Open Lecture for Public    

 Complex Analysis
  
     Nicholas Buchdahl (Univ. of Adelaide) 

Title : Towards the classification of surfaces of Class VII Abstract : In the Enriques-Kodaira classification of compact complex surfaces, the surfaces of Class VII (i.e., those with first Betti number equal to 1) remain somewhat mysterious. Although there are several methods for constructing examples of these surfaces, it is not known every such surface can be constructed by these methods. The outstanding conjecture in this area is that indeed every surface of Class VII arises by means of these constructions. In recent years, significant progress has been made in tackling this problem. In particular, by using techniques from gauge theory, Andrei Teleman has been able to completely classify the surfaces of Class VII with second Betti number zero or one, and has made substantial progress in the case of second Betti number two. (The case of zero second Betti number was earlier considered by Bogomolov.) In this talk I shall describe the background to the problem and outline Teleman's program for tackling it using gauge theory, including the successes in the case of small second Betti number.

     Jisoo Byun (POSTECH) 

Title : Explicit description of special domains Abstract : In this talk, we introduce special kinds of domain in 2 dimensional complex space. We want to find all automorphism group of these domain. This gives the complete description of automorphism group of Kohn-Nirenberg domain and Fornaess domain.

    Boo Rim Choe (Korea Univ.)

Title : Recent progress on the finite-rank-product conjecture. Abstract : It has been conjectured that if a product of Toeplitz operators with function symbols, either on Hardy space or Bergman space, has finite rank, then one of the factor must be the zero operator. In this talk we survey recent results, as well as related results, towards the conjecture.

     Vladimir Ejov (Univ. of South Australia)

Title : Normal forms of real hypersurfaces of finite type with 2-dimensional          symmetries in $\mathbb C^2$ Abstract : We give a complete description of normal forms for real hypersurfaces of finite type in $\mathbb C^2$  when their local holomorphic symmetry algebras is 2-dimesional. We use the method of simultaneous normalisation of the equations and the symmetries that goes back to Lie and Cartan. Our approach leads to a unique canonical equation of the hypersurface for every type of its symmetry algebra. Moreover, in the Levi-degenerate case our construction implies convergence of the transformation to the normal form. This is a joint work with M. Kolar and G. Schmalz.

     Adam Harris (Univ. of New England)

Title: Special deformations of complex cone singularities Abstract: Let M be the total space of a holomorphic line bundle with negative curvature over a compact complex manifold. The canonical "blowing down" of the zero section yields a complex space X exemplifying a well-known class of isolated singularities. In this talk we discuss special solutions of the Kodaira-Spencer equation having a certain rotational symmetry, and the associated deformations of complex structure they represent. Moreover we will look at the question of how these deformations might relate to the space of deformations of X.

     Alexander Isaev (Australian National Univ.)

Title : Affine equivalence of spherical tube hypersurfaces Abstract : We consider hypersurfaces in complex (n+1)-dimensional space CR-equivalent to a Levi non-degenerate hyperquadric. Such hypersurfaces are often called spherical, or, more precisely, (k,n-k)-spherical, where (k,n-k) is the signature of the quadric's Levi form. The Cartan-Tanaka-Chern-Moser theory yields that the sphericity condition can be expressed as a (rather complicated) system of partial differential equations for the defining function of the hypersurface. For tube hypersurfaces this system significantly simplifies, which allows one to efficiently study spherical tube hypersurfaces up to affine equivalence. In particular, a complete affine classification exists for k=n, n-1, n-2. Furthermore, it has been known since the 1980s that the number of affine equivalence classes is infinite in the following cases: (i) k=n-2, n>6; (ii) k=n-3, n>6; (iii) k<le n-3, and finite for all other values of k and n, except possibly for k=3, n=6. The exceptional case k=3, n=6 remained unresolved until 2008 when Fels and Kaup gave an example of a family of (3,3)-spherical tube hypersurfaces that contains infinitely many pairwise affinely non-equivalent elements. In my talk I will discuss the example due to Fels and Kaup and give an overview of the previously known results.

     Ilya Kossovskiy (Australian National Univ.)

Title : Authomorphisms of degenerate quadratic hypersurfaces in $CP^n$. Abstract : We consider 2 kinds of degenerate quadratic hypersurfaces in $\mathbb{C}^n$, extended to $\mathbb{CP}^n$ - degenerate hyperquadrics and quadratic cones, and find out under what conditions the automorphisms of extended surfaces turn out to be the automorphisms of $\mathbb{CP}^n$.

