Understanding the parameter spaces of dynamical systems has long been the dream of the greatest mathematicians. Even Newton asked: what initial conditions(positions, velocities masses) lead to a stable solar system?
 There are exceedingly few cases where we can answer such questions: no one knows anything about the parameter space for the 3-body problem. But for the simplest nonlinear dynamical system, z 7! z2 + c with parameter c, we do understand the parameter space.
 The crucial object in parameter space is the Mandelbrot set: it features some very delicate combinatorics, which can be written exactly.
 In my lecture I will attempt to describe these combinatorial laws, and sketch where they come from.

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 "Modular forms and projective invariants - J.Igusa(1967)".

In this talk, we will investigate the algebraic construction of the Jacobian of a hyperelliptic curves.

The main theme of the three lectures is to explain how syzygystratifications can be used to describe the birational geometry of curves with or without a level structure. The first lecture will deal with the basics of Koszul cohomology and the statements of the Green respectively Green-Lazarsfeld secant conjectures. Voisin's solution to the generic Conjecture will be sketched. In the second lecture, I will describe the implications of Green's Conjecture to the moduli space of curves, and how using the moduli space, one can prove Green's Conjecture for curves on arbitary K3 surfaces. Finally in the third lecture I will present three wide-ranging generalizations to Green's Conjecture which have been
recently used to compute the Kodaira dimension of the moduli space of curves with a torsion point of order p in its Jacobian variety.

 Averaging methods for ordinary differential equations is an
old idea going back to Laplace.
Recently along with my collaborators a new approach to averaging based
on Young measures has been introduced which both simplifies some of
the older and tedious approaches and allows for new applications to problems
arising is the KdV equations, flocking models due to Cucker and Smale,
and the Kuramoto system of oscillators.

The main theme of the three lectures is to explain how syzygystratifications can be used to describe the birational geometry of curves with or without a level structure. The first lecture will deal with the basics of Koszul cohomology and the statements of the Green respectively Green-Lazarsfeld secant conjectures. Voisin's solution to the generic Conjecture will be sketched. In the second lecture, I will describe the implications of Green's Conjecture to the moduli space of curves, and how using the moduli space, one can prove Green's Conjecture for curves on arbitary K3 surfaces. Finally in the third lecture I will present three wide-ranging generalizations to Green's Conjecture which have been
recently used to compute the Kodaira dimension of the moduli space of curves with a torsion point of order p in its Jacobian variety.

KMRS 집중 강연

일시: 2013년 3월 5일, 7일, 12일, 14일, 19일(화,목) 14시30분~15시 45분

 In 1956 John Nash established the existence of global smooth embedding of an n-dimensional Riemannian manifold M^n into m-dimensional Euclidean space for m <= n(3n + 11)/2 if M^n is compact, m <= n(n + 1)(3n + 11)/2 if M^n is non-compact.(See John Nash, Annals of Math. 65, 1956) In Nash's case the system of PDE's is under determined with many more unknowns than equations. In the determined case when m = n(n+1)=2 Nash's (1956) theorem does not apply and we can search for smooth local embeddings. These lectures will discuss this problem from the view of an applied mathematician, i.e.,

The main theme of the three lectures is to explain how syzygystratifications can be used to describe the birational geometry of curves with or without a level structure. The first lecture will deal with the basics of Koszul cohomology and the statements of the Green respectively Green-Lazarsfeld secant conjectures. Voisin's solution to the generic Conjecture will be sketched. In the second lecture, I will describe the implications of Green's Conjecture to the moduli space of curves, and how using the moduli space, one can prove Green's Conjecture for curves on arbitary K3 surfaces. Finally in the third lecture I will present three wide-ranging generalizations to Green's Conjecture which have been
recently used to compute the Kodaira dimension of the moduli space of curves with a torsion point of order p in its Jacobian variety.

KMRS 집중 강연

일시: 2013년 3월 5일, 7일, 12일, 14일, 19일(화,목) 14시30분~15시 45분

 In 1956 John Nash established the existence of global smooth embedding of an n-dimensional Riemannian manifold M^n into m-dimensional Euclidean space for m <= n(3n + 11)/2 if M^n is compact, m <= n(n + 1)(3n + 11)/2 if M^n is non-compact.(See John Nash, Annals of Math. 65, 1956) In Nash's case the system of PDE's is under determined with many more unknowns than equations. In the determined case when m = n(n+1)=2 Nash's (1956) theorem does not apply and we can search for smooth local embeddings. These lectures will discuss this problem from the view of an applied mathematician, i.e.,

The derived category of bounded complexes of coherent sheaves on an algebraic variety is an interesting invariant of the algebraic variety. There is more symmetry than the varieties themselves in the sense that there are different varieties with equivalent derived categories. There is a surprising parallelism between the minimal model program and the semi-orthogonal decompositions of derived categories. I will review some old and new results in this direction.

