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European Mathematical Society Publishing House
2024-03-28 13:50:32
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=PRIMS&vol=40&iss=3&update_since=2024-03-28
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
40
2004
3
Schubert Varieties and the Fusion Products
Boris
Feigin
Independent University of Moscow, MOSCOW, RUSSIAN FEDERATION
E.
Feigin
Independent University of Moscow, MOSCOW, RUSSIAN FEDERATION
For each A ∈ ℕn we define a Schubert variety shA as a closure of the SL2(ℂ[t])orbit in the projectivization of the fusion product MA. We clarify the connection of the geometry of the Schubert varieties with an algebraic structure of MA as sl2 ⊗ ℂ[t] modules. In the case, when all the entries of A are different, shA is a smooth projective complex algebraic variety. We study its geometric properties: the Lie algebra of the vector fields, the coordinate ring, the cohomologies of the line bundles. We also prove that the fusion products can be realized as the dual spaces of the sections of these bundles.
Nonassociative rings and algebras
General
625
668
10.2977/prims/1145475487
http://www.ems-ph.org/doi/10.2977/prims/1145475487
A Parallel between Brownian Bridges and Gamma Bridges
Michel
Émery
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Marc
Yor
Université Paris VI, PARIS CEDEX 05, FRANCE
Some properties of the Gamma bridges (obtained by conditioning the Gamma subordinator to take a given value at a given time) are investigated; similarities with the Brownian bridges are emphasized.
Probability theory and stochastic processes
General
669
688
10.2977/prims/1145475488
http://www.ems-ph.org/doi/10.2977/prims/1145475488
Specialization of Zero Cycles
János
Kollár
Fine Hall / Princeton University, PRINCETON, UNITED STATES
Algebraic geometry
General
689
708
10.2977/prims/1145475489
http://www.ems-ph.org/doi/10.2977/prims/1145475489
Toward the Exact WKB Analysis for Higher-Order Painlevé Equations — The Case of Noumi–Yamada Systems
Yoshitsugu
Takei
Kyoto University, KYOTO, JAPAN
As the first step toward the exact WKB analysis for higher-order Painlevé equations, we study the Stokes geometry of the Noumi–Yamada system. It is shown that there are intriguing relations, similar to those for traditional Painlevé equations, between the Stokes geometry of the Noumi–Yamada system and that of its underlying Lax pair.
Ordinary differential equations
Special functions
General
709
730
10.2977/prims/1145475490
http://www.ems-ph.org/doi/10.2977/prims/1145475490
Temperature as Order Parameter of Broken Scale Invariance
Izumi
Ojima
Kyoto University, KYOTO, JAPAN
In algebraic quantum field theory the (inverse) temperature is shown to be a macroscopic order parameter to parametrize mutually disjoint thermal sectors arising from the broken scale invariance under renormalization-group transformations. This is accomplished in a mathematical formalism for the consistent treatment of explicitly broken symmetries such as broken scale invariance, on the basis of a clearcut criterion for the symmetry breakdown in a unified scheme for sectors proposed recently by the author.
Quantum theory
Statistical mechanics, structure of matter
General
731
756
10.2977/prims/1145475491
http://www.ems-ph.org/doi/10.2977/prims/1145475491
Crystal Bases for Quantum Classical Algebras and Nakajima’s Monomials
Seok-Jin
Kang
Seoul National University, SEOUL, SOUTH KOREA
Jeong-Ah
Kim
Seoul National University, SEOUL, SOUTH KOREA
Dong-Uy
Shin
Korea Institute for Advanced Study, SEOUL, SOUTH KOREA
Using Nakajima’s monomials, we construct a new realization of crystal bases for finite dimensional irreducible modules over quantum classical algebras. We also give an explicit bijection between the monomial realization and the Young tableau realization of crystal bases.
