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Publications of the Research Institute for Mathematical Sciences
Publ. Res. Inst. Math. Sci.
PRIMS
0034-5318
1663-4926
General
10.4171/PRIMS
http://www.ems-ph.org/doi/10.4171/PRIMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Research Institute for Mathematical Sciences, Kyoto University
47
2011
1
Preface
General
1
9
10.4171/PRIMS/47.1.1
http://www.ems-ph.org/doi/10.4171/PRIMS/47.1.1
Professor Mikio Sato and Microlocal Analysis
Masaki
Kashiwara
Kyoto University, KYOTO, JAPAN
Takahiro
Kawai
Kyoto University, KYOTO, JAPAN
microlocal analysis, hyperfunctions, microfunctions
We describe the impact of microlocal analysis on mathematical sciences and the role Prof. Mikio Sato played in its creation and development.
Several complex variables and analytic spaces
Commutative rings and algebras
Algebraic geometry
Partial differential equations
11
17
10.2977/PRIMS/29
http://www.ems-ph.org/doi/10.2977/PRIMS/29
Mikio Sato and Mathematical Physics
Barry
McCoy
State Univ of New York at Stony Brook, STONY BROOK, UNITED STATES
Ising model, statistical mechanics
I present the deep and lasting contributions of Mikio Sato to the mathematical physics of statistical mechanics and random matrix theory.
History and biography
General
19
28
10.2977/PRIMS/30
http://www.ems-ph.org/doi/10.2977/PRIMS/30
A Family of Calabi–Yau Varieties and Potential Automorphy II
Tom
Barnet-Lamb
Brandeis University, WALTHAM, UNITED STATES
David
Geraghty
Institute for Advanced Study, PRINCETON, UNITED STATES
Michael
Harris
Tour 15-25, 4, PARIS CEDEX 05, FRANCE
Richard
Taylor
Harvard University, CAMBRIDGE, UNITED STATES
automorphic representation, Galois representation, Dwork family, Sato–Tate conjecture
We prove new potential modularity theorems for n-dimensional essentially self-dual l-adic representations of the absolute Galois group of a totally real eld. Most notably, in the ordinary case we prove quite a general result. Our results suffice to show that all the symmetric powers of any non-CM, holomorphic, cuspidal, elliptic modular newform of weight greater than one are potentially cuspidal automorphic. This in turns proves the Sato–Tate conjecture for such forms. (In passing we also note that the Sato–Tate conjecture can now be proved for any elliptic curve over a totally real eld.)
Number theory
General
29
98
10.2977/PRIMS/31
http://www.ems-ph.org/doi/10.2977/PRIMS/31
dg-Methods for Microlocalization
Stéphane
Guillermou
Université de Grenoble I, SAINT-MARTIN D'HERES, FRANCE
D-modules, microdifferential operators, microlocalization
For a complex manifold $X$ the ring of microdifferential operators $\mathcal{E}_X$ acts on the microlocalization $\mu hom(F,\mathcal{O}_X)$ for $F$ in the derived category of sheaves on $X$. Kashiwara, Schapira, Ivorra, Waschkies proved as a byproduct of their new microlocalization functor for ind-sheaves, $\mu_X$, that $\mu hom(F,\mathcal{O}_X)$ can in fact be defined as an object of $\mathrm{D}(\mathcal{E}_X)$: this follows from the fact that $\mu_X \mathcal{O}_X$ is concentrated in one degree. In this paper we prove that the tempered microlocalization $T - \mu hom(F,\mathcal{O}_X)$ and in fact $\mu_X \mathcal{O}_X^t$ also are objects of $\mathrm{D}(\mathcal{E}_X)$. Since we don't know whether $\mu_X \mathcal{O}_X^t$ is concentrated in one degree we built resolutions of $\mathcal{E}_X$ and $\mu_X \mathcal{O}_X^t$ such that the action of $\mathcal{E}_X$ is realized in the category of complexes (and not only up to homotopy). To define these resolutions we introduce a version of the de Rham algebra on the subanalytic site which is quasi-injective. We prove that some standard operations in the derived category of sheaves can be lifted to the (non-derived) category of dg-modules over this de Rham algebra. Then we built the microlocalization in this framework.
