Dimitri Shlyakhtenko
University of California-Los Angeles, CA·Mathematics & Statistics
In quantum mechanics, two properties of a system often cannot be measured at the same time. Mathematically, this shows up as non-commutativity: the quantities you can observe (called observables) form an algebra where the order of multiplication matters.
- Fields
Mathematics & Statistics
Physics
- Topics
von Neumann algebras
free probability
free entropy
- Grants
- 2 known grants, through Nov 2030
What they work on
- 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.
- Strengthen the connection between free probability, a probability theory for non-commuting variables, and random matrix theory, the study of large matrices with random entries.
- Define and study a notion of dimension for von Neumann algebras based on how optimal transportation distances behave. These distances measure the cost of moving one distribution into another.
- Apply free information theory techniques, including free entropy, to settle structural questions about von Neumann algebras, the algebras that describe quantum observables.
More about the research
Von Neumann algebras, introduced in the 1930s, are the mathematical framework for these quantum observables. A central question is how much of ordinary calculus, probability and analysis carries over to this non-commutative world. The research treats quantum observables as random variables and borrows ideas from information theory to study the structure of these algebras.
Why it matters
This is basic mathematics. It deepens the understanding of non-commutative algebras that model quantum systems. It also links free probability to random matrix theory, a field with many applications. Through IPAM, the PI also supports collaborations that apply mathematics to areas such as medical imaging, materials science, cryptography and artificial intelligence.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
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.
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.
The school
University of California-Los Angeles: 84 labs on PI Pages, $1.9B a year on research, 803 research doctorates a year.
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