Hugo Woerdeman
Drexel University, PA·Mathematics & Statistics
The order of operations usually matters: you wash clothes before drying them, not after. In mathematics, matrices capture this kind of order dependence, called noncommutativity.
- Fields
Mathematics & Statistics
Physics
- Topics
free analysis
noncommutative rational functions
noncommutative polynomials
- Grants
- 1 known grant, through May 2027
What they work on
- Resolve open problems on when noncommutative rational expressions in bounded operator variables become singular or vanish, combining complex analysis, representation theory and operator theory.
- Use control-theoretic tools to detect geometrically and structurally when a noncommutative rational function is a composition of simpler ones, and study tensor-rational functions in computational complexity.
- Determine when noncommutative polynomials can take low-rank values, as a step toward noncommutative algebraic geometry and approximate zero sets.
- Develop positivity and optimization methods for noncommutative polynomials in several operator variables without dimension limits, and apply them to nonlinear Bell inequalities and self-testing in quantum information.
More about the research
Many modern problems ask questions about whole families of matrices whose answers should not depend on matrix size. Such problems arise in quantum technologies, complex control systems and optimization. Free analysis, where "free" means size-free, studies functions whose inputs are matrices or operators. This project focuses on the simplest such functions, noncommutative polynomials and rational functions. They are built using only variables and arithmetic, yet many of their basic properties remain unknown.
Why it matters
This is fundamental mathematics. It aims to deepen understanding of functions of noncommuting matrix variables, a common language for quantum, control and optimization problems. New optimization methods in operator variables could eventually inform how quantum networks are analyzed and how quantum devices are certified for secure communication.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Tao Mei
Texas A&M University-College Station, TX·Mathematics & Statistics
Determine the Lp-approximation properties of group von Neumann algebras, which are operator algebras built from discrete groups, that is, how well these infinite-dimensional objects can be approximated by simpler pieces.
Cesar Cuenca
Ohio State University, OH·Mathematics & Statistics
Describe the large-scale behavior of continuous and discrete beta-ensembles (Boltzmann distributions generalizing random matrix eigenvalues and random partitions) at different temperature limits, using new versions of free probability.
Dimitri Shlyakhtenko
University of California-Los Angeles, CA·Mathematics & Statistics
Develop methods based on partial differential equations (equations describing change) for non-commutative settings, extending tools from classical analysis to von Neumann algebras using free probability.
The school
Drexel University: 11 labs on PI Pages, $170M a year on research, 282 research doctorates a year.
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