Partial differential equations
Several complex variables and analytic spaces
General
99
140
10.2977/PRIMS/32
http://www.ems-ph.org/doi/10.2977/PRIMS/32
Asymptotic Equivariant Index of Toeplitz Operators on the Sphere
Louis
Boutet de Monvel
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Toeplitz operators, index, equivariant K-theory, contact manifolds
We illustrate the equivariant asymptotic index described in [6, 8] in the case of spheres $\mathbb{S}^{2N-1}\subset\mathbb{C}^N$, equipped with a unitary action of a compact group, for which this theory is more explicit. The article is mostly a review article, except for the last section (§5) in which we describe conjecturally some very natural generators of the relevant K-theory for a torus action on a sphere, generalizing in our Toeplitz operator context the generators proposed by M. F. Atiyah [2] for the transversally elliptic pseudodi erential theory.
$K$-theory
Several complex variables and analytic spaces
Differential geometry
Global analysis, analysis on manifolds
141
151
10.2977/PRIMS/33
http://www.ems-ph.org/doi/10.2977/PRIMS/33
WKB analysis of higher order Painlevé equations with a large parameter. II. Structure theorem for instanton-type solutions of $(P_J)_m$ (J= I, 34, II-2 or IV) near a simple $P$-turning point of the first kind
Takahiro
Kawai
Kyoto University, KYOTO, JAPAN
Yoshitsugu
Takei
Kyoto University, KYOTO, JAPAN
higher order Painlevé equations, Painlevé hierarchy, exact WKB analysis, instanton-type solutions, P-turning points
This is the third one of a series of articles on the exact WKB analysis of higher order Painlev é equations $(P_J)_m$ with a large parameter (J = I, II, IV; m = 1; 2; 3;…); the series is intended to clarify the structure of solutions of $(P_J)_m$ by the exact WKB analysis of the underlying overdetermined system (DSLJ)m of linear diff erential equations, and the target of this paper is instanton-type solutions of $(P_J)_m$. In essence, the main result (Theorem 5.1.1) asserts that, near a simple P-turning point of the rst kind, each instanton-type solution of (PJ )m can be formally and locally transformed to an appropriate solution of (PI)1, the classical (i.e., the second order) Painlevé-I equation with a large parameter. The transformation is attained by constructing a WKB-theoretic transformation that brings a solution of (DSLJ)m to a solution of its canonical form (DCan) (§5.3).
Ordinary differential equations
Special functions
General
153
219
10.2977/PRIMS/34
http://www.ems-ph.org/doi/10.2977/PRIMS/34
Regular holonomic $\mathscr{D}{[\mspace{-1mu}[\hbar]\mspace{-1mu}]}$-modules
Andrea
D'Agnolo
Università di Padova, PADOVA, ITALY
Stéphane
Guillermou
Université de Grenoble I, SAINT-MARTIN D'HERES, FRANCE
Pierre
Schapira
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
D-modules, deformation-quantization, Riemann–Hilbert correspondence, torsion pairs
We describe the category of regular holonomic modules over the ring $\mathscr{D}{[\mspace{-1mu}[\hbar]\mspace{-1mu}]}$ of linear differential operators with a formal parameter $\hbar$. In particular, we establish the Riemann-Hilbert correspondence and discuss the additional $t$-structure related to $\hbar$-torsion.
Several complex variables and analytic spaces
Functional analysis
General
221
255
10.2977/PRIMS/35
http://www.ems-ph.org/doi/10.2977/PRIMS/35
On Certain Arithmetic Functions M (s; z1; z2) Associated with Global Fields: Analytic Properties
Yasutaka
Ihara
Kyoto University, KYOTO, JAPAN
L-function, arithmetic function, critical point, Plancherel equality, probabilistic inequality.
The arithmetic functions in the title arose from value-distribution theories related to L-functions of global elds. They are "complexi cations" of the Fourier duals of the corresponding density functions. We shall study their complex analytic properties including analytic continuations and the limit behaviors at the critical point s = 1/2.
