## REPORTS

### Forum

#### Casimir Force, Casimir Operators and the Riemann Hypothesis

2010.01.19

Casimir Force, Casimir Operators and the Riemann Hypothesis

■Date:

9-13 November, 2009

■Place:

Nishijin Plaza, Fukuoka, Japan

■Sponsored by:

Global COE Program “Math-for-Industry”

■Supported by:

MEXT, Japan

■Invited Speakers

CIUCU, Mihai (Indiana University : Algebraic Combinatorics & Statistical Physics)

CONSANI, Caterina

(Johns Hopkins University : Algebraic Geometry & Riemann Hypothesis)

DENINGER, Christopher (University of Münster : Riemann Hypothesis)

HARAN, Shai (Technion, Israel : Riemann Hypothesis, Geometry & Arithmetic)

HECKMAN, Gert (University of Nijmegen : Harmonic Analysis)

HOWE, Roger

(Yale University : Invariant Theory, Representation Theory & Harmonic Analysis)

ITOH, Minoru (Kagoshima University : Representation Theory & Invariant Theory)

IWASA, Yoh (Kyushu University : Mathematical biology)

KIMOTO, Kazufumi

(University of the Ryukyus : Zeta Functions & Representation Theory)

KUROKAWA, Nobushige (TIT : Zeta functions, Riemann Hypothesis & Casimir Force)

MICHALOWSKI, Stefan (OECD/ GSF : OECD/GSF Mathematics in Industry)

OCHIAI, Hiroyuki (Nagoya University : Harmonic Analysis & Zeta Functions)

PARK, Jinsung (KIAS : Zeta Regularization & Global Analysis)

PEVZNER, Misha (University of Reims, CNRS (Moscow), IPMU (Tokyo) : Representation Theory)

SATO, Hisayoshi (Hitachi, Ltd., Systems Development Laboratory : Cryptography)

TAKAGI, Tsuyoshi (Future University Hakodate : Cryptography)

TOKITA, Kei (Osaka University : Mathematical biology & Statistical Physics)

VERBITSKIY, Evgeny

(Philips Research : Applications of Mathematics in Natural sciences and Medicine)

WENG, Lin (Kyushu University : Zeta Functions and Algebraic Geometry)

ZHU, Chengbo (National University of Singapore : Representation Theory)

Abstract

BOURGUIGNON, Jean-Pierre (CNRS-IHÉS : Differential Geometry & Global Analysis)

Title: Mathematics and Industry: Towards a Challenging New Cooperation

Abstract: More and more sectors of the economy are relying on the use of advanced modelling and information intensive techniques. In both cases recent mathematics is involved to deal with the new challenges that appear in this context.

The purpose of the lecture is to highlight the basis for the new paradigm and to give a few examples of situations showing that the classical distinction between fundamental and applied research has to be thought over.

CAPASSO, Federico (Harvard University : Solid State Physics & Nano-mechanics)

Title: Casimir-Lifshitz forces: vacuum fluctuations, quantum levitation and the future of nanomachines

Abstract: Attractive forces exist between any uncharged surfaces in vacuum due to quantum mechanical fluctuations (zero point energy). Known as Casimir-Lifshitz forces, they can be tailored by suitable choice of the materials and their shape and even turned into repulsive by interleaving a suitable liquid. Measurements of these exotic forces will be presented. These results have implications for future scaled-down MicroElectroMechanicalSystems (MEMS) opening the door to new actuators, nanoscale position sensors and frictionless bearings based on quantum levitation. The talk will conclude with a brief discussion of future exciting possibilities such as the vacuum torque and the “holy grail” of quantum electrodynamics, light generation by “shaking the vacuum”

CIUCU, Mihai (Indiana University : Algebraic Combinatorics & Statistical Physics)

Title: A Casimir force in dimer systems

Abstract: In earlier work we showed that the correlation of holes in a sea of dimers on a planar lattice is governed, in the limit of large separation between the holes, by Coulomb's law of two dimensional electrostatics. When the holes have zero charge, new effects arise as their geometrical sizes grow large. We present exact calculations in some concrete situations which show a Casimir type force emerging this way.

CONSANI, Caterina

(Johns Hopkins University : Algebraic Geometry & Riemann Hypothesis)

Title: Schemes over F_1 and zeta functions.

Abstract: The talk will overview my recent work, in collaboration with A. Connes, on a theory of schemes over the `field of characteristic one', the computation of the zeta function of an arbitrary Noetherian scheme over such a `field' and the implication for the description of the counting function for the hypothetical curve \overline{Spec(Z)} over F_1.

DENINGER, Christopher (University of Münster : Riemann Hypothesis)

Title: Number theory and foliations

Abstract: We will explain certain analogies between number theory and analysis on foliated spaces. In particular we prove an analogue of the Riemann hypotheses in the dynamical context. We will also discuss the foliation analogue of Arakelov theory developed by Kopei.

HARAN, Shai (Technion, Israel : Riemann Hypothesis, Geometry & Arithmetic)

Title: The Geometry of Generalized Rings

Abstract: In it I will give a new language for geometry. This language makes arithmetic and curves look much similar.

