Final Report Summary - DIMENSION (High-Dimensional Phenomena and Convexity)
Consider a solid unit cube in three dimensions. Suppose that at each point of the cube, we measure two physical quantities such as temperature and pressure. Thus, with every point in the cube, we associate a measurement which consists of a pair of real numbers. We make the reasonable assumption that these measurements vary continuously with the point's position in the cube.
These measurements induce a partition of the cube into "fibers" in the following way: Two given points in our solid cube belong to the same fiber if and only if their two measurements are identical. Say, the temperature is the same at these two points, as well as the pressure. Typically, each fiber is a certain curve in our solid, three-dimensional cube. Could it be that all of these curves have a very small length? A theorem proven in this project states that this scenario is impossible and we are guaranteed that there will always exist a fiber whose length is at least one unit.
This theorem was conjectured earlier by Guth, and our proof relies on deep ideas by Almgren and Gromov. Let us discuss an example. Consider the specific case where the solid unit cube is parallel to the x,y,z axes in the three-dimensional space. Assume that the measurement at a point (x,y,z) in the unit cube is the pair of real numbers that are the first two coordinates of the point, the numbers x and y. In this specific case, each fiber corresponds to a line-segment of length one that is parallel to the z-axis. This picture is consistent with our result, which states that even for far more complicated measurements, at least one fiber should have a length not smaller than one.
These measurements induce a partition of the cube into "fibers" in the following way: Two given points in our solid cube belong to the same fiber if and only if their two measurements are identical. Say, the temperature is the same at these two points, as well as the pressure. Typically, each fiber is a certain curve in our solid, three-dimensional cube. Could it be that all of these curves have a very small length? A theorem proven in this project states that this scenario is impossible and we are guaranteed that there will always exist a fiber whose length is at least one unit.
This theorem was conjectured earlier by Guth, and our proof relies on deep ideas by Almgren and Gromov. Let us discuss an example. Consider the specific case where the solid unit cube is parallel to the x,y,z axes in the three-dimensional space. Assume that the measurement at a point (x,y,z) in the unit cube is the pair of real numbers that are the first two coordinates of the point, the numbers x and y. In this specific case, each fiber corresponds to a line-segment of length one that is parallel to the z-axis. This picture is consistent with our result, which states that even for far more complicated measurements, at least one fiber should have a length not smaller than one.