미국의 “전국 수학 교사 협의회”(National Council of Teachers of Mathematics, NCTM)는 1989년부터 〈학교 수학의 교육과정과 평가 규준〉(1989), 〈수학 가르침(교수)의 전문성 규준〉(1991), 〈학교 수학의 평가(시험) 규준〉(NCTM, 1995), 〈학교 수학의 원리와 규준〉(2000)을 출판하여 미국의 수학 교육 의 전망(목표, 나아갈 길)과 규준(실행 지침)을 제시하였다. 수학 교사들로 구성된 미국의 NCTM은 학생, 학부모, 학교 행정가 등 많은 사람들과 힘을 합하여 모든 학생들에게 수준 높은 수학 교육을 받을 수 있는 여건(환경, 기회)을 조성하는 데 구심점의 역할을 하였다. 한편 많은 관련 단체들은 여러 배경과 능력을 가진 학생들이 전문성을 지닌 교사(특수 교사를 일컫는 밀이 아니다. 수학 교과를 이해하고 수학의 전문성과 특수성을 가르칠 수 있는 일반 교사를 일컫는 말이다.)로부터 미래를 대비해 평등하고, 진취적이며, 지원이 잘 이루어지고, 공학 도구(IT)가 잘 갖춰진 환경에서 중요한 수학적 아이디어를 이해하면서 학습할 수 있는 수학 교실(미국에서는 우리나라처럼 수학 교사가 수학 시간에 학생의 방(교실: Homeroom)에 찾아가지 않고 학생들이 선생의 방(수학 교실: Classroom)을 찾아온다. 전형적인 수학 교실의 사진은 2쪽에 나와 있다.)을 만들기 위해 함께 힘썼다. NCTM에서 출간한 여러 규준들은 우리나라의 제6차와 제7차 교육과정에도 큰 영향을 미쳤다. 이 글에서는 NCTM(2000)에서 제시한 학습 원리를 간단히 살펴본 다음 이를 중심으로 현재 미국 수학 교육의 교수ㆍ학습 이론의 동향을 살펴본다.

미국 NCTM의 Principles and Standards for School Mathematics에 나타난 수학과 교수,학습의 이론

Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?)

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Introduction to the Minimal Model Problem

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

We formulate an extension of Maldacena’s AdS/CFT conjectures to the case of branes located at singular points in the ambient transverse space. For singularities which occur at finite distance in the moduli space of M or F theory models with spacetime-filling branes, the conjectures identify the worldvolume theory on the p-branes with a compactification of M or IIB theory on AdSp+2 × HD−p−2. We show how the singularity determines the horizon H, and demonstrate the relationship between global symmetries on the worldvolume and gauge symmetries in the AdS model. As a first application, we study some singularities relevant to the D3-branes required in four-dimensional F -theory. For these we are able to explicitly derive the low-energy field theory on the worldvolume and compare its properties to predictions from the dual AdS model. In particular, we examine the baryon spectra of the models and the fate of the Abelian factors in the gauge group. October 1998 ∗ On leave from Department of Particle Physics, Weizmann Institute of Science, Rehovot, Israel Spacetime-filling branes have emerged as an essential feature of string and M-theory compactifications in at least three contexts: (1) new branches of the heterotic string in six dimensions with “extra” tensor multiplets, which can be represented by a Hořava– Witten-type compactification of M-theory on (S/Z2) × K3 but with extra spacetimefilling M5-branes representing the extra tensor multiplets [1,2,3]; (2) F -theory models in four dimensions (which can be regarded as compactifications of the IIB string with D7branes included) which in general require spacetime-filling D3-branes to cancel a tadpole anomaly [4]; and (3) M-theory models in three dimensions, which require spacetime-filling M2-branes to cancel a similar tadpole anomaly [4]. In each of these cases, the spacetimefilling brane meets the compactifying space at a single point, and the string or M-theory remains finite near the brane. Remarkably, this short list of branes (M5, D3, and M2) is precisely the list of branes for which a certain scaling limit is expected to lead to a “boundary” conformal field theory in the recent AdS/CFT conjectures [5,6,7]. In fact, the scaling limit can be taken even when the space transverse to the branes is curved, as in the compactification scenarios above. The details of the metric far from the location y0 of the brane in the transverse space become irrelevant; for the purposes of studying the scaling limit, the metric on the compactifying space can be approximated by some metric on its tangent space Ty0 at y0. In the scaling limit, the rescaled supergravity metric approaches a metric of the form AdSp+2 × S in which the anti-de Sitter space has been formed out of the worldvolume of the brane and the radial direction within Ty0 , and S k is the unit sphere within Ty0 . Maldacena’s conjecture proposes that the M or string theory on this space AdSp+2 × S, with N units of flux of the supergravity k-form field strength through S, is dual to a specific conformal field theory on the boundary of AdSp+2. The conjecture applies to the large N limit when a large number of these branes have been brought together; in the compactification context, fairly large values of N can be obtained by bringing together all available branes in a given model. Virtually all points in the compactifying space have identical behavior in this scaling limit. That situation changes, however, if we consider a compactifying space which itself 1 This last requirement excludes consideration of F -theory models in eight dimensions with the D7-brane being spacetime-filling. 2 The exception is the four-dimensional F -theory models, where points located along the D7branes behave differently; in particular, the string coupling becomes infinite at such points. The behavior of D3-branes at such points has recently been determined [8,9,10], and we will not

