Ilia Smilga: Homepage

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Photo I am currently a CNRS post-doc at the Institut des Hautes Études Scientifiques, financed by the ERC project "DiGGeS" headed by Fanny Kassel.

Office number: 1N10
Physical address:
Institut des Hautes Études Scientifiques
35 route de Chartres
91440 Bures-sur-Yvette
France
Phone: (+33) 1 60 92 66 42
E-mail: lastname AT ihes.fr (current)
       or firstname.lastname AT normalesup.org (permanent)

My research is structured along two main axes. I started with a geometric problem, which is the subject of the papers 4 to 9 in the list below and of my Ph.D. thesis, defended on 12 November 2014 under the supervision of Yves Benoist and entitled "Tilings of the affine space". More precisely, the problem consists in studying discrete groups of affine transformations acting properly on the affine space, and in particular counterexamples to the Milnor conjecture – i.e. groups of this kind that are free, or that contain a free subgroup.
The study of these groups naturally led me to a question in representation theory: to classify, for a given semisimple real Lie group, the set of representations in which the restricted Weyl group acts nontrivially on the subspace of vectors invariant by the Levi group L (the centralizer of a maximal split torus). Papers 1 to 3 in the list below are dedicated to this question.

Here is my CV (latest update on 5 January 2020).

Publications and preprints

  1. I. Smilga. New sequences of non-free rational points, submitted.

  2. I. Smilga. Representations having vectors fixed by a Levi subgroup, submitted.
  3. I. Smilga. Action of the restricted Weyl group on the L-invariant vectors of a representation, in V. Dobrev, editor, Proceedings of the XIII International Workshop "Lie Theory and Its Applications in Physics" (Varna, Bulgaria, June 2019), volume 335 of Springer Proceedings in Mathematics and Statistics, pages 365–372. Springer Singapore, 2020.
  4. B. Le Floch, I. Smilga. Action of Weyl group on zero-weight space, Comptes Rendus Mathématique 356:8 (2018) 852–858

  5. J. Danciger, T. A. Drumm, W. M. Goldman, I. Smilga. Proper actions of discrete groups of affine transformations (a survey paper), accepted in the special volume Dynamics, Geometry, Number Theory: the Impact of Margulis on Modern Mathematics, to be published by the University of Chicago Press.
  6. I. Smilga. Construction of Milnorian representations, Geometriae Dedicata 206 (2020) 55–73
  7. I. Smilga. Proper affine actions: a sufficient criterion, to appear in Mathematische Annalen.
  8. I. Smilga. Proper affine actions in non-swinging representations, Groups, Geometry and Dynamics 12:2 (2018) 449–528
  9. I. Smilga. Proper affine actions on semisimple Lie algebras, Annales de l'Institut Fourier 66:2 (2016) 785–831
  10. I. Smilga. Fundamental domains for properly discontinuous affine groups, Geometriae Dedicata 171 (2014) 203–229

  11. I. Smilga. Harmonic functions on the Sierpinski triangle, 2012 (preprint).

Other productions

Teaching

I currently have no teaching load, but have done a lot of teaching previously. See my CV for an exhaustive list of courses I have taught.