• 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

更新日期：2020-05-19
• 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

更新日期：2020-03-04
• 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

更新日期：2020-01-29
• 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.

更新日期：2020-01-18
• 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.

更新日期：2019-12-28
• 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

更新日期：2019-06-21
• 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

更新日期：2019-06-21
• 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

更新日期：2019-06-21
• 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

更新日期：2019-05-30
• 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

更新日期：2019-05-27
• 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

更新日期：2019-03-05
• 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.

更新日期：2019-02-18
• 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.

更新日期：2019-01-28
• 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

更新日期：2018-08-02
• 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.

更新日期：2018-07-15
• 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

更新日期：2018-03-02
• 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

更新日期：2018-02-14
• 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

更新日期：2017-11-30
• 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

更新日期：2017-08-21
• 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

更新日期：2017-08-21
• 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

更新日期：2017-07-31
• Jpn. J. Math. (IF 1.636) Pub Date : 2017-07-31
Fabrizio Catanese

Kodaira fibred surfaces are remarkable examples of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications

更新日期：2017-07-31
• Jpn. J. Math. (IF 1.636) Pub Date : 2017-01-15
Sylvain Carpentier

For a rational differential operator $${L=AB^{-1}}$$, the Lenard–Magri scheme of integrability is a sequence of functions $${F_n, n \geq 0}$$, such that (1) $${B(F_{n+1})=A(F_n)}$$ for all $${n \geq 0}$$ and (2) the functions $${B(F_n)}$$ pairwise commute. We show that, assuming that property (1) holds and that the set of differential orders of $${B(F_n)}$$ is unbounded, property (2) holds if and only

更新日期：2017-01-15
• Jpn. J. Math. (IF 1.636) Pub Date : 2016-12-21
Akshay Venkatesh

We recall some unusual features of the cohomology of arithmetic groups, and propose that they are explained by a hidden action of certain motivic cohomology groups.

更新日期：2016-12-21
• Jpn. J. Math. (IF 1.636) Pub Date : 2016-08-29
Cédric Villani

Synthetic theory of Ricci curvature bounds is reviewed, from the conditions which led to its birth, up to some of its latest developments.

更新日期：2016-08-29
• Jpn. J. Math. (IF 1.636) Pub Date : 2016-08-09
Shing-Tung Yau

We survey the theory of complex manifolds that are related to Riemann surface, Hodge theory, Chern class, Kodaira embedding and Hirzebruch–Riemann–Roch, and some modern development of uniformization theorems, Kähler–Einstein metric and the theory of Donaldson–Uhlenbeck–Yau on Hermitian Yang–Mills connections. We emphasize mathematical ideas related to physics. At the end, we identify possible future

更新日期：2016-08-09
• Jpn. J. Math. (IF 1.636) Pub Date : 2016-06-29
Anatoly M. Vershik

The survey covers several topics related to the asymptotic structure of various combinatorial and analytic objects such as the path spaces in graded graphs (Bratteli diagrams), invariant measures with respect to countable groups, etc. The main subject is the asymptotic structure of filtrations and a new notion of standardness. All graded graphs and all filtrations of Borel or measure spaces can be

更新日期：2016-06-29
• Jpn. J. Math. (IF 1.636) Pub Date : 2016-06-29
Renzo Cavalieri

This survey grew out of notes accompanying a cycle of lectures at the workshop Modern Trends in Gromov–Witten Theory, in Hannover. The lectures are devoted to interactions between Hurwitz theory and Gromov–Witten theory, with a particular eye to the contributions made to the understanding of the Double Ramification Cycle, a cycle in the moduli space of curves that compactifies the double Hurwitz locus

更新日期：2016-06-29
• Jpn. J. Math. (IF 1.636) Pub Date : 2016-05-16
Alice Guionnet

We describe the Schwinger–Dyson equation related with the free difference quotient. Such an equation appears in different fields such as combinatorics (via the problem of the enumeration of planar maps), operator algebra (via the definition of a natural integration by parts in free probability), in classical probability (via random matrices or particles in repulsive interaction). In these lecture notes

更新日期：2016-05-16
• Jpn. J. Math. (IF 1.636) Pub Date : 2016-04-20
Vaughan F. R. Jones

Groups have played a big role in knot theory. We show how subfactors (subalgebras of certain von Neumann algebras) lead to unitary representations of the braid groups and Thompson’s groups $${F}$$ and $${T}$$. All knots and links may be obtained from geometric constructions from these groups. And invariants of knots may be obtained as coefficients of these representations. We include an extended introduction

更新日期：2016-04-20
• Jpn. J. Math. (IF 1.636) Pub Date : 2016-04-07
Masaki Kashiwara

This is a survey paper on the Riemann–Hilbert correspondence on (irregular) holonomic $${\mathscr{D}}$$-modules, based on the 16th Takagi Lectures (2015/11/28). In this paper, we use subanalytic sheaves, an analogous notion to the one of indsheaves.