 KMRS 집중 강연

일시: 2013년 3월 5일, 7일, 12일, 14일, 19일(화,목) 14시30분~15시 45분

 In 1956 John Nash established the existence of global smooth embedding of an n-dimensional Riemannian manifold M^n into m-dimensional Euclidean space for m <= n(3n + 11)/2 if M^n is compact, m <= n(n + 1)(3n + 11)/2 if M^n is non-compact.(See John Nash, Annals of Math. 65, 1956) In Nash's case the system of PDE's is under determined with many more unknowns than equations. In the determined case when m = n(n+1)=2 Nash's (1956) theorem does not apply and we can search for smooth local embeddings. These lectures will discuss this problem from the view of an applied mathematician, i.e.,

 I will show some easy calculations concerning examples on the topics of the colloquium talk, e.g., Fourier-Mukai transform, grade restriction windows for GIT quotients, etc.

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.

 KMRS 집중 강연

일시: 2013년 3월 5일, 7일, 12일, 14일, 19일(화,목) 14시30분~15시 45분

 In 1956 John Nash established the existence of global smooth embedding of an n-dimensional Riemannian manifold M^n into m-dimensional Euclidean space for m <= n(3n + 11)/2 if M^n is compact, m <= n(n + 1)(3n + 11)/2 if M^n is non-compact.(See John Nash, Annals of Math. 65, 1956) In Nash's case the system of PDE's is under determined with many more unknowns than equations. In the determined case when m = n(n+1)=2 Nash's (1956) theorem does not apply and we can search for smooth local embeddings. These lectures will discuss this problem from the view of an applied mathematician, i.e.,

We discuss about a result of Littlewood  on the horizontal distribution of the zeros of the Riemann zeta-function ζ(s) in the critical strip and further we discuss about the progress made on the zeros of  ζ(s) locally in the neighbourhood of the critical line. (An old work of mine jointly done with Professor K. Ramachandra).

Let V be a projective hypersurface of fixed degree and dimension which has only isolated singular points. We show that, if the sum of the Milnor numbers at the singular points of V is large, then V cannot have a point of large multiplicity, unless V is a cone. As an application, we give an affirmative answer to a conjecture of Dimca and Papadima.

Let V be a projective hypersurface of fixed degree and dimension which has only isolated singular points. We show that, if the sum of the Milnor numbers at the singular points of V is large, then V cannot have a point of large multiplicity, unless V is a cone. As an application, we give an affirmative answer to a conjecture of Dimca and Papadima.

 KMRS 집중 강연

일시: 2013년 3월 5일, 7일, 12일, 14일, 19일(화,목) 14시30분~15시 45분

 In 1956 John Nash established the existence of global smooth embedding of an n-dimensional Riemannian manifold M^n into m-dimensional Euclidean space for m <= n(3n + 11)/2 if M^n is compact, m <= n(n + 1)(3n + 11)/2 if M^n is non-compact.(See John Nash, Annals of Math. 65, 1956) In Nash's case the system of PDE's is under determined with many more unknowns than equations. In the determined case when m = n(n+1)=2 Nash's (1956) theorem does not apply and we can search for smooth local embeddings. These lectures will discuss this problem from the view of an applied mathematician, i.e.,

Given a degeneracy locus arising from a map of vector bundles, the Thom-Porteous formula allows to express its Chow ring fundamental class as a polynomial in the Chern classes of the two bundles. In this talk I will present the geometry involved in the universal case which represents the core of one of the proofs of the formula. Moreover, I will explain how it is possible to generalize this result to other oriented cohomology theories as, for instance, the graded Grothendieck ring of vector bundles, connective K-theory and algebraic cobordism.

Let I be a homogeneous ideal of a polynomial ring. The generic initial ideal play a fundamental role in the investigation of many algebraic, homological, combinatorial and geometric properties of the ideal I itself. By definition, the generic initial ideal is the initial ideal of I with respect to a monomial term order after performing a generic change of coordinates. In this talk, we explore the generic initial ideal methods in the study of syzygies and regularities and we introduce some recent interesting results in this area.

Let I be a homogeneous ideal of a polynomial ring. The generic initial ideal play a fundamental role in the investigation of many algebraic, homological, combinatorial and geometric properties of the ideal I itself. By definition, the generic initial ideal is the initial ideal of I with respect to a monomial term order after performing a generic change of coordinates. In this talk, we explore the generic initial ideal methods in the study of syzygies and regularities and we introduce some recent interesting results in this area.

Let I be a homogeneous ideal of a polynomial ring. The generic initial ideal play a fundamental role in the investigation of many algebraic, homological, combinatorial and geometric properties of the ideal I itself. By definition, the generic initial ideal is the initial ideal of I with respect to a monomial term order after performing a generic change of coordinates. In this talk, we explore the generic initial ideal methods in the study of syzygies and regularities and we introduce some recent interesting results in this area.

