In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

In this talk, we will survey the book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

In this talk, we will survey the article "Modular forms and projective invariants - J.Igusa(1967)".

In this talk, we will survey the article "Class fields over real quadratic fields and Hecke operators - G.Shimura(1972)".

▶ Date: May 15 ~ July 3

▶ Time: Thur. & Fri., 10:00-12:00 (Exercise session: 15:00-17:00)

▶ Description:

Many models in the sciences and engineering can be described by non-linear polynomial equations. This course offers an introduction to both theoretical and computational methods for working with such models. It is aimed at graduate students from across the mathematical sciences (Mathematics, EECS, Statistics, Physics, etc).

▶ Syllabus:

Each week of the semester is about a different topic in non-linear algebra, according to the schedule below. Auditors interested in a particular topic are welcome to attend just that week. Enrolled students will attend all weeks.

- Gröbner Basics, Elimination, Decomposing Varieties, Sparse Polynomial Systems, Semidefinite Programming, Moments and Sums of Squares,Representations and Invariants, Tensors and their Rank, Orbitopes, Maximum Likelihood, Numerical Algebraic Geometry, Nash Equilibria, Chemical Reaction Networks, Tropical Algebra

In tropical mathematics, the sum of two numbers is their minimum, and the product of two numbers is their usual sum. Many results familiar from high school algebra and geometry, including the formula for solving quadratic equations and the fact that two lines meet in one point, continue to hold in the tropics. In this lecture we learn how to draw tropical curves and why biologists might care about this.

▶ Date: May 15 ~ July 3

▶ Time: Thur. & Fri., 10:00-12:00 (Exercise session: 15:00-17:00)

▶ Description:

Many models in the sciences and engineering can be described by non-linear polynomial equations. This course offers an introduction to both theoretical and computational methods for working with such models. It is aimed at graduate students from across the mathematical sciences (Mathematics, EECS, Statistics, Physics, etc).

▶ Syllabus:

Each week of the semester is about a different topic in non-linear algebra, according to the schedule below. Auditors interested in a particular topic are welcome to attend just that week. Enrolled students will attend all weeks.

- Gröbner Basics, Elimination, Decomposing Varieties, Sparse Polynomial Systems, Semidefinite Programming, Moments and Sums of Squares,Representations and Invariants, Tensors and their Rank, Orbitopes, Maximum Likelihood, Numerical Algebraic Geometry, Nash Equilibria, Chemical Reaction Networks, Tropical Algebra

The Outerplanar Diameter Improvement problem asks, given a graph G and an integer D, whether it is possible to add edges to G in a way that the resulting graph is outerplanar and has diameter at most D. We provide a dynamic programming algorithm that solves this problem in polynomial time. Outerplanar Diameter Improvement demonstrates several structural analogues to the celebrated and challenging Planar Diameter Improvement problem, where the resulting graph should, instead, be planar. The complexity status of this latter problem is open.

▶ Date: May 15 ~ July 3

▶ Time: Thur. & Fri., 10:00-12:00 (Exercise session: 15:00-17:00)

▶ Title:

▶ Description:

Many models in the sciences and engineering can be described by non-linear polynomial equations. This course offers an introduction to both theoretical and computational methods for working with such models. It is aimed at graduate students from across the mathematical sciences (Mathematics, EECS, Statistics, Physics, etc).

▶ Syllabus:

Each week of the semester is about a different topic in non-linear algebra, according to the schedule below. Auditors interested in a particular topic are welcome to attend just that week. Enrolled students will attend all weeks.

- Gröbner Basics, Elimination, Decomposing Varieties, Sparse Polynomial Systems, Semidefinite Programming, Moments and Sums of Squares,Representations and Invariants, Tensors and their Rank, Orbitopes, Maximum Likelihood, Numerical Algebraic Geometry, Nash Equilibria, Chemical Reaction Networks, Tropical Algebra

There will be three intensive lectures on May 22ndand 23rdby Profoessor Tzavaras. Seniors and Graduate students who are interested in Analysis and PDE could enjoy the lectures and are invited.

▶Lecture 2 - 11am, May 23, 2014

The equations of polyconvex elasticity; approximation via variational minimization schemes

There will be three intensive lectures on May 22ndand 23rdby Profoessor Tzavaras. Seniors and Graduate students who are interested in Analysis and PDE could enjoy the lectures and are invited.

