I am currently a postdoc in Masaryk University in Brno, under the supervision of Jan Slovak.
Before I worked as a Postdoc in Charles University in the group of Réamonn Ó Buachalla

and before that in the Institute of Mathematics of the Czech Academy of Sciences, in the group of Karen Strung
and before that I defended my PhD thesis in October 2019 under the supervision of Ludwik Dabrowski.
 

My main research interests are noncommutative differential geometry of quantum homogeneous spaces.

In particular I work with Drinfeld-Jimbo quantum groups and together with my collaborators we have been interested in the following topics:

-Quantum Geometric Representation Theory. In classical Differential Geometry, the aim of Geometric Representation Theory is to give geometric constructions of representation theoretic objects via vector bundles, sheaves, D modules and so on. Since the representation theory carries over, it is just natural to ask the same kind of questions in the quantum world. The first result that carries over is the celebrated Borel-Weil theorem, which gives us a way to realize the irreducible representations of a quantum group as holomorphic sections of certain line bundles over a quantum flag manifold. Together with Ó Buachalla and Diaz-Garcia, we proved this result for irreducible quantum flag manifolds, using the celebrated Heckenberger and Kolb calculus. In a series of forthcoming paper, we aim to extend the scope of qGRT, allowing to the geometric construction of objects like quantum Verma modules and Nichols algebras. At the same time, it will be interesting to see how to geometrically encode the monoidal structure of the representation theory of quantum groups.
 

- Differential calculi for quantum flag manifolds.  Giving the right q deformations of the classical De-Rham complexes of homogeneous spaces is a non-trivial task which is deeply connected to the possibility of using these differential tools, in conjuction with Lie theory, to study the representation theory of our quantum groups. In the recent year, together with the group of Prague and others ( in primis my collaborator Sugato Mukhopadhyay), we are addressing this problem. Following an idea of  Ó Buachalla, we aim to generalize the construction of |Heckenberger and Kolb beyond the hermitian symmetric setting. Quite surprisingly this connects with the well-established construction of PBW basis for quantum enveloping algebra given by Luzstig, where one makes use of the different reduced decompositions of the longest element of the Weyl group to define the quantum root vectors.

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-Quantum fiber bundles with homogeneous fibre. Nowadays we completely understand quantum principal bundle in the framework of Brzezinski and Majid. In this context, a central role is played by a quantum Atiyah sequence. Now that we have a easy of generalizing Hopf-Galois extensions to pairs of quantum homogeneous spaces. We can now look at differential structures and generalize the Atiyah sequence.

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