We introduce the Milnor type $K$-group attached to some algebraic groups including additive groups over a perfect field as an extension of Somekawa's $K$-group. We give some descriptions of this $K$-group associated to some special algebraic groups. Furthermore, we will also explain some relations to the higher dimensional class field theory.

Isogeometric Analysis (IGA) is a non-standard numerical method for partial differential equations (PDEs), which was introduced by T. J. R. Hughes in [1]. In the isogeometric framework, the ultimate goal is to adopt the geometry description from a Computer Aided Design (CAD) parametrization, and use it for the analysis, that is, within the PDE solver. Non-uniform rational B-splines (NURBS) are a standard in CAD community mainly because they are extremely convenient of the representation of free-form surfaces and there are very efficient algorithms to evaluate them, to refine and derefine them. In IGA, those same basis functions (that represent the CAD geometry) are also used as the basis for the discrete solution space of PDEs, thus following an isoparametric paradigm. IGA methodologies have been studied and applied in fields as diverse as fluid dynamics, structural mechanics and electromagnetics.

Domain decomposition methods are a major area of recent research in numerical analysis for PDEs. They provide robust, parallel and scalable preconditioned iterative methods for the large linear systems arising in discretizaton of the continuous problems.

In this talk, we propose overlapping additive Schwarz (OAS) methods for elliptic problems in Isogeometric Analysis. We construct OAS preconditioners both in the parametric space and in the physical space and also prove that our proposed methods in multi-dimensions are scalable. Moreover, we present a set of numerical experiments, including the case with discontinuous coefficients, which is in complete accordance with the theoretical developments.

*석학초청강연* 

Bar codes are ubiquitous -- they are used to identify products in stores, parts in a warehouse, and books in a library, etc. In this talk, the speaker will describe how information is encoded in a bar code and how it is read by a scanner. The presentation will go over how the decoding process, from scanner signal to coded information, can be formulated as an inverse problem. The inverse problem involves finding the "word" hidden in the signal. What makes this inverse problem, and the approach to solve it, somewhat unusual is that the unknown has a finite number of states.

* 4시부터 자연과학동 토론실(1401호)에서 다과가 준비됩니다.

In the first three lectures on “From Boltzmann to Euler” I derived the Euler equation of gas dynamics from the Boltzmann equation. In these two lectures I will show that surprisingly the equations of the classical problem of embedding a 2 dimensional Riemannian manifold into 3 dimensional Euclidean space can be posed in the form Euler equations. This allows us to prove existence of solutions to the embedding problem.

지난 3번의 “From Boltzmann to Euler” 주제의 강연에서 Boltzmann 방정식으로부터 기체 방정식인 Euler 방정식을 유도하였다. 이 두 번의 강연에서는 놀랍게도 Riemannian 2차원 구조를 3차원 Euclidean 공간에 투영하는 고전적인 문제가 Euler 방정식으로 재구성 되어 진다는 것을 보인다. 이를 통해서 투영 (embedding) 문제의 해의 존재성을 보일 수 있다.

In the first three lectures on “From Boltzmann to Euler” I derived the Euler equation of gas dynamics from the Boltzmann equation. In these two lectures I will show that surprisingly the equations of the classical problem of embedding a 2 dimensional Riemannian manifold into 3 dimensional Euclidean space can be posed in the form Euler equations. This allows us to prove existence of solutions to the embedding problem.

지난 3번의 “From Boltzmann to Euler” 주제의 강연에서 Boltzmann 방정식으로부터 기체 방정식인 Euler 방정식을 유도하였다. 이 두 번의 강연에서는 놀랍게도 Riemannian 2차원 구조를 3차원 Euclidean 공간에 투영하는 고전적인 문제가 Euler 방정식으로 재구성 되어 진다는 것을 보인다. 이를 통해서 투영 (embedding) 문제의 해의 존재성을 보일 수 있다.

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.