     Finnur Larusson (Univ. of Adelaide)

Title : Gromov's Oka Principle Abstract : The Oka Principle refers to a cluster of results in complex analysis and geometry saying, roughly speaking, that analytic problems on Stein manifolds that can be formulated cohomologically or even homotopically have only topological obstructions to their solutions. The Oka Principle has its roots in work of Oka himself in the 1930s and matured in deep work of Grauert around 1960. It was significantly expanded in a seminal 1989 paper of Gromov and has, since 2000, been further developed by Forstneric and others. We will give an overview of the basic theory of Gromov's Oka Principle, some of its applications, and connections with abstract homotopy theory, and conclude with results proved in 2009 by Forstneric and by the speaker.

     Jaesung Lee (Sogang Univ.)

Title : A volume mean values in the unit disc and polydisc. Abstract : Let $D$ be the open unit disc of $\Bbb C$ and $m$ be a radial measure on $D$ with $m(D)=1$. \\ If $u \in L^{1}(D,m)$ is harmonic, then $u\circ \varphi$ is also harmonic for every $\varphi  \in \hbox{Aut}(D)$. \\ Thus $u\circ \varphi$ satisfies a volume mean value property ; $$ u \big(  \varphi(0)  \big) \ = \ \int_{D} \ u \circ \varphi \ dm \quad \hbox{for \ every} \ \varphi \in \hbox{Aut}(D), \leqno(0.1) $$ Here, we examine under what condition of $u$ and $m$ satisfying $(0.1)$ type of equation forces $u$ to be harmonic. Especially, for $c>-1$, let $\nu_{c}$ denote a weighted radial measure on $\mathbb{C}$ normalized so that $\nu_{c}(D)=1.$ If $u$ is harmonic and integrable with respect to $\nu_{c}$, then $\int_D (u\circ\psi)\ d\nu_{c}=u(\psi(0))$ for every $\psi \in {\hbox{Aut}(D)}$. Equivalently $u$ is invariant under the weighted Berezin transform; $B_cu=u$. Conversely, does the invariance under the weighted Berezin transform imply the harmonicity of a function? Here, we show that for any $1 \le p< \infty$ and $c_1, c_2 >-1$, a function $f \in L^{p}(D^{2},\ \nu_{c_1} \times \nu_{c_2})$ which is invariant under the weighted Berezin transform; $B_{c_1,c_2}f=f$ needs not be $2-$harmonic.

     Kang-Hyurk Lee (KIAS)

Title : Upper semi-continuity of automorphism groups under the deformation of almost complex structures. Abstract : We consider a strongly pseudoconvex domain in almost complex manifold and a smooth deformation of almost complex structure up to the boundary. In case of bounded domain in $\mathbb C^n$ and the integrable deformation of the standard complex structure, the automorphism group of the deformation is Lie group isomorphic to a subgroup of the original one. This is the theorem of Greene and Krantz in 1982. We will give its generalization to the almost complex setting. This is a joint work with J. Byun and H. Gaussier.

     Jongdo Park (KIAS)

Title : Recent problems of Cartan-Hartogs domains Abstract : In this talk, we introduce Cartan-Hartogs domains whose base domains are bounded symmetric domains. The Bergman kernel functions, Einstein-Kaehler metric, and Caratheodory extremal mappings for Cartan-Hartogs domains have been studied by Weiping Yin and Guy Roos. We review the recent results related to these topics.

 Geometric Analysis
 
     Maria Athanassenas (Monash Univ.)

Title : Recent developments in capillarity                                            Abstract : Capillary surfaces are liquid-air or liquid-liquid interfaces, and are mathematically modelled as prescribed mean curvature surfaces -- a second order, nonlinear, elliptic PDE. We will present two topics in which there have been exciting developments since 2000.                                                    (a) A new model incorporating compressibility has been introduced. We will describe how the prescribed mean curvature equation has changed, the results, and their similarities and differences to those of the imcompressible model.      (b) We will give an overview of the resolved and open problems concerning behaviour in domains with corners.

     Jaigyoung Choe (KIAS) 

Title : First eigenvalue of the Laplacian on minimal surfaces in S3 Abstract : Let 1 be the rst nontrivial eigenvalue of the Laplacian on a compact surface without boundary. We show that 1 = 2 on compact embedded minimal surfaces in S3 which are invariant under a nite group of reections and whose fundamental piece is simply connected and has less than six edges. In particular 1 = 2 on compact embedded minimal surfaces in S3 that are constructed by Lawson, by Karcher-Pinkall-Sterling and by Kapouleas-Yang.

     Michael Eastwood (Australian National Univ.)

Title : Higher symmetries of the Laplacian Abstract : Which linear differential operators preserve harmonic functions? Even on Euclidean space, this is a deceptively simple question. The answer may be expressed in terms of conformal geometry and the AdS/CFT correspondence. If time permits, I shall also discuss this same question with regard to other familiar differential operators such as the Dirac operator.

