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  • Rank and duality in representation theory
    Jpn. J. Math. (IF 1.636) Pub Date : 2020-05-19
    Shamgar Gurevich, Roger Howe

    There is both theoretical and numerical evidence that the set of irreducible representations of a reductive group over local or finite fields is naturally partitioned into families according to analytic properties of representations. Examples of such properties are the rate of decay at infinity of “matrix coefficients” in the local field setting, and the order of magnitude of “character ratios” in

  • Transgressions of the Euler class and Eisenstein cohomology of GL N (Z)
    Jpn. J. Math. (IF 1.636) Pub Date : 2020-03-04
    Nicolas Bergeron, Pierre Charollois, Luis E. Garcia

    These notes were written to be distributed to the audience of the first author’s Takagi Lectures delivered June 23, 2018. These are based on a work-in-progress that is part of a collaborative project that also involves Akshay Venkatesh. In this work-in-progress we give a new construction of some Eisenstein classes for GLN (Z) that were first considered by Nori [41] and Sczech [44]. The starting point

  • Recent results on the Kobayashi and Green-Griffiths-Lang conjectures
    Jpn. J. Math. (IF 1.636) Pub Date : 2020-01-29
    Jean-Pierre Demailly

    The study of entire holomorphic curves contained in projective algebraic varieties is intimately related to fascinating questions of geometry and number theory—especially through the concepts of curvature and positivity which are central themes in Kodaira’s contributions to mathematics. The aim of these lectures is to present recent results concerning the geometric side of the problem. The Green-Griffiths-Lang

  • Hankel transform, Langlands functoriality and functional equation of automorphic L -functions
    Jpn. J. Math. (IF 1.636) Pub Date : 2020-01-18
    Bảo Châu Ngô

    This is a survey on recent works of Langlands’s work on functoriality conjectures and related works including the works of Braverman and Kazhdan on the functional equation of automorphic L-functions. Efforts have been made to carry out in complete generality the construction of the L-monoid, and certain a kernel which is, we believe, related to the elusive Hankel kernel.

  • Double Yangian and the universal R -matrix
    Jpn. J. Math. (IF 1.636) Pub Date : 2019-12-28
    Maxim Nazarov

    We describe the double Yangian of the general linear Lie algebra glN by following a general scheme of Drinfeld. We also describe the centre of the Yangian by using its Hopf algebra structure, and provide a proof of the analogue of the Poincaré—Birkhoff—Witt theorem for the Yangian based on its representation theory. This proof extends to the double Yangian.

  • Abundance of minimal surfaces
    Jpn. J. Math. (IF 1.636) Pub Date : 2019-06-21
    Fernando Codá Marques

    This article is concerned with the existence theory of closed minimal hypersurfaces in closed Riemannian manifolds of dimension at least three. These hypersurfaces are critical points for the area functional, and hence their study can be seen as a high-dimensional generalization of the classical theory of closed geodesics (Birkhoff, Morse, Lusternik, Schnirel’mann,…). The best result until very recently

  • The twin prime conjecture
    Jpn. J. Math. (IF 1.636) Pub Date : 2019-06-21
    James Maynard

    The Twin Prime Conjecture asserts that there should be infinitely many pairs of primes which differ by 2. Unfortunately this long-standing conjecture remains open, but recently there has been several dramatic developments making partial progress. We survey the key ideas behind proofs of bounded gaps between primes (due to Zhang, Tao and the author) and developments on Chowla's conjecture (due to Matomäki

  • An operadic approach to vertex algebra and Poisson vertex algebra cohomology
    Jpn. J. Math. (IF 1.636) Pub Date : 2019-06-21
    Bojko Bakalov, Alberto De Sole, Reimundo Heluani, Victor G. Kac

    We translate the construction of the chiral operad by Beilinson and Drinfeld to the purely algebraic language of vertex algebras. Consequently, the general construction of a cohomology complex associated to a linear operad produces the vertex algebra cohomology complex. Likewise, the associated graded of the chiral operad leads to the classical operad, which produces a Poisson vertex algebra cohomology

  • Singularities in mixed characteristic. The perfectoid approach
    Jpn. J. Math. (IF 1.636) Pub Date : 2019-05-30
    Yves André

    The homological conjectures, which date back to Peskine, Szpiro and Hochster in the late 60’s, make fundamental predictions about syzygies and intersection problems in commutative algebra. They were settled long ago in the presence of a base field and led to tight closure theory, a powerful tool to investigate singularities in characteristic p. Recently, perfectoid techniques coming from p-adic Hodge

  • Brownian geometry
    Jpn. J. Math. (IF 1.636) Pub Date : 2019-05-27
    Jean-François Le Gall

    We present different continuous models of random geometry that have been introduced and studied in recent years. In particular, we consider the Brownian sphere (also called the Brownian map), which is the universal scaling limit of large planar maps in the Gromov-Hausdorff sense, and the Brownian disk, which appears as the scaling limit of planar maps with a boundary. We discuss the construction of

  • Information complexity and applications
    Jpn. J. Math. (IF 1.636) Pub Date : 2019-03-05
    Mark Braverman

    This paper is a lecture note accompanying the 19th Takagi Lectures lectures in July 2017 at Kyoto University. We give a high-level overview of information complexity theory and its connections to communication complexity.We then discuss some fundamental properties of information complexity, and applications to direct sum theorems and to exact communication bounds. We conclude with some open questions

  • Takagi Lectures on Donaldson–Thomas theory
    Jpn. J. Math. (IF 1.636) Pub Date : 2019-02-18
    Andrei Okounkov

    These are introductory notes on Donaldson–Thomas counts of curves in threefolds and their connections with other branches of mathematics and mathematical physics. They are based on my 2018 Takagi Lectures at The University of Tokyo.

