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  3. Yukinobu Toda

InaRIS Fellow (2025-)

Yukinobu Toda

Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo Institutes for Advanced Study, The University of TokyoProfessor*Profile is at the time of the award.

2025InaRISScience & Engineering

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Research topics
Developing new research areas through categorical Donaldson-Thomas theory
Keyword
Summary
The Donaldson-Thomas invariant is a virtual count of geometric objects, such as spheres or donut-type surfaces, in a complex Calabi-Yau 3-fold, and is an important research subject both in mathematics and theoretical physics. On the other hand, a ‘category’ is an abstract mathematical notion, which is regarded as a kind of community of mathematical objects. So far I used categories to elucidate several properties of the invariants and proved some conjectures. In this research, I will construct categories which recover the invariants, regard them as non-commutative spaces, and explore a new research area which connects several mathematical fields through them.

　


Citation

Algebraic geometry is a field of mathematics that studies algebraic varieties, which are created by gluing geometric figures defined as the zero-point sets of polynomial systems. For instance, parabolas, hyperbolas, and ellipses are examples of algebraic curves, which are one-dimensional algebraic varieties. Algebraic geometry is a field in which theoretical research is actively conducted; however, it also appears in applied mathematics such as code and cryptography theory, and theoretical physics. The field in which the number of algebraic curves in an algebraic variety is investigated is known as enumerative geometry, and this has been extensively studied for a long time in algebraic geometry. The Donaldson–Thomas (DT) invariant, an integer-valued invariant that represents the number of algebraic curves in an algebraic variety known as a complex three-dimensional (3D) Calabi–Yau manifold, has garnered wide attention, and the research on it has progressed because it is related not only to mathematics but also to the superstring theory in theoretical physics.

In contrast, category theory is a mathematical theory that abstractly considers mathematical objects and the relationships between them. This theory can be assumed to represent the characteristic of mathematics regarding which we gain a deeper understanding by abstractly considering things. Dr. Toda is a leading researcher in DT invariants and has tackled the study of DT invariants by employing the original idea of utilizing a category known as the derived category of coherent sheaves on algebraic varieties, proving and proposing various theorems and formulas, including the DT/PT correspondence, Bayer–Macri–Toda inequality, and Maulik–Toda theory.

To obtain a deeper understanding of equations between numbers in mathematics, theoretically elucidating the reasons why the equations hold is necessary; therefore, elevating the equations between numbers to isomorphisms between algebraic objects and equivalences between categories is important. In the case of integer-valued DT invariants, the cohomological DT invariants, which are the algebraic objects cogenerating them, have already been defined. Dr. Toda's research proposal aims to construct a category known as the “DT category” to categorize cohomological DT invariants, deepening the formulas obtained thus far, and formulating and proving them in the language of categories. Furthermore, he aims to create a new research field that could be described as categorical geometry by studying the relationship between categories, such as DT categories and birational geometry, McKay correspondence, geometric representation theory, symplectic geometry, geometric Langlands correspondence, and mirror symmetry. Dr. Toda himself has already made research progress on a special type of 3D Calabi–Yau manifolds known as local algebraic surfaces. This is a significant, spectacular and fascinating research proposal that should be realized.

Dr. Toda has led the world with his original ideas in the study of DT invariants. With the support of the InaRIS Fellowship for 10 years, it is expected that his research on DT categories will progress and that research with a wide impact on various fields of mathematics and theoretical physics will be developed.


Message from Fellow


My research is motivated by pure mathematical interest, so it is not clear how it will be useful in the society at this moment. I am very grateful to be supported for such basic research over a long period of 10 years, and also feel a responsibility for it. Through the support of InaRIS, I would like to explore a new mathematical theory which fundamentally changes the notion of counting, and will change our thought on this research field in 10 years later. I also would like to contribute to internationalizations of this research and nurturing a new generation.

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