Plenary Speakers

• Alejandro Adem (Univ. of British Columbia & PIMS)

 TopologyTitle: Commuting matrices and spaces of homomorphisms Abstract: Let $G$ denote a Lie group, and consider the space of all commuting $n$-tuples of elements in $G$. In this talk we will discuss basic topological properties of spaces such as these, and how they relate to bundle theory, group cohomology and other interesting topological invariants.

• Jinho Baik (Univ. of Michigan)

 AlgebraTitle: Random permutations and random matrices Abstract: We consider the length of the longest increasing subsequence of a random permutation as the size of the permutation tends to infinity. This combinatorial/probabilistic question turns out to be related to the theory of random matrices. Also related are mathematical problems such as integrable systems, random tiling and a probability on partitions. We will discuss such various aspects of random permutations.

• Heisuke Hironaka (1970 Field Medalist, Harvard Univ. & Seoul Nat'l Univ.)

 Title: A general theory of algebra-geometric singularities

Invited Speakers

• Habib Ammari (Ecole Polytechnique & CNRS)

 Applied MathematicsTitle: Asymptotic spectral analysis with applications in imaging, effective medium theory, and optimal design Abstract: The aim of this talk is to present an asymptotic theory for eigenvalue problems using layer potential techniques with applications in the fields of inverse problems, band-gap structures, and optimal design. Our general approach combines layer potentials techniques with the elegant theory of Ghoberg and Sigal on meromorphic operator-valued functions.

• Michèle Artigue (Université de Paris VII)

 Mathematical Education Title: Digital technologies and mathematics education Abstract: In 1985, the first international study launched by ICMI, the International Commission on Mathematical Instruction, was devoted to the influence of computers on mathematics and mathematics education. Since that time, the educational affordances of digital technologies have been extensively discussed; research has substantially developed; experiments and institutional incentives have multiplied, and digital technologies themselves have dramatically evolved. Where are we now? Reflecting on the different projects I have been involved in that area since the eighties, from middle school to university, and on the first outcomes of the ICMI Study 17 launched some years ago for revisiting technological issues, I will try to address this question, and to point out some of the main challenges that we have to face regarding technology and mathematics education.

• Bernard R. Hodgson (Université Laval)

 Mathematical logic and foundationsTitle: Some views on ICMI at the dawn of its second century Abstract: The International Commission on Mathematical Instruction (ICMI) was established in 1908 during the Fourth International Congress of Mathematicians held in Rome, with the eminent German mathematician Felix Klein being appointed as its first President. This year, marking the centennial of the Commission, thus provides an opportunity to reflect on ICMI, on its mission, actions, programmes of activities, and publications. The purpose of this talk is to present an overview of ICMI, both from an historical perspective and also as regards its functioning today as a commission of the International Mathematical Union. Part of this talk is based on material developed in the preparation of the symposium celebrating the first century of ICMI and organised last March in Rome at the very venue that witnessed the birth of ICMI, the magnificent Accademia dei Lincei. This will allow reviewing some highlights in the life of ICMI and also providing glimpses at some key figures in the history of the Commission. But within the limit of this talk one can hardly do justice to this remarkable century-long saga. In the second part of the talk I will look at current actions of ICMI and discuss some of the main issues that it is facing at the beginning of its second century, especially as regards the fostering of the development of mathematics education in less affluent parts of the world. In particular I wish to stress, among the activities organised by ICMI, the prominent character of the International Congress on Mathematical Education and examine some of the challenges and prospects, both for the host country and for the region, as it will soon be taking place for a second time in Asia.

• Hoon Hong  (North Carolina State Univ. & KIAS)

 Algebra Title: Subresultants in Roots Abstract: Subresultant, a generalization of resultant, plays a fundamental role in computational algebra and computational algebraic geometry. Subresultants are defined in terms of the coefficients of the polynomials. The definition is convenient for computation (algebra), but not so useful for reasoning (geometry). In this talk, we review various "nice" expressions for subresultants that are useful for reasoning. In particular, we show several ways to express the subresultants in terms of the roots of the polynomials. We hope that these expressions will be found useful in discovering interesting geometric properties, such as geometric basis for gap structure, root separation bound for non-square-free polynomials, etc

• Masaki Kashiwara (Kyoto Univ. & RIMS)

 Algebra Title: Crystal bases and combinatorics Abstract: The crystal basis is introduced through the consideration at $q=0$, the temperature absolute $0$, of the quantum groups. It combines representation theory and combinatorics. In this talk, after introducing the notion of crystal bases, we focus the role of a crystal basis as a combinatorial tool controlling the representations of affine Hecke algebras.

• In-Sook Kim (SungKyunKwan Univ.)

