Marie Curie grants to David Jekel and Harold Nieuwboer

The Marie Skłodowska-Curie Action’s Postdoctoral Fellowships aim to support researchers’ careers and foster excellence in research. The Postdoctoral Fellowships target PhD researchers who wish to conduct their research activities abroad, acquire new skills and develop their careers. The grants help researchers gain experience in other countries, disciplines and non-academic sectors.

The monetary value of each grant is a bit more than 1.8 million DKK.

David Jekel is American and obtained his PhD in 2020 from the University of California, Los Angeles, with Dimitri Shlyakhtenko. Then he had an NSF postdoc position at the University of California, San Diego, with Todd Kemp for three years. He finally spent one year as a postdoc at the Fields Institute and York University (Toronto, Canada) working with Ilijas Farah.

David has been working as a postdoc in the Department for Mathematical Sciences, UCPH, since 1 June 2024, where he joined the operator algebras group in the department’s section for Analysis and Quantum. His supervisor is Magdalena Musat.

Free Information Geometry

Davids’s project will develop an information geometry free probability theory. 

Information geometry refers to the synthesis of optimal transport together with measures of information such as entropy, which has shaped much recent work in partial differential equations, optimization, and data analysis. 

Free probability is a theory of non-commuting random variables that describes the large-n behaviour of many families of n x n random matrices. Free probability has had applications in data analysis, communication, finance, and many other topics where matrices appear.  It also has deep applications to the structure of von Neumann algebras, which serve as a non-commutative analogue of probability spaces, an old and challenging subject with connections to quantum mechanics, geometric group theory, ergodic theory, and more.

Information geometry has already motivated many corresponding results in free probability theory, but several deep questions remain open concerning the relationship between optimal transport and entropy in free probability, and whether it accurately describes the large-n limit of the classical information geometry for random matrices.

The project aims to

• show that free entropy is concave along optimal transport geodesics,
• establish the existence of momentum measures in free probability,
• give an optimal control formulation of free entropy, and
• exhibit counterexamples to regularity properties for optimal transport through connections with quantum information.

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Harold Nieuwboer received his PhD from the University of Amsterdam (The Netherlands) in January 2024, supervised by Michael Walter and Eric Opdam. His thesis was titled "Classical and quantum algorithms for scaling problems". Before his PhD, Harold studied pure mathematics at the Vrije Universiteit Amsterdam (The Netherlands ) and the University of Cambridge (United Kingdom).

Harold started on 1 January 2024 as a postdoc in our department’s Analysis & Quantum Section, associated with the Quantum for Life Centre. His supervisors are Matthias Christandl and Laura Mancinska from the Centre for the Mathematics of Quantum Theory.

AsympTensorPolytope

Harold calls his project “Asymptotic properties of moment polytopes of tensors”. He will study the behaviour of moment polytopes of families of tensors, as well as their ability to prove bounds on (asymptotic) tensor rank and sub-rank. He aims to determine properties of the moment polytopes of the unit tensors of varying ranks, as well as matrix multiplication tensors, in particular, whether they are distinct from the generic polytopes of their respective format.

To achieve this, Harold will use a combination of the various descriptions of moment polytopes, which come in representation-theoretic, symplectic-geometric, intersection-theoretic, or more combinatorial forms. In particular, he will study the behaviour of moment polytopes undertaking direct sums and Kronecker products of tensors, as well as recently obtained computational results.

The project has the potential to prove new bounds on the complexity of various tensors, as well as further our understanding of Strassen's asymptotic spectrum of tensors. The techniques here can also be extended to understand other settings, such as symmetric or antisymmetric tensors (bosonic or fermionic systems), algebras and quiver representations. As a result, the AsympTensorPolytope project will have an impact in other contexts such as the complexity of matrix multiplication, quantum information theory and combinatorics.