NSF-BSF: Metric space-valued hard analysis

Throughout mathematics, science, and engineering, one needs to understand, manipulate and use, a rich and expressive zoo of geometric objects. It has long been clear that one can characterize important properties of objects by studying mappings between them. For example, suppose that one is given a collection of points on a high dimensional curved surface (such surfaces could be expressing many phenomena of interest, such as symmetries in mathematics, a range of phenomena and theories in physics or biology, or extrapolations from massive datasets that model statistical predictions about the rest of the respective “universes”), or vertices within huge and intricate networks that represent pairwise interactions and connections between members of some community (which can consist of, say, people, websites, proteins). Computing efficient estimates of numerical characteristics of collections of points (for example, the average of their distances) is a common challenge. It turns out that this could be done very efficiently by imposing an external geometric/combinatorial structure (called an expander graph with respect to the given ambient geometry) on n vertices, thinking of the given n points as a mapping from that geometric structure into the ambient geometry, and proving an analytic inequality called a metric Poincaré inequality for any such mapping. When the target space has a linear structure, one classically (and very powerfully) proves such estimates by decomposing a function into a sum of frequencies and studying how those can interact. However, for more general nonlinear/curved geometries, concepts like “sum” and “frequency” lose meaning, and the above classical strategy becomes (seemingly) nonsensical. In this project, the investigators will devise creative ways to make sense of such issues, namely, some of those that for centuries were studied in the linear context but seemed irrelevant when functions take values in more unwieldy, but ubiquitously useful, geometries. Those will be used to tackle longstanding open questions. For example, a novel notion of “metric frequency decomposition” will be developed and fundamental questions about it studied, suitable answers to which will solve an old question about the aforementioned “average distance estimation/geometric expander” issue whenever the target geometry is non-positively curved (namely, all triangles are indented inwards; this covers a well-studied, rich and useful class for which this question has remained a mystery for many years). Multiple concrete open questions of the above nature (how to make sense of, and use, analysis of mappings that take values in nonlinear geometries), will be addressed by introducing and developing new concepts and methodologies, and devising their applications to both mathematics and computer algorithms. Can one meaningfully define a useful version of (Fourier) “spectrum” for functions that take their values in certain metric spaces of interest (as an especially important example, those that are curved non-positively)? This endeavor may initially seem nonsensical, as classically such decompositions into frequencies are inherently a superposition of linear operations (summations and multiplications by scalars). Thus, one must come up with an innovative definition to merely take the first step here; such a definition will be introduced as part of the research on this project. After formulating such a definition, can one prove quantitative control on how much the size of a metric space-valued function can increase after removing some of its Fourier spectrum? This project will study such an estimate, which implies that there are arbitrarily large graphs of bounded degree that are simultaneously expanders with respect to every metric space of nonpositive curvature; the question whether such graphs exist is a longstanding open question. Other aspects of the project include proving a range of analytic inequalities that can serve as invariants that are obstructions to existence of well-behaved embeddings into “nicer” spaces. Also, longstanding questions about the multi-scale structure of metric spaces (e.g., availability of an optimal “skeleton” that has a useful multi-scale hierarchical structure and suitably “mimics” the original space) will be investigated as part of the project. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.