     Rod Gover (Univ. of Auckland)

Title : Differential complexes in conformal geometry Abstract : The de Rham complex is a prototype for a large class of sequences of differential operators often called (generalised) Bernstein-Gelfand-Gelfand BGG sequences. Conformal manifolds admit such sequences and on locally conformally flat manifolds the sequences are elliptic complexes (in fact locally exact). These complexes are closely linked to the complexes constructed and treated by Spencer, Gasqui, Goldschmidt and others starting in the 1950's, in relation to deformations of geometric structures and the study of overdetermined PDE. A basic example is a complex which controls the deformation theory of conformally flat manifolds. On conformally curved manifolds the BGG sequences no longer form complexes and so have limited application. It turns out that certain related sequences of differential operators do yield conformally invariant elliptic complexes on classes of curved conformal manifolds and hence yield global conformal invariants. The study of these ``detour complexes'' is in its early stages and there are many open problems. An introduction to the basic ideas will be sketched with some results and applications described. The latter includes links to a certain invariant of Branson and Orsted known as the Q-curvature.

     Seongtag Kim (Inha Univ.)

Title : A study of complete Bach-at manifolds Abstract : Let (M, g) be a Riemannian four-manifold with Weyl curvature W. Then the square of Weyl curvature functional F(g) = RM jWj2dVg is conformal invariant, i.e., F(g) = F(ewg). A critical metric of F is called a Bach-at metric. Let Rij be the Ricci curvature Rij of g. The bach tensor Bij is dened by Bij rkrlWkijl + 1 2RklWkijl, which is conformal invariant. It is known that (M, g) is Bach-at if and only if Bij = 0. Many important four-manifolds are Bach-at. For examples, Einstein metrics, conformally Einstein metrics and self-dual metrics are Bach-flat.

     Young Ho Kim (Kyungpook National Univ.) 

Title : Differential geometry on ruled submanifolds in a Minkowski space Abstract : Ruled submanifolds of finite type in a Lorentz-Minkowski space are introduced. A ruled submanifold of the null scroll type which is called a NS-ruled submanifold is introduced and classify NS-ruled submanifolds with finite type. Also, we show that a special NS-ruled submanifold called a BS-ruled submanifold is characterized by the minimal polynomial of the shape operator associated with the mean curvature vector field.

     Jeong Hyeong Park (Sungkyunkwan Univ.)

Title : When are the tangent sphere bundles of a Riemannian manifold eta-Einstein? Abstract : An Einstein manifold is a manifold where the Ricci operator has just a single eigenvalue. If the Ricci operator has 2 eigenvalues of multiplicities $(m-1,1)$, then we get an $eta$-Einstein structure. For an Einstein manifold of dimension $mge3$, the eigenvalue doesn't change. However, for an $eta$-Einstein manifold, the eigenvalues can change. We study the geometry of a tangent sphere bundle of a Riemannian manifold $(M,g)$. The condition 'Einstein' for tangent sphere bundles of constant radius is extremely rigid, so, it is an interesting problem to determine when a tangent sphere bundle is eta-Einstein with the standard contact metric structure. Let $M$ be an $n$-dimensional Riemannian manifold and $T_r M$ be the tangent bundle of $M$ of constant radius $r$. The main theorem is that $T_r M$ equipped with the standard contact metric structure is $eta$-Einstein if and only if $M$ is a space of constant sectional curvature $frac{1}{r^2}$ or $frac{n-2}{r^2}$. (This is joint work with K. Sekigawa)

     Gerd Schmalz (Univ. of New England)

Title : Cartan's homogeneous CR-manifolds revisited Abstract : In 1932 Elie Cartan classified the homogeneous 3-dimensional CR-manifolds. This was based on Bianchi's classification of 3-dimensional Lie algebras and the fact that these Lie algebras admit a representation by holomorphic mappings on the 2-dimensional complex projective space. Consequently, the homogeneous manifolds appear as orbits of the 3-dimensional Lie group actions. Our approach is similar, but purely local and exploits normal forms for the infinitesimal automorphisms that generate the respective symmetry algebras as well as for the germs of the CR-manifolds. This is joint work with Vladimir Ezhov and Martin Kolar.

     Hae-Yong Shin (Chung-Ang Univ.)

Title : Ruled minimal surfaces in homogeneous spaces Abstract : In the Euclidean space E3, the planes and the helicoids are the only ruled minimal surfaces and they are the only surfaces in R3 which is both minimal for the standard Riemannian metric and maximal for the standard Lorentzian metric. In this talk, we will list all the ruled minimal surfaces in the product spaces S2 R, H2 R and show that these surfaces are the only minimal surfaces which is also maximal with respect to the standard Lorentzian metric on those product spaces. We also prove the similar results for the Heisenberg group H3 with left invariant Riemannian and Lorentzian metric. This is joint work with Y.W. Kim, S.E. Koh, S.D. Yang and H.Y. Lee.

