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Mathematics - Japan

Research topics


Random Matrices and Applications

The main objective of my research project is to attack problems in Random Matrix Theory, and in particular, to initiate the study some aspects of this theory in the context of random tensors. This field is emerging and is expected to have ramifications in Quantum Information Theory, Operator Algebras and in particular in Free Probability Theory.

The reason for a renewed need of the nascent theory of random tensors is that these objects start to become important models in theoretical physics, quantum information theory and statistics (and big data, machine learning). On the other hand, this theory has remained in its infancy from the analysis point of view because beyond combinatorics, there was little or no mathematical technology available to tackle these kinds of questions.
In Random Matrix Theory, many very powerful mathematical tools are available: complex analysis, matrix integrals, orthogonal polynomials, etc, but these tools fail to work for higher tensors. The only tool that was used with some success to date was combinatorics.

All this is about to change. With collaborators, we initiated recently techniques that allow to combine new tools of algebra, analysis and combinatorics (matrix valued non backtracking theory) and that is expected to work for tensors. I plan to work in this direction while at the Collegium, and interact with the many experts of operator algebras, probability theory, random matrix theory and free probability theory in the Lyon area.

Activities / Resume


Benoît Collins is associate professor at Kyoto University’s mathematics department. He studied at Ecole Normale Supérieure de Paris and Sorbonne Université (Paris 6) from which he got a PhD in mathematics in 2003. Before arriving in Kyoto in 2014, he has held visiting positions in Japan, and permanent positions at CNRS (Lyon 1) and in Canada (Ottawa).

His primary field of research is probability theory and operator algebras. In the last ten years, he has devoted a substantial amount of his research energy to the field quantum information. He is associate editor of four international research journals and has more than 65 peer reviewed published papers. He has supervised dozens of graduate students and postdoctoral fellows, many of whom have become leading academic researchers. Finally, he contributed to the scientific community in many ways, such as being involved as organizer of world events in random matrix theory, quantum information theory and operator algebras.


Selected Publications – the following is a subjective selection of publications by importance, covering most of my research interests (one paper per research theme):

  • Random graphs, Operator Algebra, Combinatorics: Bordenave, Collins. Eigenvalues of random lifts and polynomials of random permutation matrices, Annals of Mathematics Pages 811-875 from Volume 190 (2019), Issue 3
  • Quantum Algebra, Operator Algebras: Brannan, Collins. Dual bases in Temperley-Lieb algebras, quan- tum groups, and a question of Jones, Quantum Topol. 9 (2018), no. 4, 715-748
  • Representation theory, Random Matrix Theory: Collins, Novak, Sniady. Semiclassical asymptotics of GLN (C) tensor products and quantum random matrices, Sel. Math. New Ser. 24 (2018), no. 3, 2571-2623
  • Quantum Information Theory, Probability theory, Free probability theory: Belinschi, Collins, Nechita. Eigenvectors and eigenvalues in a random subspace of a tensor product, Invent. Math. 190 (2012), no. 3, 647-697
  • Random Matrix Theory, Free probability theory: Collins, Male. The strong asymptotic freeness of Haar and deterministic matrices Ann. Sci. Ec. Norm. Supe ́r. (4) 47 (2014), no. 1, 147-163
  • Operator Theory, Intersection theory: Bercovici, Collins, Dykema, Li, Timotin. Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor, J. Funct. Anal. 258 (2010), no. 5, 1579-1627
  • Quantum groups, integrable systems: Banica, Collins, Zinn-Justin. Spectral analysis of the free orthog- onal matrix, Int. Math. Res. Not. IMRN 2009, no. 17, 3286-3309
  • Quantum groups, homology: Collins, H’artel, Thom. Homology of free quantum groups C. R. Math. Acad. Sci. Paris 347 (2009), no. 5-6, 271-276
  • Matrix Integrals, Mathematical physics, combinatorics: Collins, Guionnet, Maurel-Segala. Asymptotics of unitary and orthogonal matrix integrals, Adv. Math. 222 (2009), no. 1, 172-215
  • Random Matrix Theory, Weingarten calculus: Collins. Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability Int. Math. Res. Not. 2003, no. 17, 953-982