▶Lecture 3 - 2:30pm, May 23, 2014 

Diffusive limits from Euler equations with friction to gradient flows

There will be three intensive lectures on May 22ndand 23rdby Profoessor Tzavaras. Seniors and Graduate students who are interested in Analysis and PDE could enjoy the lectures and are invited. 

▶Lecture 1 - 2:30pm, May 22, 2014

The relative entropy method and its relation to the structure of the equations of thermomechanics

Many problems involving phase transitions have a variational formulation.
Treatment of these problems with the aid of tools from PDE , dynamical systems, and geometry as well as the calculus of variations leads to many interesting results and open questions. We will survey the results and methods and also mention several open questions.

Glioblastoma is the most common and the most aggressive type of brain cancer. The median survival time from the time of diagnosis is approximately one year. Invasion of glioma cells from the core tumor into the surrounding brain tissue is a major reason for treatment failure: these migrating cells are not eliminated in surgical resection and cause tumor recurrence. Variations are seen in number of invading cells, and in the extent and patterns of migration. Cells can migrate diffusely and can also be seen as clusters of cells distinct from the main tumor mass. This kind of clustering is also evident in vitro using 3-D spheroid models of glioma invasion. This has been reported for U87 cells stably expressing the constitutively active EGFRVIII mutant receptor, often seen expressed in glioblastoma. In this case the cells migrate as clusters rather than as single cells migrating in a radial pattern seen in control wild type U87 cells. Several models have been suggested to explain the different modes of migration, but none of them, so far, has explored the important role of cell-cell adhesion. We develop a mathematical model which includes the role of adhesion and provides an explanation for the various patterns of cell migration. It is shown that, depending on adhesion, haptotactic, and chemotactic parameters, the migration patterns exhibit a gradual shift from branching to dispersion, as has been reported experimentally. Recently, the miR-451-AMPK-mTOR signaling network was shown to play a significant role in regulation of cell proliferation and migration in glioblastoma. Oncolytic virus could be also a great way of killing glioma cells. We discuss how one use these models to test hypothesis on killing infiltration glioma cells through the network of extracellular matrix and other normal cells, leading to better therapeutic treatment options.

*Joint work with Avner Friedman (Dept of Mathematics, Mathematical Biosciences Institute, The Ohio State University), Balveen Kaur (Dardinger Laboratory for Neuro-Oncology and Neurosciences, Ohio State University), Sean Lawler (Harvard medical school, BWH), E.A. Chiocca (Harvard medical school, BWH), Soyeon Roh (University of Michigan-Ann Arbor).

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

In this talk, we will survey the book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

In this talk, we will survey the article "Modular forms and projective invariants - J.Igusa(1967)".

In this talk, we will survey the article "Class fields over real quadratic fields and Hecke operators - G.Shimura(1972)".

▶ Date: May 15 ~ July 3

▶ Time: Thur. & Fri., 10:00-12:00 (Exercise session: 15:00-17:00)

▶ Description:

Many models in the sciences and engineering can be described by non-linear polynomial equations. This course offers an introduction to both theoretical and computational methods for working with such models. It is aimed at graduate students from across the mathematical sciences (Mathematics, EECS, Statistics, Physics, etc).

▶ Syllabus:

Each week of the semester is about a different topic in non-linear algebra, according to the schedule below. Auditors interested in a particular topic are welcome to attend just that week. Enrolled students will attend all weeks.

- Gröbner Basics, Elimination, Decomposing Varieties, Sparse Polynomial Systems, Semidefinite Programming, Moments and Sums of Squares,Representations and Invariants, Tensors and their Rank, Orbitopes, Maximum Likelihood, Numerical Algebraic Geometry, Nash Equilibria, Chemical Reaction Networks, Tropical Algebra

2차방정식이 4000년 전 바빌로니아에서 풀린 뒤 3,4 차 방정식은 1545 년에야 간신히 풀려서 발표되었다.

그 이후 280년이 지난 뒤 아벨과 갈로아는 5차 이상의 방정식은 대수적으로 풀 수 없음을 보였다. 본 강연에서는 이 300 년간 수학자들이 고차 방정식 문제를 해결하기 위해 어떻게 노력했는가 그 역사를 되돌아볼 계획이다.

The general theory of exact relations

(sections 17.1, 17.2, 17.3, 17.4)

Grabovsky realized that if an exact relation holds for arbitrary composites it must at least hold for laminate materials and this imposes stringent algebraic constraints on the form an exact relation can take. In the right coordinates an exact relation must be a linear relation, and must remain linear under certain non-linear transformations. This provides necessary conditions for an exact relation to hold. By extending these ideas to general composites, using series expansions, we (Grabovsky, Sage and myself) found sufficient algebraic conditions for an exact relation to hold. This final lecture will review this general theory, which encompasses all known exact relations, and provides new ones.

