Mathematics

In the 17th century, Newton found differential and integral calculus, giving a language and method to describe the law of dynamics in nature. 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 a mathematician’s original concepts. Particularly in recent years the interaction between mathematics and physics has been in full flow.

Gauge theory, quantum field theory, general relativity, superstring theory and the theory of integrable systems in physics have provided major influences in the development of mathematics such as algebraic geometry, differential geometry, topology, representation theory, algebraic analysis and number theory. A large scale development has been newly emerging.

This close collaboration between mathematics and physics is particularly important for advancing the study of the concept of space and universe that have been developed by scientists such as Kepler, Newton, Gauss, Riemann, Maxwell, Einstein and many others.

For the past twenty years, methods of quantum field theory have had a major influence on mathematics. Since quantum field theory treats the differential and integral calculus of an infinite number of degrees of freedom, the rigorous development of quantum field theory in mathematics has yet to be established. Nevertheless, in these twenty years, a lot of concepts arising from quantum field theory such as quantum groups have had a major influence on modern mathematics and physics.

Mathematicians at IPMU are working to develop modern mathematics by closely working with physicists. The following are the fields of mathematics studied at IPMU. We divided the fields into geometry and algebra.

Geometry:

Geometric objects we study in mathematics include several kinds of spaces, such as topological spaces, differentiable manifolds, symplectic manifolds, complex manifolds and algebraic varieties. Recently these various branches of geometries are deeply connected and influence each other. For instance, mirror symmetry is a conjectural duality between symplectic manifolds and algebraic varieties, which was found by the duality between different types of string theories. One of the research focus of our geometry group is to invent and investigate the mathematical notions which describe the mirror symmetry, and give some applications to the geometric problems we are interested in.

In the theory of mirror symmetry, a Calabi-Yau 3-fold plays an important role. A Calabi-Yau 3-fold is a complex manifold of real dimension 6 with a Ricci flat metric. In string theory, the spacetime is expected to be 10-dimensional, and the extra 6-dimensional space is expected to take the form of a Calabi-Yau 3-fold. On a Calabi-Yau 3-fold, we can define the quantum invariant counting Riemann surfaces on it, called Gromov-Witten (GW) invariant. One of the ways to describe the mirror symmetry is to establish the relationship between GW-invariants and the period map on the mirror manifold. In our group, S. Galkin studies GW-invariant, K. Saito studies the period map, and they develop these theories.

Another way to describe the mirror symmetry is to use the homological algebra proposed by M. Kontsevich. It is stated as an equivalence of triangulated categories between derived category of coherent sheaves and derived Fukaya category on the mirror manifold. In our group, A. Bondal develops the theory of triangulated categories, and describes the structure of several triangulated categories, e.g. to show the existence of the exceptional collections. The development of this theory is relevant in understanding the mirror symmetry.

On a Calabi-Yau 3-fold, we can define another quantum invariant, called Donaldson-Thomas (DT) invariant. It counts D-branes in terms of string theory, and is expected to be equivalent to the GW-invariant. (GW/ DT correspondence). The DT-theory depends on a choice of a stability condition on the derived category, and the set of stability conditions form a complex manifold, which is expected to be a stringy Kahler moduli space. Understanding DT-invariants and the structure of the space of stability conditions is important in connection with string theory, and Y. Toda studies these theories. Also the theory of quantum invariants of low dimensional manifolds has begun with the study of quantum theory such as integrable systems, soliton equations and the conformal field theory. These quantum invariants turn out to have a deep connection with GW-theory, and T. Kohno studies these invariants.

Algebra:

Algebra is a collection of branches of mathematics, which studies the system of numbers such as integers and polynomials. Some examples of the branches are set theory, group theory, (commutative) ring theory, (algebraic) number theory, category theory, algebraic geometry, combinatorics and representation theory. Of course, each branch may not be fully contained in algebra, and may lie in between geometry.

Algebra studied at IPMU includes homological algebra and category theory. Homological algebra began as a study of homology groups of topological spaces. K-theory is an example of cohomology theories. Recall that in connection with string theory, an interesting and basic example is that an element of a K-group of a certain topological space has a physical interpretation. This enables us to use the powerful machinery of homological algebra to the study of string theory.

Nowadays, a basic algebraic invariant associated with a geometric object is a triangulated category. For example, this appears from an algebraic variety as the derived category of coherent sheaves. The notion of triangulated category is so abstract that they appear everywhere in mathematics. We know that some non-commutative geometry is better described in this language. Recent research is focused on finding a more complicated structure than that of a triangulated category. Differential graded categories and model categories are examples of objects that are equipped with more structure than a triangulated category. We seek to reveal the algebraic structure common to various phenomena (which may or may not look unrelated) occurring in mathematics and physics.

Another basic example of an algebraic structure is a group or a group action. A group describes the symmetry of things. Groups are everywhere in mathematics from Galois groups in number theory to mapping class groups in topology. Study of groups, or representation theory, will then lead to the explanation of the phenomena caused by the symmetry. Let us give a list of those groups (or algebras) our researchers are interested in, just to give an idea on how diverse we are. The groups (or algebras) that appear are vertex operator algebras, Lie groups (algebras), braid groups, Galois groups, and mapping class groups. We refer to the table below of group members for more information on how each deals with the group in his research.

We certainly hope to go the other direction. An example question would be if the product structure in K-theory has an interpretation. We can ask if it has a physical interpretation. The “distance” of algebra from physics, compared with geometry, is greater, in the sense that many of the problems in physics are first stated using (quantum) field theory. While geometry is used to describe the universe rather directly as if taking a picture, algebra tends to seek for the exact laws behind the phenomena.

Group Members