Theory | Non-Archimedean Analysis and Applications

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: Christoph Kriegler: U Clermont
Title: Functional calculus and semilinear evolution equations for the Taibleson operator on non-Archimedean local fields
Abstract: For any non-Archimedean local field \(\mathbb{K}\) and any integer \( n \geq 1 \) , we show that the Taibleson operator admits a bounded \( H^{\infty} \) \( (\Sigma_{\theta} \)) functional calculus on the Bochner space \( L^p ( \mathbb{K}^n, Y ) \)for any  U M D Banach function space Y and any angle \( \theta \gt 0 \) , where \( \Sigma_{\theta} = {z~\in \mathbb{C}^*: \lvert (\mathrm{arg}~z) \rvert \lt \theta} \) and \( 1 \lt p \lt \infty \). Moreover, we prove that it even admits a bounded H\"ormander functional calculus of order \( \frac{3}{2} \). In our study, we explore harmonic analysis on locally compact Spector-Vilenkin groups and establish the R-boundedness of a family of convolution operators. Our results contribute to the theory of functional calculi for operators acting on vector-valued \( L^p \)-spaces over totally disconnected spaces. As an application, we obtain maximal regularity results and well-posedness for a class of evolution equations driven by the Taibleson operator.

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: Oleksandra Antoniouk: American University Kyiv
Title: Properties of the solutions to the linear and nonlinear equations in the non-Archimedean case
Abstract: This talk presents recent results on linear and nonlinear pseudo-differential equations over non-Archimedean fields. We discuss the well-posedness and qualitative properties of solutions to several classes of evolution and boundary value problems. In the linear setting, we consider the non-Archimedean Neumann problem and describe the existence, uniqueness, and regularity of weak and strong solutions. In the nonlinear setting, we focus on p-adic counterparts of the porous medium equation and more general multidimensional nonlinear pseudo-differential evolution equations. The methods combine functional-analytic techniques with the theory of non-Archimedean pseudo-differential operators, providing a unified framework for the analysis of linear and nonlinear models. Connections with diffusion processes on ultrametric spaces and directions for further research are also outlined.

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: Sarthak Parikh: IIT Delhi
Title: TBA
Abstract: TBA

Speaker: Mariia V. Serdiuk: Taras Shevchenko National University of Kyiv, Ukraine
Title: Radial Solutions of Pseudo-differential Equations on the Field of p-adic Numbers
Abstract:   The theory of p-adic mathematical physics concerned with the mappings from \(t\in\mathbb{Q}_p\) to \(t\in\mathbb{C}\) is well developed [Albeverio et.al.]. A typical linear operator is the Vladimirov-Taibleson operator \(D^{\alpha,n}\), which can be seen as a kind of an elliptic operator.

In a paper by A. N. Kochubei [Kochubei et.al.] a right inverse to the operator \(D^{\alpha},\;\alpha>0\) was found. It turns out that this permits to reduce the \(p\)-adic Cauchy problem for radial functions to an integral equation whose properties resemble those of classical Volterra equations.

Let \( \alpha \gt 0 \), \(\gamma \gt 0\). We consider the problem

\begin{equation}\label{equation1}\tag{1}\
|t|_p^{\gamma}(D^{\alpha}u)(|t|_p)=f(|t|_p,u(|t|_p)),\;0\neq t\in\mathbb{Q}_p,\;u(0)=u_0.
\end{equation}

We suppose that the function \(f:p^{\mathbb{Z}}\times\mathbb{R}\to\mathbb{R}\) satisfies the conditions

\begin{equation}\\
|f(|t|_p,x)|\leq M, \label{equation2}\tag{2}
\end{equation}

\begin{equation}\\
|f(|t|_p,x)-f(|t|_p,y)|\leq F|x-y|, \label{equation3}\tag{3}
\end{equation}

for all \(t\in\mathbb{Q}_p,\;x,y\in\mathbb{R}\) and some constants \(M \) ,\(F \) independent on \(t \), \(x \), \(y \).

With the problem \ref{equation1} we associate the integral equation

\begin{equation}\\
u(|t|_p)=u_0+I^{\alpha}\left[|\cdot|_p^{-\gamma}f(|\cdot|_p,u(|\cdot|_p))\right](|t|_p). \label{equation4}\tag{4}
\end{equation}

We call a solution \(u\) to the equations \ref{equation4}, if it exists, a mild solution to the Cauchy problem \ref{equation1}.

Theorem 1.
Suppose that \(\gamma<\min(1,\alpha)\) and the conditions \ref{equation2}, \ref{equation3} hold. Then the problem \ref{equation1} has a unique local mild solution, that is the integral equation \ref{equation4} has a solution \(u(|t|_p)\) defined for \(|t|_p\leq q^N\), where \(N\in\mathbb{Z}\) is sufficiently negative, and any other solution \(\overline{u}(|t|_p)\), if it exists, coincides with \(u\) for \(|t|_p\leq p^K\), where \(K\leq N\).