 Invited Speakers:

Sung Yong Kim (KAIST)

Hyundae Lee (Inha Univ.)

Won-Kwang Park (Kookmin Univ.)

Sanghyeon Yu (KAIST)

 Let S be an immersed stable hypersurface of constant mean curvature in a wedge bounded by two hyperplanes in R^n. Suppose that S meets those two hyperplanes in constant contact angles and is disjoint from the edge of the wedge. We will show that if the boundary of S is embedded for n=3, or if the boundary of S is convex for n=4, then S is part of the sphere.

▶ Date: May 15 ~ July 3

▶ Time: Thur. & Fri., 10:00-12:00 (Exercise session: 15:00-17:00)

▶ Description:

Many models in the sciences and engineering can be described by non-linear polynomial equations. This course offers an introduction to both theoretical and computational methods for working with such models. It is aimed at graduate students from across the mathematical sciences (Mathematics, EECS, Statistics, Physics, etc).

▶ Syllabus:

Each week of the semester is about a different topic in non-linear algebra, according to the schedule below. Auditors interested in a particular topic are welcome to attend just that week. Enrolled students will attend all weeks.

- Gröbner Basics, Elimination, Decomposing Varieties, Sparse Polynomial Systems, Semidefinite Programming, Moments and Sums of Squares,Representations and Invariants, Tensors and their Rank, Orbitopes, Maximum Likelihood, Numerical Algebraic Geometry, Nash Equilibria, Chemical Reaction Networks, Tropical Algebra

Composite materials can have properties unlike any found in nature, and in this case they are known as metamaterials. Recent attention has been focussed on obtaining metamaterials which have an interesting dynamic behavior. Their effective mass density can be anisotropic, negative, or even complex. Even the eigenvectors of the effective mass density tensor can vary with frequency. Within the framework of linear elasticity, internal masses can cause the effective elasticity tensor to be frequency dependent, yet not contribute at all to the effective mass density at any frequency. One may use coordinate transformations of the elastodynamic equations to get novel unexpected behavior. A classical propagating wave can have a strange behavior in the new abstract coordinate system. However the problem becomes to find metamaterials which realize the behavior in the new coordinate system. This can be solved at a discrete level, by replacing the original elastic material with a network of masses and springs and then applying transformations to this network. The realization of the transformed network requires a new type of spring, which we call a torque spring. The forces at the end of the torque spring are equal and opposite but not aligned with the line joining the spring ends. We show how torque springs can theoretically be realized.

We briefly survey the game of cops-and-robbers on graphs and its variants in the fi nite case and then concenrate on in finite graphs, stressing the diff erence between the fi nite and the in finite. Along the way we show (time allowing) how to construct in finite vertex transitive graphs from any graphs and point out some strange properties of the construction. We also suggest several open problems, both fi nite and infi nite. The talk is based on work with A. Bonato, C.Tardif and R.E. Woodrow.

 Let $d geq 3$ be an odd positive integer and let $f(x_1, ldots, x_n, x_{n+1}), n geq d,$ be a weighted homogeneous polynomial of degree $2d$ with respect to the weights ${rm wt}(x_1)=cdots={rm wt}(x_n) =1$ and ${rm wt}(x_{n+1}) =2$. Let $X^f$ be a Veronese double cone of dimension $n$ associated to a general choice of $f.$ This is an $n$-dimensional Fano manifold of Picard number 1 with index $n+2-d$.

In this talk, I will describe the variety of minimal rational tangents $mathcal C_xsubsetmathbb P T_x(X)$ at a general point $x$ of $X^f$ and show that it is not smooth if $2d leq n$.

A number of natural graph problems are known to be W-hard to solve exactly when parameterized by standard widths (treewidth or clique-width). At the same time, such problems are typically hard to approximate in polynomial time. In this talk we will present a natural randomized rounding technique that extends well-known ideas and can be used to obtain FPT approximation schemes for several such problems, evading both polynomial-time inapproximability and parameterized intractability bounds.

Following the work of Fujita, Angehrn and Siu, Helmke obtained an effective bounds for the global generation of the adjoint line bundles. In this talk, we will introduce his method. By carefully analyzing upper bound of deficit function, we obtain a Kawamata-type result on projective 5-folds. More precisely, we show that the adjoint line bundle is globally generated with bound 7. 

