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.

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.

1. 수학을 왜 배우는가?
2. 외국수학연구소의 국방관련 프로젝트 소개
3. 국방무기체계에서 수학과 국방분야의 접목사례 소개
4. 국방부문에서 수학을 요구되는 분야 소개
5. 제안 및 토의

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.

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.

PET imaging can yield quantitative information about a radiotracer’s spatial and temporal distribution within the body. The ideal PET radiotracer will allow the detection of some changes at a very early stage of a disease or changes with treatment of that disease.  In an ideal situation, the measure will be both quantitative and sensitive.  However, in a clinical setting, it is less important for the tracer to be a quantitative measure than it is to be sensitive to the change.  An arterial input function is typically measured by acquiring discrete arterial blood samples, usually from a radial artery. However the placement of the arterial catheter and frequent blood draws during the scan is also very difficult and is usually not performed in a clinical setting.  These constraints limit the full quantification of the PET study. I will introduce the alternative to use an image-derived input function (IDIF) using the carotid artery. Also, I like to discuss how to quantify brain PET images with different input functions.

Belief propagation (BP) is a popular message-passing algorithm for computing a maximum-a-posteriori assignment in a graphical model. It has been shown that BP can solve a few classes of Linear Programming (LP) formulations to combinatorial optimization problems including maximum weight matching and shortest path. However, it has been not clear what extent these results can be generalized to. In this talk, I first present a generic criteria that BP converges to the optimal solution of given LP, and show that it is satisfied in LP formulations associated to many classical combinatorial optimization problems including maximum weight perfect matching, shortest path, network flow, traveling salesman, cycle packing and vertex cover. Using the criteria, we also construct the exact distributed algorithm, called Blossom-BP, solving the maximum weight matching problem over arbitrary graphs. In essence, Blossom-BP offers a distributed version of the celebrated Blossom algorithm (Edmonds '1965) jumping at once over many sub-steps of the Blossom-V (most recent implementation of the Blossom algorithm due to Kolmogorov, 2011). Finally, I report the empirical performance of BP for solving large-scale combinatorial optimization problems. This talk is based on a series of joint works with Sungsoo Ahn (KAIST), Michael Chertkov (LANL), Inho Cho (KAIST), Dongsu Han (KAIST) and Sejun Park (KAIST).

The celebrated theorem of Kuratowski characterizes those graphs that require at least one crossing when drawn in the plane, by exhibiting the complete list of topologically-minimal such graphs. As it is very well known, this list contains precisely two such 1-crossing-critical graphs: $K_5$ and $K_{3,3}$. The analogous problem of producing the complete list of 2-crossing-critical graphs is significantly harder. In fact, in 1987, Kochol exhibited an infinite family of 3-connected 2-crossing-critical graphs. In the talk, I will discuss the current status of the problem, including our recent work, which includes: (i) a description of all 3-connected 2-crossing-critical graphs that contain a subdivision of the M"obius Ladder $V_{10}$; (ii) a proof that there are only finitely many 3-connected 2-crossing-critical graphs not containing a subdivision of $V_{10}$; (iii) a description of all 2-crossing-critical graphs that are not 3-connected; and (iv) a recipe on how to construct all 3-connected 2-crossing-critical graphs that do not contain a subdivision of $V_{8}$.

I will study spheres in the context of A^1-homotopy theory.  In particular, I will try to explain why the A^1-homotopy of spheres is algebro-geometrically very rich and has bearing on a number of concrete problems.  Again, to keep the discussion concrete, I will mention applications to vector bundles on smooth varieties.  

실토릭공간은 (Z_2)^n 작용이 있는 n차원 위상공간입니다. 이들은 토릭다양체 등의 고전적인 대상과 밀접한 관련이 있어서 다양한 분야에서 오랫동안 연구되어 왔습니다. 하지만 토릭 다양체들과는 다르게 이들의 위상적 성질에 대해서는 많이 알려진 바가 없습니다. 이들 중 대부분은 simply connected 가 되지 않는 등 위상적 구조가 복잡해서 위상적 불변값을 계산하기도 쉽지 않기 때문입니다.

Industrial mathematics is a term used to describe a broad range of applied mathematics topics, with the common factor that the work is motivated by some problem of practical interest. In this talk I will give a brief introduction to industrial mathematics and then illustrate it with three more detailed examples.

• Football motion through the air. This project came from a question posed by a South African premiership team. Simply put the question was, can we choose a football that will disadvantage a visiting team. The answer was yes and the teams results improved significantly after this work was completed.

• Phase change. The mathematical description of the change of phase of a substance, for example from liquid to solid, is well established. However, in certain situations the standard formulations break down. I will describe our recent work on the melting of nanoparticles and solidification of a supercooled liquid.

• Flow in carbon nanotubes. Carbon nanotubes are viewed as one of the most exciting new materials with applications in electronics, optics, materials science and architecture. One unusual property is that liquid flows through nanotubes have been observed up to five orders of magnitude faster than predicted by classical fluid dynamics. I will describe a model for fluid flow in a CNT and show that the theoretical limit is closer to 50 times the classical value. This result is in keeping with later experimental and molecular dynamics papers.

