John Adamski, PhD

Math

"God made the integers, all the rest is the work of man." -- Leopold Kronecker

Research

My research interests lie in analysis, dynamical systems, and ergodic theory. My PhD dissertation contains three main parts. First, I studied the topological conjugations between pairs of dual circle endomorphisms that preserve Haar measure with a property called bounded geometry that can be obtained from each other through a process of "shuffling" the lengths of the intervals at each level of their respective Markov partitions. Then I wrote some code to generate random, arbitrary examples of such circle maps. The algorthm I used led naturally to the observation that a certain martingale determined by such circle maps converges almost everywhere. In the last part, I applied this observation to a "special class" of circle endomorphisms that preserve Haar measure with bounded geometry to show that if any two are conjugate by a symmetric homeomorphism, then the homeomorphism must be the indentity. In joint work with colleagues, we have since obtained the same result without the restriciton to a "special class". This work has applications to quasiconformal mappings. Now, my research is focused on applying this work to advancing the results of Furstenberg and Rudolph on measures simultaneously invariant under a semigroup of circle endomorphisms.

Education

Publications and Preprints

  1. "Symmetric rigidity for circle endomorphisms with bounded geometry". Submitted for publication (2021).
    https://arxiv.org/abs/2101.06870
  2. "Symmetric rigidity for circle endomorphisms with bounded geometry and their dual maps". PhD thesis, The Graduate Center. CUNY Academic Works (2020).
    https://academicworks.cuny.edu/gc_etds/3790

Slides for talks

  1. Symmetric rigidity for circle endomorphisms with bounded geometry and their dual maps
    The Graduate Center, CUNY (4/15/2020).