  • Sharp threshold phenomena in statistical physics
    Jpn. J. Math. (IF 1.636) Pub Date : 2019-01-28
    Hugo Duminil-Copin

    This text describes the content of the Takagi Lectures given by the author in Kyoto in 2017. The lectures present some aspects of the theory of sharp thresholds for Boolean functions and its application to the study of phase transitions in statistical physics.

  • Renormalisation of parabolic stochastic PDEs
    Jpn. J. Math. (IF 1.636) Pub Date : 2018-08-02
    Martin Hairer

    We give a survey of recent result regarding scaling limits of systems from statistical mechanics, as well as the universality of the behaviour of such systems in so-called cross-over regimes. It transpires that some of these universal objects are described by singular stochastic PDEs. We then give a survey of the recently developed theory of regularity structures which allows to build these objects

  • Equivalence of formulations of the MKP hierarchy and its polynomial tau-functions
    Jpn. J. Math. (IF 1.636) Pub Date : 2018-07-15
    Victor G. Kac, Johan W. van de Leur

    We show that a system of Hirota bilinear equations introduced by Jimbo and Miwa defines tau-functions of the modified KP (MKP) hierarchy of evolution equations introduced by Dickey. Some other equivalent definitions of the MKP hierarchy are established. All polynomial tau-functions of the KP and the MKP hierarchies are found. Similar results are obtained for the reduced KP and MKP hierarchies.

  • Spectral asymptotics for Kac–Murdock–Szegő matrices
    Jpn. J. Math. (IF 1.636) Pub Date : 2018-03-02
    Alain Bourget, Allen Alvarez Loya, Tyler McMillen

    Szegő’s First Limit Theorem provides the limiting statistical distribution of the eigenvalues of large Toeplitz matrices. Szegő’s Second (or Strong) Limit Theorem for Toeplitz matrices gives a second order correction to the First Limit Theorem, and allows one to calculate asymptotics for the determinants of large Toeplitz matrices. In this paper we survey results extending the First and Second Limit

  • Hilbert schemes of lines and conics and automorphism groups of Fano threefolds
    Jpn. J. Math. (IF 1.636) Pub Date : 2018-02-14
    Alexander G. Kuznetsov, Yuri G. Prokhorov, Constantin A. Shramov

    We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds of Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold of index 1 and genus 10. We also show that the action of the automorphism group

  • Categorification of invariants in gauge theory and symplectic geometry
    Jpn. J. Math. (IF 1.636) Pub Date : 2017-11-30
    Kenji Fukaya

    This is a mixture of survey article and research announcement. We discuss instanton Floer homology for 3 manifolds with boundary. We also discuss a categorification of the Lagrangian Floer theory using the unobstructed immersed Lagrangian correspondence as a morphism in the category of symplectic manifolds. During the year 1998–2012, those problems have been studied emphasizing the ideas from analysis

  • The size of infinite-dimensional representations
    Jpn. J. Math. (IF 1.636) Pub Date : 2017-08-21
    David A. Vogan

    An infinite-dimensional representation π of a real reductive Lie group G can often be thought of as a function space on some manifold X. Although X is not uniquely defined by π, there are “geometric invariants” of π, first introduced by Roger Howe in the 1970s, related to the geometry of X. These invariants are easy to define but difficult to compute. I will describe some of the invariants, and recent

  • Algebraic representations and constructible sheaves
    Jpn. J. Math. (IF 1.636) Pub Date : 2017-08-21
    Geordie Williamson

    I survey what is known about simple modules for reductive algebraic groups. The emphasis is on characteristic p > 0 and Lusztig’s character formula. I explain ideas connecting representations and constructible sheaves (Finkelberg–Mirković conjecture) in the spirit of the Kazhdan–Lusztig conjecture. I also discuss a conjecture with S. Riche (a theorem for GL n ) which should eventually make computations

  • Conformal embeddings of affine vertex algebras in minimal W -algebras II: decompositions
    Jpn. J. Math. (IF 1.636) Pub Date : 2017-07-31
    Dražen Adamović, Victor G. Kac, Pierluigi Möseneder Frajria, Paolo Papi, Ozren Perše

    We present methods for computing the explicit decomposition of the minimal simple affine W-algebra \({W_k(\mathfrak{g}, \theta)}\) as a module for its maximal affine subalgebra \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) at a conformal level k, that is, whenever the Virasoro vectors of \({W_k(\mathfrak{g}, \theta)}\) and \({\mathscr{V}_k(\mathfrak{g}^\natural)}\) coincide. A particular emphasis is

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