It will showcase research from the Mathematics Departments of Nanjing University (China) and University of Rome Tor Vergata (Italy).
Singularities in Hamilton-Jacobi equations: A story of mathematical collaboration between Nanjing and Tor Vergata
Piermarco Cannarsa (University of Rome Tor Vergata)
Hamilton-Jacobi equations are nonlinear partial differential equations with profound links to optimization and control theory. Their solutions are typically not differentiable, and the formation and propagation of gradient singularities (or shocks) have been the subject of extensive research over the past few decades. A significant body of this work has emerged from a fruitful collaboration between researchers in Nanjing and Tor Vergata, as well as from other Chinese and Italian institutions. This talk will highlight some of the main achievements of this joint effort, particularly regarding the global behavior of singularities and their connections with dynamical systems and topology. I will also outline some prospective directions for future collaborative research.
How sampling distribution affects gradient-free optimization
Liyuan Cao (Nanjing University)
Gradient-free optimization is often used to optimize functions where only function values are accessible. It primarily refers to gradient descent methods where the ‘gradients’ are directional derivatives estimated via finite differences using randomly sampled points. The sampling distribution is often chosen as Gaussian, uniform on sphere, Haar, or uniform along the coordinate directions. This choice has been shown to affect the algorithms' numerical efficiency. It is presented in this talk a theoretical analysis of the algorithms' performance under different sample distribution. The result provides a guidance on how to choose the distribution.
The strong Green function rigidity conjecture
Xuezhang Chen (Nanjing University)
Using the spectrum theory of self-adjoint elliptic operators, we derive the explicit formula of Green function for GJMS operator (including fractional one) through asymptotic solutions. We emphasize the extrinsic geometric interpretation of the Green function and raise the Strong Green Function Rigidity Conjecture (SGFRC):
Let Mn ⊂ ℝn+1 be a closed embedded hypersurface with induced metric g. Fix Q ∈ M, the Green function G(·,Q) for GJMS operator is exactly the one for the GJMS operator on (𝕊n,g𝕊n). Then M is a sphere.
We manage to answer SGFRC affirmatively for the conformal Laplacian and Paneitz operator. Our strategy is related to the Positive Mass Theorem and the classification of complete flat Euclidean hypersurfaces due to Hartman-Nirenberg. The above conjecture has a close connection with the mass of Euclidean hypersurfaces under inversion. This is based on joint work with Yalong Shi, and a project in progress with Jiaxue Gan and Yalong Shi.
Special metrics in Kähler geometry
Eleonora Di Nezza (University of Rome Tor Vergata)
Over the past two decades, a powerful variational framework has emerged for studying special metrics on Kähler manifolds, including Kähler-Einstein and constant scalar curvature Kähler metrics. In this talk, I will introduce the central ideas of this approach, highlight the key players in the theory, and discuss some of the most recent advances in the field.
Some steps toward general nonabelian Hodge correspondence
Pengfei Huang (Nanjing University)
The celebrated nonabelian Hodge theory, attributed to the fundamental contributions of Donaldson, Corlette, Hitchin, and Simpson, explores the correspondence between Higgs bundles and local systems through the intermediary objects known as integrable connections. However, such correspondence is not well-understood for general varieties as the base or for general groups as the structure group. In this talk, we will start the story from a quick review of the classical results. Then we will talk about meromorphic G-connections with regular and irregular singularities, and show the associated nonabelian Hodge correspondences. Finally we will demonstrate the construction of moduli spaces of filtered (Stokes) G-local systems. Based on some joint work in collaboration with G. Kydonakis, H. Sun, L. Zhao, and with H. Sun.
Equivariant localisation in representation theory
Martina Lanini (University of Rome Tor Vergata)
Equivariant localisation is a technique that turns global geometric questions into local computations around fixed points of a group action. If the group is a torus, a particularly elegant form of this principle is the Goresky–Kottwitz–MacPherson (GKM) theory, which describes equivariant cohomology through simple combinatorial data encoded in a labelled graph. In this talk we will introduce the GKM approach and show how it provides useful insights into spaces that play a central role in representation theory, such as flag varieties. The goal is to give a flavour of how localisation methods create unexpected links between geometry, combinatorics, and algebra.