***** KAIST Discrete Math Seminar *****

DATE: October 19, Friday
TIME: 4PM-5PM
PLACE: E6-1, ROOM 1409
SPEAKER: Eun Jung Kim (김은정), CNRS, LAMSADE, Paris, France.
TITLE: Linear kernels and single-exponential algorithms via protrusion decompositions
http://mathsci.kaist.ac.kr/~sangil/seminar/entry/20121019/

A t-treewidth-modulator of a graph G is a set X⊆V(G) such that the treewidth of G-X is at most some constant t-1. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs G that come equipped with a t-treewidth-modulator. This decomposition, called a protrusion decomposition, is the cornerstone in obtaining the following two main results. We first show that any parameterized graph problem (with parameter k) that has finite integer index and is treewidth-bounding admits a linear kernel on H-topological-minor-free graphs, where H is some arbitrary but fixed graph. A parameterized graph problem is called treewidth-bounding if all positive instances have a t-treewidth-modulator of size O(k), for some constant t. This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus [Bodlaender et al., FOCS 2009] and H-minor-free graphs [Fomin et al., SODA 2010]. Our second application concerns the Planar-F-Deletion problem. Let F be a fixed finite family of graphs containing at least one planar graph. Given an n-vertex graph G and a non-negative integer k, Planar-F-Deletion asks whether G has a set X⊆ V(G) such that |X|≤k and G-X is H-minor-free for every H∈F. Very recently, an algorithm for Planar-F-Deletion with running time 2^O(k) n log^2 n (such an algorithm is called single-exponential) has been presented in [Fomin et al., FOCS 2012] under the condition that every graph in F is connected. Using our algorithm to construct protrusion decompositions as a building block, we get rid of this connectivity constraint and present an algorithm for the general Planar-F-Deletion problem running in time 2^O(k) n^2. This is a joint work with Alexander Langer, Christophe Paul, Felix Reidl, Peter Rossmanith, Ignasi Sau, and Somnath Sikdar.

Informations on future talks at KAIST Discrete Math Seminar can be found at : http://mathsci.kaist.ac.kr/~sangil/seminar/
DiscreteMath mailing list http://mathsci.kaist.ac.kr/mailman/listinfo/discretemath

In 1967 Coburn showed that C ∗ -algebras generated by a single non-unitary isometry on a Hilbert space don’t depend on the particular choice of the isometry. And R. G. Douglas proved that the C ∗ - algebra AΓ generated by a non-unitary one-parameter semigroup of isometries is canonically unique for a subgroup Γ of the real number group R. A. Nica called it the uniqueness property, which means to some extend that C ∗ -algebras generated by non-unitary isometries on a Hilbert space don’t depend on the particular choice of the isometries. Since Coburn, many operator algebraists have extended Coburn’s result cosistently. Toeplitz algebra, Cuntz algebra, Wiener-Hopf C ∗ -algebra W(G, M ) for a discrete group G with a semigroup M are their outcomes. We can see that if the Wiener-Hopf C ∗ -algebra W(G, M ) of a partially ordered group G with the positive cone M has the uniqueness property, then (G, M ) is weakly unperforated. We also can see that the extented Coburn’s result of the Wiener-Hopf C ∗ -algebra W(G, M ) depends on the order structure of the semigroup M .

In this lecture Boltzmann equation is introduced. The intention of this introduction is to introduce it without the complication of collision operators, but with essential building blocks of the theory that guide us to PDEs such as Euler.

In his 1900 address to the International Congress of Mathematics in Paris, D. Hilbert as part of his 6th Problem (dealing with the axiomitization at mechanics) proposed to develop “mathematically the limiting process which leak from the atomistic view to the laws of motion of continua.” More precisely with the context of gas dynamics this challenge is passage (rigorously) from the Boltzmann equation for rarefied gas dynamics to the compressible Euler equations of continuum mechanics as Knudsen number $eps$ approaches zero. We will discuss it in the second part of this lecture series.

요약:본 강연에서는 충돌 연산자의 복잡한 분석을 통하지 않고 Kinetic 이론에 실제적으로 사용되고 관련 PDE와 연관을 맺게하는 기본적인 성질을 이용하여 볼쯔만 방정식을 소개하고자 한다. 또한 1900년 파리의 세계수학자 대회에서 D. Hilbert가 발표한 문제들 중 제6문제에서 제시한 연속체 역학의 원자이론을 통한 수렴에 대한 문제를 소개하고 그러한 접근법의 분제점을 소개하고자 한다.

We will show how it is possible to calculate the fundamental group of a large class of projective surfaces admitting a Genus-2 fibration. As an application we will verify the Shafarevich Conjecture for holomorphic convexity for such surfaces.