     Young Jin Suh (Kyungpook National Univ.)

Title : Commuting Ricci Tensor, Einstein Manifolds and Related Topics Abstract : In this talk, first we introduce the full expression of the curvature tensor of a real hypersurface M in complex two-plane Grass-mannians G2(Cm+2) from the equation of Gauss. Next we derive a new formula for the Ricci tensor of M in G2(Cm+2). Finally we prove that a Hopf hypersurface in G2(Cm+2) with commuting Ricci tensor is locally congruent to a tube of radius r over a totally geodesic G2(Cm+1) and give some classifications of generalized Einstein hypersurfaces in G2(Cm+2) and related topics on geometric analysis of hypersurfaces in dual complex two-plane Grassmannians.

 Non-linear Analysis
  
    Yeol Je Cho (Gyeongsang National Univ.)

Title : On modifed iterative method for generalized equilibrium problems and fixed points problems Abstract : In this paper, we present a new iterative method for finding a common element of the set of solutions of generalized equilibrium problems and the set of fixed points of a strict pseudo-contractive mapping in Hilbert spaces. Furthermore, we prove that the proposed iterative method converges strongly to the common element under some mild conditions imposed on algorithm parameters.

     Sever S. Dragomir (Victoria Univ.)

Title: Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces Abstract : The main aim of the present paper is to survey various inequalities for convex functions of selfadjoint operators in Hilbert spaces obtained recently by the author. Amongst these, some reverses of the Jensen¡¯s operator inequality, the Slater operator inequality and related results are presented.

     Jong Soo Jung (Dong-A Univ.)

Title : Strong convergence theorems for nonexpansive semigroups in banach spaces Abstract : In this talk, we consider implicit and explicit iterative schemes for finding a common fixed point of the nonexpansive semigroup in Banach spaces. Under suitable conditions, strong convergence of the sequences generated by iterative schemes to a common fixed point of the nonexpansive semigroup, which is a solution of certain variational inequality, are obtained. The main results improve and complement the previous corresponding results.

     Jung Im Kang (National Institute for Mathematical Sciences)

Title : On Amenability of Semigroups Abstract : In the 1940's and 1950's, the subject of amenability, essentially begun with Lebesgue, was studied by Mahlon M. Day, and his 1957 paper on amenable semigroups is a major landmark. In this talk, we will investigate amenability for semigroups, in particular, semi-topological semigroups. Also, we will discuss characterizations, especially, fixed point properties of several important semigroups in terms of amenability.

     Do Sang Kim (Pukyong National Univ.)* Young Min Kang (Pukyong National Univ.)

Title : Nonsmooth Multiobjective Programming Involving Support Functions Abstract : In this talk, we consider the nonsmooth multiobjective programming problem involving locally Lipschitz functions and support functions. Two types of Karush-Kuhn-Tucker optimality conditions are introduced. We give sufficient Karush-Kuhn-Tucker optimality conditions by using generalized convexity assumptions and certain regularity conditions. Furthermore, we formulate the Wolfe type dual and Mond-Weir type dual problems and establish duality theorems between our primal problem and dual problems under generalized convexity and regularity conditions.

     Gwang Hui Kim (Kangnam Univ.)

Title : On the superstability of the pexiderized cosine type functional equation Abstract : The aim of this paper is to investigate the stability problem for the pexiderized cosine type functional equations $f(x+y)+f(x-y)=2g(x)h(y), f(x+y)+g(x-y)=2f(x)g(y), f(x+y)+g(x-y)=2g(x)f(y)$ under the conditions : $|f(x+y)+f(x-y)-2g(x)h(y)| \leq \varphi(x)$, $|f(x+y)+g(x-y)-2f(x)g(y)| \leq \varphi(x)\; {\text or}\; \varphi(y) $, $|f(x+y)+f(x-y)-2g(x)h(y)| \leq \varphi(x)\; {\text or}\; \varphi(y)$. As a consequently we have generalized the results of stability for the cosine(d'Alembert) and the Wilson functional equations by J. Baker,  P. G{\v a}vruta, R. Badora and R. Ger, Pl.~Kannappan, and G. H. Kim.

     Jong Kyu Kim (Kyungnam Univ.)

Title : convergence theorems for multi-step iteration schemes with errors for  symptotically quasi nonexpansive type nonself mappings Abstract : In this paper, a strong convergence theorem for multi step iteration scheme with errors for asymptotically quasi-nonexpansive type nonself mappings is established in a real uniformly convex Banach space. Our results extend the corresponding results of Wangkeeree [12], Xu and Noor [13] and many others.

     Tae-Hwa Kim (Pukyong National Univ.)

