2010 Global KMS International Conference

 ±âÁ¶°­¿¬  (Plenary Lecture)

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Paul H. Rabinowitz Edward Burr Van Vleck Professor of Mathematics and a Vilas Research Professor at the University of Wisconsin, Madison, USA Distinguished Visiting Professor of Mathematics at POSTECH, Korea   PH.D. New York University, 1966 M.A. New York University, 1963 B.A. New York University, 1961   University of Wisconsin, Professor of Mathematics, 1971-present University of Wisconsin, Associate Professor,         1969-1971 Stanford University, Assistant Professor,                 1966-1969 Stanford University, Instructor,                                January-September 1966

¡¡       Plenary Lecture  ||  11:20-12:10, October 22(Fri), 2010          It all started with Moser     We survey work on a class of nonlinear elliptic partial differential equations that was initiated by Moser. Methods     from the calculus of variations, dynamical systems, and geometry as well as PDE have been used to find a rich     variety of solutions.

 
 
 Çѱ¹°úÇлó ¼ö»ó±â³ä °­¿¬ (Special Lecture by  the Korean Science Award Winner) 
        

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°­Çö¹è  Hyeonbae Kang ÀÎÇÏ´ëÇб³ Á¤¼®¼®Á±³¼ö / Jungseok Chair Professor of Inha University, Korea   2010³â Çѱ¹°úÇлó ¼ö»ó / Korean Science Award Winner (Presidential Award)   PH.D. University of Wisconsin, Madison, 1989 M.S. Seoul Nat'l University, 1984 B.S. Seoul Nat'l University, 1982   Inha University, Jungseok Chair Professor,                          2008.9-present Seoul Nat'l University, Assistant/Associate/Full Professor,   1997.9-2008.8

¡¡       Special Lecture by the Korean Science Award Winner  ||  10:30-11:20, October 23(Sat), 2010               Generalized Polarization Tensors: Mathematics and Applications     The generalized polarization tensors (GPT) are geometric and physical quantities associated with inclusions.     They appear naturally in multipolar expansions of electric potentials near infinity in the presence of the inclusion.     In this talk I will discuss about optimal bounds for GPTs, Kang-Milton's solution to the Polya-Szego conjecture, and     connection to the Dirichlet-to-Neumann map. I then discuss their usage for reconstruction of fine details of the     shape of the inclusion. I will also mention briefly on their connection to the cloaking (invisibility) which attracts much     attention these years.

  
 
 ÃÊû°­¿¬ (Invited Lectures)
 

¡¡Carsten Carstensen            Professor of Humboldt University, Germany       Invited Lecture_Applied Mathematics  ||  15:30-16:10, October 22(Fri), 2010        A posteriori error estimator competition for 2nd-order Partial Differential Equations    Five classes of up to 13 a posteriori error estimators compete in three second-order model cases, namely the    conforming and non-conforming first-order approximation of the Poisson-Problem plus some conforming obstacle    problem. Since it is the natural first step, the error is estimated in the energy norm exclusively -- hence the    competition has limited relevance. The competition allows merely guaranteed error control and excludes the question    of the best error guess. Even nonsmooth problems can be included. For a variational inequality, Braess considers    Lagrange multipliers and some resulting auxiliary equation to view the a posteriori error control of the error in the    obstacle problem as computable terms plus errors and residuals in the auxiliary equation. Hence all the former a    posteriori error estimators apply to this nonlinear benchmark example as well and lead to surprisingly accurate    guaranteed upper error bounds. This approach allows an extension to more general boundary conditions and a    discussion of efficiency for the affine benchmark examples. The Luce-Wohlmuth and the least-square error estimators    win the competition in several computational benchmark problems. Novel equilibration of nonconsistency residuals and    novel conforming averaging error estimators win the competition for Crouzeix-Raviart nonconforming finite element    methods. Our numerical results provide sufficient evidence that guaranteed error control in the energy norm is indeed    possible with efficiency indices between one and two. Furthermore, accurate error control is slightly more expensive    but pays off in all applications under consideration while adaptive mesh-refinement is sufficiently pleasant as accurate    when based on explicit residual-based error estimates. Details of our theoretical and empirical ongoing investigations    will be found in the papers quoted below.

  

¡¡Youri Egorov         Professor of Paul Sabatier University, Toulouse, France    Invited Lecture_Analysis  ||  15:30-16:10, October 22(Fri), 2010        On the tallest column    A new approach is proposed to the study of the classical problem about the highest column of given volume stated    by L. Euler. The diculty of the problem is related to the presence of the continuous spectrum for the corresponding    Sturm-Liouville operator. The existence and the uniqueness of the solution is proved for the rst time. The method is    based on the study of critical points of a suitable nonlinear functional.

      

¡¡Baohua Fu          Professor of Chinese Academy of Science, China    Invited Lecture_  ||  15:30-16:10, October 22(Fri), 2010         Geometry of nilpotent orbit closures     Nilpotent orbits play an important role in the study of representation theory of a reductive group. Their closures    provide very nice examples of symplectic singularities. I'll give a survey on the study of birational geometry of    nilpotent orbit closures, from the viewpoint of Minimal Model Program. A list of open problems will be discussed at    the end.

