One can say that the history of mathematics is more or less equivalent to the history of mankind attacking problems in reason or knowledge. Mathematics abstracts the truth behind generalities. It began as a useful tool in everyday physics in the sense that we count numbers, we measure length and volume. Then as seen in Euclidean geometry of Euclid or in the problem of determining the existence or non-existence of roots in algebraic equations, the practical side became lost and mathematics becomes pure pursuit of knowledge of human kind. Diophantine equations, Fermat’s lifework, do not affect our life at all whether they had integral solutions or not.
In the 17th century, Newton found differential and integral calculus, giving a language and method to describe the law of dynamics in nature, so then mathematics encountered physics. This is a good example of mathematics providing the scientific community, and sometimes society in general, with a common language and method to describe phenomena in their study. This in turn helps to establish mathematician’s original concepts. Particularly in recent years the interaction between mathematics and physics has been in full flow.
This interaction is more important than ever, and mathematics of the 20th century developed through an enormous influence from physics. Gauge theory, quantum field theory, general relativity, and superstring theory in physics have provided major impetus in the field of mathematics such as algebraic geometry, differential geometry, topology, representation theory, algebraic analysis and number theory.
Although the influence is great, mathematicians have been and will be working on those problems in mathematics which arose and to be solved purely within mathematics. A prominent recent example is the resolution of Fermat’s last theorem. Notably, there exist examples where those tools developed purely mathematically have interesting applications in physics.
The Kavli IPMU is a very exceptional research environment for mathematicians, in that there always are physicists nearby. The mathematicians at are to develop both physics and mathematics through much discussions with the physicists. However, the emphasis is on that the true interest of mathematics is in mathematics, so the researchers undertake their research with their top priority in the field of mathematics.
Geometry is a collection of branches of mathematics, which studies mathematically defined geometric objects. For instance, geometric objects include topological spaces, differentiable manifolds, Riemannian manifolds, symplectic manifolds, complex manifolds and algebraic varieties. These are not only geometric figures but also have important mathematical structures, which make the theories behind them very rich. Also the geometry plays an important role in describing our universe, and has contributed much to physics. For instance, Riemannian geometry is necessary in developing Einstein’s theory of general relativity. On the other hand, by the influence of string theory, it has been found recently that there are deep connections among these geometric theories. The most significant example is the mirror symmetry, which predicts a duality between symplectic manifolds and algebraic varieties. These have been developed as different mathematical fields, but are now expected to be equivalent by the duality between the two different types of string theories. The mirror symmetry is one of the themes of the study of geometry at .
In the theory of mirror symmetry, a Calabi-Yau 3-fold plays an important role. It is an algebraic manifold of complex dimension three (real dimension six), with a Ricci flat metric. In string theory, our universe is considered to be 10 dimensional, and a Calabi-Yau 3-fold appears as an extra six dimensional space. The mirror symmetry predicts an equivalence between period integrals on a Calabi-Yau 3-fold and a curve counting theory (Gromov-Witten theory) on its mirror manifold. In , there are specialists in each side, and the mirror symmetry is studied vigorously. K. Saito has studied the theory of periods for a long time, and found the notion of Frobenius structures on the deformation spaces of the singularities. This Frobenius structure is now an essential tool in describing the mirror symmetry. T. Milanov is an expert in Gromov-Witten theory, and studies the Frobenius structures determined by Gromov-Witten invariants.
Another way to describe the mirror symmetry is to use the homological algebra. This was proposed by Kontsevich in 1994, and it is formulated as an equivalence between the derived category of coherent sheaves on a Calabi-Yau 3-fold and the derived Fukaya category on its mirror manifold. A. Bondal is a leading expert in derived categories of coherent sheaves, and invented the notions of exceptional collections, enhancement of the derived category. These notions are now essential in the study of derived categories. Y. Toda studies stability conditions on derived categories, and Donaldson-Thomas type invariants counting stable objects in the derived categories. This corresponds to “BPS state counting” in string theory, and an interesting research subject also in string theory.
