Adam Abrams

Employment

From 2020 to 2024 I was an instructor and researcher at the Wroclaw University of Science and Technology (Politechnika Wrocławska). From 2018 to 2020 I was a research postdoc at the Institute of Mathematics PAN. I recieved my Ph.D. in Mathematics from The Pennsylvania State University in 2018.

Research area

There is a classical connection between real “minus continued fractions” \[ a_0 - \dfrac1{a_1 - \dfrac1{a_2 - \dfrac1{\ddots}}} \]

a₀ -

1

a1 - 1 a2 - 1 ⋱

and geodesic flow on the “modular surface” $\mathrm{PSL}(2,\mathbb Z) \backslash {\mathcal H}^2$. This overlap of number theory, geometry, and dynamical systems has inspired much of my current research program, which essentially consists of three settings:

1. symbolic coding systems and their natural extensions,
2. entropies ($h_{\mu}$ and $h_{\rm top}$) of piecewise monotone circle maps,
3. preservation and destruction of normality.

My individual published work focuses on Gauss-like maps for complex continued fractions and on a discrete family of generalized Bowen–Series boundary maps. Jointly with Svetla Katok and Ilie Ugarcovici, I have used a variety of tools and techniques to obtain results on geodesic flow on quotients of ${\mathcal H}^2$ by Fuchsian groups. Jointly with Tomasz Downarowicz, I have analyzed subsequences of normal sequences (generic points for shifts) along deterministic sets of indices.

See my Research Statement (pdf) for details. You can also download my CV ąs a pdf.