A widespread phenomenon in microorganisms and cells is their movement depending on a certain chemical signals. We introduce several type of models taking into account of influence of chemical stimuli, and other kind of alignment models in the absence of chemicals as well. In this talk, we discuss mainly a chemotaxis-fluid model with nonlinear diffusion which describes a nonlinear diffusive aggregation process of swimming bacteria in an incompressible fluid.

 A fake projective plane is a nonsingular algebraic surface of general type with the same Berri numbers as the projective plane. The universal cover of such a surface is a complex 2-dimensional ball, the structure of the fundamental groups and their actions on the ball are now known. However, no purely geometric construction of any fake projective plane has been found. In my talks I will discuss an attempt to find such a construction via the theory of elliptic surfaces. All the needed background will be explained in the lectures.

In this lecture Boltzmann equation is introduced. The intention of this introduction is to introduce it without the complication of collision operators, but with essential building blocks of the theory that guide us to PDEs such as Euler.

In his 1900 address to the International Congress of Mathematics in Paris, D. Hilbert as part of his 6th Problem (dealing with the axiomitization at mechanics) proposed to develop “mathematically the limiting process which leak from the atomistic view to the laws of motion of continua.” More precisely with the context of gas dynamics this challenge is passage (rigorously) from the Boltzmann equation for rarefied gas dynamics to the compressible Euler equations of continuum mechanics as Knudsen number $eps$ approaches zero. We will discuss it in the second part of this lecture series.

요약:본 강연에서는 충돌 연산자의 복잡한 분석을 통하지 않고 Kinetic 이론에 실제적으로 사용되고 관련 PDE와 연관을 맺게하는 기본적인 성질을 이용하여 볼쯔만 방정식을 소개하고자 한다. 또한 1900년 파리의 세계수학자 대회에서 D. Hilbert가 발표한 문제들 중 제6문제에서 제시한 연속체 역학의 원자이론을 통한 수렴에 대한 문제를 소개하고 그러한 접근법의 분제점을 소개하고자 한다.

The Disjoint-Paths Problem asks, given a graph G and a set of pairs of terminals (s1,t1),…,(sk,tk), whether there is a collection of k pairwise vertex-disjoint paths linking si and ti, for i=1,…,k. In their f(k)n^3 algorithm for this problem, Robertson and Seymour introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k) there is an "irrelevant" vertex whose removal creates an equivalent instance of the problem. This fact is based on the celebrated Unique Linkage Theorem, whose — very technical — proof gives a function g(k) that is responsible for an immense parameter dependence in the running time of the algorithm. In this paper we prove this result for planar graphs achieving g(k)=2^O(k). Our bound is radically better than the bounds known for general graphs. Moreover, our proof is new and self-contained, and it strongly exploits the combinatorial properties of planar graphs. We also prove that our result is optimal, in the sense that the function g(k) cannot become better than exponential. Our results suggest that any algorithm for the Disjoint-Paths Problem that runs in time better than 2^(2^o(k))n^O(1) will probably require drastically different ideas from those in the irrelevant vertex technique

 A fake projective plane is a nonsingular algebraic surface of general type with the same Berri numbers as the projective plane. The universal cover of such a surface is a complex 2-dimensional ball, the structure of the fundamental groups and their actions on the ball are now known. However, no purely geometric construction of any fake projective plane has been found. In my talks I will discuss an attempt to find such a construction via the theory of elliptic surfaces. All the needed background will be explained in the lectures.

길이가 가장 짧은 것은 직선이고, 넓이가 가장 작은 것은 극소곡면이라고 부른다.

극소곡면은 어떠한 성질을 가지고 있는가? 극소곡면의 역사와 최근 연구결과에 관해서 알아보자.

In this lecture Boltzmann equation is introduced. The intention of this introduction is to introduce it without the complication of collision operators, but with essential building blocks of the theory that guide us to PDEs such as Euler.

In his 1900 address to the International Congress of Mathematics in Paris, D. Hilbert as part of his 6th Problem (dealing with the axiomitization at mechanics) proposed to develop “mathematically the limiting process which leak from the atomistic view to the laws of motion of continua.” More precisely with the context of gas dynamics this challenge is passage (rigorously) from the Boltzmann equation for rarefied gas dynamics to the compressible Euler equations of continuum mechanics as Knudsen number $eps$ approaches zero. We will discuss it in the second part of this lecture series.