Number theory
Algebraic geometry
Probability theory and stochastic processes
General
257
305
10.2977/PRIMS/36
http://www.ems-ph.org/doi/10.2977/PRIMS/36
Donaldson = Seiberg–Witten from Mochizuki's Formula and Instanton Counting
Lothar
Göttsche
International Centre for Theoretical Physics, TRIESTE, ITALY
Hiraku
Nakajima
Kyoto University, KYOTO, JAPAN
Kota
Yoshioka
Kobe University, KOBE, JAPAN
Donaldson invariants, Seiberg{Witten invariants, instanton counting
We propose an explicit formula connecting Donaldson invariants and Seiberg–Witten invariants of a 4-manifold of simple type via Nekrasov's deformed partition function for the N = 2 SUSY gauge theory with a single fundamental matter. This formula is derived from Mochizuki's formula, which makes sense and was proved when the 4-manifold is complex projective. Assuming our formula is true for a 4-manifold of simple type, we prove Witten's conjecture and sum rules for Seiberg–Witten invariants (superconformal simple type condition), conjectured by Mariño, Moore and Peradze.
Algebraic geometry
Manifolds and cell complexes
Quantum theory
General
307
359
10.2977/PRIMS/37
http://www.ems-ph.org/doi/10.2977/PRIMS/37
Painlevé Functions in Statistical Physics
Craig
Tracy
University of California at Davis, DAVIS, UNITED STATES
Harold
Widom
University of California at Santa Cruz, SANTA CRUZ, UNITED STATES
Painlevé function, ASEP, Bethe Ansatz, KPZ scaling
We review recent progress in limit laws for the one-dimensional asymmetric simple exclusion process (ASEP) on the integer lattice. The limit laws are expressed in terms of a certain Painlevé II function. Furthermore, we take this opportunity to give a brief survey of the appearance of Painlevé functions in statistical physics.
Ordinary differential equations
Probability theory and stochastic processes
Statistical mechanics, structure of matter
General
361
374
10.2977/PRIMS/38
http://www.ems-ph.org/doi/10.2977/PRIMS/38
On the Voros Coefficient for the Whittaker Equation with a Large Parameter – Some Progress around Sato's Conjecture in Exact WKB Analysis
Tatsuya
Koike
Kyoto University, KYOTO, JAPAN
Yoshitsugu
Takei
Kyoto University, KYOTO, JAPAN
exact WKB analysis, Whittaker equation, Voros coefficient, alien derivative, Stokes automorphism
Generalizing Sato's conjecture for the Weber equation in exact WKB analysis, we explicitly determine the Voros coeffi cient of the Whittaker equation with a large parameter. By using our results we also compute alien derivatives of WKB solutions of the Whittaker equation at the so-called xed singular points of their Borel transform.
Ordinary differential equations
General
375
395
10.2977/PRIMS/39
http://www.ems-ph.org/doi/10.2977/PRIMS/39
2
On M. Sato's Classification of Some Reductive Prehomogeneous Vector Spaces
Tatsuo
Kimura
University of Tsukuba, IBARAKI, JAPAN
Yukimi
Ishii
University of Tsukuba, IBARAKI, JAPAN
Inhyun
Ryu
University of Tsukuba, IBARAKI, JAPAN
Michio
Hamada
University of Tsukuba, IBARAKI, JAPAN
Yoshiteru
Kurosawa
University of Tsukuba, IBARAKI, JAPAN
Masaya
Ouchi
University of Tsukuba, IBARAKI, JAPAN
Tomohiro
Kamiyoshi
University of Tsukuba, IBARAKI, JAPAN
prehomogeneous vector spaces, classification
Under some condition, M. Sato classified reductive prehomogeneous vector spaces of the form $(G_{0}\times G,\Lambda _{1} \otimes \rho ,V(n)\otimes V)$. In this paper, under the other condition, we classify the prehomogeneous vector spaces of the same form. We consider everything over the complex number field $\mathbb{C}$.