更新日期：2016-04-07
• Jpn. J. Math. (IF 1.636) Pub Date : 2016-03-25
Michael J. Hopkins

The history and solution of the Kervaire invariant problem is discussed, along with some of the future prospects raised by its solution.

更新日期：2016-03-25
• Jpn. J. Math. (IF 1.636) Pub Date : 2015-11-17
Peter Scholze

This note explains some of the author’s work on understanding the torsion appearing in the cohomology of locally symmetric spaces such as arithmetic hyperbolic 3-manifolds.The key technical tool was a theory of Shimura varieties with infinite level at p: As p-adic analytic spaces, they are perfectoid, and admit a new kind of period map, called the Hodge–Tate period map, towards the flag variety. Moreover

更新日期：2015-11-17
• Jpn. J. Math. (IF 1.636) Pub Date : 2015-08-19
Ciprian Manolescu

We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg–Witten Floer homology, and its application to the triangulation conjecture.

更新日期：2015-08-19
• Jpn. J. Math. (IF 1.636) Pub Date : 2015-08-18
Luc Illusie

I discuss four unpublished letters of Deligne (one on Hodge theory, two on Euler–Poincaré characteristics and ramification of $${\ell}$$-adic sheaves, one on generalized divisors), and sketch some of the developments they generated.

更新日期：2015-08-18
• Jpn. J. Math. (IF 1.636) Pub Date : 2015-06-26
Maria Gorelik; Victor G. Kac

In the paper we consider the problem of computation of characters of relatively integrable irreducible highest weight modules L over finite-dimensional basic Lie superalgebras and over affine Lie superalgebras $${\mathfrak{g}}$$. The problem consists of two parts. First, it is the reduction of the problem to the $${\overline{\mathfrak{g}}}$$-module F(L), where $${\overline{\mathfrak{g}}}$$ is the associated

更新日期：2015-06-26
• Jpn. J. Math. (IF 1.636) Pub Date : 2014-12-19
Eiichi Bannai; Hajime Tanaka

This paper is Appendix of the paper of T. Ceccherini-Silberstein, F. Scarabotti and F. Tolli [8].We pay close attention on a special condition related to Gelfand pairs. Namely, we call a finite group G and its automorphism $${\sigma}$$ satisfy Condition ($${\bigstar}$$) if the following condition is satisfied: if for $${x,\,y\,\in G}$$, $${x\,\cdot\, x^{-\sigma}}$$ and $${y\,\cdot\, y^{-\sigma}}$$

更新日期：2014-12-19
• Jpn. J. Math. (IF 1.636) Pub Date : 2014-12-18
Tullio Ceccherini-Silberstein; Fabio Scarabotti; Filippo Tolli

The aim of the present paper is to expose two contributions of Mackey, together with a more recent result of Kawanaka and Matsuyama, generalized by Bump and Ginzburg, on the representation theory of a finite group equipped with an involutory anti-automorphism (e.g. the anti-automorphism $${g \mapsto g^{-1}}$$). Mackey’s first contribution is a detailed version of the so-called Gelfand criterion for

更新日期：2014-12-18
• Jpn. J. Math. (IF 1.636) Pub Date : 2014-12-06
Gang Tian

This is an expository paper on Kähler metrics of positive scalar curvature. It is for my Takagi Lectures at RIMS in November of 2013. In this paper, I first discuss the Futaki invariants, the K-stability and its relation to the K-energy. Next I will outline my work in 2012 on the existence of Kähler–Einstein metrics on K-stable Fano manifolds. Finally, I will present S. Paul’s work on stability of

更新日期：2014-12-06
• Jpn. J. Math. (IF 1.636) Pub Date : 2014-07-31
Yuval Z. Flicker