요약:

본 강연에서는 충돌 연산자의 복잡한 분석을 통하지 않고 Kinetic 이론에 실제적으로 사용되고 관련 PDE와 연관을 맺게하는 기본적인 성질을 이용하여 볼쯔만 방정식을 소개하고자 한다. 또한 1900년 파리의 세계수학자 대회에서 D. Hilbert가 발표한 문제들 중 제6문제에서 제시한 연속체 역학의 원자이론을 통한 수렴에 대한 문제를 소개하고 그러한 접근법의 분제점을 소개하고자 한다.

We consider the failure of the integral Hodge
conjecture and its relation to recent work of Colliot-Thelene
and Voisin.

Controlling the electromagnetic properties of materials, beyond the limit that is attainable with naturally existing substances, has become a reality with the advent of metamaterials.  The collection of structured artificial ‘atoms’ has promised a vast variety of otherwise unexpected physical phenomena, among which the experimental realization of a negative refractive index has been one of the main foci thus far.   From the perspective of manipulating a refractive index, however, expanding the refractive index further into a positive high regime is in high demand as this will complete the whole spectra of achievable refractive index and provide more design flexibility for transformation optics.  In the first part of the talk, I will show that a broadband, extremely high index of refraction can be realized from large-area, freestanding, flexible terahertz metamaterials composed of strongly coupled unit cells.  In the second part of the talk, I will present the experimental demonstration of electrically controllable light-matter interactions of an unprecedented degree in a hybrid material/metamaterial system, which consists of artificially constructed two-dimensional meta-atoms (metamaterial) and naturally occurring two-dimensional carbon atoms (graphene). The exotic electrical and optical properties of graphene, when enhanced by the strong resonance of meta-atoms, lead to a very strong interaction between massless Dirac fermions and photons such that persistent switching (‘photonic memory’) and fast linear modulation (‘photonic modulator’) of low-energy photons are realized in the extreme subwavelength-scale (below λ/1,000,000). Our work, we believe, will push the limit of wavelength-scale photonics (~λ-scale) and subwavelength-scale plasmonics (~λ/100-scale) further into the regime of photon manipulation within the extreme subwavelength scale (~λ/1,000,000-scale), which will have a significant impact on broader disciplines of physics, material sciences, electrical engineering, nano-sciences, and mechanical engineering.

In this lecture we will study the finite element approximation of
eigenvalue problems  for compact operators. The general error estimates
for Galerkin approximations of the eigenvalues and eigenvectors are
presented. More specifically, the analysis would be extended to the

multiscale and the discontinuous Galerkin finite element methods.
In the end, we present several numerical experiments ranging from
acoustics to electromagnetics.

I talk about how to construct the Gibbs measure for the
isothermal Falk model and explain how the Gibbs measure works for the
proof of global existence of solution.
This approach was first introduced by Bourgain in 1994 and was improved
by Tzvetkov, Burq-Tzvetkov and Oh to apply it to many other nonlinear
dispersive equations.
The isothermal Falk model is a nonlinear dispersive equation which
describes shape memory alloys.
In the PDE case, the support space of the Gibbs measure is larger than
the energy space.
Especially, for the isothermal Falk model, the support space of the
Gibbs measure contains discontinuous shear strains, though they are
excluded from the energy space.
In this respect, the global existence theorem based on the Gibbs measure
seems natural from a physical point of view.
This is a joint work with Shuji Yoshikawa, Ehime University.

IV. A DICTIONARY BETWEEN MAPPING CLASS GROUPS AND RIGHT-ANGLED ARTIN GROUPS VIA CURVE COMPLEXES

In this lecture, we will primarily be discussing the results of [1], together with appropriate background. The general principle we would like to explore is that right-angled Artin groups behave a lot like mapping class groups from the point of view of their actions on their extension graphs and curve complexes respectively. 

[1] Sang-hyun Kim and Thomas Koberda. Actions of right-angled Artin groups on quasi–trees. In preparation.

[2] Sang-hyun Kim and Thomas Koberda. Embedability of right-angled Artin groups. Preprint.

[3] Thomas Koberda. Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups. To appear in Geom. Funct. Anal.

In this lecture we will study the finite element approximation of
eigenvalue problems  for compact operators. The general error estimates
for Galerkin approximations of the eigenvalues and eigenvectors are
presented. More specifically, the analysis would be extended to the

multiscale and the discontinuous Galerkin finite element methods.
In the end, we present several numerical experiments ranging from
acoustics to electromagnetics.