Number theory
Group theory and generalizations
General
397
418
10.2977/PRIMS/40
http://www.ems-ph.org/doi/10.2977/PRIMS/40
Asymptotic Behaviour of Variation of Pure Polarized TERP Structure
Takuro
Mochizuki
Kyoto University, KYOTO, JAPAN
harmonic bundle, TERP structure, new supersymmetric index
The purpose of this paper is twofold. One is to give a survey of our study on reductions of harmonic bundles, and the other is to explain a simple application in the study of TERP structures. In particular, we investigate the asymptotic behaviour of the "new supersymmetric index" for variation of pure polarized TERP structure.
Several complex variables and analytic spaces
Algebraic geometry
General
419
534
10.2977/PRIMS/41
http://www.ems-ph.org/doi/10.2977/PRIMS/41
Gelfand-Zetlin basis, Whittaker Vectors and a Bosonic Formula for the ${\mathfrak{sl}}_{n+1}$ Principal Subspace
Boris
Feigin
Independent University of Moscow, MOSCOW, RUSSIAN FEDERATION
Michio
Jimbo
Rikkyo University, TOKYO, JAPAN
Tetsuji
Miwa
Kyoto University, KYOTO, JAPAN
difference Toda Hamiltonian, quantum groups, fermionic formulas, bosonic formulas
We derive a bosonic formula for the character of the principal space in the level $k$ vacuum module for $\widehat{\mathfrak{sl}}_{n+1}$, starting from a known fermionic formula for it. In our previous work, the latter was written as a sum consisting of Shapovalov scalar products of the Whittaker vectors for $U_{v^{\pm1}}(\mathfrak{gl}_{n+1})$. In this paper we compute these scalar products in the bosonic form, using the decomposition of the Whittaker vectors in the Gelfand-Zetlin basis. We show further that the bosonic formula obtained in this way is the quasi-classical decomposition of the fermionic formula. %for $U_{v^{\pm1}}(\mathfrak{gl}_{n+1})$ % modules computing it %by using the decomposition %of the Whittaker vectors in the Gelfand-Zetlin basis. %We show that the bosonic formula obtained in this way %is the quasi-classical decomposition of the fermionic formula.
Associative rings and algebras
General
535
551
10.2977/PRIMS/42
http://www.ems-ph.org/doi/10.2977/PRIMS/42
Baker–Akhiezer Modules on the Intersections of Shifted Theta Divisors
Koji
Cho
Kyushu University, FUKUOKA, JAPAN
Andrey
Mironov
Siberian Branch of the Russian Academy of Sciences, NOVOSIBIRSK, RUSSIAN FEDERATION
Atsushi
Nakayashiki
Tsuda College, TOKYO, JAPAN
abelian variety, theta function, D-module, commuting differential operators
The restriction, on the spectral variables, of the Baker–Akhiezer (BA) module of a g-dimensional principally polarized abelian variety with the non-singular theta divisor to an intersection of shifted theta divisors is studied. It is shown that the restriction to a k-dimensional variety becomes a free module over the ring of di erential operators in k variables. The remaining g–k derivations de ne evolution equations for generators of the BA-module. As a corollary new examples of commutative rings of partial di fferential operators with matrix coe cients and their non-trivial evolution equations are obtained.
Algebraic geometry
General
553
567
10.2977/PRIMS/43
http://www.ems-ph.org/doi/10.2977/PRIMS/43
Wall Crossing and M-Theory
Mina
Aganagic
University of California, BERKELEY, UNITED STATES
Hirosi
Ooguri
California Institute of Technology, PASADENA, UNITED STATES
Cumrun
Vafa
Harvard University, Cambridge, UNITED STATES
Masahito
Yamazaki
California Institute of Technology, PASADENA, UNITED STATES
string theory, bound state, Fock space, Calabi–Yau manifold
We study BPS bound states of D0 and D2 branes on a single D6 brane wrapping a Calabi-Yau 3-fold $X$. When $X$ has no compact 4-cyles, the BPS bound states are organized into a free field Fock space, whose generators correspond to BPS states of spinning M2 branes in M-theory compactified down to 5 dimensions by a Calabi-Yau 3-fold $X$. The generating function of the D-brane bound states is expressed as a reduction of the square of the topological string partition function, in all chambers of the Kähler moduli space.
