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.

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 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 address the problem of local well-posedness of two types of abelian Chern-Simons gauge theories, namely the Chern-Simons-Dirac (CSD) and the Chern-Simons-Higgs (CSH) equations, under the Lorenz gauge condition. One of our main contributions is the uncovering of a special structure (null structure) of (CSD) which, combined with the standard machinery of $X^{s,b}$ spaces, allows us to obtain local well-posedness of (CSD) for initial data $a_{mu}, psi in H^{1/4+epsilon}_{x}$. Moreover, it is observed that the same techniques applied to (CSH) lead to an improvement over the previous result due to Selberg-Tesfahun (2012). This is a joint work with H. Huh.

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.

The first application of Szemeredi’s regularity method was the following celebrated Ramsey-Turan result proved by Szemeredi in 1972: any K4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollobas and Erdos gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K4-free graph on N vertices with independence number o(N) and (1/8 – o(1)) N^2 edges. Starting with Bollobas and Erdos in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this work, we give nearly best-possible bounds, solving the various open problems concerning this critical window.

More generally, it remains an important problem to determine if, for certain applications of the regularity method, alternative proofs exist which avoid using the regularity lemma and give better quantitative estimates. In their survey on the regularity method, Komlos, Shokoufandeh, Simonovits, and Szemeredi surmised that the regularity method is likely unavoidable for applications where the extremal graph has densities in the regular partition bounded away from 0 and 1. In particular, they thought this should be the case for Szemeredi’s result on the Ramsey-Turan problem. Contrary to this philosophy, we develop new regularity-free methods which give a linear dependence, which is tight, between the parameters in Szemeredi’s result on the Ramsey-Turan problem.

Joint work with Jacob Fox and Yufei Zhao.

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.

Bridson, Martin R.; Haefliger, André. Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften 319. Springer-Verlag, Berlin, 1999의 Chapter I.5, "Some Basic Constructions"에 대해 발표하는 김상현 교수님 지도 대학원생 랩세미나입니다. 발표 내용은 다음과 같은 대상에 대한 정의 및 기본적인 성질입니다.

- κ-cones

- Spherical joins

- Quotient pseudometrics

- Gluing along isometric subspaces

- Gromov-Hausdorff convergence, Gromov's theorem

- Ultralimits and asymptotic cones

발표자는 대학원 1학년차인 이돈성입니다.

The cube graph Q_n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2ⁿ vertices. The Ramsey number r(Q_n, K_s) is the minimum N such that every graph of order N contains the cube graph Q_n or an independent set of order s. Burr and Erdős in 1983 asked whether the simple lower bound r(Q_n, K_s) ≥ (s-1)(2ⁿ -1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.
Joint work w/ David Conlon, Jacob Fox, and Benny Sudakov