We give an introduction to the theory of Chow groups and explain classical and new results obtained by considering the Ceresa cycle.

III. RIGHT-ANGLED ARTIN SUBGROUPS OF RIGHT-ANGLED ARTIN GROUPS

In this lecture, we will discuss the primary results of [2]. In that article, the authors develop a general theory for determining when there exists an embedding A(X) -> A(Y)

for two graphs X and Y.

[1] Sang-hyun Kim and Thomas Koberda. Actions of right-angled Artin groups on quasi–trees. In preparation.

[2] Sang-hyun Kim and Thomas Koberda. Embedability of right-angled Artin groups. Preprint.

[3] Thomas Koberda. Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups. To appear in Geom. Funct. Anal.

II. RIGHT-ANGLED ARTIN SUBGROUPS OF MAPPING CLASS GROUPS

In this lecture, we will discuss the primary result of [3], which roughly says that if we take any collection of mapping classes, say {f1,...,fk} and replace them by sufficiently high powers {f1^N,...,fk^N}, they generate a right-angled Artin subgroup of the mapping class group of the expected type. Unless otherwise noted, all examples and statements can be found with proof (or appropriate reference) in [3].

[1] Sang-hyun Kim and Thomas Koberda. Actions of right-angled Artin groups on quasi–trees. In preparation.

[2] Sang-hyun Kim and Thomas Koberda. Embedability of right-angled Artin groups. Preprint.

[3] Thomas Koberda. Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups. To appear in Geom. Funct. Anal.

I will give a quick conceptual introduction to motivic homotopy theory of Morel and Voevodsky, in such a way that those who know what derived functors in homological algebra can understand this subject expressed in terms of "homotopical algebra" of Quillen. In motivic homotopy theory, one hopes to do some homotopy theory using algebraic varieties as one does the usual homotopy theory for topological spaces. We use "motivic weak-equivalences" in this subject.

Next, I will explain some "descent theorems" in motivic homotopy theory. These theorems will allow us to handle the motivic weak-equivalences a bit better. 

I will sketch some applications of these machines.

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 1935 Erdős and Szekeres showed that every “sufficiently large” set of points in general position in the plane contains a “large” subset which is in convex position. Since then many mathematicians have tried to determine good bounds for “sufficiently large” in terms of “large”, as well as given numerous generalizations and refinements. In this talk I will survey this famous problem and extend it to a natural object which we call generalized wiring diagram. This unifies several proposed generalizations, and as a result we will settle several conjectures in this area.

This is joint work with Michael Dobbins and Alfredo Hubard.

A way to study the geometry of a homogeneous variety under a semi-simple algebraic group is to investigate its Chow group of algebraic cycles modulo the rational equivalence relation. In general, the problem of determining the Chow group of a projective homogeneous variety reduces to computing the torsion. In this talk, we discuss the latter problem including the cases of Severi-Brauer varieties and Spin-flags. 

아인슈타인의 브라운 운동에 관한 연구는 균일한 조건에서의 확산이 mean free path과 collision time interval time에 의해 주어진다는 것을 보였다. 

A graph G is called H-saturated if it does not contain any copy of H, but for any edge e in the complement of G the graph G+e contains some H. The minimum size of an n-vertex H-saturated graph is denoted by sat(n,H). We prove sat(n,C_k) = n + n/k + O((n/k²) + k²) holds for all n≥k≥3, where C_k is a cycle with length k.
Joint work with Zoltan Füredi.

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.

When a prey population is infected, we study a predator-prey system with a ratio-dependent functional response under no-flux boundary condition. First, all nonnegative equilibria are  investigated, and then conditions which gives a stability at these equilibria are  found. Especially, disease-free and biological control states are discussed in view of  biological interpretations. Lastly,  the existence of nonconstant positive steady-states is studied. The methods employed  are  a comparison principle for a parabolic problem and Leray-Schauder Theorem. 

TBA

This is a prequel to Agol's lecture series on Virtual Haken Conjecture. We survey basic facts on cube complexes and discuss how those facts are related to the study of subgroups of right-angled Artin groups. The following notions will be defined while doing so: hyperbolic group, relatively hyperbolic group, cube complex, non-positive curvature, right-angled Artin groups, Salvetti complex, local isometry and pi-1-injectivity, hyperplane, special cube complex and subgroup separability.

There are no prerequsites for this talk, except for algebra and topology at undergraduate level.