Algebraic geometry
General
569
584
10.2977/PRIMS/44
http://www.ems-ph.org/doi/10.2977/PRIMS/44
Algebraic Analysis of Minimal Representations
Toshiyuki
Kobayashi
University of Tokyo, TOKYO, JAPAN
minimal representations, hyperfunction, branching law, reductive group, generalized Fourier transform, holomorphic semigroup, conservative quantity, ${\mathcal{D}}$-module
Small representations of a group yield large symmetries in the representation space. Analysis of minimal representations utilizes large symmetries in their geometric models, and serves as a driving force in creating new interesting problems that interact with other branches of mathematics. This article discusses the following three topics that arise from minimal representations of the inde nite orthogonal group: 1. construction of conservative quantities for ultra-hyperbolic equations, 2. quantitative discrete branching laws, 3. deformation of the Fourier transform, with emphasis on the prominent role of Sato's ideas in algebraic analysis.
Topological groups, Lie groups
Functional analysis
Differential geometry
Global analysis, analysis on manifolds
585
611
10.2977/PRIMS/45
http://www.ems-ph.org/doi/10.2977/PRIMS/45
Microlocal Geometry and Valued Fields
François
Loeser
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
non-archimedean geometry, valued fields, Milnor fiber, tangent cones, microlocal geometry
This is an overview of some recent connections between non-archimedean geometry and microlocal analysis, with some emphasis on the motivic Milnor fiber and the properties of tangent cones.
Mathematical logic and foundations
Number theory
Algebraic geometry
General
613
627
10.2977/PRIMS/46
http://www.ems-ph.org/doi/10.2977/PRIMS/46
The Laplace Transform of the Cut-and-Join Equation and the Bouchard–Mariño Conjecture on Hurwitz Numbers
Bertrand
Eynard
CEA Saclay, GIF-SUR-YVETTE CEDEX, FRANCE
Motohico
Mulase
University of California at Davis, DAVIS, UNITED STATES
Bradley
Safnuk
Central Michigan University, MOUNT PLEASANT, UNITED STATES
Hurwitz numbers, cut-and-join equation, linear Hodge integral, topological recursion, Lambert curve
We calculate the Laplace transform of the cut-and-join equation of Goulden, Jackson and Vakil. The result is a polynomial equation that has the topological structure identical to the Mirzakhani recursion formula for the Weil–Petersson volume of the moduli space of bordered hyperbolic surfaces. We fi nd that the direct image of this Laplace transformed equation via the inverse of the Lambert W-function is the topological recursion formula for Hurwitz numbers conjectured by Bouchard and Mariño using topological string theory.
Algebraic geometry
Combinatorics
Quantum theory
General
629
670
10.2977/PRIMS/47
http://www.ems-ph.org/doi/10.2977/PRIMS/47
3
Matroids on Convex Geometries: Subclasses, Operations, and Optimization
Yoshio
Sano
Kyoto University, KYOTO, JAPAN
matroid, convex geometry, independent set, spanning set
A matroid-like structure de ned on a convex geometry, called a cg-matroid, was introduced by S. Fujishige, G. A. Koshevoy, and Y. Sano [Matroids on convex geometries (cg-matroids), Discrete Math. 307 (2007) 1936{1950]. In this paper, we continue the study of cg-matroids and extend the theory of cg-matroids. We give some characterizations of cg-matroids by axioms. Strict cg-matroids are a special subclass of cg-matroids which have nice properties. We de ne another subclass of cg-matroids, called co-strict cg-matroids, which also have good properties. Moreover, we consider operations on cg-matroids such as restriction and contraction. These operations are closely related to subclasses of cg-matroids. We also consider an optimization problem on cg-matroids, which reveals the relation between the greedy algorithm and cg-matroids.
