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The conference is dedicated to the memory of the outstanding topologist Corresponding Member of National Academy of Sciences Volodymyr Vasylyovych Sharko to commemorate his contributions to the Morse theory, K-theory, L2-theory, homological algebra, low-dimensional topology and dynamical systems. The conference will be held at Institute of Mathematics of National Academy of Sciences of Ukraine, Kyiv, on 25-28 September 2019. Participants invited to present their current research and outline future directions in the theory and frontier applications.