Date/Time : 04 Feb 2026 15:00
Venue : IMS Executive Seminar Room (S17 Level 3)
Speaker : Zhang Pingwen, Wuhan University and Peking University

Real numbers underpin mathematical research and the understanding of natural phenomena. The real number set comprises rational numbers (a measure-zero set) and irrational numbers (a full-measure set), implying that systems governed by irrational numbers are more prevalent. However, current computers cannot exactly store or represent irrational numbers, rendering numerical computations for irrational-dominated systems susceptible to Diophantine errors—incurred when approximating irrationals with rationals. Such errors significantly compromise numerical results. In this talk, we analyze the mechanism of Diophantine error and its impact on numerical computations. We also develop an approximation theory for multi-dimensional quasiperiodic functions with Diophantine frequencies when approximated by periodic functions. To avoid Diophantine errors, we propose two novel algorithms: the projection method and the finite points recovery method, which extend high-precision numerical computation from the rational to the real domain. Leveraging the ergodic and arithmetic properties of irrational numbers, we establish the mathematical foundations of these algorithms. Furthermore, we present their applications to several irrational-dominated scientific problems to discover new physical phenomena, including quasicrystals and their phase transitions, interfacial structures in materials, and quasiperiodic Schrödinger operators.

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Date/Time : 04 Feb 2026 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Gunter Schuetz, University of Lisbon

The symmetric Dyson exclusion process (SDEP) is a stochastic interacting particle system with exclusion and long-range, Coulomb-type interactions that emerge both as the maximal-activity limit of the symmetric simple exclusion process and as a discrete version of Dyson’s Brownian motion on the unitary group. Exploiting an exact ground-state (Doob) transform, we map the stochastic generator of the SDEP onto the spin-1/2 XX quantum chain, which in turn admits a free-fermion representation. This allows for proving an explicit formula for the time evolution of the expected microscopic particle density. At macroscopic scales we conjecture that the SDEP displays ballistic (Eulerian) scaling and non-local hydrodynamics where the current is a genuinely non-local functional of the density given by a Hilbert transform. This non-local one-field description is equivalent to a local complex Burgers equation for finite particle density. Closed evolution formulas allow us to solve the melting of single and double block initial states, producing limit shapes and arctic curves that agree with large-scale Monte Carlo simulations. The model thus offers a tractable example of emergent non-local hydrodynamics driven by long-range interactions.

(joint work with J. Dubail and A. Zahra)

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