Uniform parameterization and regularity in real geometry

The problem of finding regular parameterizations for the solutions of polynomial equations with differentiable coefficients is ubiquitous in analysis and geometry. It is, for instance, central for the perturbation theory of linear operators and for the Cauchy problem of hyperbolic partial differential equations. The goal of the project is the application of the techniques, which we developed for the solution of this problem, to various unsolved problems in analysis and geometry. The focus is on the quest for quantitative bounds for the Hausdorff measure of the zero sets of smooth functions vanishing of finite order and for the volume of tubular neighborhoods thereof. Of particular in- terest in this context are the nodal sets of Laplace eigenfunctions. While these problems are well understood in the analytic category, for smooth functions new methods are required. An important tool of real geometry is the uniform parameterization of sets that are definable in an o-mimimal structure. Definable sets provide a general framework for real geometry and they are also studied in model theory. Since these sets incorporate a natural tameness and have many good topological, geometric, and metric properties, they often find applications in other fields, e.g. number theory. Recent years brought spectacular results on the number of rational points in definable sets. One essential ingredient in this context are geometric parameterizations of the sets with uniform control on the partial derivatives up to some finite order. As part of the project I plan to adapt our parameterization techniques to the definable setting and hence to refine the known methods of regular parameterization. Beside the general interest in good parameterizations of definable sets, we expect applications for the number of rational points and beyond. A fundamental result in smooth analysis states that functions on open domains are smooth if and only if their compositions with smooth curves in the domain are smooth. For functions on closed domains that is not necessarily true, it depends on the geometry of the domain. In recent work I showed that the result holds on closed subanalytic sets (with some natural topological assumptions). Subanalytic sets form an important family of definable sets. It is known that this useful characterization of smoothness is not true for general definable sets. But I expect that it holds for sets definable in polynomially bounded o-minimal structures. Analogous questions can be asked for real analyticity and ultradifferentiability. Furthermore, it is important to understand the natural topological and bornological properties of the associated function spaces. A difficult problem in singularity theory is to understand the geodesics (i.e. length minimizing curves) on singular spaces with respect to the inner geodesic distance. It is known that on suban- alytic sets the limits of secants of geodesics exist. But it is an open question whether the limits of tangents of geodesics exist, or in other words, whether the geodesics are continuously differentiable. There is a striking similarity with the regularity problem for geodesics in sub-Riemannian geometry. In that case the geodesics are either solutions of a differential equation and, consequently, their regularity is clear, or else the regularity is not fully understood. However, there are exciting recent developments in some particular cases.

The project focused on the regularity theory of solutions of polynomial equations that depend on parameters. Significant progress was made, yielding optimal and decisive results that provide a comprehensive understanding. We now have a clear grasp of the optimal regularity of these solutions under minimal conditions on the coefficients. Additionally, we proved that the map "coefficients-to-solutions" is bounded and continuous relative to natural topologies. By generalizing these techniques, we obtained lifting theorems for complex representations of finite groups. These theorems not only extend the results for polynomials but also enhance our understanding within a broader context. As an application, we established that the zero sets of differentiable functions, given appropriate control over their derivatives, exhibit remarkable properties similar to those of polynomials. Specifically, we obtained effective estimates of the size of these zero sets and good local parameterizations. Building on an influential result in analysis, which states that a function on an open set is smooth if it respects smooth curves, we identified a broad class of closed sets that admit an analogous theorem. This work uncovers a subtle relationship between the analytic properties of a function and the geometric characteristics of its domain. In particular, we established a precise link between the loss of derivatives and the sharpness of singularities on the boundary of the domain. Similar results were also obtained in the real analytic category. Whitney's extension problem has garnered significant attention since Fefferman's solution in the early 2000s. In our project, we focused on the geometric aspects of the problem and resolved a partial case of a related conjecture. Specifically, we characterized the partially defined functions in Euclidean space that are restrictions of globally defined functions of a certain regularity (C^1 with a given modulus of continuity for the first derivatives) and that are definable in an o-minimal expansion of the real field. This was accomplished by proving a definable Lipschitz selection theorem for specific set-valued maps, which also led to other interesting applications. In addition, we showed the uniformity and boundedness of the extension by establishing a uniform bounded version of the definable Whitney jet extension theorem. In another line of research, we studied extension problems within the context of ultradifferentiable functions, often arriving at optimal solutions. Ultradifferentiable functions form classes of indefinitely differentiable functions with growth constraints on the infinite sequence of derivatives, generalizing the Cauchy bounds for analytic functions. We also explored a broad range of problems in ultradifferentiable analysis, in great generality. This included studying nonlinear conditions for ultradifferentiability, investigating functional-analytic properties of spaces of ultradifferentiable functions, and applying these results to microlocal analysis.