Towards Catastrophe theory for Khovanov-Rozansky homology

The Catastrophe theory is a powerful tool of mathematical physics to study complicated dynamical systems. It commonly operates with systems of differential equations, but is close in spirit to homological calculus in topology. A kind of hybrid of the two fields is known as a cohomological quantum field theory (CQFT). Such models seem to be very interesting and profound, and they might be useful in various applications as a new tool of the Catastrophe theory. Our goal is to use the knot homology calculus to develop a “Cohomological Catastrophe theory”, which would be associated with a family of constructively defined CQFT models related by a kind of evolution “regular” in the moduli space but special points of “catastrophes” in the knot homology. In the talk we summarize our current knowledge on the Catastrophes for explicitly studied CQFT models associated with the Khovanov–Rozansky homology of several knot families.