# Smooth Functions on Rough Spaces and Fractals with Connections to Curvature Functional Inequalities (22w5080)

## Organizers

(University of Connecticut)

(Texas A&M)

(Waseda University)

(University of Cincinnati)

## Description

The Banff International Research Station will host the "Smooth Functions on Rough Spaces and Fractals with Connections to Curvature Functional Inequalities" workshop in Banff from November 20 - 25, 2022.

Many physical problems, like describing the flow of heat or the oscillation of waves, depend on the geometry of the underlying space: for example waves are different on a sphere than on a plane or a surface shaped like a saddle because of the way they are curved. Mathematically, some of these problems can be described using analysis (via differential equations) and probability (for example, heat flow can be thought of as a macroscopic effect due to suitably averaged random behavior in many small particles). One can then think of the geometry of a space as constraining the behavior of these analytically or probabilistically defined mathematical objects, or conversely that understanding the behavior of the latter objects reveals something about the underlying geometry. Over many years, sophisticated mathematical tools have been developed to understand and describe this interplay of analysis, geometry and probability in smooth settings, such as smooth curved surfaces.

However, many physical settings do not look like smooth surfaces: lungs are not spherical, the power grid is not linear, and the boundary of a turbulent flow does not look like a surface. Some of these spaces are rough and others have some (statistical) self-similarity or fractal structure. In the past few decades mathematicians have built analytic and probabilistic tools for studying physical problems on some spaces like these, but the nature and role of geometry in these contexts is not yet clear. Studying this and related questions is primarily about understanding the properties of functions that should be thought of as smooth'' in these settings. The purpose of this workshop is to bring together experts and beginning researchers, whose work involves smooth functions in rough and fractal spaces, for the purpose of sharing perspectives and techniques and with the goal of developing some frameworks for addressing significant problems in this area.

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).