Algebraically integrable domains (22rit259)


(University of Alberta)

Mark Agranovsky (Bar-Ilan University)

Alexander Koldobsky (University of Missouri)

(Kent State University)


The Banff International Research Station will host the "Algebraically Integrable Domains" workshop in Banff from July 24, 2022 to July 31, 2022.

The main theme of the proposed research activities concerns
algebraically integrable domains. The first study of such domains goes
back to Newton and is connected to Kepler's laws of planetary motion.
Let $D$ be an infinitely smooth bounded domain in $\mathbb R^n$. For a non-zero
vector $u$ in $\mathbb R^n$ and a real number $t$, consider the cut-off volumes of
$D$, i.e., the volumes of parts of $D$ on both sides of the hyperplane
perpendicular to $u$ at distance $t$ from the origin. In Lemma XXVIII of
his Principia, Newton proved that there are no domains $D$ in
$\mathbb R^2$ for which the cut-off function is algebraic (such domains are called
algebraically integrable).
Three centuries later, Arnold asked for extensions of Newton's result to other
dimensions and non-convex domains; see problems 1987-14, 1988-13, and
1990-27 in his book ``Arnold's problems". Arnold's problem in even dimensions
was solved by Vassiliev, who showed that there are no
algebraically integrable bounded domains with infinitely smooth
boundaries in $\mathbb R^{2n}$. It is still an open problem whether in odd
dimensions the only algebraically integrable smooth domains are

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).