HECKMAN, Gert (University of Nijmegen : Harmonic Analysis)

Title: Hyperbolic structures associated with hypergeometric equations.

Abstract: The projective line minus the points 0,1 and infinity has a hyperbolic structure with ramification order p, q and r at the three points in case 1/p+1/q+1/r<1. This result was obtained in the 19th century by Schwartz and Klein in their study of the Euler-Gauss hypergeometric equation. We explain the multivariable generalization of this theorem in the context of root systems. This gives a unification of higher dimensional ball quotients found earlier for weighted point configurations on the projective line (by Deligne and Mostow), for cubic surfaces (by Allcock, Carlson and Toledo), for quartic curves (by Kondo) and for rational elliptic surfaces (by Heckman and Looijenga).

HOWE, Roger

(Yale University : Invariant Theory, Representation Theory & Harmonic Analysis)

Title: Maxwell, Casimir, and dual pairs.(Title has been changed.)

Abstract: (This talk will survey the structure of a certain class of lattice cones and their applications to some of the basic counting problems of representation theory and invariant theory.)

ITOH, Minoru (Kagoshima University : Representation Theory & Invariant Theory)

Title: On Extensions of the Tensor Algebra

Abstract: We introduce a natural extension of the tensor algebra. On this algebra, we can consider two types of interesting operators: left multiplications by vectors and derivations by covectors. I will talk about some applications of this algebra and these operators to representation theory and invariant theory.

IWASA, Yoh (Kyushu University : Mathematical biology)

Title: Modeling morphogenesis in development.

Abstract: In the development of animals and plants (multicellular organisms), the life starts from a single cell, named fertilized egg, goes through cell division, differentiation, deformation including cell migration and rearrangement, eventually form complex structures. To understand this process, mathematical models and computer simulations have been useful to understand this process of development. In this talk, I will first explain modeling of the formation of limb bud in vertebrates (chicks and mice). Then I will speak on the traveling wave of gene expression in zebrafish somitogenesis.

KIMOTO, Kazufumi

(University of the Ryukyus : Zeta Functions & Representation Theory)

Title: Arithmetics derived from the non-commutative harmonic oscillator

Abstract: The non-commutative harmonic oscillator (NCHO) is a parametric family of differential operators regarded as a non-trivial higher-dimensional version of the ordinary quantum harmonic oscillator. Its spectral zeta function essentially gives the Riemann zeta function if we take special parameters (or NCHO is "commutative"). We discuss several arithmetic problems which derive from the study of "anomaly caused by the non-comutativity" in special value formulas for the spectral zeta function.

KUROKAWA, Nobushige (TIT : Zeta functions, Riemann Hypothesis & Casimir Force)

Title: Absolute zeta functions, absolute Riemann hypothesis and absolute Casimir energies

Abstract: I will talk on the absolute zeta functions constructed over the field F1 with one element. I will show analyticities, functional equations and the associated Riemann hypothesis. Special values at negative integers are considered to be absolute Casimir energies.

MICHALOWSKI, Stefan (OECD/ GSF : OECD/GSF Mathematics in Industry)

Title: Practical steps for strengthening the links between industry and academic mathematics

Abstract: Given the increasingly intimate connection between science, innovation, and mathematics, it is natural to enquire whether the interface between these three domains is functioning in an optimal way. The OECD Global Science Forum - a committee of senior science policy officials - convened a workshop that brought together mathematicians, representatives of industry, and officials of science funding agencies. The written report from the event contains analyses, findings and recommendations that national administrations, academic institutions, and high-technology companies can use to establish new programmes and mechanisms for bridging the mathematics/industry divide.

OCHIAI, Hiroyuki (Nagoya University : Harmonic Analysis & Zeta Functions)

Title: Zeta functions and Casimir energies on infinite symmetric groups.

Abstract: We define the Casimir energies of permutations of the natural numbers by using the analytic continuation of the zeta functions associated with such permutations. We discuss the analytic properties of such zeta functions and compute the explicit values of Casimir energies for several examples. This is a joint work with Nobushige Kurokawa.

PARK, Jinsung (KIAS : Zeta Regularization & Global Analysis)

Title: Spectral invariants and dynamical zeta function over hyperbolic manifolds

Abstract: The relation of spectral invariants - eta invariant and analytic torsion - to special values of dynamical zeta functions - Selberg zeta function and Ruelle zeta function - has been studied by Millson, Fried and Moscovici-Stanton and some others for closed hyper - bolic manifolds. A recent development in global analysis provides us with accessible ways to extend these to noncompact hyperbolic manifolds. In this talk, we will explain some recent results in these problems for non-compact hyperbolic manifolds - finite volume hyperbolic manifolds with cusps and infinite volume convex co-compact hyperbolic manifolds. Some related problems and possible applications will be also discussed.

PATTERSON, Samuel (University of Göttingen : Number Theory & Zeta functions)

Title: The Riemann Hypothesis - pro and contra.