 Analysis Title: Bifurcation problems for nonlinear equations Abstract: Using nonlinear spectral theory, V\"ath (2005) obtained the existence of a global branch of solutions $(\lambda,u)$ for the boundary value problem\begin{equation*}\begin{cases}\ - \Delta_pu = \mu_0|u|^{p-2}u + f(\lambda,x,u) \quad & \text{in} \ \Omega \\\ u = 0 \quad & \text{on} \ \partial\Omega \end{cases}\end{equation*}when $\mu_0$ is not an eigenvalue of $- \Delta_p$.Here $\Delta_pu=\text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian. In fact, most of other papers dealt with global bifurcation at the first eigenvalue of the $p$-Laplacian like e.g. Del Pino and Man\'asevich (1991). In this talk, the goal is to study the global behavior of the solution set for the boundary value problem\begin{equation*}\begin{cases}\ -\text{div}\,(a(x) |\nabla u|^{p-2}\nabla u) =\mu_0b(x)|u|^{p-2}u + f(\lambda,x,u) \quad & \text{in} \ \Omega \\ \ u=0 \quad & \text{on} \ \partial \Omega \tag{B}\end{cases}\end{equation*}when $\mu_0$ is not an eigenvalue of the divergence form \begin{equation*}\begin{cases}\ -\text{div}\,(a(x) |\nabla u|^{p-2}\nabla u)= \mu_0b(x)|u|^{p-2}u \quad & \text{in} \ \Omega \\\ u= 0 \quad & \text{on} \ \partial \Omega. \end{cases}\end{equation*}Here $1To solve our problem (B) in the weak sense, we consider the corresponding nonlinear integral operators.The main tool is a bifurcation result on noncompact components of solutions for nonlinear operator equations. To do this, we give some related properties of these integral operators in the weighted Sobolev spaces, with the aid of nonlinear spectral theory for homogeneous operators. Moreover, in case that$\Omega=\Bbb R^N$and$1Finally, global bifurcation for equations involving nonhomogeneous operators is also discussed.

• Jin Ho Kwak (POSTECH)

 Mathematics for Information Sciences Title: Enumerating chiral maps on surfaces with a given underlying graph Abstract: Two $2$-cell embeddings $\imath : X \to S$ and $\jmath : X \to S$ of a connected graph $X$ into a closed orientable surface $S$ are {\em congruent} if there are an orientation-preserving surface homeomorphism $h$ on $S$ and a graph automorphism $\gamma$ of $X$ such that $\imath h =\gamma\jmath$. A $2$-cell embedding $\imath : X \to S$ of a graph $X$ into a closed orientable surface $S$ is sometimes described combinatorially by a pair $(X;\rho)$ called a map, where $\rho$ is a product of disjoint cycle permutations each of which is the permutation of the dart set of $X$ initiated at the same vertex following the orientation of $S$. The {\em mirror image} of a map $(X;\rho)$ is the map $(X;\rho^{-1})$, and one of the corresponding embeddings is called the {\em mirror image} of the other. A $2$-cell embedding of $X$ is {\em reflexible} if it is congruent to its mirror image. Mull et al. [Proc. Amer. Math. Soc. 103(1988) 321-330] developed an approach for enumerating the congruence classes of $2$-cell embeddings of graphs into closed orientable surfaces. In this paper we introduce a method for enumerating the congruence classes of reflexible $2$-cell embeddings of graphs into closed orientable surfaces, and apply it to the complete graphs, the bouquets of circles, the dipoles and the wheel graphs to count their congruence classes of reflexible or nonreflexible (called chiral) embeddings.

• Anthony Lau (Univ. of Alberta & CMS)

 Functional Analysis Title: Separation and extension properties for positive definite functions for groups Abstract: A well known theorem of Hahn Banach asserts that if E is a closed subspace of a Banach space and F is a closed subspace of E, then each continuous linear functional on F can be extended to a continuous linear functional on E. As a consequence, any element in E but not in F can be separated by continuous linear functional on E. In my talk, I should discuss similar properties for closed subgroups of a locally compact group and their relations with the invariant complementation problem for the group von Neumann algebra of a locally compact group.

• Chang-Ock Lee (KAIST)

 Applied Mathematics Title: A dual iterative substructuring method with a penalty term Abstract: Iterative substructuring methods with Lagrange multipliers are considered for second order elliptic problems in two and three dimensions, which are variants of the FETI-DP methods. The standard FETI-DP formulation is associated with the saddle-point problem which is induced from the minimization problem with a constraint for imposing the continuity across the interface. Starting from the slightly changed saddle-point problem by addition of a penalty term with a positive penalization parameter $\eta$, we propose a dual substructuring method which is implemented iteratively by the conjugate gradient method. For $\eta=0$, the proposed method is reduced to the FETI-DP method which is one of the most advanced dual substructuring methods. Performance of such a dual iterative substructuring method is directly connected with the condition number of a relevant dual system. For the preconditioned FETI-DP with the optimal Dirichlet preconditioner, it is well-known that the condition number is bounded by a polylogarithmic factor: $(1+\log(H/h))^{2}$ in two dimensions and $(H/h)(1+\log(H/h))^{2}$ in three dimensions. To the contrary, in spite of the absence of any preconditioners, it is shown that the proposed method is numerically scalable in the sense that for a large value of $\eta$, the condition number of the resultant dual problem is bounded by a constant independent of both the subdomain size $H$ and the mesh size $h$ in three dimensions as well as two dimensions. In addition, particular attention is paid to computational issues to improve the practical efficiency of proposed methods and the theoretical bounds are verified by numerical results. Finally, the parallel scalability is analyzed based on parallel numerical experiments which are implemented on the 10 node IBM p595 at the KISTI Supercomputing Center.