The game of SET is a popular card game in which the objective is to form Sets (triplets of cards that match in a particular sense) using cards from a special deck. For more details regarding the game, please visit the official website: http://www.setgame.com/.We analyze the computational complexity of some variations of the game of SET, presenting positive as well as hardness results in the classical and parameterized sense. Along the way, we make interesting connections of these generalizations of the game with other combinatorial problems, like Perfect Multi-Dimensional Matching, Set Packing, Independent Edge Dominating Set, and a two-player game played on graphs called Arc Kayles.

Examples of exact relations and links between effective tensors

(sections 3.1, 3.2, 3.3, 5.1, 5.3, 6.2)

 Here we give examples of some of the many exact, microstructure independent, relations that have been found for the effective moduli of composites. These include the Keller-Dykhne-Mendelson exact relations for the conductivity of two-dimensional composites; the exact relation of Hill for the effective Lame modulus of a composite with constant shear modulus; the exact relation of Levin linking the effective bulk modulus and effective thermal expansion coefficient in two phase composites, and the exact relations of Milgrom and Shtrikman for the effective moduli of thermoelectric and other coupled field problems.

Exact formulae for the effective tensors of laminates and series expansions for the effective tensor(sections 9.2, 9.3, 12.1, 14.1, 14.9)

 One of the simplest composites is a laminate of the consitutent phases. The formula giving the effective tensor of a laminate is non-linear, but following ideas of Backus, Tartar and myself, reduces to a linear average in the right coordinates, dependent on the direction of lamination. For more general composites one can expand the effective tensor in a series expansion in powers of the contrast between the phases, and some expansions have especially fast convergence. Following the ideas of Moulinec and Suquet these series expansions lead to numerical schemes for computing the effective moduli of both linear and non-linear composites

Trial-to-trial variability in the neural response to the same stimuli has been observed in a wide range of neuronal systems.  Such response variability and noisiness may degrade the fidelity of information transmission and computation in the neural systems.  In the first part of the talk, I will discuss the effect of noise on the network structure in memory circuits that store stimulus value in a graded manner (Lim and Goldman, Neural Comp., 2012).  Using information-theoretic measure, I compared the performance of two prominent classes of memory networks, feedback-based attractor networks and feedforward networks under different conditions.

In the second part of the talk, I will discuss the statistical properties of noise-induced phenomena in spontaneously active networks having a relaxation character (Lim and Rinzel, J. Comp. Neuro., 2010).  I have developed mathematical tools to show the relation of the slow process of relaxation dynamics and the statistical properties of noisy neuronal activities.  This analysis was then used to develop criteria by which to distinguish among different slow negative feedback mechanism in the rat respiratory central-pattern-generator circuit.

In the brain, massive interactions between neurons through synapses give rise to rich dynamics and have been thought to be critical for brain computation.  In this talk, I will discuss recurrent network models for working memory that refers to an ability to maintain information on a time scale of seconds.  Persistent neural activity in the absence of stimulus has been identified as a neural correlate of working memory, and it has been suggested that network interactions must be used to prolong the duration of persistent activity.  Using dynamical systems theory and control theories, I found a new mechanism for generating persistent activity based on the principle of corrective feedback both in spatially homogeneous networks (Lim and Goldman, Nat. Neurosci., 2013) and in spatially structured networks (Lim and Goldman, J. Neurosci., in press).  Several advantages of this new network model compared to previous models will also be discussed. 

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

In this talk, we will survey the book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

In this talk, we will survey the article "Modular forms and projective invariants - J.Igusa(1967)".

In this talk, we will survey the article "Class fields over real quadratic fields and Hecke operators - G.Shimura(1972)".

In this lecture,  we explore the emergence of the notion of compactness within its historical beginning through rigor versus intuition modes in the treatment of Dirichlet`s principle. We emphasize on the intuition in Riemann`s statement on the principle criticized by Weierstrass`requirement of rigor followed by Hilbert`s restatement again criticized by Hadamard, which pushed the ascension of the notion of compactness in the analysis of PDEs. A brief overview of some techniques and problems involving compactness is presented illustrating the importance of this notion.

Compactness is discussed here to raise educational issues regarding  rigor vs intuition in mathematical studies.  The concept of compactness advanced rapidly after  Weierstrass's famous criticism of Riemann's use of the Dirichlet principle. The rigor of Weierstrass contributed to establishment of the concept of compactness, but such a focus on rigor blinded  mathematicians to big pictures. Fortunately, Poincare and Hilbert defended Riemann's use of the Dirichlet principle and found a balance between rigor and intuition. There is no theorem without rigor, but we should not be a slave of rigor.  Rigor (highly detailed examination with toy models) and intuition (broader view with  real models) are essentially complementary to each other.