Quantum theory
Group theory and generalizations
General
757
791
10.2977/prims/1145475492
http://www.ems-ph.org/doi/10.2977/prims/1145475492
Heat Kernel Estimates and Parabolic Harnack Inequalities on Graphs and Resistance Forms
Takashi
Kumagai
Kyoto University, KYOTO, JAPAN
We summarize recent work on heat kernel estimates and parabolic Harnack inequalities for graphs, where the time scale is the β-th power of the space scale for some β ≥ 2. We then discuss self-adjoint operators induced by resistance forms. Using a resistance metric, we give a simple condition for detailed heat kernel estimates and parabolic Harnack inequalities. As an application, we show that on trees a detailed two-sided heat kernel estimate is equivalent to some volume growth condition.
Probability theory and stochastic processes
Potential theory
General
793
818
10.2977/prims/1145475493
http://www.ems-ph.org/doi/10.2977/prims/1145475493
The Geometry of Anabelioids
Shinichi
Mochizuki
Kyoto University, KYOTO, JAPAN
Algebraic geometry
General
819
881
10.2977/prims/1145475494
http://www.ems-ph.org/doi/10.2977/prims/1145475494
Correspondence and Analyticity
Henry
Stapp
Lawrence National Laboratory, BERKELEY, UNITED STATES
The analyticity properties of the S-matrix in the physical region are determined by the correspondence principle, which asserts that the predictions of classical physics are generated by taking the classical limit of the predictions of quantum theory. The analyticity properties deducible in this way from classical properties include the locations of the singularity surfaces, the rules for analytic continuation around these singularity surfaces, and the analytic character (e.g., pole, logarithmic, etc.) of these singularities. These important properties of the S-matrix are thus derived without using stringent locality assumptions, or the Schrödinger equation for temporal evolution, except for freely moving particles. Sum-over-all-paths methods that emphasize paths of stationary action tend to produce the quantum analogs of the contributions from classical paths. These quantum analogs are derived directly from the associated classical properties by reverse engineering the correspondence-principle connection.
Quantum theory
Functional analysis
General
883
903
10.2977/prims/1145475495
http://www.ems-ph.org/doi/10.2977/prims/1145475495
Functional Equations and Fusion Matrices for the Eight Vertex Model
Klaus
Fabricius
University of Wuppertal, WUPPERTAL, GERMANY
Barry
McCoy
State Univ of New York at Stony Brook, STONY BROOK, UNITED STATES
We derive sets of functional equations for the eight vertex model by exploiting an analogy with the functional equations of the chiral Potts model. From these equations we show that the fusion matrices have special reductions at certain roots of unity. We explicitly exhibit these reductions for the 3, 4 and 5 order fusion matrices and compare our formulation with the algebra of Sklyanin.
Statistical mechanics, structure of matter
General
905
932
10.2977/prims/1145475496
http://www.ems-ph.org/doi/10.2977/prims/1145475496
Filtrations on Chow Groups and Transcendence Degree
Morihiko
Saito
Kyoto University, KYOTO, JAPAN
Chow group, Deligne cohomology, cycle map
For a smooth complex projective variety X defined over a number field, we have filtrations on the Chow groups depending on the choice of realizations. If the realization consists of mixed Hodge structure without any additional structure, we can show that the obtained filtration coincides with the filtration of Green and Griffiths, assuming the Hodge conjecture. In the case the realizations contain Hodge structure and etale cohomology, we prove that if the second graded piece of the filtration does not vanish, it contains a nonzero element which is represented by a cycle defined over a field of transcendence degree one. This may be viewed as a refinement of results of Nori, Schoen, and Green–Griffiths–Paranjape. For higher graded pieces we have a similar assertion assuming a conjecture of Beilinson and Grothendieck’s generalized Hodge conjecture.
Algebraic geometry
General
933
948
10.2977/prims/1145475497
http://www.ems-ph.org/doi/10.2977/prims/1145475497
A Cabling Formula for the 2-Loop Polynomial of Knots
Tomotada
Ohtsuki
Kyoto University, KYOTO, JAPAN
Knot, 2-loop polynomial, Kontsevich invariant, cabling
The 2-loop polynomial is a polynomial presenting the 2-loop part of the logarithm of the Kontsevich invariant of knots. We show a cabling formula for the 2-loop polynomial of knots. In particular, we calculate the 2-loop polynomial for torus knots.