Heat equation from a deterministic dynamics
Carlangelo Liverani (University of Rome Tor Vergata)
I'll describe a derivation of the heat equation in the thermodynamics limit, with a diffusive scaling, from purely deterministic dynamics satisfying Newton's equations under an external, time-dependent, external field. The talk is based on work in collaboration with G. Canestrari and S. Olla.
Brownian motions and Dirichlet forms on self-similar fractals
Hua Qiu (Nanjing University)
The theory of Laplacians on fractals is closely related to Brownian motion (from a probabilistic perspective) and Dirichlet form (from an analytical perspective) theories on fractals. In this talk, I will introduce our recent progress on the analytic construction of Dirichlet forms on unconstrained Sierpinski carpets, which are natural extension of Sierpinski carpets allowing cells to live off grids. This is based on recent joint works with Shiping Cao.
On the power of smoothness in isogeometric analysis
Hendrik Speleers (University of Rome Tor Vergata)
Isogeometric analysis is a well-established paradigm to improve interoperability between geometric modeling and numerical simulation. It is based on smooth spline representations and shows important advantages over classical C0 finite element analysis. In particular, the higher smoothness enables a higher accuracy per degree of freedom. This superior performance has been observed numerically since long time, and is recently also theoretically supported by error estimates with constants that are explicit, not only in the mesh size, but also in the polynomial degree and the smoothness. Moreover, the isogeometric approach based on maximally smooth spline spaces over uniform grids turns out to be an excellent choice for addressing eigenvalue problems: it gives a very good approximation of the full spectrum, except for a very small portion of spurious outliers.
In this talk we review some recent results on explicit error estimates for approximation with highly smooth splines. These estimates are sharp or close to sharp in several interesting cases and are actually good enough to cover convergence to eigenfunctions of classical differential operators under so-called k-refinement. We also discuss how subspaces of maximally smooth splines, which are optimal in the sense of Kolmogorov n-widths, can be selected to make outlier-free isogeometric discretizations. The talk is based on joint work with Carla Manni and Espen Sande.
Understanding neural networks with reproducing kernel Banach spaces
Stefano Vigogna (University of Rome Tor Vergata)
Understanding the theoretical properties of neural networks is key to explaining their practical success. Functional analysis, in particular, sheds light on their inductive bias by identifying which classes of functions they can efficiently represent and learn. I propose reproducing kernel Banach spaces as a natural functional framework for studying neural networks. In this setting, networks can be expressed through Banach space dualities, and their complexity characterized by sparsity-inducing norms and metrics, such as total variation or the Wasserstein distance. I will present representer theorems that establish the finite-architecture of empirical networks, as well as a result of statistical generalization for Banach space feature learning.
k-Yamabe problem and its related Sobolev inequalities
Wei Wei (Nanjing University)
In this talk, we will present some recent progress on the σk Sobolev inequalities in a more general cone, and the corresponding existence of σk Yamabe problem. And new Yamabe problem related to Q curvature and scalar curvature are proposed. These are joint works with Prof. Y. X. Ge and Prof. G. F. Wang.
The onset of instability for zonal stratospheric flows
Hao Zhu (Nanjing University)
In this talk, we discuss some qualitative aspects of the dynamics of the Euler equation on a rotating sphere that are relevant or stratospheric flows. Zonal flow dominates the dynamics of the stratosphere and for most known planetary stratospheres the observed flow pattern is a small perturbation of an n-jet. Since the 1-jet and the 2-jet are stable, the main interest is the onset of instability for the 3-jet. We prove that the 3-jet is linearly unstable if and only if the rotation rate belongs to a critical interval. Turning to the nonlinear problem, we prove that linear instability implies nonlinear instability and that, as the rotation rate goes to infinity, nearby traveling waves change gradually from a cat's eyes streamline pattern to a profile with no stagnation points. This talk is based on a joint work with Profs. Adrian Constantin, Pierre Germain and Zhiwu Lin.