Dr. Illia Karabash

Research

Research interests

• Analysis, Optimization, Probability
• Spectral Optimization, Structural Optimization
• Wave equations, Resonances, Photonics
• Liquid Film Equations
• Random Schrödinger operators
• Random Fourier Series
• Optimal Control, Hamilton–Jacobi–Bellman equations
• Degenerate Parabolic Equations
• Nonlinear/Non-selfadjoint Spectral Theory

Main papers:

• M. Eller, I.M. Karabash, M-dissipative boundary conditions and boundary tuples for Maxwell operators, preprint arXiv:2110.04586 [math.AP], 2021, 38 p., submitted. Preprint
• M. Eller, I.M. Karabash, Euler-Lagrange equations for full topology optimization of the Q-factor in leaky cavities, to appear in the proceedings of the conference ``Days on Diffraction 2021'' (St.Petersburg, 31.05-4.06.2021), 2021, 7 p., IEEE. Preprint
• S. Albeverio, I.M. Karabash, Asymptotics of random resonances generated by a point process of delta-interactions, In: Albeverio S., Balslev A., Weder R. (eds.) “Schrödinger Operators, Spectral Analysis and Number Theory. In Memory of Erik Balslev”, Springer Proc. in Math. & Stat., Vol. 348, 2021. doi.org/10.1007/978-3-030-68490-7_2. Preprint
• I.M. Karabash, H. Koch, I.V. Verbytskyi, Pareto optimization of resonances and minimum-time control, Journal de Mathématiques Pures et Appliquées 136 (2020), 313-355. doi.org/10.1016/j.matpur.2020.02.005. Preprint
• S. Albeverio, I.M. Karabash, Generic asymptotics of resonance counting function for Schrödinger point interactions. In: Kurasov P., Laptev A., Naboko S., Simon B. (eds.), ``Analysis as a Tool in Mathematical Physics: in Memory of Boris Pavlov'', Oper. Theory Adv. Appl., Vol. 276, pp. 80-93, Birkhäuser, Cham, 2020. doi.org/10.1007/978-3-030-31531-3_8. Preprint
• S. Albeverio, I.M. Karabash, On the multilevel internal structure of the asymptotic distribution of resonances, J. Differential Equations 267 (2019), 6171-6197. doi.org/10.1016/j.jde.2019.06.020. Preprint
• I.M. Karabash, J. Prestin, Recovery of periodicities hidden in heavy-tailed noise, Math. Nachr. 291 (2018), pp. 86-102. doi.org/10.1002/mana.201600361. Preprint
• I.M. Karabash, O.M. Logachova, I.V. Verbytskyi, Nonlinear bang-bang eigenproblems and optimization of resonances in layered cavities, Integral Equations Operator Theory 88 (2017).
• I.M. Karabash, Pareto optimal structures producing resonances of minimal decay under L^1-type constraints, J. Differential Equations 257 (2014).
• I. Karabash, Optimization of quasi-normal eigenvalues for 1-D wave equations in inhomogeneous media; description of optimal structures, Asymptotic Analysis 81 (2013).
• E. Braverman, I. Karabash, Structured stability radii and exponential stability tests for Volterra difference systems, Comput. Math. Appl. 66 (2013).
• I. Karabash, Optimization of quasi-normal eigenvalues for Krein-Nudelman strings, Integral Equations Operator Theory 75 (2013).
• E. Braverman, I. Karabash, Bohl-Perron-type stability theorems for linear difference equations with infinite delay. J. Difference Equ. Appl. 18 (2012), no. 5.
• P. Binding, P.J. Browne, I. Karabash, Sturm-Liouville problems for the p-Laplacian on a half-line. Proc. Edinb. Math. Soc. (2) 53 (2010), no. 2.
• I. Karabash, A. Kostenko, M. Malamud, The similarity problem for J-nonnegative Sturm-Liouville operators, J. Differential Equations 246 (2009).
• M. Chugunova, I. Karabash, S.G. Pyatkov, On the nature of ill-posedness of the forward-backward heat equation, Integral Equations Operator Theory 65 (2009).
• I. Karabash, C. Trunk, Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), no. 3, 483–503.
• I. Karabash, A. Kostenko, Indefinite Sturm-Liouville operators with the singular critical point zero, Proc. Roy. Soc. Edinburgh 138A (2008).
• I. Karabash, M. Malamud, Indefinite Sturm-Liouville operators (sgn x)(-d^2/dx^2 +q) with finite-zone potentials, Oper. Matrices 1 (2007).
• I. Karabash, S. Khassi, On the similarity between a J-selfadjoint Sturm-Liouville operator with operator potential and a selfadjoint operator, Math. Notes 78 (2005), no.3-4.
• I. Karabash, J-selfadjoint ordinary differential operators similar to selfadjoint operators, Methods Funct. Anal. Topology 6 (2000), no.2.

