Graduate Student

Max Planck Institute for Mathematics

Vivatsgasse 7

53111 Bonn, Germany

Office 404

leoxardo(at)mpim-bonn.mpg.de (replace x with n, if you are not a bot)

I'm a third year Ph.D. student in the IMPRS, Bonn, under the supervision of Geordie Williamson.

I am particularly interested in the following topic:

- Hecke algebras and their categorification via Soergel bimodules
- Kazhdan-Lusztig Combinatorics
- Perverse sheaves
- Hodge Theory
- Flag manifolds and the geometry of Schubert varieties
- Koszul Duality
- Affine Grassmannians and Geometric Satake Correspondence
- Quantum groups and KLR algebras
- Nilpotent Orbits and Springer Correspondence
- Representations of Algebraic Groups

- The Néron-Severi Lie Algebra of a Soergel module

*To appear in Transformation Groups*

Following Looijenga and Lunts, we introduce for any Soergel module a semisimple Lie algebra generated by all the Lefschetz operators and their adjoints. This algebra, called the Néron-Severi Lie algebra, can be used to provide a simple proof of the well-known fact that a Schubert variety is rationally smooth if and only if its Betti numbers satisfy Poincaré duality. We also determine a large set of elements, for finite Coxeter group, where the Neron-Severi Lie algebra coincides with the full Lie algebra of endomorphism preserving the intersection form. - The Hard Lefschetz Theorem in positive characteristic for the Flag Varieties

*To appear in International Mathematical Research Notices*

We show that the Hard Lefschetz theorem holds for the flag variety of some reductive group in characteristic p, if p is larger than the number of positive roots. The converse also holds, a part from 3 exceptional cases. One can also deduce an algebraic proof of the Hard Lefschetz theorem in the already known characteristic 0 case.

- My Master Thesis: The Erweiterungssatz for the
Intersection Cohomology of Schubert Varieties (written under the supervision of Prof. Luca Migliorini)

The main result contained is the categorification of the Hecke algebra via perverse sheaves and Soergel modules - My Posters (displayed at BIGS annual Poster Exhibition): 2015, 2016