Efren Ruiz
University of Hawaii at Manoa, HI·Mathematics & Statistics
Some mathematical systems repeat patterns at every scale, much like fractals. Others evolve step by step according to simple rules.
- Fields
Mathematics & Statistics
- Topics
operator algebras
C*-algebras
graph algebras
- Grants
- 1 known grant, through Aug 2028
What they work on
- Develop new invariants for subshifts of finite type, symbol-sequence dynamical systems, by studying the graph algebras attached to them and how these algebras interact.
- Characterize structure-preserving isomorphisms between operator algebras attached to self-similar groups, groups with fractal-like self-repeating patterns, to uncover new dynamical phenomena.
- Generalize shift equivalence and strong shift equivalence to self-similar groups using self-similar groupoids, to understand isomorphisms and Morita equivalences of their algebras.
More about the research
Mathematicians can attach algebraic objects called operator algebras to such systems, and the algebra can reveal hidden structure in the original system. This project studies how symmetry and complexity interact in these systems. It focuses on two kinds. Shifts of finite type are dynamical systems built from sequences of symbols with simple allowed transitions. Self-similar groups are groups that show self-repeating patterns. These systems appear in data encryption, statistical physics, neural biology and fractal geometry.
Why it matters
This is pure mathematics. It aims to deepen understanding of how symmetry and complexity interact in symbolic dynamical systems and self-repeating groups, using algebra as a lens. The systems studied appear in data encryption, statistical physics, neural biology and fractal geometry. The new invariants and equivalences could eventually inform work in those areas.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Denis Osin
Vanderbilt University, TN·Mathematics & Statistics
Study rigidity properties of acylindrically hyperbolic groups by addressing auxiliary questions and conjectures connecting analytic and geometric properties, motivated by open problems on quasi-isometric and measure equivalence rigidity.
Corey Jones
North Carolina State University, NC·Mathematics & Statistics
Classify approximately finite-dimensional actions of amenable tensor categories on approximately finite-dimensional C*-algebras using K-theoretic invariants, extending earlier results for fusion categories.
Srivatsav Kunnawalkam Elayavalli
University of Maryland, MD·Mathematics & Statistics
Apply Voiculescu's free entropy theory, a measure of size for noncommuting variables, to determine the structure of free products and ultrapowers of finite von Neumann algebras.
The school
University of Hawaii at Manoa: 23 labs on PI Pages, $376M a year on research, 182 research doctorates a year.
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