为促进学科交流融合、拓宽师生学术视野、释放科研创新活力，助力人大数学学科走向一流，中国人民大学设立“人大数学时间”，以专题研讨、高端学术论坛为载体，搭建数学思想充分碰撞、优秀人才不断涌流、创造活力竞相迸发的舞台。“人大数学时间”将持之以恒，久久为功，立志通过交流与创新、提出重大问题，引领数学学科及相关领域的创新与发展，成为对我国数学发展有贡献意义的平台。以下为“人大数学时间I”第五十七期信息：

议程：

9月7日（星期一）14:00

Luca Barbieri Viale教授报告及前沿问题探讨

地点：立德楼902

题目：The Infinitesimal Structure of Quantum Information

主讲专家：Luca Barbieri Viale，米兰大学数学系教授

摘要：

In this talk, we explain a foundational mathematical bridge between quantum information theory and higher-order nilpotent algebras. By leveraging the gauge-invariant density operator formalism, we show that these real algebras—canonically associated with Grothendieck’s non-reduced infinitesimal spectra—provide a complete, non-singular, and rigidly isometric symbolic container for the true physical degrees of freedom of a quantum state space, demonstrating that every quantum state can be faithfully regarded as a non-reduced multiple point. We show that the Liouville-von Neumann dynamics map to flat algebraic flows, proving that the non-commutative vector flow is induced by a density embedding. As an instance of Bohr’s correspondence principle, when the dimension of the quantum system increases, this family of finite-dimensional configuration spaces converges to a Cauchy-complete power series ring Quinfinity. Within this topological envelope, the dual Archimedean Fréchet and non-Archimedean completions systematically absorb the shifting affine background of the trace, linearizing the macroscopic phase space and aligning the Fubini-Study geometry identically with the classical Fisher-Rao manifold.

报告人简介：

Prof. Luca Barbieri Viale is Full Professor of Algebra at the University of Milan. He earned his Ph.D. from the University of Milan & Genoa (1990) under C. Pedrini, with a research stay in Paris attending courses by J.-P. Serre and P. Cartier. His research focuses on motives, algebraic cycles, and K-theory, with publications in the Annals of Mathematics and Advances in Mathematics. He has held visiting positions at IHÉS, TIFR Mumbai, and Rutgers. He served as Director of the Department of Mathematics "F. Enriques" (2017–2020) and the Doctoral School in Mathematics at Milan. His research articles have appeared in the most prestigious journals in mathematics, including the Annals of Mathematics, Advances in Mathematics, Mathematische Annalen, Crelle's Journal, and the Duke Mathematical Journal. He has collaborated with prominent mathematicians such as Joseph Ayoub, B. Kahn, V. Srinivas, A. Huber, L. Lafforgue, and Fields Medalist Laurent Lafforgue.