The study of the geometry in does not restrict to the mirror symmetry, and the geometry is studied from various viewpoint. T. Kobayashi studies the action of discrete groups on non-Riemannian homogeneous spaces, and develops an original theory from a new geometric viewpoint. He discovered that local rigidity may fail even in higher dimensions for indefiniteRiemannian symmetric spaces, and is challenging to develop spectral theory in connection with their deformation theory. Also T. Kohno studies quantum invariants on low dimensional manifolds, which are related to integrable systems and the conformal field theory. He revealed a quantum group symmetry for homological representations of braid groups and described the image of quantum representations of mapping class groups. In this way, there are various studies of the geometry at , and we aim to have a further breakthrough by combining these studies.
Algebra was originally the field of mathematics that studied numbers and equations. In fact, in the Middle Ages, a central problem in algebra was solving algebraic equations of higher degrees. In the 19th century, E ́variste Galois discovered a symmetry hidden in such equations, which led to the invention of the notion of a group. After this discovery, the focus of the field of algebra shifted to understanding deep hidden structures.
Homological algebra is one such example. Homological algebra stems from geometry and was originally a way to count the number of holes of, say, doughnuts. Once geometric notions are abstracted to homological algebra, many new invariants which do not necessarily have an obvious geometric interpretation appear naturally. Derived categories are one such example. Derived categories were originally invented as a tool for efficiently computing (co)homologies, but today they are considered interesting in their own right. Mathematicians at the are pursuing a number of interests in the field of derived categories. For example, one question of interest is how much information on a given variety the derived category of coherent sheaves over that variety possesses. This question is currently with the relation to mirror symmetry in mind being actively studied by A. Bondal and Y. Toda, as was written in Geometry section above. We are also trying to construct additional frameworks, such as differential graded algebra for example, to encode more information on a variety by enhancing the structures on its derived category or its abstraction, the triangulated category.
Another field of interest is groups or group actions. Groups were originally a language for describing symmetries behind solutions of equations, but nowadays, they are an indispensable tool not only in mathematics but also in many areas of physics such as crystallography or gauge theory. Often, groups are studied through their representation theory. K. Saito considers sets of vanishing cycles coming from geometry as a generalization of root systems, and applies the corresponding theory of infinite dimensional Lie algebras and its representations to geometry. In T. Kobayashi’s new theory of global analysis for minimal representation and visible action on complex varieties, which are his recent themes, group appear as “leit motiv”, and the theory is expanded in a way special cases are connected to infinite dimensional representation theory. A. Tsuchiya studies the algebraic structures of logarithmic conformal field theory such as the representation theory of the chiral algebra and conformal blocks. T. Kohno studies monodromy representations of braid groups appearing in conformal field theory from the point of view of hypergeometric integrals.
Some researchers at the also study equations, one of the original goals of algebra. One of the ultimate goals of modern arithmetic is a framework called the Langlands program, which predicts that two seemingly completely different looking sets of representations coming from arithmetic coincide in a natural sense. This vast conjecture includes even Fermat’s last theorem as a very small part. It is therefore not surprising that this program is thought to be very difficult to realize, however a number of simpler variants exist. S. Kondo and T. Abe are working on the local Langlands program and the Langlands program for function fields for example. Yet another variant of the Langlands program, for which a connection to S-duality in physics has been observed, is the geometric Langlands program. The Langlands program and its variants have therefore become very active and important modern topics of research.
As one can see, we study many branches of algebra here at the . Algebra may not necessarily directly deal with understanding the universe and moreover most of the algebraists at the are interested in mathematics in and of itself, not in its application to other fields. However, when the efforts of practitioners of different disciplines align, the benefit to all fields involved is huge. Our daily efforts in understanding the structures ubiquitous to mathematics feed directly into efforts directed at understanding the most fundamental phenomena of nature. We therefore await the next great breakthrough rivaling Galois’ discovery of groups.
- Tomoyuki Abe
- Alexey Bondal
- Scott Huai-Lei Carnahan
- Kentaro Hori
- Yohsuke Imagi
- Yasuyuki Kawahigashi
- Toshiyuki Kobayashi
- Toshitake Kohno
- Satoshi Kondo
- Todor Milanov
- Hirosi Ooguri
- Kyoji Saito
- Yuji Tachikawa
- Yukinobu Toda
- Akihiro Tsuchiya
- Mikhail Verbitsky
- Alexander Voronov
- James Michael Wallbridge
- Marcus Werner