Abstract: What we know as the Riemann Hypothesis was put forward more or less as an aside in Riemann's 1859 paper presented to the Berlin Akademie. It took on its present status, and name, around 50 years later. In this talk, which will be partly historical, I shall discuss what speaks for its truth, and what against it.

PEVZNER, Misha (University of Reims, CNRS (Moscow), IPMU (Tokyo) : Representation Theory)

Title: Composition formulas for the Weyl calculus and phase space representations.

Abstract: The celebrated Moyal star-product formula is an asymptotic counter-part of composition formula for the Weyl symbolic calculus. It may also be derived from the covariance of the Weyl quantization map under the Heisenberg group. Considering the covariance of the Weyl calculus with respect to the symplectic group we find a new type of composition formulas related with the spectral decomposition of the unitary action of the symplectic group Sp(n,R) on the space of square-integrable functions on its minimal co-adjoint orbit.

SATO, Hisayoshi (Hitachi, Ltd., Systems Development Laboratory : Cryptography)

Title: Indifferentiability of Cryptographic Constructions

Abstract: The security notion of indifferentiability from random oracles is very important to assure that there exists no structual flaw in the cryptographic scheme. In this presentation, several recent results on the indifferentiability will be intorduced.

SCHUURMANS, Martin (EIT : Solid State Physics & Biomedical Engineering)

Title: Casimir Lessons, Innovation and Technology.

Abstract: The European vision on Innovation and Technology as proposed by EIT will be presented and connected to the life and work of Hendrik Brugt Gerhard Casimir (1909-2000), who has been recognized (Pake Price of the American Physical Society 1999) as an excellent leader of industrial research at Royal Philips Electronics and for fundamental contributions to the foundations of quantum mechanics and Solid State Physics. The author has enjoyed private and scientific contacts with Henk Casimir, has had an active scientific career in solid state physics and has been a leader of Research and Development related activities at Philips Electronics. The over time changed role of curiosity driven (fundamental) research for society will be discussed.

TAKAGI, Tsuyoshi (Future University Hakodate : Cryptography)

Title: Pairing-Based Cryptography and its Security Analysis

Abstract: We present a short overview of pairing-based cryptography, which is an extension of conventional public-key cryptography such as RSA cryptosystem and elliptic curve cryptography. We then explain the security of pairing-based cryptography - how to solve the discrete logarithm problems over finite fields by the function field sieve.

TAN, Eng Chye (National University of Singapore : Representation Theory & Harmonic Analysis)

Opening Address

Abstract: I would like to share the typical government's and the university's perspective on the desired functions of a Mathematics Department. Being a top-rate academic Department is a given, but in a knowledge economy, there has been more demands. This is especially crucial for a country like Singapore, since we are small and do not have the luxury of scale. I would like to give a macro perspective what we hope Singapore Mathematics Departments could do, over and above the traditional functions of teaching and academic research. This should apply to all Departments in a developed country, if they hope to continue to sustain the high level of (research) support and to attract strong students.

TOKITA, Kei (Osaka University : Mathematical biology & Statistical Physics)

Title: Random matrices and their application to mathematical biology

Abstract: I will first sketch early application of random matrices to mathematical biology and the succeeding controversy on the stability of a large and complex ecosystem which is well known as the "paradox of ecology", and will secondly present recent theoretical treatments on the "random community models".

VERBITSKIY, Evgeny

(Philips Research : Applications of Mathematics in Natural sciences and Medicine)

Title: Mathematics in the industrial environment: Dutch perspective

Abstract: After a short historic overview, I will discuss "best practices" of applied mathematics in industrial environments. I will also discuss approaches to training students in industrial mathematics used in the Netherlands.

WENG, Lin (Kyushu University : Zeta Functions and Algebraic Geometry)

Title: Symmetries and the Riemann Hypothesis

Abstract: Naturally associated to pairs (G,P) of reductive groups and their maximal parabolics are abelian zeta functions. These zetas, governed by huge symmetries, including that of Weyl, are expected to satisfy a standard functional equation and the Riemann Hypothesis. In this talk, we are going to talk about them, together with their relations to non-abelian high rank zetas. Limited examples for SL, SO, Sp and G2, and confirmations of the associated Riemann Hypothesis, due to Haseo Ki, Jeff Lagarias, Masatoshi Suzuki, Tsukasa Hayashi and myself, will be presented as well.

ZHU, Chengbo (National University of Singapore : Representation Theory)

Title: Multiplicity one theorems and the Casimir operator

Absract: Let G be one of the classical groups GL(n, R), GL(n, C), O(p,q), O(n,C), U(p,q), and let G' be respectively the subgroup GL(n-1, R), GL(n-1, C), O(p,q-1), O(n-1,C), U(p,q-1), embedded in G in the standard way. We consider the class of irreducible admissible smooth Fréchet representations of moderate growth, for G and G'. The multiplicity one theorems in the title assert that any representation of G' in this class occurs (as a quotient) with multiplicity at most one in any representation of G in the same class. We will discuss the crucial role of the Casimir operator in its proof. The talk represents joint work with Binyong Sun of the Chinese Academy of Sciences.

■Information:web site