Combinatorics
Convex and discrete geometry
Operations research, mathematical programming
General
671
703
10.2977/PRIMS/48
http://www.ems-ph.org/doi/10.2977/PRIMS/48
Local Duality and Polarized Hodge Modules
Christian
Schnell
Stony Brook University, STONY BROOK, UNITED STATES
polarized Hodge module, local duality, Cohen–Macaulay property, holonomicity, characteristic variety, de Rham complex
We establish a relationship between the graded quotients of a filtered holonomic ${\mathcal{D}}$-module, their duals as coherent sheaves, and the characteristic variety, in case the filtered $\mathcal{D}$-module underlies a polarized Hodge module on a smooth algebraic variety. The proof is based on M.~Saito's result that the associated graded module is Cohen-Macaulay, and on local duality for the cotangent bundle. The result plays a role in the study of Néron models for families of intermediate Jacobians, recently constructed by the author.
Several complex variables and analytic spaces
Algebraic geometry
General
705
725
10.2977/PRIMS/49
http://www.ems-ph.org/doi/10.2977/PRIMS/49
Fundamental Theorems for the Log Minimal Model Program
Osamu
Fujino
Kyoto University, KYOTO, JAPAN
log minimal model program, log canonical pairs, vanishing theorems, non-vanishing theorem, base point free theorem, rationality theorem, cone theorem, lengths of extremal rays
In this paper, we prove the cone theorem and the contraction theorem for pairs $(X, B)$, where $X$ is a normal variety and $B$ is an effective $\mathbb R$-divisor on $X$ such that $K_X+B$ is $\mathbb R$-Cartier.
Algebraic geometry
General
727
789
10.2977/PRIMS/50
http://www.ems-ph.org/doi/10.2977/PRIMS/50
The Dirac-Hardy and Dirac-Sobolev inequalities in $L^1$
Alexander
Balinsky
University of Wales Cardiff, CARDIFF, UNITED KINGDOM
W. Desmond
Evans
University of Wales Cardiff, CARDIFF, UNITED KINGDOM
Tomio
Umeda
University of Hyogo, HIMEJI, JAPAN
Dirac–Sobolev inequalities, Dirac–Hardy inequalities, zero modes, Sobolev inequalities, Hardy inequalities
Dirac-Sobolev and Dirac-Hardy inequalities in $L^1$ are established in which the $L^p$ spaces which feature in the classical Sobolev and Hardy inequalities are replaced by weak $L^p$ spaces. Counter examples to the analogues of the classical inequalities are shown to be provided by zero modes for appropriate Pauli operators constructed by Loss and Yau.
Partial differential equations
General
791
801
10.2977/PRIMS/51
http://www.ems-ph.org/doi/10.2977/PRIMS/51
Néron Models of Green–Griffiths–Kerr and log Néron Models
Tatsuki
Hayama
National Taiwan University, TAIPEI, TAIWAN
Néron model, log mixed Hodge structure, admissible normal function, intermediate Jacobian
For a variation of Hodge structure over a punctured disk, Green, Gri ths and Kerr introduced a Néeron model which is a Hausdor ff space that includes values of admissible normal functions. On the other hand, Kato, Nakayama and Usui introduced a N éron model as a logarithmic manifold using log mixed Hodge theory. This work constructs a homeomorphism between these two models.
Algebraic geometry
General
803
824
10.2977/PRIMS/52
http://www.ems-ph.org/doi/10.2977/PRIMS/52
4
Representations of Tame Quivers and Affine Canonical Bases
Zongzhu
Lin
Kansas State University, MANHATTAN, UNITED STATES
Jie
Xiao
Tsinghua University, BEIJING, CHINA
Guanglian
Zhang
Shanghai Jiao Tong University, SHANGHAI, CHINA
affine canonical bases, tame quivers, perverse sheaves.
There are several diff erent ways to construct affi ne canonical bases, in addition to approaches by Lusztig and Kashiwara. In this paper we present a diff erent approach to canonical bases via Hall algebras and representations of tame quivers over fi nite fields. The main idea is to tensor together integral bases constructed for cyclic quivers and Kronecker quivers with those from the preinjective and preprojective parts of tame quiver representations. Several diff erent bases: a PBW type basis, a monomial basis, and a bar-invariant basis are constructed and their relations to the canonical basis are discussed. The result also answers a question by Nakajima.
