Research:

My research has primarily been in complex dynamics and, more recently, in analytic number theory.

More specifically, my research in complex dynamics has focused on local discrete holomorphic dynamics in several complex variables. The latter is the study of iterates of germs of holomorphic self-maps of Cn at a fixed point. I have focused on the interesting case when the germ f fixes the origin and is tangent to the identity, so it can be locally expressed as:

f(z)=z+(higher degree terms).
In dimension one, the Leau-Fatou flower theorem is a beautiful result describing the dynamics of f near the origin — in particular, it describes domains that are attracted and repelled to the origin by f and points inside the domains are attracted and repelled along particular directions — and is an inspiration for work in higher dimensions.

My research has focused on generalizing the Leau-Fatou flower theorem to higher dimensions. In particular, determining when there is a domain whose points are attracted to the fixed point. If one exists, we ask other questions such as: Are points inside the domain of attraction converging to the fixed point tangentially to a particular direction? Can we holomorphically conjugate f on the domain of attraction to a simpler map (like translation)? Is the domain of attraction a Fatou-Bieberbach domain?

Publications:

Universal systole bounds for arithmetic locally symmetric spaces. (joint with J. Meyer and B. Linowitz). Proceedings of the American Mathematical Society (accepted May 2021). Proceeding of the AMS version forthcoming and arXiv version.

Systole inequalities up congruence towers for arithmetic locally symmetric spaces (joint with J. Meyer and B. Linowitz). Communications in Analysis and Geometry (accepted Feb 2021). CAG version forthcoming and arXiv version.

Interesting examples ℂ2 in of maps tangent to the identity without domains of attraction. Journal of Fractal Geometry (2019). JFG version and arXiv version.

Attracting domains of maps tangent to the identity in two complex variables with characteristic direction of multiple degrees. Journal of Geometric Analysis (2015). JGA version and arXiv version.

Attracting domains of maps tangent to the identity whose only characteristic direction is non-degenerate. Int. J. Math., 24 (2013). IJM version and arXiv version.

On the existence of attracting domains for maps tangent to the identity. Ph.D. Thesis, University of Michigan (2013). Available here.

Research Group at UCR:

I maintain the website for the Fractals, Dynamics, & Mathematical Physics Group at UCR. This website also contains a list of conferences in these research areas. If you would like to add a conference to the list, email me at slapan "at" ucr "dot" edu.

Research-Related Videos:

Here is a talk I gave at the Interactions between continuous and discrete holomorphic dynamical systems in Banff, 2012 on Attracting domains of certain maps tangent to the identity:

Here is a video I made for the Dance Your Ph.D. competition in 2012 to explain (some of) my thesis: