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Osamu Saeki

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Topology of DNA Knots

Osamu Saeki (Unit Leader)
Degree: Doctor of Science (the University of Tokyo)
Research Interests: Topology
Unit: Space & Flow

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Figure: Doughnut and a mug are identical in topology

Topology is the study of extremely soft geometry where an object is considered to be made of something soft like rubber, and continuously deformed object is considered to be identical to the original object. In other words, topology is a field in pure mathematics where its research objects are the properties of geometrical objects, which do not change as objects are continuously deformed. For instance, a mug and doughnut are considered to be the same in topology. Each of them has one hole, which is a typical quantity that does not change by continuous deformation.

As such, topology provides a powerful tool for analyzing flexible objects. One extreme are knots and links created upon tying a knot, which is an important action in daily life and has been strongly associated with humans since the primitive age. In fact, it is known that wild gorillas can tie a knot. Moreover, it has recently been elucidated that these knots have a lot to do with research into DNA (deoxyribonucleic acid).

Figure: Example of an enzyme-mediated DNA recombination

DNA carries hereditary information in the shape of a twisted thread in a cell of living organisms, and often assumes a ring form. Biological observations have shown that strands of ring-shaped DNA are knotted and linked with each other. Although certain enzymes are responsible for these knots and links, limitation in experimental techniques has prevented disclosure of the details of this mechanism. In the late 1980s, mathematicians C. Ernst and D. W. Sumners used the most recent knot theory of the day in topology to elucidate the mechanism of an enzyme. Although it is often said that knot theory has its origin in the electromagnetism of Gauss and vortex atom hypothesis of Lord Kelvin in the 19th century, chemists and physicists seem to have forgotten about knots, and only mathematicians have maintained this topic as a research interest. My group is making extensive use of knot theory, which is one of the most actively researched fields in modern mathematics, to analyze DNA recombination mediated by an enzyme called topoisomerase from a mathematical point of view. Our goal is to apply the analysis to industrial technologies.

In addition, we are working on singularities of differentiable mappings. In particular, we have made a number of discoveries for specific cases where singular points of mappings between smooth objects deeply reflect the topological properties of the objects. Moreover, I am a worldwide recognized authority on the theory of inverse images of points (called singular fibers) and recently published a first book that formulates the theory.

Besides the theory of singularities of differentiable mappings, we have been working on a broad range of topics in topology, including primary obstruction to topological embeddings, separation properties of codimension one mappings, topology of isolated singularities of complex hypersurfaces, fibered knots, 4-dimensional manifolds, embeddings of codimension one, differential geometric invariants of space curves, and unknotting numbers of knots. Additionally, my group has been actively investigating asymptotic behavior of generalized Fibonacci sequences as well as the zeros of analytical functions.

My diverse research has greatly impacted my students, and is exemplified by the broad range of research thesis topics of my Master’s and Ph.D. students. In fact, one of my Master’s students wrote a dissertation on DNA knots. In addition, some of my students have contributed to industrial technologies, and in fact, one of my Ph.D. students is currently employed as a member of the research staff in the Program to Produce Mathematics Ph.D. and New Master’s Required by the Industrial Technology, which is part of the graduate level training reform program of Kyushu University’s Graduate School of Mathematics.

As the leader of the “Space & Flow” Unit, I would like to deepen the relationship between mathematics and industrial technologies.

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