Keywords:
 functional analysis and operator algebras (C*algebras, von Neumann algebras)

quantum symmetries (compact quantum groups, compact matrix quantum groups, BanicaSpeicher quantum groups/ easy quantum groups, classification of quantum subgroups, quantum automorphism groups of finite graphs)
 free probability theory (de Finetti theorems, noncommutative distributions, bifree probability)
 combinatorics (set partitions, noncrossing partitions, finite graphs)
My research is in functional analysis, with links to algebra, combinatorics and (free) probability theory. One of the main motivations is to study structures in mathematics when noncommutativity of the multiplication plays a role, for instance when dealing with matrices in linear algebra, operators on a Hilbert space in functional analysis, or with observables in quantum physics.
In such noncommutative theories, symmetries are no longer given only by groups, but by more general objects: by quantum groups. In 1987, Woronowicz gave an analytic definition based on C*algebras. In the 1990's, Wang introduced free quantum versions of the orthogonal group O_{n}, the unitary group U_{n} (both over the complex numbers) and the symmetric group S_{n}. His work was extended and put into a systematic framework  the one of BanicaSpeicher quantum groups (also called easy quantum groups)  by Banica and Speicher in 2009. Their approach is based on the combinatorics of set partitions. These partitions also appear in Voiculescu's free probability theory. There are deep links between BanicaSpeicher quantum groups and free probability theory for instance via de Finetti theorems.
Amongst other things, a main focus of my past research has been on the classification of BanicaSpeicher quantum groups. This has been completed in the orthogonal case (partially joint work with Raum) and has been begun in the unitary case (joint work with Tarrago). Furthermore, I investigated the combinatorial nature of the representation theory of BanicaSpeicher quantum groups and proved a de Finetti theorem for bifreeness (both in joint work with Freslon). New examples of partially quantized quantum groups (similar to the graph product of groups) have been given by Speicher and myself, building on partially quantized real spheres. This can be seen as the quantum isometry groups of some quantum spheres in the sense of Connes's noncommutative geometry. With Gabriel, I investigated fixed point algebras of the actions of BanicaSpeicher quantum groups on the Cuntz algebra and we computed their Kgroups. With my PhD student Stefan Jung, I am working on quantum spaces fitting with BanicaSpeicher quantum groups. With my PhD student Simon Schmidt, I am working on quantum automorphism groups of finite graphs. With my PhD student Daniel Gromada, I am developping computer algebraic methods for the classification of compact matrix quantum groups. With my PhD student Laura Maaßen, I am exploring the links between certain reflection groups and quantum groups.
Here is a little (slightly outdated) introduction into BanicaSpeicher/easy quantum groups and their links to free probability:
What actually are BanicaSpeicher quantum groups?
Member (PI) of the SFBTRR 195
Publications
Preprints
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 Weber, M.
Partition C*algebras
arXiv:1710.06199 [math.OA, math.CO, math.QA], 19 pages + 11 pages of appendix and references (2017).
 Weber, M.
Partition C*algebras II  links to compact matrix quantum groups
arXiv:1710.08662 [math.OA, math.CO, math.QA], 27 pages (2017).
 Cébron, Guillaume; Weber, M.
Quantum groups based on spatial partitions
arXiv:1609.02321 [math.QA, math.OA], 32 pages (2016).
 Speicher, Roland; Weber, M.
Quantum groups with partial commutation relations
arXiv:1603.09192 [math.QA, math.OA], 44 pages (2016).
Monograph
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Voiculescu, DanVirgil; Stammeier, Nicolai; Weber, M. (eds)
Free probability and operator algebras
Münster Lecture Notes in Mathematics European Mathematical Society (EMS) 132 pages Zürich, 2016

Chapter in monographs
 Weber, M.
Basics in free probability, 6 pages
in Free probability and operator algebras, ed. by DanV. Voiculescu, N. Stammeier, M. Weber, EMS, 2016.
 Weber, M.
Easy quantum groups, 23 pages
in Free probability and operator algebras, ed. by DanV. Voiculescu, N. Stammeier, M. Weber, EMS, 2016.
in peer reviewed journals
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 Schmidt, Simon; Weber, M.
Quantum symmetries of graph C*algebras
to appear in Canadian Mathematical Bulletin
arXiv:1706.08833 [math.OA, math.FA], 18 pages (2017).
 Tarrago, Pierre; Weber, M.
The classification of tensor categories of twocolored noncrossing partitions
Journal of Combinatorial Theory, Series A, Vol. 154, Feb 2018, 464506
arXiv:1509.00988 [math.CO, math. QA], 40 pages (2015).
 Weber, M.
Introduction to compact (matrix) quantum groups and BanicaSpeicher (easy) quantum groups
Notes of a lecture series at IMSc Chennai, India, 2015
to appear in Indian Academy of Sciences. Proceedings. Mathematical Sciences, pp 153, 2017.
 Mai, Tobias; Speicher, Roland; Weber, M.
Absence of algebraic relations and of zero divisors under the assumption of finite full nonmicrostates free entropy dimension
Advances in Mathematics, Vol. 304, 2 January 2017, pages 10801107.
See also: arXiv:1502.06357 [math.OA], 25 pages (2015).
(we extended the former version arXiv:1407.5715 substantially)
 Tarrago, Pierre; Weber, M.
Unitary easy quantum groups: the free case and the group case
International Mathematics Research Notes, 18, 1 Sept 2017, 57105750.
See also: arXiv:1512.00195 [math. QA, math.OA], 39 pages (2015).
 Gabriel, Olivier; Weber, M.
Fixed point algebras for easy quantum groups
SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) 12 (2016), 097, 21 pages.
See also: arXiv:1606.00569 [math.OA, math.KT], 21 pages (2016).
 Freslon, Amaury; Weber, M.
On bifree De Finetti theorems
Annales Mathématiques Blaise Pascal, 23(1), 2151, 2016.
See also: arXiv:1501.05124 [math.PR, math.OA, math.QA], 16 pages (2015).
 Raum, Sven; Weber, M.
The full classification of orthogonal easy quantum groups
Communications in Mathematical Physics, 341(3), 751779, Feb 2016.
See also: arXiv:1312.3857 [math.QA], 38 pages (2013).
 Raum, Sven; Weber, M.
Easy quantum groups and quantum subgroups of a semidirect product quantum group
Journal of Noncommutative Geometry, Vol. 9(4), 12611293, 2015
See also: arXiv:1311.7630 [math.QA], 26 pages (2013).
 Freslon, Amaury; Weber, M.
On the representation theory of partition (easy) quantum groups
Journal für die reine und angewandte Mathematik [Crelle's Journal], Vol. 2016, Issue 720 (Nov 2016), 2016.
See also: arXiv:1308.6390 [math.QA], 42 pages (2013).
 Raum, Sven; Weber, M.
The combinatorics of an algebraic class of easy quantum groups
Infinite Dimensional Analysis, Quantum Probability and related topics Vol 17, No. 3, 2014.
See also: arXiv:1312.1497 [math.QA], 16 pages (2013).
 Weber, M.
On the classification of easy quantum groups
Advances in Mathematics, Volume 245, 1 October 2013, pages 500533.
See also: arXiv:1201.4723 [math.OA], 39 pages (2012).
 Weber, M.
On C*Algebras Generated by Isometries with Twisted Commutation Relations
Journal of Functional Analysis, Volume 264, Issue 8, pages 19752004, 2013.
See also: arXiv:1207.3038 [math.OA], 35 pages (2012).
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