These are purely expository notes of Opdam’s analysis [O1] of the trace form τ(f) = f(e) on the Hecke algebra H = C c (I\G/I) of compactly supported functions f on a connected reductive split p-adic group G which are biinvariant under an Iwahori subgroup I, extending Macdonald’s work. We attempt to give details of the proofs, and choose notations which seem to us more standard. Many objects of harmonic

更新日期：2014-07-31
• Jpn. J. Math. (IF 1.636) Pub Date : 2014-07-11
Alexander Lubotzky

Expander graphs in general, and Ramanujan graphs in particular, have been of great interest in the last four decades with many applications in computer science, combinatorics and even pure mathematics. In these notes we describe various efforts made in recent years to generalize these notions from graphs to higher dimensional simplicial complexes.

更新日期：2014-07-11
• Jpn. J. Math. (IF 1.636) Pub Date : 2014-05-23
Anne-Marie Aubert; Paul Baum; Roger Plymen; Maarten Solleveld

This expository paper first reviews some basic facts about p-adic fields, reductive p-adic groups, and the local Langlands conjecture. If G is a reductive p-adic group, then the smooth dual of G is the set of equivalence classes of smooth irreducible representations of G. The representations are on vector spaces over the complex numbers. In a canonical way, the smooth dual is the disjoint union of

更新日期：2014-05-23
• Jpn. J. Math. (IF 1.636) Pub Date : 2014-02-15
Hee Oh

We give an overview of various counting problems for Apollonian circle packings, which turn out to be related to problems in dynamics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared for the 13th Takagi Lectures given at RIMS, Kyoto in the fall of 2013.

更新日期：2014-02-15
• Jpn. J. Math. (IF 1.636) Pub Date : 2014-02-05
Laurent Lafforgue

The main purpose of this paper is to show that some type of explicit nonlinear Poisson formulas, which is implied by Langlands’ functoriality principle, allows to build “kernels” of automorphic transfer. So, Langlands’ functoriality principle is equivalent to these nonlinear Poisson formulas.

更新日期：2014-02-05
• Jpn. J. Math. (IF 1.636) Pub Date : 2013-09-24
Alberto De Sole; Victor G. Kac

We develop a rigorous theory of non-local Poisson structures, built on the notion of a non-local Poisson vertex algebra. As an application, we find conditions that guarantee applicability of the Lenard–Magri scheme of integrability to a pair of compatible non-local Poisson structures. We apply this scheme to several such pairs, proving thereby integrability of various evolution equations, as well as

更新日期：2013-09-24
• Jpn. J. Math. (IF 1.636) Pub Date : 2013-09-24
Robert Seiringer

We present an overview of mathematical results on the low temperature properties of dilute quantum gases, which have been obtained in the past few years. The presentation includes a discussion of Bose–Einstein condensation, the excitation spectrum for trapped gases and its relation to superfluidity, as well as the appearance of quantized vortices in rotating systems. All these properties are intensely

更新日期：2013-09-24
• Jpn. J. Math. (IF 1.636) Pub Date : 2013-03-20
Alberto De Sole; Victor G. Kac

It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology

更新日期：2013-03-20
• Jpn. J. Math. (IF 1.636) Pub Date : 2013-03-20
Narutaka Ozawa

In his celebrated paper in 1976, A. Connes casually remarked that any finite von Neumann algebra ought to be embedded into an ultraproduct of matrix algebras, which is now known as the Connes embedding conjecture or problem. This conjecture became one of the central open problems in the field of operator algebras since E. Kirchberg’s seminal work in 1993 that proves it is equivalent to a variety of

更新日期：2013-03-20
• Jpn. J. Math. (IF 1.636) Pub Date : 2012-11-17
Igor E. Shparlinski

We give a survey of a variety of recent results about the distribution and some geometric properties of points (x, y) on modular hyperbolas $${xy \equiv a\;(\mod m)}$$. We also outline a very diverse range of applications of such results, discuss multivariate generalisations and suggest a number of open problems of different levels of difficulty.

