NAA---Arithmetic & Quantum Systems

Date & Time: Monday, September 28, 2026 | 16:30 - 19:00
Place: tba
Chairs: Oleksandra Antoniouk, Angel Alfredo Moran Ledezma

Abstract 
The  mini symposium aims at bringing together researchers in non-archimedean analysis and applications, in order to present the state of the art of new developments in the mathematical theory of non-archimedean analysis, as well as applications in the Sciences, Engineering and Economics. Having initially diffusion processes over the p-adic number field, now non-linear dynamics (e.g. an analogue of Navier-Stokes or the porous medium equation), as well as other p-adic domains like p-adic analytic manifolds are now included. First applied and further developped in mathematical physics, the methods are now part of computer science, biology, earth sciences and other domains. Bringing together researchers on the theory side aims at fostering new insights and ideas towards a deeper understanding of general hierarchical structures within and outside mathematics. 

Confirmed Speakers
 

Speaker: Eric Whyman: Brandeis University
Title: The Green's function on the Tate curve and the local height
Abstract: The genus one -adic string worldsheet action gives rise to a dual boundary action on the Tate curve. We show that the two-point function of this dual action agrees with the Néron local height function on the Tate curve.

Speaker: Toka Diagana: (University of Alabama, Huntsville)
Title: Recurrent Structures and Nonlocal Dynamics over the \(p\)-adic 
Abstract: We develop a theory for studying recurrent phenomena over the \(p\)-adic field \( \mathbb{Q}_p\), with particular emphasis on Stepanov almost periodicity and its interaction with nonlocal evolution equations. The ultrametric geometry of \(\mathbb{Q}_p\), together with its compact-open subgroup structure and Haar measure, provides a natural setting in which recurrence can be formulated through uniform local \(L^q\)-averages rather than pointwise translation estimates. We establish fundamental structural properties of Stepanov almost periodic functions on \(\mathbb{Q}_p\), including a Bochner-type compactness characterization and approximation by \(p\)-adic trigonometric polynomials. A uniformly regular subclass is introduced that provides a bridge between Stepanov and Bohr almost periodicity, allowing averaged recurrence to be upgraded to uniform recurrence. These results lead naturally to stability principles for nonlinear compositions and convolutions. As an application, we consider semilinear equations driven by the Vladimirov fractional operator, \(D^\alpha u+\lambda u=f+F(u)\), and use the convolution structure of the resolvent, together with compactness and Schauder's fixed-point theorem, to obtain almost periodic mild solutions. The resulting theory connects ultrametric geometry, recurrent function spaces, harmonic analysis, and nonlocal dynamics, and provides a basis for the study of broader classes of recurrent nonlinear phenomena in non-archimedean settings. 

Speaker: Patrick Erik Bradley: KIT
Title: Extracting number-theoretic quantities via p-adic diffusion
Abstract:An overview on recent work of diffusion on p-adic analytic manifolds is given, and how spectral methods can be used to extract number theoretic information contained in p-adic analytic manifolds underlying algebraic curves.

Speaker: Zoran Rakic: University of Belgrade
Title: Path integrals on real, p-adic, and adelic spaces
Abstract: We study path integrals in ordinary, p-adic and adelic quantum mechanics for systems determined by wide class of Lagrangians. The corresponding probability amplitudes \({\cal K}(x^{''},t^{''};x',t')\) for two-dimensional systems with quadratic Lagrangians are found. The obtained expressions are generalized to any finite-dimensional spaces. These exact general formulas are presented in the form which is invariant under interchange of the number fields \({\mathbb R} \longleftrightarrow{\mathbb Q}_p\) and \({\mathbb Q}_p \longleftrightarrow {\mathbb Q}_{p'} \, ,\, p\neq p'\). This invariance shows the fundamental role of adelic path integral in mathematical physics of quantum phenomena.

Speaker: Vincenzo Parisi: (IIT Genoa)
Title: Entangled states in p-adic Hilbert spaces
Abstract: Building on our recent construction of the tensor product of p-adic Hilbert spaces [1], we investigate the entanglement of -adic quantum systems. We begin by discussing the projective and injective tensor products of two -adic Hilbert spaces H and K and we show that, in sharp contrast with the standard complex case, the two associated norms coincide; accordingly, we then characterize the most relevant classes of operators acting on the composite Hilbert space H \(\otimes \) K--- notably, the bounded, nuclear, and trace-class/Hilbert-Schmidt operators. We then turn to the set D(H) of density operators on a p-adic Hilbert space. After characterizing its \(\mathbb{Q_p}\)-convex structure, we describe its extreme-point set Ext (D (H )) by means of a non-Archimedean generalization of the Kreĭn-Milman theorem. Finally, moving to the bipartite setting, we characterize the separable and entangled states of D(H \(\otimes \) K) , for finite-dimensional Hilbert spaces H and K, and prove a p-adic analogue of Carathéodory's theorem, adapted to the non-Archimedean framework. Throughout, we highlight both the analogies with, and the nontrivial differences from, the standard complex theory.

References
[1] P. Aniello, L. Guglielmi, S. Mancini, and V. Parisi, “The tensor product of p-adic Hilbert spaces”, J. Math. Phys., 033503 (2026).

Speaker: Sarthak Parikh: IIT Delhi
Title: Holography on biregular trees
Abstract: Biregular trees arise as Bruhat-Tits buildings associated with the unitary group over p-adic numbers, corresponding to subcomplexes of the Bruhat–Tits building for PGL3 over the unramified quadratic extension of p-adic numbers. We introduce a discrete analog of AdS/CFT holography on such spaces, and highlight its novel features compared to the regular tree (PGL2) case.