Department Seminars & Colloquia
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Kollar—Shepherd-Barron—Alexeev (KSBA) have given a general construction that provides a geometric compactification for the moduli space of varieties of general type. Unfortunately, even in relatively simple cases (e.g. surfaces of general type with small invariants) it is difficult to understand the boundary points and the structure of this KSBA compactification. Thus, it is natural to try to compare the KSBA construction with other constructions, in particular with Hodge theoretic constructions of the moduli space. The Hodge theoretic construction has the advantage of having a lot of structure (of arithmetic and representation theoretic nature), but except a few cases (essentially abelian varieties and K3s) it is highly transcendental. In this talk, I will report on joint work with P. Griffiths, M. Green and C. Robles on the study of the moduli and periods of H-surfaces (Horikawa surfaces). The H-surfaces are surfaces of general type with p_g=2, q=0, K^2=2. They are essentially the simplest case where both the KSBA and Hodge theoretic construction are non-trivial. Considering and comparing the two approaches gives a rich picture which suggests an important role for the period map in the study of moduli spaces beyond the classical cases of abelian varieties and K3s.
We will discuss a toy model of heavy tails and show how this does not follow central limit behavior. We will then see how this relates to models in physics including random matrices. In the random matrix setting, we equate limiting spectral distributions (LSD) to spectral measures of rooted graphs. The LSD result also includes matrices with i.i.d. entries (up to self-adjointness) having infinite second moments, but following central limit behavior. In this case, the rooted graph is the natural numbers rooted at one, so the LSD is well-known to be the semi-circle law.
We will discuss local estimates for the evolutions of strictly convex hypersur- faces by Gauss curvature. We will address geometric cut-off functions and an ap- plication of the Euler’s formula to the Pogorelov type estimate.
2. The uniqueness of fully nonlinear evolutions of complete non-compact hypersurfaces.
It is well-known that the heat equation ut “ uxx, defined on Rˆr0,Tq, does not have a unique solution even for the trivial initial data u0pxq “ 0. However, we can observe that the Mean curvature flow has the unique solution for the trivial initial data; a hyperplane. We will discuss the comparison principle and Jensen’s approximate solutions to show the uniqueness of the complete convex solution of fully nonlinear flows. Special emphasis will be given to the Mean curvature flow.
We will begin with explaining the Poincare-Dulac normal form idea to prove the local well-posedness of nonlinear dispersive equations. Later, we will discuss with a particular example, quadratic derivative NLS. We develop an infinite iteration scheme of normal form reductions for dNLS. By combining this normal form procedure with the (modified) Cole-Hopf transformation, we prove unconditional global well-posedness in L^2(T), and more generally in certain Fourier-Lebesgue spaces FL^{s,p}(T), under the mean-zero and smallness assumptions. With this example, we observe a relation between normal form approach and canonical nonlinear transform.
In this talk, we study the kinetic Fokker-Planck equation in general multi-dimensional bounded domains with inflow boundary condition. There has not been many results on the regularity of solutions when the spatial domain has a boundary. We will discuss the global well-posedness, interior and boundary regularity for the Fokker-Planck case, and compare it with some other kinetic equations
Day 1. Introductory and the coercivity estimate
In this talk, we discuss the mean field quantum fluctuation dynamics for a system of infinitely many fermions with delta pair interactions in the vicinity of an equilibrium solution (the Fermi sea) at zero temperature in two and three dimensions. Our work extends some recent important results of M. Lewin and J. Sabin, who address the corresponding problem for more regular pair interactions. This is a joint work with Thomas Chen and Natasa Pavlovic at University of Texas at Austin.
In applications, the optimal solution is extremely important
because it can directly impact the efficiency of allocated resources.
This talk focuses on establishing numerical methods for
visibility-related optimal control problems. We address problems under
limited sensing ability which arises as an essential part in many
scientific fields, e.g., robotic path planning, unmanned automatic
vehicles, and designs of the surveillance system. We formulate the
problem using the level set framework and find the solution using an
optimization method with SDEs. After some modeling issues are addressed,
numerical results are presented. We will also discuss open problems.
I first review basic results of Koszul cohomology. More precisely, I consider the effect of projections and hyperplane section methods on sygyzies of sections rings. I also discuss Green's duality theorem and Green's vanishing theorem. Finally, I present several questions on Koszul cohomology.
Recently, Okounkov bodies have become a very interesting and useful tool to understand the positivity of divisors (or line bundles). In the first lecture, we review the construction and basic properties of Okounkov bodies following Lazasfeld-Mustata. In the second lecture, we study some of the recent results on Okounkov bodies.
I will define the two Hecke actions on the dual of the formal affine
Demazure algebra. Then I will define the push-pull operators of the
oriented cohomology and define perfect pairings on the equivariant
cohomology of complete and partial flag varieties. If time permits, I
will talk about a parallel construction which gives the formal affine
Hecke algebra.
