CAREER: Equivariant Analysis: At the Crossroads of Descriptive Set Theory, Dynamical Systems, and Classical Analysis

This project aims to further develop "equivariant analysis," a subject that may be described informally as "analysis without the choice of an origin." Many classical constructions in mathematics tacitly depend on an arbitrary starting point, such as a marked vertex, a basepoint, or an origin of coordinates. This dependence persists even when the end result looks symmetric, leaving no trace of the point from which it was built. Equivariant analysis makes precise the question of which classical analytic theorems survive when no such choice is permitted: it asks that a construction be carried out across an entire space by a rule that is applied uniformly and singles out no point as the origin. This places the subject at the crossroads of descriptive set theory, dynamical systems, and classical analysis, much as descriptive combinatorics connects classical combinatorics with mathematical logic and ergodic theory. The intrinsic value of this viewpoint lies in the rigorous theory it provides for "random" analytic objects, such as random entire functions and random solutions of partial differential equations. On the pedagogical side, the project is also valuable for the mathematical toolkit it builds in the students who carry it out. The techniques it relies on are drawn from ergodic theory, the dynamics of group actions, probabilistic reasoning, and geometric constructions such as Voronoi and Delaunay decompositions. These are also the tools used in modern machine learning and in the perception and control of autonomous systems. Ergodicity has been used as a direct design principle governing how autonomous robots explore and actively sense their surroundings; the mixing and ergodic-averaging theory of Markov chains is useful for the convergence analysis of stochastic gradient descent, which trains today's AI models; and Voronoi-type partitions govern multi-robot sensor coverage and motion planning. Work on this project will therefore leave students with concrete, transferable analytical skills that apply in machine learning and autonomous systems, among other areas. The project focuses on Borel equivariant maps from free Borel or probability measure-preserving actions of the complex plane or Euclidean space into spaces of analytic objects: entire and meromorphic functions, divisors and principal parts, Radon measures, subharmonic and caloric functions, etc. In recent work with M. Sodin and A. Wennman, we established equivariant analogues of the Weierstrass and Mittag-Leffler theorems and isolated a general lifting theorem from which these results follow. Broadly speaking, the central aim of the proposed work is to understand when an analytic equation admits equivariant solution maps; that is, when a fixed analytic or differential operation, applied to an unknown analytic object, can be made to reproduce prescribed data in a way that varies measurably and respects the symmetry of the underlying space. The project advances this line of work on several fronts. Existing lifting theorems work well when all orbits of the action share the same stabilizer; treating general actions will require extending the lifting theorem to actions of Borel fields of Polish groups. The existence of equivariant solution maps appears to be closely tied to Runge-style approximation theorems, and the project will investigate whether Borel actions of Euclidean space admit Borel toasts whose regions are Runge domains for a given class of functions. Finally, the viewpoint of equivariant analysis is closely connected to the study of invariant measures on spaces of analytic objects, which leads naturally to the notion of a random entire function, a random harmonic function, and so on. In some cases, such functions are known to have unusual growth rates, and the project will investigate whether these rates can serve as invariants of dynamical systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using t NSF Award ID: 2540790 | Program: 01003031DB NSF RESEARCH & RELATED ACTIVIT,01002930DB NSF RESEARCH & RELATED ACTIVIT,01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Kostyantyn Slutskyy | Institution: Iowa State University, AMES, IA | Award Amount: $239,160 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2540790 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2540790.html