Nonassociative rings and algebras
Associative rings and algebras
General
825
885
10.2977/PRIMS/53
http://www.ems-ph.org/doi/10.2977/PRIMS/53
Noncommutative Topological Entropy of Endomorphisms of Cuntz Algebras II
Adam
Skalski
Polish Academy of Sciences, WARSZAWA, POLAND
noncommutative topological entropy, Cuntz algebra, polynomial endomorphisms
A study of noncommutative topological entropy of gauge invariant endomorphisms of Cuntz algebras began in our earlier work with J.\,Zacharias is continued and extended to endomorphisms which are not necessarily of permutation type. In particular it is shown that if $\mathsf{H}$ is an $N$-dimensional Hilbert space, $V$ is an irreducible multiplicative unitary on $\mathsf{H} \otimes \mathsf{H}$ and $F: \mathsf{H} \otimes \mathsf{H} \to \mathsf{H} \otimes \mathsf{H}$ is the tensor flip, then the Voiculescu entropy of the Longo's canonical endomorphism $\rho_{VF} \in {\textrm{End}}(\mathcal{O}_N)$ is equal to $\log N$.
Functional analysis
Dynamical systems and ergodic theory
General
887
896
10.2977/PRIMS/54
http://www.ems-ph.org/doi/10.2977/PRIMS/54
Periodicity for Mumford–Morita–Miller Classes of Surface Symmetries
Toshiyuki
Akita
Hokkaido University, SAPPORO, JAPAN
surface symmetry, Mumford–Morita–Miller class, group cohomology
We prove periodicity for mod p Mumford–Morita–Miller classes of surface symmetries and thereby for nite subgroups of mapping class groups. As an application, we obtain a couple of vanishing results for mod $p$ Mumford–Morita–Miller classes for surface symmetries.
Algebraic topology
Group theory and generalizations
Manifolds and cell complexes
General
897
909
10.2977/PRIMS/55
http://www.ems-ph.org/doi/10.2977/PRIMS/55
On Some Properties of Universal Sigma-Finite Measures Associated with a Remarkable Class of Submartingales
Joseph
Najnudel
Universität Zürich, ZÜRICH, SWITZERLAND
Ashkan
Nikeghbali
Universität Zürich, ZÜRICH, SWITZERLAND
martingale, submartingale, penalization
In a previous work, we associated with any submartingale $X$ of class $(\Sigma)$, defined on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$ satisfying some technical conditions, a $\sigma$-finite measure $\mathcal{Q}$ on $(\Omega, \mathcal{F})$, such that for all $t \geq 0$, and for all events $\Lambda_t \in \mathcal{F}_t$: $$ \mathcal{Q} [\Lambda_t, g\leq t] = \mathbb{E}_{\mathbb{P}} [\mathbb{1}_{\Lambda_t} X_t],$$ where $g$ is the last hitting time of zero by the process $X$. In this paper we establish some remarkable properties of this measure from which we also deduce a universal class of penalisation results of the probability measure $\mathbb{P}$ with respect to a large class of functionals. The measure $\mathcal{Q}$ appears to be the unifying object in these problems.
Probability theory and stochastic processes
General
911
936
10.2977/PRIMS/56
http://www.ems-ph.org/doi/10.2977/PRIMS/56
Asymptotic Property of Divergent Formal Solutions in Linearization of Singular Vector Fields
Masafumi
Yoshino
Hiroshima University, HIROSHIMA, JAPAN
divergent series, small denominators, asymptotic analysis
We study asymptotic properties of divergent formal solutions appearing in the linearization problem of a singular vector field without a Diophantine condition or existence of additional first integrals. We give an asymptotic meaning to divergent formal solutions constructed from a singular perturbative solution (cf. [6]).
Ordinary differential equations
Dynamical systems and ergodic theory
General
937
958
10.2977/PRIMS/57
http://www.ems-ph.org/doi/10.2977/PRIMS/57