My main contribution here is the development of an efficient approximation technique for eigenvalue surfaces near multiple eigenvalues – diabolical and exceptional points – and its application to the analytical computation of the geometric phase around exceptional points in non-Hermitian problems of classical and quantum physics. I proposed experimental verification of these findings to the group of Prof. A. Richter at TU-Darmstadt which resulted in our joint work published in Phys. Rev. Lett. in 2011 [5]. The series of these works on non-Hermitian Hamiltonians [1-4] is now actively cited in both physical and mathematical communities.

[1] **A.P. Seyranian, O.N. Kirillov, A.A. Mailybaev** (2005) *Coupling of eigenvalues of complex matrices at diabolic and exceptional points. *Journal of Physics A: Mathematical and General. 38(8): 1723-1740

[2] **O.N. Kirillov, A.A. Mailybaev, A.P. Seyranian** (2005) *Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation.* Journal of Physics A: Mathematical and General, 38(24): 5531-5546

[3] **A.A. Mailybaev, O.N. Kirillov, A.P. Seyranian** (2005) *Geometric phase around exceptional points.* Physical Review A, 72: 014104

[4] **A.A. Mailybaev, O.N. Kirillov, A.P. Seyranian** (2006) *Berry phase around degeneracies.* Doklady Mathematics, 73(1): 129-133

[5] **B. Dietz, H. L. Harney, O.N. Kirillov, M. Miski-Oglu, A. Richter, F. Schaefer** (2011) *Exceptional Points in a Microwave Billiard with Time-Reversal Invariance Violation. *Physical Review Letters, 106(15): 150403