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/1999/0003/0001/ATMP-1999-0003-0001-a001.pdf

Non-spherical horizons, I

Let E be an elliptic curve which is defined over Q and has stable reduction modulo a given prime p. Assuming that E is modular, one can associate to E a p-adic L-function Lp(E, s). (See [-Mz-SwD, A-V, Vi, Mz-T-T] for its construction in various cases.) This function is defined by a certain interpolation property and is analytic for seZp. In this paper, we will assume that E has split multiplicative reduction at p. Under this assumption the interpolation property implies that Lp(E, 1)=0. We will prove a formula for Ep(E, 1) which was discovered experimentally by Mazur, Tate, and Teitelbaum [Mz-T-T]. By Tate's p-adic urfiformization theory, there is a p-adic integer q~:epZp (which we refer to as the Tate period for E) and a p-adic analytic isomorphism

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p -adic L -functions and p -adic periods of modular forms

-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture

This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.

Modular Forms and Fermat's Last Theorem

Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues.

We classify isolated terminal cyclic quotient singularities in dimension three, and isolated Gorenstein terminal cyclic quotient singularities in dimension four In addition, we give a new proof of a combinatorial lemma of G K White using Bernoulli functions Let X be a smooth algebraic variety over C, and let wx be the canonical bundle of X For each n > O, if Ii(X, WOn) =# 0, there is a natural pluricanonical map On: X -PI'(X, wOn)* An algebraic variety is of general type if on is a birational map for n sufficiently large For a variety of general type, the pluricanonical images On(X) are the most natural birational modets of X to study Canonical singularities are the singularities which may occur in the pluricanonical models of varieties of general type In dimension 1, the pluricanonical models are smooth so there are no canonical singularities; in dimension 2 the canonical singularities coincide with the classical rational double points One characterization of the rational double points is as quotient singularities: if G is any finite subgroup of S1(2, C), then the quotient C2/G has a rational double point, and every rational double point is analytically isomorphic to such a quotient singularity Reid and Shepherd-Barron [10], and independently Tai [14], have given a condition for quotient singularities to be canonical in arbitrary dimensions (although not all canonical singularities are quotient singularities in dimensions greater than two) Terminal singularities are a class of canonical singularities which play an important role in birational geometry (as evidenced by recent work of Mori [8], Reid [12], and Tsunoda [15]) In this note we study cyclic quotient singularities which are terminal In dimension three we explicitly describe all isolated terminal cyclic quotient singularities, while in dimension four, we describe isolated terminal cyclic quotient singularities which are also Gorenstein The description uses a combinatorial lemma due to G K White [17]; we have given a new proof of this lemma (Corollary 14 below) using Bernoulli functions Received by the editors March 31, 1983 1980 Mathematics Subject Classification Primary 14B05; Secondary IOA40, 14L30, 32B30, 52A25

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Glenn Stevens

Papers

p -adic L -functions and p -adic periods of modular forms

-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture

Modular Forms and Fermat's Last Theorem

Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues.

Terminal quotient singularities in dimensions three and four