자연과학동(E6-1) Room 1409
ASARC Seminar
Sung Rak Choi (IBS, POSTECH)
Introduction to Okounkov bodies, II
Recently, Okounkov bodies have become a very interesting and useful tool to understand the positivity of divisors (or line bundles). In the first lecture, we review the construction and basic properties of Okounkov bodies following Lazasfeld-Mustata. In the second lecture, we study some of the recent results on Okounkov bodies.
In this talk I will introduce the definition of formal affine Demazure
algebra and sketch the proof of the structure theorem. Taking its
dual, we get the algebraic replacement of $T$-equivariant oriented
cohomology of complete flag variety. I will also mention the proof of
the generalized Borel Isomorphism.
I will introduce the concept of oriented cohomology in the sense of
Levine and Morel, and the work of Kostant and Kumar on algebraic
construction of singular cohomology and Grothendieck group of flag
varieties. Then I will introduce the formal group algebra of
Calmes-Petrov-Zainoulline, which is the algebraic replacement of
$T$-equivariant oriented cohomology of a point.
Stable surfaces are the two-dimensional analogue of stable curves: they are exactly the singular surfaces one needs to compactify the Giesecker moduli space of surfaces of general type (over the complex numbers). I will first give some examples illustrating the general picture both for curves and surfaces. Then I will dive deeper into the technical complications that arise, explain a glueing result of Kollar, that allows to deal with non-normal surface, and illustrate the theory with some applications.
Calculating the residues for rational integrals in complex variables is a classical problem in mathematics. It is directly related to questions on algebraic cycles, their cohomology classes, and the Abel–Jacobi map. In this talk I will present joint work with Michael Hopkins in which we use topological cohomology theories to shed some new light on the Abel–Jacobi map.
Stable surfaces are the two-dimensional analogue of stable curves: they are exactly the singular surfaces one needs to compactify the Giesecker moduli space of surfaces of general type (over the complex numbers). I will first give some examples illustrating the general picture both for curves and surfaces. Then I will dive deeper into the technical complications that arise, explain a glueing result of Kollar, that allows to deal with non-normal surface, and illustrate the theory with some applications.
Stable surfaces are the two-dimensional analogue of stable curves: they are exactly the singular surfaces one needs to compactify the Giesecker moduli space of surfaces of general type (over the complex numbers). I will first give some examples illustrating the general picture both for curves and surfaces. Then I will dive deeper into the technical complications that arise, explain a glueing result of Kollar, that allows to deal with non-normal surface, and illustrate the theory with some applications.
1901년도 발표된 Alfred Young의 대칭군의 불변량에 관한 연구논문에는 자연수들의 (특정한) 배열이 소개되었다. 이제 Young tableau라고 불리는 이 배열들은 대칭군의 표현론, 대칭함수이론은 물론 슈베르트계산이론에서도 중요한 역할을 하고 있다.
대수적 조합론의 중심에 위치하고 있는 Young tableaux의 역할을, Littlewood-Richardson rule, Robinson-Schensted Algorithm, hook formula를 포함한 역사적으로 중요한 몇 가지 주제를 통해서 살펴본다.
자연과학동(E6) Room 1501
Colloquium
Dohan Kim (Seoul National Univ.)
Recent History of Korean Mathematics
When Korea achieved its independence in 1945 there were less than 10 bachelors in Mathematics in Korea. In 1960’s GDP per capita of Korea was less than 70 dollars, almost same as poor countries in Africa. We present a leisurely talk on the development of Korean Mathematics since 1945, especially on the behind stories how we could succeed in hosting ICM 2014 in Seoul.
Also, a comparison between the research trends of South and North Korean Mathematics is provided, based on our analysis of the statistics of almost all the articles published by both Korean mathematicians from 2001 to 2010.
자연과학동 E6-1, ROOM 1409
Discrete Math
Sungsoo Ahn (School of Electrical Engineering, KAIST)
Minimum Weight Perfect Matching via Blossom Belief Propagation
Max-product Belief Propagation (BP) is a popular message-passing algorithm for computing a Maximum-A-Posteriori (MAP) assignment over a distribution represented by a Graphical Model (GM). It has been shown that BP can solve a number of combinatorial optimization problems including minimum weight matching, shortest path, network flow and vertex cover under the following common assumption: the respective Linear Programming (LP) relaxation is tight, i.e., no integrality gap is present. However, when LP shows an integrality gap, no model has been known which can be solved systematically via sequential applications of BP. In this paper, we develop the first such algorithm, coined Blossom-BP, for solving the minimum weight matching problem over arbitrary graphs. Each step of the sequential algorithm requires applying BP over a modified graph constructed by contractions and expansions of blossoms, i.e., odd sets of vertices. Our scheme guarantees termination in O(n2) of BP runs, where n is the number of vertices in the original graph. In essence, the Blossom-BP offers a distributed version of the celebrated Edmonds’ Blossom algorithm by jumping at once over many sub-steps with a single BP. Moreover, our result provides an interpretation of the Edmonds’ algorithm as a sequence of LPs.