2018年度年会(於:東京大学)

SUMMARY: My research interests are mainly in discrete countable groups, their actions on measure spaces and the associated orbit equivalence relations. I will discuss historical backgrounds of studying them and recent development. The story ranges over unitary representations of topological groups, the theory of operator algebras and rigidity aspects of simple Lie groups and some countable groups with fascinating features. I will explain how these are related toward recent results on rigidity in orbit equivalence and von Neumann algebras.

SUMMARY: As the Gauss-Bonnet theorem shows, there are relations between curvature and topology of Riemannian manifolds, and studying such relations has been one of the main problems in Differential geometry. The theory of Gromov-Hausdorff convergence of Riemannian manifolds is known to be an effective method in this direction. In this theory, we consider a certain family of closed Riemannian manifolds with a lower curvature bound. The lower curvature bound ensures the precompactness of the family, and one can expect that some topological, geometric or analytic invariants should be bounded on that family. Moreover, it is natural to ask deeper information on manifolds themselves through the convergence. The theory of collapsing Riemannian manifolds answers this question and turns out to be quite effective in some cases. For example, the theory of collapsing Riemnnian three-manifolds was used in Perelman’s work on the geometrization conjecture to determine the topology of the collapsed part of a closed three-manifold under the Ricci flow. In this lecture, after a brief survey of the development of the collapsing theory for closed manifolds, I will mainly focus on recent development of the theory of collapsing Riemannian manifolds with boundary, based on joint works with Zhilang Zhang.

SUMMARY: Discrete convex analysis is a unified framework of discrete optimization and two concepts, called L-convexity and M-convexity, play important roles in this framework. Structural results of discrete convex analysis include the conjugacy theorem between L-convexity and M-convexity, separation theorems for L-convex/L-concave functions and for M-convex/M-concave functions, and the Fenchel-type discrete duality theorem. Algorithmic aspects of L-convex and M-convex functions have also been discussed in discrete mathematics. In mathematical economics, discrete convex analysis has been applied to models with indivisible commodities. In this talk, I will cover these fundamental topics and recent developments.

SUMMARY: Gaisi Takeuti has left us numerous distinguished works in various fields of the foundations of mathematics. Among them, “proof theory” can be proclaimed to have been the matter of his foremost concern. It was deeply related to his desire (dream) to understand the true nature of “sets”.

He formulated second order logic (and higher order ones), which is a formalized framework for mathematics, and posed a conjecture, known as “Takeuti’s fundamental conjecture”, which would provide many logical facts, including the consistency, of the formal systems. An attempt of (partially) proving the conjecture is called, as a catchword, “cut elimination” or “consistency proof”.

The fundamental conjecture claims that its proofs must be performed within “Hilbert’s Program”, that is, the metamathematical proofs of formal systems must abide by the “finitist viewpoint”. Formalism and finitist viwrpoint are the important thoughts behind Takeuti’s mathematical achievements.

In order to prove the fundamental conjecture, Takeuti created and developed a theory of constructive well-ordered structures of notations, called “ordinal diagrams”. Applying this theory, he made a monumental advance of the consistency proof.

I will introduce Takeuti’s proof-theoretical results along with the background thoughts, which determined the direction of “Takeuti’s proof theory”.

SUMMARY: The strong Lefschetz property for Artinian Gorenstein algebras is a ring-theoretic abstraction of the Hard Lefschetz Theorem for compact Kähler manifolds.

Suppose that \(A=\bigoplus _{i=0}^d A_i\) is a graded Artinian Gorenstein algebra. We say that \(A\) has the strong Lefschetz property if there exists a linear element \(l \in A\) such that the multiplication map \[\times l^{d-2i}: A_i \to A_{d-i}\] is bijective for all \(i=0,1,2, \ldots , [d/2]\).

This can be defined for graded vector spaces and basic properties can be derived as properties of an endomorphism of graded vector spaces. As a consequence, it can be proved that “almost all” Artinian Gorenstein algebras have the strong Lefschetz property. It gives us new problems to ask (1) what classes of Gorenstein algebras have the strong/weak Lefschetz property without exception and (2) what Gorenstein algebras fail to have the strong/weak Lefschetz property. I would like to speak about known results, methods to prove them, applications and problems of the Lefschetz properties of Artinian Gorenstein algebras.

SUMMARY: This talk provides an overview on matroid parity, which was introduced in the 70s as a common generalization of matching and matroid intersection. In particular, we present a combinatorial, deterministic, polynomial-time algorithm for the weighted linear matroid parity problem.

SUMMARY: We will review algebraic and category-theoretic structures in 3-dimensional topology. The most well-known structures of such kind are topological quantum field theories, which are vector-space-valued functors on the category of closed surfaces and 3-dimensional cobordisms. We will mainly focus on the category Cob of once-punctured surfaces and cobordisms between them, introduced by Crane and Yetter and by Kerler, and some of its subcategories. The category Cob has a structure of a braided monoidal category, and it is equipped with a Hopf algebra object. We will consider braided subcategories sLCob and LCob of Cob, which are called the category of special Lagrangian and Lagrangian cobordisms, respectively, where sLCob is also a subcategory of LCob. The category sLCob may be identified with the opposite of the category H of handlebodies and “disc-based” embeddings. We will also discuss functors on these categories, which may be regarded as functorial 3-manifold invariants.

SUMMARY: By the work of Brylinsiki-Kashiwara and Belinson-Bernstein we can localize representations of complex simple Lie algebras on the flag manifolds. There is also a similar theory in positive characteristics due to Bezrukavnikov-Mirkovic-Rumynin.

In this talk I would like to talk about the corresponding results for quantized enveloping algebras using the quantized flag manifolds. The quantized flag manifold is not an algebraic variety in the ordinary sense; it is a non-commutative scheme equipped with non-commutative ring of functions. Nevertheless, we have the notion of \(D\)-modules on the quantized flag manifolds, by which we can localize representations of quantized enveloping algebras on the quantized flag manifolds.

SUMMARY: The nonlinear Hille–Yosida theorem in Hilbert spaces was established in 1967 by Kōmura. After his pioneering work, a generation theorem of semigroups of contractions in general Banach spaces was proved in 1971 by Crandall and Liggett. The notion of semigroups of Lipschitz operators in Banach spaces was introduced by Kobayashi in 1990’s as a nonlinear analogue of strongly continuous semigroups of bounded linear operators. Metric-like functionals, not the metrics induced by norms, play an important role in characterizing such semigroups with continuous generators. What happens if we avoid using not only norms but also linear structures? I will talk about mutational equations described by ‘transitions’ and ‘mutations’, which are mathematical tools extending the concept of differential equations to the case of metric spaces. The mutational analysis was initiated by Aubin in 1990’s to analyze varying shapes, and has been recently developed by Lorenz to give a unified way to various types of evolution equations. Their abstract results can be extended to apply to quasilinear evolution equations due to Kato.