Title : Implicit Iteration Methods with Errors for non-Lipschitzian families with Perturbed Mappings Abstract : In this paper, motivated and inspired by recent works on implicit iteration methods, we prove necessary and sufficient conditions for strong convergence of the eventually implicit iteration methods with errors to a common fixed point of a family which is continuous total asymptotically nonexpansive (in brief, TAN) on $q$-uniformly smooth Banach spaces with a perturbed mapping $F$, $1<q\leq 2$, under some suitable control conditions of parameters. Some applications to viscosity approximation methods or to the eventually implicit algorithms with errors for a finite family of continuous TAN self mappings in real Banach spaces are also added. These improve and extend the corresponding results due to Zeng and Yao [{\em Nonlinear Anal.,} {\bf 64} (2006), 2507--2515] for a finite family of nonexpansive self mappings in Hilbert space settings and Chidume and Ofoedu [{\em J. Math. Anal. Appl.,} {\bf 333} (2007), 128-141] for a finite family of $N$ continuous TAN self mappings in real Banach spaces.

     Gue Myung Lee (Pukyong National Univ.)* Le Anh Tuan (Ninh Thuan College of Pedagogy)

Title : On $\epsilon$-Optimality Theorems for Convex Set-Valued Optimization Problems Abstract : In this talk, $\epsilon$-subgradients for convex set-valued maps are defined. We prove an existence theorem for $\epsilon$-subgradients of convex set-valued maps. Also, we give $\epsilon$- optimality theorems for an $\epsilon$-solution of a convex set-valued optimization problem.

     Choonkil Park(Hanyang Univ.)

Title : Fuzzy stability of an additive-quadratic-cubic-quartic functional equation Abstract : Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic  functional equation \begin{eqnarray*}  f(x+2y)+f(x-2y)=4f(x+y)+4f(x-y) - 6 f(x) + f(2y) + f(-2y) - 4f(y) - 4 f(-y)  \end{eqnarray*}   in fuzzy Banach spaces.   Keywords: fuzzy Banach space, fixed point, generalized Hyers-Ulam stability, additive-quadratic-cubic-quartic  functional equation. This work was supported by Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00041).

     Sehie Park (Seoul National Univ.)

Title : From Simplices to KKM Spaces - A brief history of the KKM theory Abstract : We review briefly the history of the KKM theory from the original KKM theorem on simplices in 1929 to the birth of the new KKM spaces by the following steps. (1) We recall equivalent formulations of the Brouwer fixed point theorem and the KKM theorem. (2) We summarize Fan's foundational works on the KKM theory from 1960s to 1980s. (3) We note that, in 1983-2005, basic results in the theory were extended to convex spaces by Lassonde, to $H$-spaces by Horvath, and to $G$-convex spaces due to Park. (4) In 2006, we introduced the concept of abstract convex spaces $(E,D;\Gamma)$ on which we can construct the KKM theory. Moreover, abstract convex spaces satisfying an abstract form of the KKM theorem and its `open' version are called {\it KKM spaces}. Now the KKM theory becomes the study of KKM spaces. (5) Finally, we introduce the KKM space versions of the von Neumann minimax theorem, the von Neumann intersection lemma, the Nash equilibrium theorem, and the Himmelberg fixed point theorem.

     Xialong Qin (Gyeongsang National Univ.)* Shin Min Kang (Gyeongsang National Univ.)

Title : Iterative processes with errors for $m$-accretive operators in Banach spaces Abstract : In this paper, we study the convergence of paths for continuous pseudocontractions in a real Banach space by viscosity approximation methods. As applications, we consider the problem of finding zeros of $m$-accretive operators by an iterative process with errors. Strong convergence theorems of the iterative method are established in a real Banach space.

    Ta Quang Son (Nhatrang College of Education)* Do Sang Kim (Pukyong National Univ.)

Title : Lagrange Multiplier Characterizations of Solution Sets of a Class of Nonconvex Semi-Infinite Programs Abstract : In this paper we deal with characterizations of solution sets of a class nonconvex semi-infinite programing problems. Based on a semiconvexity property applied to functions involved, several types of characterizations of solution sets of the problems are given. These types of characterizations are extended from those of solution sets of some classes of convex problems. Although the problems in consideration are nonconvex, their characterizations of solution sets (in symbolic forms) are similar to those of some convex problems known before.

 Partial Differential Equations
 
     Inkyung Ahn (Korea Univ.)

Title : Upper-Lower Solution Method for Nonlinear  Systems and its Applications Abstract :  We report on the upper-lower solution technique for nonlinear elliptic and parabolic  systems without  assumptions of quasi-monotonicity. An application is also described involving  the existence of positive steady states of a certain interaction system arising in medical science.

     Hyeong-Ohk Bae (Ajou Univ.)