I will describe the A^1-homotopy category and basic constructions therein.  Motivated by geometry and classical homotopy theory, it is natural to attempt to study homotopy of maps between two smooth varieties by simply replacing the unit interval by A^1.  I will explain how this ``naive" notion of homotopy is problematic in general, but still a very useful guide in a number of situations of interest.  To keep the discussion concrete, I will focus on the study of vector bundles on smooth affine varieties.  

The clique-width parameter provides a rough measure of the complexity of structure in (classes of) graphs. A well-known result of Courcelle, Makowsky and Rotics shows that many problems on graphs which are NP-hard in general can be solved in polynomial time in any class of graphs of bounded clique-width. Unlike the better-known treewidth graph parameter, clique-width respects the induced subgraph ordering, and in particular it can handle dense graphs. However, also unlike treewidth there is no known characterisation of the minimal classes of graphs which have unbounded clique-width.

We give an introduction of our study on nonsingular projective 3-folds of general type whose geometric genus is 2 or 3. The main aspect is the characterization of the birationality of Phi_6 and Phi_5 respectively.

A quasi-phantom category is an admissible category in the bounded derived category of a smooth projective variety having trivial Hochschild homology and finite Grothendieck group. If, in addition,
the Grothendieck group vanishes, then we call such a category a phantom. In these talks I will first give an introduction to derived categories, semiorthogonal decompositions etc., before explaining how
the first example of a quasi-phantom category, namely in the bounded derived category of the classical Godeaux surface, was constructed. To conclude I will describe some of the subsequent developments and discuss possible questions.

After initiated by the work of Böhning, Graf von Bothmer, and Sosna, there have been enumerous results on exceptional collection of maximal length on surfaces of general type. In this talk, we explain our recent result on exceptional collection of maximal length on the surfaces with Kodaira dimension 1. Also, we prove that the orthogonal complement of the collection is nonzero phantom. This is a joint work with Yongnam Lee.

A quasi-phantom category is an admissible category in the bounded derived category of a smooth projective variety having trivial Hochschild homology and finite Grothendieck group. If, in addition,
the Grothendieck group vanishes, then we call such a category a phantom. In these talks I will first give an introduction to derived categories, semiorthogonal decompositions etc., before explaining how
the first example of a quasi-phantom category, namely in the bounded derived category of the classical Godeaux surface, was constructed. To conclude I will describe some of the subsequent developments and discuss possible questions.

A quasi-phantom category is an admissible category in the bounded derived category of a smooth projective variety having trivial Hochschild homology and finite Grothendieck group. If, in addition,
the Grothendieck group vanishes, then we call such a category a phantom. In these talks I will first give an introduction to derived categories, semiorthogonal decompositions etc., before explaining how
the first example of a quasi-phantom category, namely in the bounded derived category of the classical Godeaux surface, was constructed. To conclude I will describe some of the subsequent developments and discuss possible questions.

L 함수에 관한 이해는 오일러(Euler) 시대부터 21세기 현대 정수론에 이르는 핵심문제이다. 본 강연에서는 현대 정수론의 이정표를 제시해주는 랑글란즈(Langlands) 프로그램에 관해 소개 하겠다.

참석하고자 하시는 분은 아래 링크를 통해 사전 등록을 해주시면 감사하겠습니다^^

https://goo.gl/l1fqZ3

Although biological processes are undeniably complex, there are underlying mathematical principles that govern the operation of many of these. In this talk, I will show how the combination of chemical reactions with positive feedback coupled with diffusion underlies that operation of many systems, including signaling networks, pattern forming developmental processes, and measurement-based decisions. I will also show how mathematical modeling and analysis leads to an improved understanding of emergent and collective behaviors in cell biology.

Let S be a complete intersection of a smooth quadric 3-fold Q and a hypersurface of degree d in P4.
We analyze GIT stability of S with respect to the natural G = SO(5,C)-action. We prove that if d > 4 and S
has at worst semi-log canonical singularities then S is G-stable. Also, we prove that if d > 3 and S has at worst
semi-log canonical singularities then S is G-semistable.

주식/지수 파생상품의 이론가 산출에 쓰이는 변동성 데이터에 대해 소개한다. 특히 옵션의 시장 가격 데이터로부터 내재변동성 및 로컬 변동성 곡면을 산출해내는 방법을 단계별로 설명할 예정이다. 실제 시장 데이터에는 다양한 방식으로 노이즈가 개입될 수 있는데, 이런 노이즈 데이터를 적절히 필터링 해야 할 필요가 있다. 또한 필터링 된 후 남은 데이터가 변동성 곡면을 만들어 내기에 충분치 않을 수도 있다. 이와 같은 변동성 데이터 관련 이슈를 소개하고 그 해결책에 대해 논의한다.

 In Bondy and Murty’s book the authors wrote that Tutte conjectured the wheels have the fewest spanning trees out of all 3-connected graphs on fixed number of vertices. The statement can easily be shown to be false and the corrected version, where we fix the number of edges and consider only the planar graphs, were also found to be false. We prove that if we consider the cycles instead of spanning trees then the wheels are indeed extremal. We also establish a lower bound for the number of spanning trees and suggest the prisms as possible extremal graphs.