Although elliptic PDEs  have been used widely, it seems that its precise definition has been overlooked. How can we  understand coefficients of elliptic PDEs and their solutions? Indeed, the determination of coefficients of elliptic PDEs has been studied by many distinguished scientists, including Maxwell, Poisson, Faraday, Rayleigh, Fricke, Lorentz, and so on. But their studies were mostly restricted to very simplified models, and this subject is not well understood. In this lecture, we discuss this fundamental issue in PDE  by reviewing math history.

A complex normal variety $X$ is called a symplectic variety if it admits a holomorphic symplectic 2-form $omega$ on the regular part $X_{reg}$ and $omega$ extends to a holomorphic 2-form on a resolution $Y$ of $X$. Compared with the compact case, there are a lot of examples of affine symplectic varieties. They are not only interesting objects in algebraic geometry, but also play important roles in geometric representation theory.
The aim of this talk is to characterize the nilpotent variety of a complex semisimple Lie algebra among affine symplectic varieties. The main result is that if $(X, omega)$ is an affine singular symplectic variety embedded in an affine space as a complete intersection of homogeneous polynomials and $omega$ is homogeneous, then $(X, omega)$ coincides with the nilpotent variety
$N$ of a complex semisimple Lie algebra together with the Kostant-Kirillov 2-form $omega_{KK}$.
The proof of the main result uses the theory of Poisson deformation, holomorphic contact geometry, Mori theory and some elementary representation theory. 

 The aim of this talk is to introduce some theory of algebraic geometry to Commutative Ring Theory and to translate some properties of singularities to the language of Commutative Ring Theory over fields of positive characteristic.
 The contents includes the following topics.
   (1) Resolution of singularities and rational singularities.
   (2) Positive characteristic counterpart of rational singularities and log terminal singularities.
   (3) Construction of normal graded rings from projective varieties and Q- divisors.
   (4) Ideal theory of integrally closed ideals and cycles on the resolution.

In this talk, I will present deformations of compact holomorphic Poisson manifolds on the basis of Kodaira-Spencer's analytic deformation theory and extend their "Theorem of existence" for deformations of complex structures in the context of holomorphic Poisson deformations. I will discuss infinitesimal version of deformations of compact holomorphic Poisson manifolds and describe a differential graded Lie algebra governing holomorphic Poisson deformations of a compact holomorphic Poisson manifold in the language of "functor of Artin rings".

A complex normal variety $X$ is called a symplectic variety if it admits a holomorphic symplectic 2-form $omega$ on the regular part $X_{reg}$ and $omega$ extends to a holomorphic 2-form on a resolution $Y$ of $X$. Compared with the compact case, there are a lot of examples of affine symplectic varieties. They are not only interesting objects in algebraic geometry, but also play important roles in geometric representation theory.
The aim of this talk is to characterize the nilpotent variety of a complex semisimple Lie algebra among affine symplectic varieties. The main result is that if $(X, omega)$ is an affine singular symplectic variety embedded in an affine space as a complete intersection of homogeneous polynomials and $omega$ is homogeneous, then $(X, omega)$ coincides with the nilpotent variety
$N$ of a complex semisimple Lie algebra together with the Kostant-Kirillov 2-form $omega_{KK}$.
The proof of the main result uses the theory of Poisson deformation, holomorphic contact geometry, Mori theory and some elementary representation theory.

A complex normal variety $X$ is called a symplectic variety if it admits a holomorphic symplectic 2-form $omega$ on the regular part $X_{reg}$ and $omega$ extends to a holomorphic 2-form on a resolution $Y$ of $X$. Compared with the compact case, there are a lot of examples of affine symplectic varieties. They are not only interesting objects in algebraic geometry, but also play important roles in geometric representation theory.
The aim of this talk is to characterize the nilpotent variety of a complex semisimple Lie algebra among affine symplectic varieties. The main result is that if $(X, omega)$ is an affine singular symplectic variety embedded in an affine space as a complete  intersection of homogeneous polynomials and $omega$ is homogeneous, then $(X, omega)$ coincides with the nilpotent variety
$N$ of a complex semisimple Lie algebra together with the Kostant-Kirillov 2-form $omega_{KK}$.
The proof of the main result uses the theory of Poisson deformation, holomorphic contact geometry, Mori theory and some elementary  representation theory.

