Looking Forward to illuminating the multidimensional geometry of a phase transition

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Prof. Gady Kozma

Prof. Gady Kozma: "We aim to understand how geometry affects phase transitions. It's important to me to study possibilities in three-dimensional space that can be applied to the real world, and not just in high-dimensional space."


Changing Paths

What are the chances a stumbling drunk will find his way home? The drunk has only a two-dimensional field to navigate. In contrast, a disoriented bird must search for its nest in a three-dimensional framework and is therefore much less likely to find its way home. Such a problem – after proper mathematical formulation – involves a mixture of geometry and probability. These are the kinds of problems studied by Prof. Gady Kozma, who recently joined the Institute's Mathematics Department.

Kozma evaluates the paths between two points in imaginary worlds that have various dimensions, and he checks to see how these paths depend on the number of dimensions. Such worlds resemble a city undergoing constant roadwork in which various routes are continually being closed and reopened. He erases, masks and restores different segments of the landscape, testing the significance of these changes for objects navigating within the space (a process called percolation).

Of particular interest are paths in the critical phase – the point at which a system changes its qualitative behavior. For such paths between two points in a multidimensional world there's a clear boundary: In the critically reduced system, if the underlying system contains six dimensions or less, one can get from one point to the other via more than one path. Above six dimensions, however, only one such path exists. Studies in many dimensions have applications in the real, three-dimensional world: They help scientists better understand the processes that take place in phase transitions between different physical states.

Prof. Gady Kozma's research is supported by the Sieff Prize.