Lists of papers in databases:

• mathscinet.ams.org
• arXiv.org
• scholar.google.com

Recent conference and seminar talks:

• 11/06/2021, Asymptotics of random and deterministic resonances, Seminar of Functional Analysis Group, Institute for Applied Mathematics, University of Bonn (online)
• 04/06/2021, Asymptotic distribution of random and deterministic resonances on rough and fine levels, Online Seminar of the project ``From Modeling and Analysis to Approximation'', Institute of Mathematics, Universität zu Lübeck
• 01/06/2021, Optimization of resonances in photonic crystals and related Hamilton–Jacobi–Bellman equations, International Conference ``Days on Diffraction 2021'', St.Petersburg, Russia (online)
• 07/03/2020, Asymptotics of random resonances generated by a point process, Workshop ``Analytical Modeling and Approximation Methods``, Humboldt-Universität zu Berlin
• 20/01/2020, Pareto maximization of Q-factor in photonic crystals, Oberseminar Angewandte Analysis, TU Dortmund
• 28/06/2019, Hamilton-Jacobi and Euler-Lagrange equations arising in optimization of resonances, Oberseminar Analysis, Mathematische Physik & Dynamische Systeme, TU Dortmund
• 17/06/2019, High-Q optical cavities and deviations from the periodicity in optimal structures, Oberseminar, Institute of Mathematics, Universität zu Lübeck
• 28/05/2019, Nonlinear Maxwell equations arising in optimization of resonances for composite cavities, Seminar of Nonlinear PDE Group, Karlsruher Institut für Technologie (KIT)
• 29/04/2019, Non-Hermitian spectral optimization with applications to resonances in leaky cavities, Oberseminar Nonlinear Analysis, Universität zu Köln
• 26/04/2019, Optimization of resonances and dynamical systems of billiard-type, Seminar of Functional Analysis Group, Institute for Applied Mathematics, University of Bonn
• 09/04/2019 , Optimization of resonances in composite structures, Seminar of Trimester Program ''Evolution of Interfaces'', Hausdorff Research Institute for Mathematics (HIM), University of Bonn

Recent teaching

• WS 2021/22 Dozent für das Proseminar zu Analysis I/II (LA Gym) (hybrid), (TU Dortmund)
• SS 2021 Dozent für das Seminar zu Analysis III (LA Gym) (digital), (TU Dortmund)
• WS 2020/21 "Analysis I für LA Gymnasium/BK " (hybrid), Korrekturen, Aufsicht der Klausur, Korrektur der Klausur (TU Dortmund)
• SS 2020 Übungsleiter für "Zufällige Graphen" (TU Dortmund)
• WS 2019/20 Übungsleiter für "Lokalkonvexe Methoden der Analysis" (TU Dortmund)
• WS 2018/19 Dozent für die Vorlesung V5B2 "Selected topics in Analysis and PDE - Geometric Optimal Control" (Universität Bonn)

Kontakt

Adresse

TU Dortmund
Fakultät für Mathematik
Lehrstuhl IX
Vogelpothsweg 87
44227 Dortmund

Sie finden uns auf dem sechsten Stock des Mathetowers.

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