Title : Pressure Representation and Boundary Regularity of the Navier-Stokes Equations with Slip Boundary Condition Abstract : We first represent the pressure in terms of the velocity in $\mathbb{R}^3_+$. Using this representation we prove that a solution to the Navier-Stokes equations is in $L^\infty(\mathbbR^3_+\times (0,\infty))$ under the critical assumption that $u \in L^{r,r^{'}}_{loc}$, $\frac{3}{r}+\frac{2}{r^{'}}\le 1$ with $r\ge 3$, while for $r=3$ the smallness is required. In \cite{choe98}, a boundary $L^\infty$ estimate for the solution is derived if the pressure on the boundary is bounded. In our work, we remove the boundedness assumption of the pressure. Here, our estimate is local. Indeed, employing Moser type iteration and the reverse H\"older inequality, we find an integral estimate for $L^\infty$-norm of $\mathbf {u}$.

     Soohyun Bae (Hanbat Univ. & POSTECH)

Title : On semilinear elliptic equations with supercritical exponent Abstract : Semilinear elliptic equations with supercritical exponent has been studied under some monotonicity of the coefficient function. When the exponent is bigger than the critical Sobolev exponent, various conditions without monotonicity conditions lead to the existence of positive entire solutions. In order to characterize the solutions, we analyze the asymptotic behavior at infinity. We also discuss the existence of positive solutions with specific asymptotic behavior.

     Dongho Chae (Sungkyunkwan Univ.)

Title : Liouville theorems in fluid mechanics Abstract : In this talk we discuss Liouvillle type of theorems in the  systems of equations for incompressible/compressible fluids. For the incompressible fluids the condition of suitable integrability combined with  sign condition for the pressure integrals leads to the conclusion that the solution should be trivial. For the stationary compressible fluids the pressure condition can be omitted.  For the time dependent compressible fluids we have result that there exists no global weak solution for certain class of initial data.

     Hi Jun Choe (Yonsei Univ.)

Title : Hydrodynamic limit of binary mixture of spheres by Hi Jun Choe Abstract : In this talk we consider two species of spheres that experience elastic collision. Whe two spheres have different mass, the Boltzmann equation has Knudsen number, Mach number and Reynolds numbers involved with the mass ratio. We derive Euler, Navier-Stokes equations as Knudsen number goes to zero. Further, we will discuss the related hydrodynamic parameters.

     Kwangseok Choe (Inha Univ.)

Title : On the vortex solutions in the self-dual Chern-Simons-Higgs theory Abstract : The Chern-Simons-Higgs(CSH) equation is an elliptic equation arising in the self-dual Chern-Simons-Higgs theory. The CSH equation is known to admit two different kinds of vortex solutions : topological and non-topological solutions. We briefly review existence and qualitative properties of topological solutions. Existence of non-topological solutions will be also discussed.

     Daniel Daners (Univ. of Sydney)

Title : An isoperimetric inequality for the first eigenvalue of Robin problems Abstract : We prove an isoperimetric inequality for the first eigenvalue of the $p$-Laplace operator with Robin boundary conditions similar to the classical Faber-Krahn inequality, asserting that amongst all domains of equal volume, the ball has the smallest first eigenvalue. The method is very different from the usual symmetrisation arguments used in case of Dirichlet boundary conditions. We present a new proof based on an alternative representation of the first eigenvalue and the co-area formula.

     Yihong Du (Univ. of New England)

Title : Convergence and sharp thresholds for propagation in nonlinear diffusion problems Abstract : We consider the Cauchy problem \[ u_t=u_{xx}+f(u)\ (t>0,\;x\in \R^1), \quad u(0,x)=u_0(x)\ (x\in \R^1), \] where $f(u)$ is a locally Lipschitz continuous function satisfying $f(0)=0$. In this talk I'll explain the background of this problem and introduce some of my recent joint work with Hiroshi Matano (University of Tokyo). The first part of our results shows that any nonnegative bounded solution with compactly supported initial data converges to a stationary solution as $t\to\infty$. Moreover the limit is either a constant or a symmetrically decreasing stationary solution. The second part applies this result to the special case where $f$ is a bistable nonlinearity and the case where $f$ is a combustion type nonlinearity. Examining the behavior of a parameter-dependent solution $u_\lambda$, it is shown that there exists a sharp threshold between extinction (namely, convergence to $0$) and propagation (namely, convergence to $1$). The result holds even if $f$ has a jumping discontinuity at $u=1$.

     Joseph Grotowski (Univ. of Queensland)

Title: The A-harmonic approximation technique Abstract: we present a survey of the A-harmonic approximation approach to elliptic regularity, including a discussion of recent results on boundary regularity.

     Seung Yeal Ha (Seoul National Univ.)

