2019年度秋季総合分科会(於:金沢大学)

SUMMARY: We consider the Cauchy problem of the nonlinear partial differential equations of evolution type. This type of problem covers various models in mathematical sciences such as the nonlinear heat equation, the nonlinear Schrödinger equations, incompressible Navier–Stokes equations and drift-diffusion equations. This type of problem has a scaling invariant property and hence the Fujita–Kato principle for the solvability is applicable. Here we consider the role of critical exponents for a drift-diffusion type with related problems. In order the consider the relation of those problem, we introduce a critical function class and show maximal regularity for the heat equation with the end-point exponent. Then we derive the drift-diffusion equation from the original problem by a singular limit procedure.

SUMMARY: Fomin–Zelevinsky introduced the notion of cluster algebras and they proved (in a prticular case) that the quantum coordinate ring has a cluster algebra structure. In this talk, we discuss such a cluster algebra structures using its categorification by quiver Hecke algebras introduced by Rouquier and Khovanov–Lauda. This is a joint work with Seok-Jin Kang, Myungho Kim, Se-jin Oh and Euiyong Park.

SUMMARY: We review our mathematical and applied studies on large particle numbers (hydrodynamic) limits of stochastic ranking processes (SRP), systems of particles aligned in a line with move-to-front rules driven by point processes. On the application side, we observe that a simple version of the model, driven by the Poisson processes, explains behaviors of book ranking numbers at Amazon.co.jp. The results further imply that the main sales of the company is from top sales books, in opposition to expectations of possibilities of long-tail business model. On the mathematical study, we prove a hydrodynamic limit of SRP with position dependent intensities, allowing dependence of intensity functions of the driving processes on positon variables, which mathematically implies non-trivial stochastic dependence among the particles, complicating the studies. To overcome the difficulties, we introduce an intermediate model, SRP with flow driven intensities, which is driven by what we name ‘the point processes with last-arrival-time dependent intensities’ (PPLATDI), which, unlike Poission processes, lack independence of disjoint increments. The solutions in the hydrodynamic limit correspond to those of the systems of partial differential equations of one-dimensional fluid solved by characteristic curves, generalized to allow for non-local interaction terms, and whose solutions are found to be written by the expectations of PPLATDI.

SUMMARY: In 1974, Charles Fefferman proved that the Bergman kernel for strictly pseudoconvex domains has pole type and logarithmic type singularities. Since then many people thought that the logarithmic singularity vanishes if and only if the domain is biholomorphic to the ball —it is so called the Ramadanov conjecture. It is affirmatively solved in dimensions 2 more than 30 yeas ago, but is still open in higher dimensions. In this talk, I will explain the current status of the conjecture starting from the basic facts on CR geometry and complex Monge–Ampère equation.

SUMMARY: One of the interesting themes in complex differential geometry is to pursue a natural correspondence between objects in differential geometry and algebraic geometry. In particular, the variants of “Kobayashi–Hitchin correspondence” have been studied for a long time. The original theorem says that an algebraic vector bundle on a complex projective manifold has a Hermitian–Einstein metric if and only if it is stable. Among many variants, the most interesting is the “trinity” of Higgs bundles, flat bundles and harmonic bundles, which is a starting point of the so called non-abelian Hodge theory. After the study of the singularity, we obtained a correspondence between semisimple algebraic holonomic D-modules and polarizable pure twistor D-modules, which was applied to the study of the functoriality of the semisimplicity of algebraic holonomic D-modules.

The abstract existence theorem and the functorial property of mixed twistor D-modules imply that harmonic bundles exist ubiquitously. It is expected that they are related to concrete examples of “1-parameter family of flat bundles degenerating to a Higgs bundle” which naturally appear in various fields of mathematics, called quantum curves, quantum D-modules, etc. For that purpose, it would be useful to obtain more explicit information for some classes of twistor D-modules. For instance, we made some explicit computations for GKZ-systems and Toda equations.

More recently, by pursing an analogue of the non-abelian Hodge theory, we are interested in Kobayashi–Hitchin correspondences for monopoles with periodicity, and it turned out that they are equivalent to difference modules of various types which have not yet been intensively studied in differential geometry. It is expected that the equivalences would be a starting point of new rich studies.

SUMMARY: In this talk, I would like to present, mainly to non-specialists, some of the results concerning homogeneous open convex cones obtained during these 20 years or so by collaborating with Hideyuki Ishi, Chifune Kai, Hideto Nakashima and Takashi Yamasaki. Topics include the minimum size matrix realization with the help of weighted oriented graphs, basic relative invariants, various characterizations of symmetric cones among homogeneous open convex cones, interesting examples of homogeneous open convex cones etc.

SUMMARY: In this talk, we give a structure theorem for projective manifolds \(W_0\) with the property of admitting a 1-parameter deformation where \(W_t\) is a hypersurface in a projective smooth manifold \(Z_t\). Their structure is the one of special iterated univariate coverings which we call of normal type. We give an application to the case where \(Z_t\) is a projective space, respectively an abelian variety. We also give a characterizaton of smooth ample hypersurfaces in abelian varieties and describe an irreducible connected component of their moduli space. This is a joint work with Fabrizio Catanese.

SUMMARY: I consider the Allen–Cahn equation (or Nagumo equation) in a set \(\Omega \) of some special type. For semilinear parabolic equations, there are a lot of studies on the initial value problem in the case that \(\Omega \) is a bounded or unbounded domain. The existence and uniqueness of solutions and analysis of behavior of solution \(u\) for \(t\) grows up to infinity are important problems. In this talk I deal with the case that \(\Omega \) is a star graph which is a union of several half lines connected at the common end point (or a network of some special type) and consider the existence of time entire solutions and their structure. The “time entire” implies that a solution \(u=u(t,x)\) exists for all \(t\in (-\infty ,\infty )\). I explain some results obtained through the joint work with Y. Takazawa (Hokkaido Univ.) and Y. Morita (Ryukoku Univ.).

SUMMARY: This is joint work Jens Niklas Eberhardt. Categories of mixed \(l\)-adic sheaves and mixed Hodge modules are indispensable tools in geometric representation theory. They are used in the proof of the Kazhdan–Lusztig conjecture, uncover hidden gradings in categories of representations or categorify objects such as Hecke algebras, representations of quantum groups and link invariants, to name a few. But they are—by their nature—limited to characteristic zero coefficients.

In this talk, I will discuss a formalism of mixed sheaves with coefficients in characteristic \(p\) following ideas of Soergel, Wendt, and Virk to make use of the recent developments in the world of motivic sheaves. As an application, our work produces a geometric and graded version of Soergel’s modular category \(\mathcal {O}(G)\), consisting of rational representations of a split semisimple group \(G\) over a positive characteristic field, thereby equipping it with a full six functor formalism. In particular, one can express characters of irreducible modules of \(SL_n\) in terms of mixed motives.