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 the University of Nanjing (China) and the University of Rome Tor Vergata (Italy).

 

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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.

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How sampling distribution affects gradient-free optimization

Liyuan Cao (University of Nanjing)

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.

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The strong Green function rigidity conjecture

Xuezhang Chen (University of Nanjing)

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,g_𝕊ⁿ). 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.

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Eleonora Di Nezza (University of Rome Tor Vergata)

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Some steps toward general nonabelian Hodge correspondence

Pengfei Huang (University of Nanjing)

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.

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Martina Lanini (University of Rome Tor Vergata)

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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.

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Hua Qiu (University of Nanjing)

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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.

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Stefano Vigogna (University of Rome Tor Vergata)

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Wei Wei (University of Nanjing)

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Hao Zhu (University of Nanjing)

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