Research Interests

My long-term research interests revolve around challenges of creating theories, methods, and algorithms underpinning creation and understanding of machine intelligence, adaptation, and learning as well as helping to reveal their fundamental limitations. Development of these theoretical tools requires combination of techniques from different areas spanning concentration of measure theory, statistical learning theory, analysis, modelling and synthesis of fragile, nonlinear, chaotic, meta-stable dynamics, adaptation and adaptive control theory, synchronization (stable and critical), biologically-inspired systems for processing of the visual information, computer vision, networks of interconnected dynamical systems, analysis of dynamics of the spiking neuron models, their properties and possible functions, algorithms for machine learning and data analysis in high dimensions. For the sake of simplicity, I organized these different lines of inquiry into the following specific challenges I am addressing at the moment

1. Provably Resilient, Trustworthy, and Robust Artificial Intelligence. Discovering general principles and fundamental understanding of how to build certifiably safe, quantifiable, and trustworthy AI systems for the benefit of people.

2. Machine Learning and Data Analysis in High dimensions. Exploiting measure concentration effects for development and analysis of efficient algorithms for machine learning and data analytics in high dimensions.

3. Processes and Mechanisms of Adaptation in Complex Nonlinear Systems. Systems with nonlinear parametrization, unstable target dynamics, non-dominating (non-majorating, gentle, non-dominating) adaptation

4. Synchronization in Nonlinear Dynamical Systems. Global, partial, intermittent  synchronization in the ensembles of linearly and nonlinearly  coupled nonlinear oscillators. Study of connectivity-dependent synchronization. Adaptive and unstable, multi-stable, alternating synchrony.

5. Optimization algorithms for nonconvex and nonlinear problems. Parameter estimation and inverse problems involving systems of ordinary differential equations with nonlinear in parameter right-hand side.

6. Analysis of dynamical systems with unstable and semi-stable attractors. Extension of the method of Lyapunov functions for systems with weakly attracting  (in Milnor sense) invariant sets, first method of Lyapunov for non-hyperbolic equilibria, non-uniform small-gain theorems.

7. Neuroscience and physics of neuronal cells. Principles of neuronal processing of  information. Study of properties of the biological cells, analysis of their functions. Structural organization of the visual system, models system for robust  and adaptive processing (w.r.t modelled uncertainties) of visual information.

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Department of Mathematics
University of Leicester
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Tel.: +44 (0)116 229 7407

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