Manifolds and cell complexes
General
949
971
10.2977/prims/1145475498
http://www.ems-ph.org/doi/10.2977/prims/1145475498
From Exact-WKB towards Singular Quantum Perturbation Theory
André
Voros
CEA Saclay, GIF-SUR-YVETTE CEDEX, FRANCE
We use exact WKB analysis to derive some concrete formulae in singular quantum perturbation theory, for Schrödinger eigenvalue problems on the real line with polynomial potentials of the form (qM + gqN), where N > M > 0 even, and g > 0. Mainly, we establish the g → 0 limiting forms of global spectral functions such as the zeta-regularized determinants and some spectral zeta functions.
Ordinary differential equations
Quantum theory
General
973
990
10.2977/prims/1145475499
http://www.ems-ph.org/doi/10.2977/prims/1145475499
The Uniformity Principle on Traced Monoidal Categories
Masahito
Hasegawa
Kyoto University, KYOTO, JAPAN
The uniformity principle for traced monoidal categories has been introduced as a natural generalization of the uniformity principle (Plotkin’s principle) for fixpoint operators in domain theory. We show that this notion can be used for constructing new traced monoidal categories from known ones. Some classical examples like the Scott induction principle are shown to be instances of these constructions. We also characterize some specific cases of our constructions as suitable enriched limits.
Category theory; homological algebra
$K$-theory
General
991
1014
10.2977/prims/1145475500
http://www.ems-ph.org/doi/10.2977/prims/1145475500
Applications of Discrete Convex Analysis to Mathematical Economics
Akihisa
Tamura
Kyoto University, KYOTO, JAPAN
Discrete convex analysis, which is a unified framework of discrete optimization, is being recognized as a basic tool for mathematical economics. This paper surveys the recent progress in applications of discrete convex analysis to mathematical economics.
Operations research, mathematical programming
Game theory, economics, social and behavioral sciences
General
1015
1037
10.2977/prims/1145475501
http://www.ems-ph.org/doi/10.2977/prims/1145475501
The Second Painlevé Hierarchy and the Stationary KdV Hierarchy
Nalini
Joshi
The University of Sydney, SYDNEY, AUSTRALIA
Painlevé equations, Painlevé hierarchies, asymptotics
It is well known that soliton equations such as the Korteweg–de Vries equation are members of infinite sequences of PDEs known as hierarchies. Here we consider infinite sequences of ODEs associated with the Painlevé equations. We review methods of constructing such hierarchies, specifically the second Painlevé hierarchy, as reductions of PDE hierarchies. We also show that in the large-parameter limit, the solutions of the second Painlevé hierarchy are given by the periodic solutions of the stationary KdV hierarchy.
Special functions
Ordinary differential equations
General
1039
1061
10.2977/prims/1145475502
http://www.ems-ph.org/doi/10.2977/prims/1145475502
String and Vortex
Toshiya
Kawai
Kyoto University, KYOTO, JAPAN
We discuss how the geometry of D2-D0 branes may be related to Gromov–Witten theory of Calabi–Yau threefolds.
Quantum theory
Algebraic geometry
General
1063
1091
10.2977/prims/1145475503
http://www.ems-ph.org/doi/10.2977/prims/1145475503
On Twisted Microdifferential Modules I. Non-existence of Twisted Wave Equations
Andrea
D'Agnolo
Università di Padova, PADOVA, ITALY
Pierre
Schapira
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Using the notion of subprincipal symbol, we give a necessary condition for the existence of twisted D-modules simple along a smooth involutive submanifold of the cotangent bundle to a complex manifold. As an application, we prove that there are no generalized massless field equations with non-trivial twist on grassmannians, and in particular that the Penrose transform does not extend to the twisted case.
Partial differential equations
Algebraic geometry
Several complex variables and analytic spaces
General
1093
1111
10.2977/prims/1145475504
http://www.ems-ph.org/doi/10.2977/prims/1145475504