• Tohru Ozawa (Hokkaido Univ.)

 Analysis Title: Global existence of analytic solutions to nonlinear Schr\"odinger equations Abstract: In this talk I present my recent results on global existence of analytic solutions to cubic NLS with small Cauchy data in space dimensions greater than or equal to 2.

• Soogil Seo (Yonsei Univ.)

 Algebra Title: Truncated Euler systems over imaginary quadratic fields Abstract: Let K be an imaginary quadratic field and let $F$ be an abelian extension of K. It is known that the order of the class group Cl$F$ of $F$ is equal to the order of the quotient $U_F /El_F$ of the group of global units $U_F$ by the group of elliptic units $El_F$ of $F$. We introduce a fltration on $U_F /El_F$ made from the so-called truncated Euler systems and conjecture that the associated graded module is isomorphic, as a Galois module, to the class group. We provide evidence for the conjecture using Iwasawa theory.

• Hong-Yeop Song (Yonsei Univ.)

 Cryptography Title: Choi's orthogonal latin squares is at least 67 years earlier than Euler's Abstract: Euler's conjecture on 36 officers problem (a.k.a. orthogonal Latin squares of order 6) was first appeared in 1782, and it has been widely known that this is the origin of orthogonal Latin squares and combinatorial mathematics. The literature on Latin squares goes back to at least 300 years to the monograph Koo-Soo-Ryak by Choi Seok-Jeong (1646-1715). He uses orthogonal Latin squares of order 9 to construct a magic square and notes that he cannot find orthogonal latin squares of order 10. We will see Latin squares, orthogonal Latin squares, magic squares and their relations in the early history of combinatrial mathematics. Finally, we will see how these just have appeared in Handbook of Combinatorial Designs, published in 2006 by CRC Press.

• Roland Speicher (Queens Univ.)

 Functional Analysis Title: Invariance under quantum permutations and free probability theory Abstract: The classical de Finetti theorem says that invariance under permutations of the joint distribution of infinitely many random variables is equivalent to the fact that the variables are independent and identically distributed (with respect to the conditional expectation onto the tail algebra of the variables). We address the question whether we have a corresponding characterization in a noncommutative context. Random variables are now replaced by, in general noncommuting, operators on Hilbert spaces, their joint distribution is given by a state on the corresponding operator algebra and invariance under the permutation group is replaced by invariance under the action of the quantum permutation group. It turns out that one has indeed a noncommutative analogue of de Finetti's theorem and that this relates with Voiculescu's free probability theory. (This is joint work with Claus Koestler.)

• Ulrike Tillmann (Oxford Univ.)

 Topology Title: The cohomology of moduli spaces Abstract: In recent years major contributions to the study of the cohomology of moduli spaces of Riemann surfaces have been made by topologists. We will review some of these developments and demonstrate the power of homotopy theoretic tools by solving some natural questions in the cohomology of groups related to moduli spaces. The latter is in part joint work with Yongjin Song.

• Nicole Tomczak-Jaegermann (Univ. of Alberta)

 Functional Analysis Title: Random embeddings and other high-dimensional geometric phenomena Abstract: This talk will illustrate a geometric and probabilistic approach of Asymptotic Geometric Analysis to several high-dimensional phenomena described by a large class of random matrices. Those include Gaussian and +1/-1 matrices, more generally, subgaussian matrices, and also matrices determined by subsets of bounded orthogonal systems. We shall consider random embeddings of normed spaces (notably, of the Euclidean space) and some properties of combinatorial flavor of random 0/1 polytops. These phenomena are intimately connected to probabilistic inequalities for singular numbers of a wide class of random matrices.

• Takao Yamaguchi (Univ. of Tsukuba)

 Geometry Title: Collapsing and essential coverings of Riemannian manifolds Abstract: There are some relations between covering and topology of manifolds. For instance, the minimal number of topologically simple metric balls needed to cover a manifold represents a topological complexity of the manifold. This idea brought Cheeger and Weinstein's finiteness theorems in the non-collapsing case. In this talk, I mainly consider the collapsing case. After recalling basic problems and results about collapsing Riemannian manifolds, I will introduce the notion of an essential covering of a Riemannian manifold, and discuss the uniform boundedness of the minimal number of the balls in the essential coverings of collapsed manifolds.

Tel:82-2-565-0361  Fax:82-2-565-0364  E-mail:kms@kms.or.kr  http://www.kms.or.kr/eng