Title : Nonlinear stability of spherical self-similar ows to the Compressible Euler  equations Abstract : In this talk, we present the nonlinear stability of spherical self-similar flows arising from the uniform expansion of a spherical piston toward still gas. We show that if the perturbation of the expansion speed of a piston is sufficiently small compare to the strength of the leading shock, an entropy weak solution of the isentropic compressible Euler system exists in the self-similar regime between the spherical piston and the leading shock. Moreover, we show that the perturbed ow tends to the corresponding self-similar ow time-asymptotically. Our analysis is based on the modified Glimm scheme. This is a joint work with Wen-Ching Lien (National Cheng Kung University, Taiwan).

     Jongmin Han (Hankuk Univ. of Foreign Studies)

Title : Dynamic bifurcation of the one-dimensional Swift-Hohenberg equation Abstract : In this talk, we consider the dynamic bifurcation of the one-dimensional Swift-Hohenberg equation on a periodic interval. As the control parameter crosses the critical value, it is shown that the equation bifurcates from the trivial solution to an attractor which determines the long time dynamics of the system.

     Min-Chun Hong (Univ. of Queensland)

Title : Some geometric flows between mathematics and physics Abstract : The study of geometric flow is a very active area of modern mathematics, with diverse applications in physics and other sciences. Since Eells and Sampson introduced the heat flow for harmonic maps in 1964, many profound mathematical and practical problems have been solved by geometric flow methods. In 1985, Donaldson used the Yang-Mills heat flow to establish one of the most important theorems, the Donaldson-Uhlebeck-Yau Theorem, in holomorphic vector bundles. Recently, Perelman used the Ricci flow, introduced by Hamilton in 1982, to solve the famous Poincar\'e conjecture, one of the seven Clay Millennium Prize problems. In this talk, I will survey some important results about the Yang-Mills flow and outline new developments for two new geometric flows: the anti-self-dual connection flow (ASD flow) and Seiberg-Witten flow.

     Jaeduck Jang (Hankuk Univ. of Foreign Studies)

Title : Uniqueness of Positive Radial Solutions of  u + f(u) = 0 in ,  n   2 Abstract : We study the uniqueness of radially symmetric ground states for the semilinear elliptic partial differential equation  u + f(u) = 0 in ,  n   2. Assuming F(t) =  f(s)ds is negative in (0, A) and positive in   (A, B), we obtain the uniqueness of nonnegative solutions with u(0) = max u   (0, B) in the case where S(u) = uf'(u)/f(u) is monotonically decreasing in [A, B).

     Hyeonbae Kang (Inha Univ.)

Title : Layer potential techniques in spectral analysis with applications to imaging and optimization. Abstract : In this talk we explain how layer potential techniques can be applied to attack some spectral problems. We will focus on two problems: eigenvalue perturbation problems due to presence of small inclusions and due to small perturbation of the interface. We will derive asymptotic expansions formula of the perturbation of eigenvalues. We then show how these formulae can be used to image the small inclusion and the interface perturbation. The reconstruction method for imaging is based on new optimization approaches. This talk is based on joint works with Habib Ammari, Elena Beretta, Elisa Francini, Eunjoo Kim, Hyundae Lee, Mikyung Lim, Habib Zribi

     Kyungkeun Kang  (Sungkyunkwan Univ.)

Title : Qualitative behavior of a Keller-Segel model with non-diffusive memory Abstract : A one-dimensional Keller-Segel model with a logarithmic chemotactic - sensitivity and a non-diffusing chemical is classified with respect to its long time behavior. The strength of production of the non-diffusive chemical has a strong influence on the qualitative behavior of the system concerning existence of global solutions or Dirac-mass formation. Further, the initial data play a crucial role.

     Yong Jung Kim (KAIST)

Title : A generalization of moment problem to a complex density and its application to the heat equation                                                        Abstract : The moment problem is about positive density functions. This theory has been employed to construct a high order approximation method for solutions to the heat equation. However, its application has been limited due to the positivity assumption on the density. In this paper we extend the theory to complex density functions. Then, an arbitrary real sequence can be considered as real parts of a complex moment sequence. This extended moment problem is successfully applied to approximate solutions to the heat equation. For example, there are several asymptotic approximations of a solution to the heat equation. They show good behavior for time t  large. However, for small time, these methods show bad behaviors. We apply the extended moment problem to the moments of the solution in the backward time. Then approximation using these solutions gives a desired approximation that agrees for $t \to 0$ and for $t \to \infty$. Numerical examples that show the various properties of the method are included.

     Hyunseok Kim (Sogang Univ.)