We'll review interesting properties of graded Betti numbers in the linear strands of Betti tables and their implications on the geometry of projective varieties.

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 "Modular forms and projective invariants - J.Igusa(1967)".

In this talk, we will investigate the algebraic construction of the Jacobian of a hyperelliptic curves.

The goal of this lecture is to explain a proof of the local Langlands
correspondence for GL(2). Our strategy is a mixture of the works
by Deligne-Carayol, Harris-Taylor and Scholze.
The plan of the lecture is the following:

1. Statement and characterization of the local Langlands correspondence (=LLC).
2. Construction of LLC by the non-abelian Lubin-Tate theory.

3. Geometry of the modular/Shimura curve. Relation with Lubin-Tate space.
4. l-adic cohomology of the modular/Shimura curve.
Compatibility of LLC with cyclic base change etc.
5. End of proof: local study of the Lubin-Tate tower.

스케줄

13일 10:00-11:30 (Y. Mieda), 4:00-5:30 (Y. Mieda)

14일 10:00-11:30 (Y. Mieda), 2:00-3:30 (Dong UK Lee), 4:00-5:30 (Y. Mieda)

15일 10:00-11:30 (Y. Mieda)

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

우리나라 연간 국제투자대조표와 국제수지통계표를 이용하여 대외자산과 대외부채의 변동을 국제수지표상 대외거래를 반영한 거래요인과 미실현 자본이득, 즉 대외자산과 부채의 재평가분을 반영한 비거래요인으로 나누고 다시 설정한 구조식에 따라 대외자산∙부채의 변동요인을 추정한다. 상식적 판단과 달리 본 연구의 표본기간이며 대외수지흑자기조가 정착된 2001-2010년 기간 동안 오히려 순대외부채는 GDP대비 11%에서 14%로 증가하였는데 비거래요인이 거래요인을 압도한 결과다. 한편 우리와 유사하게 수출의존도가 높고 막대한 외환보유액을 보유한 대만은 같은 기간 동안 순대외자산이 GDP대비 67%에서 153%으로 증가하였다. 이 차이는 양국 모두에서 국채와 같은 안전자산을 매각하여 조달한 자금을 다시 주식, FDI 등 위험자산에 투자함으로써 높은 수익률을 올리는 ‘벤처자본가’로서 외국인의 투자수익률이 내국인보다 높지만 대만의 경우 비거래요인이 거래요인을 압도할 정도로 높지는 않기 때문에 일어난다.

In this talk, we will investigate the algebraic construction of the Jacobian of a hyperelliptic curves.

경제학의 시각에서 본 금융글로벌화의 Pro와 Con, 대외자산과 부채의 거래와 환율변동이 가지는 평가효과, 환율과 국내금융시장의 연계 및 중앙은행 통화정책의 파급효과, 외환위기 및 은행위기의 이론과 실제, sovereign default, 유로존위기 등 최근 국제금융의 주요 이슈를 중심으로 특강의 내용을 구성한다. 아시아금융위기 후 급속히 추진된 금융글로벌화는 소비평탄화, 위험의 분산, 투자의 효율화 등 순기능에도 불구하고 막대한 자본의 유입에 뒤이은 갑작스런 중단과 역류가 초래하는 국민경제의 불균형과 시스템위험 등 이른바 ‘자본유입의 문제’를 초래한다. 국가간 자본거래의 급팽창은 대외자산, 부채 그리고 환율에서 차지하는 경상수지의 위상이 줄어드는 대신 투자한 자산의 평가효과가 대외수지에 중요한 함의를 가지게 되며 지속가능한 수지불균형을 정당화하는 암흑물질가설을 뒷받침한다. 환율은 국내외 금융시장의 연결고리로서의 역할을 수행하나 환율제도에 따라 통화재정정책이 국민경제에 미치는 파급효과는 달라진다. 환율이 폭등하거나 고정환율제도가 와해됨으로써 발생하는 외환위기는 크게 서로 다른 두 메커니즘으로 설명되며 국민경제에 미치는 파급효과는 그 위기요인에 상당부분 의존한다. 국가부도는 채무상환 능력이 아니라 의지의 문제이며 따라서 민간부문의 부도와 다른 함의를 가진다.

We explore the impact of capital market integration on the welfare of domestic investors, in particular, with closed-form solutions to optimal asset holdings and utility changes in a simple equilibrium framework wherein agents have mean–variance utility. Our model allows us to show the welfare loss of domestic investors with inefficient portfolios from market integration. The results indicate that only efficient portfolio holdings before integration can guarantee the welfare enhancement of all domestic investors, in contrast to the extant literature, which emphasizes the beneficial effects of market integration. In addition, we decompose the welfare changes of domestic investors into two components, i.e., the correlation effect and the quantity-volatility effect, to enhance our understanding of economic implications.

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 "Modular forms and projective invariants - J.Igusa(1967)".

In this talk, we will investigate the algebraic construction of the Jacobian of a hyperelliptic curves.