A complex normal variety $X$ is called a symplectic variety if it admits a holomorphic symplectic 2-form $omega$ on the regular part $X_{reg}$ and $omega$ extends to a holomorphic 2-form on a resolution $Y$ of $X$. Compared with the compact case, there are a lot of examples of affine symplectic varieties. They are not only interesting objects in algebraic geometry, but also play important roles in geometric representation theory.
The aim of this talk is to characterize the nilpotent variety of a complex semisimple Lie algebra among affine symplectic varieties. The main result is that if $(X, omega)$ is an affine singular symplectic variety embedded in an affine space as a complete  intersection of homogeneous polynomials and $omega$ is homogeneous, then $(X, omega)$ coincides with the nilpotent variety
$N$ of a complex semisimple Lie algebra together with the Kostant-Kirillov 2-form $omega_{KK}$.
The proof of the main result uses the theory of Poisson deformation, holomorphic contact geometry, Mori theory and some elementary  representation theory.

Given trees F and T, a F-matching is a collection of disjoint copies F in T. Generalizing results from Wagner (2009), Alon et al (2011) proved that the number of F-matchings for fixed F in a random tree T whose size tends to infinity is a.a.s. a multiple of any given integer m>0. We will discuss inverse and extremal problems on the number of F-matchings.

We define and study an extended hyperbolic space which contains the hyperbolic space and de Sitter space as subspaces, and which is obtained as an analytic continuation of the hyperbolic space. The construction of the extended hyperbolic space gives rise to a complex valued geometry consistent with both the hyperbolic and de Sitter space. Such a construction inspires a new concrete insight for the study of the hyperbolic geometry and Lorentzian geometry as a unified object. We also discuss the advantage of this new geometric model as well as some of its applications.

Volume conjecture was first introduced by Kashaev late 90's and has arisen as a very important research problem in low dimensional topology since it involves many mathematical areas such as knot theory, hyperbolic geometry and quantum representation as well as theoretical physics.

In this talk I would like to give an introductory description of the conjecture for the general audience, especially with the figure eight knot complement. And then I will discuss an optimistic limit technique to obtain a new approach to knot/hyperbolic invariant of link complement.

Symbolic dynamics, a part of discrete dynamical systems, is the study of spaces consisting of infinite arrays defined by certain constraints on finite subsets. A rich theory has been developed on 1-dimensional symbolic dynamics in the last several decades, but the higher-dimensional case is more problematic and many problems occurring in multidimensional symbolic dynamics involve computability issues. In this talk, an introduction to the theory of multidimensional symbolic dynamics and the basic notion of the theory of computation will be presented. In the second part of the talk, we will see why computability problems are crucial for the development of the theory in the multidimensional actions.

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

In this talk, we will survey the article "Class fields over real quadratic fields and Hecke operators - G.Shimura(1972)".

In this talk, we will survey the article "Modular forms and projective invariants - J.Igusa(1967)".

Propensity score adjustment is a popular technique for handling unit nonresponse in samplesurveys. If the response probability depends on the study variable that is subject to missingness, estimating the response probability often relies on additional distributional assumptions about the study variable. Instead of making fully parametric assumptions about the population distribution of the study variable and the response mechanism, we propose a new approach of maximum likelihood estimation that is based on the distributional assumptions of the observed part of the sample. Since the model for the observed part of the sample can be verified from the data, the proposed method is less sensitive to failure of the assumed model of the outcomes. Generalized method of moments can be used to improve the efficiency of the proposed estimator. Results from a limited simulation study are presented to compare the performance of the proposed method with the existing methods. We also present an application of the proposed method to the exit poll for the 19th legislative election in Korea.

In this talk, we want to introduce famous geometers in Histroy of Mathe-
matics from C.F. Gauss, and B. Riemann, to S.S.Chern by way of E. Cartan.
Related to these great mathematicians some remarks on Fields medals and
Math. prizes in ICM will be remarked and some explanations about Poincare
conjecture and S.S. Chern conjecture will be given in detail.
Every minimal hypersurface in Sn+1(1) with constant scalar curvature
has the property that the squared norm of the second fundamental tensor h2
is constant.
From such a view point S.S.Chern conjectured the following problem:
Chern's conjecture For n-dimensional compact minimal hypersurfaces
in Sn+1(1) with constant scalar curvature, the value h2 of the squared norm
of the second fundamental forms should be discrete.
Finally, related to this conjecture we will report some solved problems
until now.