更新日期：2012-11-17
• Jpn. J. Math. (IF 1.636) Pub Date : 2012-11-17
Assaf Naor

We survey problems, results, ideas, and recent progress in the Ribe program. The goal of this research program, which is motivated by a classical rigidity theorem of Martin Ribe, is to obtain structural results for metric spaces that are inspired by the local theory of Banach spaces. We also present examples of applications of the Ribe program to several areas, including group theory, theoretical computer

更新日期：2012-11-17
• Jpn. J. Math. (IF 1.636) Pub Date : 2012-11-17
Yves Benoist; Jean-François Quint

Let a 0 and a 1 be two matrices in SL(2, $${\mathbb{Z}}$$) which span a non-solvable group. Let x 0 be an irrational point on the torus $${\mathbb{T}^2}$$. We toss a 0 or a 1, apply it to x 0, get another irrational point x 1, do it again to x 1, get a point x 2, and again. This random trajectory is equidistributed on the torus. This phenomenon is quite general on any finite volume homogeneous space

更新日期：2012-11-17
• Jpn. J. Math. (IF 1.636) Pub Date : 2012-03-26

This is an expository article on modern crystallography based on discrete geometric analysis, a hybrid field of several traditional disciplines: graph theory, geometry, theory of discrete groups, and probability, which has been developed in the last decade. The mathematical part relying on algebraic topology is fairly elementary, but may be still worthwhile for crystallographers who want to learn how

更新日期：2012-03-26
• Jpn. J. Math. (IF 1.636) Pub Date : 2012-03-26
Maria Gorelik; Victor G. Kac; Pierluigi Möseneder Frajria; Paolo Papi

We provide formulas for the denominator and superdenominator of a basic classical type Lie superalgebra for any set of positive roots. We establish a connection between certain sets of positive roots and the theory of reductive dual pairs of real Lie groups, and, as an application of these formulas, we recover the Theta correspondence for compact dual pairs. Along the way we give an explicit description

更新日期：2012-03-26
• Jpn. J. Math. (IF 1.636) Pub Date : 2011-12-25
Carlos Kenig

We discuss recent progress in the understanding of the global behavior of solutions to critical non-linear dispersive equations. The emphasis is on global existence, scattering and finite time blow-up. For solutions that are bounded in the critical norm, but which blow-up in finite time, we also discuss the issue of universal profiles at the blow-up time.

更新日期：2011-12-25
• Jpn. J. Math. (IF 1.636) Pub Date : 2011-12-25
Sergei Gukov

When combined with mirror symmetry, the A-model approach to quantization leads to a fairly simple and tractable problem. The most interesting part of the problem then becomes finding the mirror of the coisotropic brane. We illustrate how it can be addressed in a number of interesting examples related to representation theory and gauge theory, in which mirror geometry is naturally associated with the

更新日期：2011-12-25
• Jpn. J. Math. (IF 1.636) Pub Date : 2011-09-28
Simon Brendle

A fundamental question in Riemannian geometry is to find canonical metrics on a given smooth manifold. In the 1980s, R.S. Hamilton proposed an approach to this question based on parabolic partial differential equations. The goal is to start from a given initial metric and deform it to a canonical metric by means of an evolution equation. There are various natural evolution equations for Riemannian

更新日期：2011-09-28
• Jpn. J. Math. (IF 1.636) Pub Date : 2011-09-28
Alain Connes

In these lectures we survey some relations between L-functions and the BC-system, including new results obtained in collaboration with C. Consani. For each prime p and embedding σ of the multiplicative group of an algebraic closure of $${\mathbb {F}_p}$$ as complex roots of unity, we construct a p-adic indecomposable representation πσ of the integral BC-system. This construction is done using the identification

更新日期：2011-09-28
• Jpn. J. Math. (IF 1.636) Pub Date : 2010-12-25
Marcin Mazur; Bogdan V. Petrenko

A recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has $${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})$$. We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A, B are congruent modulo

更新日期：2010-12-25
• Jpn. J. Math. (IF 1.636) Pub Date : 2010-12-25
Mikhail Khovanov

A diagrammatic presentation of functors and natural transformations and the virtues of biadjointness are discussed. We then review a graphical description of the category of Soergel bimodules and a diagrammatic categorification of positive halves of quantum groups. These notes are a write-up of Takagi Lectures given by the author in Hokkaido University in June 2009.

更新日期：2010-12-25
• Jpn. J. Math. (IF 1.636) Pub Date : 2010-04-02
Michael Harris

This expository article is an introduction to the Langlands functoriality conjectures and their applications to the arithmetic of representations of Galois groups of number fields. Thanks to the work of a great many people, the stable trace formula is now largely established in a version adequate for proving Langlands functoriality in the setting of endoscopy. These developments are discussed in the

更新日期：2010-04-02
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