     

¡¡±è¿µÈÆ  Young-Hoon Kiem          ¼­¿ï´ëÇб³ ±³¼ö / Professor of Seoul National University, Korea    Invited Lecture_Algebra  ||  15:30-16:10, October 22(Fri), 2010    Curve counting invariants and wall crossing    This talk is about joint work with Jun Li (Stanford).    Curve counting invariants are defined as intersection numbers on moduli spaces of curves in a given projective variety    when the moduli spaces are (1)compactified and (2)equipped with perfect obstruction theories. I will explain various    compactifications which give rise to a variety of curve counting invariants such as Gromov-Witten and Donaldson-    Thomas invariants. A prominent open problem is to compare these invariants and wall crossing for objects in the    derived category has been playing a key role in recent progress. Also I will explain known techniques to handle    virtual fundamental classes obtained from perfect obstruction theories. Special emphasis will be laid on our recent    technique, called ``localization by cosection'' which tells us that if the obstruction sheaf admits a cosection, the    virtual fundamental class localizes to the locus where the cosection is not surjective. This technique has found    exciting applications in (1) a recent proof of the Katz-Klemm-Vafa conjecture (generalization of Yau-Zaslow formula)    by Maulik-Pandharipande-Thomas, (2) a calculation of GW invariants for surfaces of general type, (3) a theory of spin    curve counting by Chang-Li, (4) a wall crossing formula without Chern-Simons functional and (5) a theory of g    eneralized DT invariants via Kirwan blow-ups.

   

¡¡Miyuki Koiso         Professor of Kyushu University, Japan    Invited Lecture_Geometry  ||  15:30-16:10, October 22(Fri), 2010     ¡¡   ¡¡ ¡¡    Geometric variational problems and bifurcation theory     In the study of geometric variational problems, it is natural to ask whether each critical point is stable (that is, the    considered critical point attains a local minimum of the energy) or not. Also it is important to determine the geometric    properties of solutions and to study the structure of the set of solutions. In this talk, as one of the steps to investigate    these problems, we discuss stability, existence of bifurcation, and symmetry-breaking for solutions of variational    problems for hypersurfaces with constraint. Although our method is sufficiently general to apply various variational    problems, we mainly concentrate on hypersurfaces with constant anisotropic mean curvature in the euclidean space,    which are characterized as critical points of anisotropic surface energy with volume constraint.     Useful criteria for the stability and existence of bifurcation are given by the properties of eigenvalues and     eigenfunctions of the eigenvalue problem associated with the second variation of the energy. We will give general    methods and their applications to several concrete examples which may be interesting from both mathematical and    physical point of view.

   

¡¡ÀÌÇö´ë  Hyun Dae Lee           ÀÎÇÏ´ëÇб³ ±³¼ö / Professor of Inha University, Korea    2009³âµµ »ó»ê ÀþÀº¼öÇÐÀÚ»ó ¼ö»ó / The winner of 2009 Sangsan Prize for young Mathematicians    Invited Lecture_Applied Mathematics  ||  15:30-16:10, October 22(Fri), 2010      Asymptotic imaging of cracks    We consider cracks with Dirichlet boundary conditions. We first derive an asymptotic expansion of the boundary    perturbations that are due to the presence of a small crack. Based on this formula, we design a noniterative    approach for locating a collection of small cracks. In order to do so, we construct a response matrix from the    boundary measurements. The location and the length of the crack are estimated, respectively, from the projection    onto the noise space and the first significant singular value of the response matrix. Indeed, the direction of the crack    is estimated from the second singular vector. We then consider an extended crack with Dirichlet boundary conditions.    We rigorously derive an asymptotic expansion for the boundary perturbations that are due to a shape deformation of    the crack. To reconstruct an extended crack from many boundary measurements, we develop two methods for    obtaining a good guess. Several numerical experiments show how the proposed techniques for imaging small cracks    as well as those for obtaining good initial guesses toward reconstructing an extended crack behave.

   

¡¡ÀÌ°­Çõ  Kang-Hyurk Lee           °íµî°úÇпø ±³¼ö / Professor of KIAS, Korea    Invited Lecture_Analysis ||  15:30-16:10, October 22(Fri), 2010    On the bounded realization of unbounded models in Several Complex Variables     As the Riemann mapping theorem fails on multi-dimensional complex Euclidean spaces, it has been a fundamental    problem in Several Complex Variables to classify domains which can play the same role of model objects as the unit    disc. This is indeed the classification program for domains with non-compact automorphism group. Many theories    said that most of model domains can be realized as a so-called Siegel type domain, a generalization of the    half-plane representation of the unit disc. Although Siegel type domains are unbounded, they have many affine     automorphisms; hence this representation of models has been employed in wide area of Several Complex Variables.    In this talk, I will introduce briefly theories on this topic and then discuss problems on the realization of a Siegel type    domain as a bounded domain. Especially I will mention my current collaboration with Jisoo Byun on the smoothly    bounded realization of a Siegel type domain with real-analytic boundary.