Title : Unique solvability for the stationary Navier-Stokes system on Lipschitz domains Abstract : This is a joint work with Professor Hi Jun Choe in Yonsei University. Let be a bounded Lipschitz domain in R^3 with connected boundary . In a classical paper[1], Fabes, Kenig and Verchota proved the existence and regularity of a unique solution of the Dirichlet problem for the Stokes system with boundary data in L^2(). The Stokes system in Lipschitz domains has been further studied by Shen[2], Brown and Shen[3] and Dinso¡¤s and Mitrea[4] to obtain optimal results. The goal of the talk is to present our recent results on the solvability of the Dirichlet problem for the stationary Navier-Stokes system. The main result is the existence and regularity of a solution of the Navier-Stokes system with arbitrary boundary data in L^2(). Moreover, the uniqueness of solutions is shown under some smallness condition on the data. Our results are optimal extensions of all the previously known results in [4, 5, 6] for the Navier-Stokes system in Lipschitz domains, and proved by following the approach in the paper [7] for very weak solutions of the Navier-Stokes system in smooth domains. References [1] E. Fabes, C. Kenig and G. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J., Vol. 57, 769-793(1988). [2] Z. Shen, A note on the Dirichlet problem for the Stokes system in Lipschitz domains, Proceedings AMS, Vol. 123, No. 3, 801-811(1995). [3] R.M. Brown and Z. Shen Estimates for the Stokes operator in Lipschitz domains, Indiana Univ Math J. Vol. 44, No. 4 1183-1206(1995). [4] M. Dinso¡¤s and M. Mitrea, The stationary Navier-Stokes system in nonsmooth manifolds: the Poisson problem in Lipschitz and C2 domains, Arch. Rational Mech. Anal. 174 (2004), 1-47. [5] R. Russo, On the existence of solutions to the stationary Navier-Stokes equations, Ricerche Mat. 52 (2003), 285-348. [6] A. Russo and G. Starita, On the existence of steady-state solutions to the Navier- Stokes system for large ¡Æuxes, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), 171-180. [7] H. Kim, Existence and regularity of very weak solutions of the stationary Navier- Stokes equations, to appear in Arch. Rational Mech. Anal.

     Namkwon Kim(Chosun Univ.)

Title : Remarks on inviscid limit of Navier-Stokes equations in bounded domain Abstract : The incompressible Navier-Stokes equations in a bounded domain has been believed to converge to the Euler equations by many people. This convergence question answered affirmatively in a whole domain or with Navier friction type boundary condition. However, with the Dirichlet boundary condition, it has not been answered largely even in two dimensional domain. The difficulty underlying seems to be due to the discrepancy of the boundary conditions. In this talk, we discuss the difficulty of the question and give some criterions which assures the convergence in $L^2$.

     Seick Kim (Yonsei Univ.)

Title : Regularity of a degenerate parabolic equation appearing in Vecer's unified pricing of Asian options Abstract : Vecer derived a degenerate parabolic equation with a boundary condition characterizing the price of Asian options for both discrete and continuous arithmetic average. It is well understood that there exists a unique probabilistic solution to such a problem. However, due to degeneracy of the partial differential operator and lack of smoothness in the boundary data, the regularity of the probabilistic solution remained unclear in the case of discretely sampled Asian options. We prove the probabilistic solutions of Vecer's equation associated with discretely sampled Asian options are regular solutions.

     Jihoon Lee (Sungkyunkwan Univ.)

Title : On the behavior of the solutions to the Navier-Stokes equations near the possible singularity Abstract : We study the behavior of the suitable weak solutions to the 3-dimensional Navier-Stokes equations near the possible singularity. The behavior of the suitable weak solutions near the possible singularity to the 3-dimensional axisymmetric Navier-Stokes equations with swirl is also discussed.

     Jaiok Roh (Hallym Univ.)

Title : Decay of solutions of the Navier-Stokes equations in bounded domains Abstract : In this talk, we discuss decay in space and time for solutions of the Navier-Stokes equations in unbounded domains. We first discuss recent results of decay of solutions of the Navier-Stokes equations in exterior domain. Then we discuss new results of spatial decay of a perturbed Navier-Stokes equations in unbounded domains. We also will discuss shortly decay of solutions of the g-Navier-Stokes equations which derived to study 3D Navier-Stokes equations in thin domain.

     Inbo Sim (Univ. of Ulsan)

Title : Classification of weight functions and existence results for p-Laplacian problems Abstract : In this talk, firstly, we introduce and classify the weight functions. Secondly, we establish the sequence of eigenvalues for the half-linear type eigenvalue problem.  Finally, we employ the degree theory and global bifurcation phenomena and obtain the existence results of sign-changing solutions for nonlinear p-Laplacian problems.

     Xu-Jia Wang (Australian National Univ.)

Title : Singularity analysis for the mean curvature flow Abstract : We consider the geometry of the first time singularities of the mean curvature flow. By Huisken and Sinestrari's curvature pinching estimate, we show that a blow-up solution to the mean curvature flow of mean convex hypersurfaces is convex. We also give a classification of the convex blow-up solutions.