Department Seminars & Colloquia
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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).
E6-1, ROOM 1409
Discrete Math
Bogdan Oporowski (Louisiana State University)
Characterizing 2-crossing-critical graphs
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.
자연과학동 E6-1, ROOM 1409
Discrete Math
Robert Brignall (The Open University, Milton Keynes, UK)
Characterising structure in classes with unbounded clique-width
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.
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.
수리과학과 E6-1 Room 3435
KAIST CMC noon lectures
YoungJu Choi (POSTECH)
L function (elliptic curve, modular form and beyond)
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^^
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.
E6-1, ROOM 1409
Discrete Math
Seongmin Ok (Technical University of Denmark)
Tutte’s conjecture on minimum number of spanning trees of 3-connected graphs
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.
과거 보험계리(Actuarial science)는 대수의 법칙(the law of large numbers)에 기반한 확률론적 방법론을 사용하였으나 최근에는 금융공학 및 통계학과 결합하여 다양한 적용 사례를 보여주고 있습니다.
보험부채를 시가평가하는 IFRS4 2단계의 할인율 산출,
변액보험 보증옵션 평가 및 헤지제도 도입,
클러스터링을 이용한 대표계약 모델링,
확률론적 사망률 모형을 이용한 장수지수 산출 등 국내 보험계리 분야의 최근 동향을 소개합니다.
또한, 현재 사회문제화 되고 있는 보험사기 방지와 관련한 보험개발원의 보험사기 예측 시스템도 소개합니다
A k-colouring of a graph G is a partition of V(G) into k independent sets. The chromatic number χ(G) is defined as the smallest k so that G has a k-colouring. Alternatively, we can view colourings as homomorphisms to complete graphs, and define χ(G) to be the smallest k so that there is a homomorphism from G to Kk. The variants of χ(G) (for example, fractional chromatic number) are defined by replacing the complete graph Kkwith a representative of a different class of graphs (for example, Kneser graphs).
In this talk, we will consider the vector chromatic number of a graph. A vector colouring of a G is a map from V(G) to vectors in Rm (for some m). The goal is to map adjacent vertices to vectors that are far apart. We will look at representations of a graph on its least eigenspace as examples of vector colourings. For distance-regular graphs, these colourings are optimal. Furthermore, we give a method for determining if these colourings are unique. This leads to proofs that certain classes of distance-regular graphs are all cores.
Prandtl (1936) first employed the shock polar analysis to show that, when a steady supersonic flow impinges a solid wedge whose angle is less than a critical angle (i.e., the detachment angle), there are two possible configurations: the weak shock solution and the strong shock solution, and conjectured that the weak shock solution is physically admissible. The fundamental issue of whether one or both of the strong and the weak shocks are physically admissible has been vigorously debated over several decades and has not yet been settled in a definite manner. In this talk, I address this longstanding open issue and present recent analysis to establish the stability theorem for steady weak shock solutions as the long-time asymptotics of unsteady flows for all the physical parameters up to the detachment angle for potential flow.
This talk is based on collaboration with Gui-Qiang G. Chen (Univ. of Oxford) and Mikhail Feldman(UW-Madison ).
The first lecture is about how elliptic partial differential equations
(PDEs) may be reformulated as Fredholm second kind integral equations
for the purpose of obtaining fast and accurate numerical solutions. A
discretization scheme for integral equations, called the Nyström
scheme, is presented. Advantages with this integral equation approach
to solving elliptic PDEs are reviewed, as are current trends and
challenges. In particular, I will discuss the difficulties that arise
on domains whose boundaries contain singularities such as corners, and
how to the Recursively Compressed Inverse Preconditioning (RCIP)
method is used to combat these difficulties. Numerical examples cover
applications to electromagnetic scattering and to solid mechanics.y
The first lecture is about how elliptic partial differential equations
(PDEs) may be reformulated as Fredholm second kind integral equations
for the purpose of obtaining fast and accurate numerical solutions. A
discretization scheme for integral equations, called the Nyström
scheme, is presented. Advantages with this integral equation approach
to solving elliptic PDEs are reviewed, as are current trends and
challenges. In particular, I will discuss the difficulties that arise
on domains whose boundaries contain singularities such as corners, and
how to the Recursively Compressed Inverse Preconditioning (RCIP)
method is used to combat these difficulties. Numerical examples cover
applications to electromagnetic scattering and to solid mechanics.
금융시장에서 거래되는 상품 또는 계약의 적정가치에 대한 논의는 금융시장의 태동과 더불어 시작되어 아직도 수 많은 방법론이 나타났다가 동시에 사라져가는 소위 정답이 없는 문제라고 할 수 있다. 이러한 금융시장에서 1970년 대 이후 확률론, 편미분방정식과 같은 수리적 방법론으로 무장된 사람들이 어떠한 관점에서 금융시장을 바라보았고, 이들의 노력이 금융상품의 적정가치 발견에 있어서 어떠한 프레임을 제공하였는가에 대해서 함께 생각해 보고자 한다.
참석하고자 하시는 분은 아래 링크를 통해 사전등록을 해주시면 감사하겠습니다^^
Let $M_{lambda}$ be the $lambda$-component Milnor link. For $lambda ge 3$, we determine completely when a finite slope surgery along $M_{lambda}$ yields a lens space including $S^3$ and $S^1times S^2$, where {it finite slope surgery} implies that a surgery coefficient of every component is not $infty$. For $lambda =3$ (i.e. the Borromean rings), there are three infinite sequences of finite slope surgeries yielding lens spaces. For $lambda ge 4$, any finite slope surgery does not yield a lens space. We also discuss generalizations of our present results. Our main tools are Alexander polynomials and Reidemeister torsions.
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.
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.
E6, Room 3435
Undergrad. Colloquium
Yongjin Song (Inha University)
Mathematics, Language of Universe
수학은 수천 년간 그 지식을 축척하며 발전해 온 유일한 학문이다. 수학은 완벽한 해를 추구한다는 점에서도 다른 학문과 구별된다. 수학은 오랜 세월 동안을 차곡 차곡 탑을 쌓듯이 발전해 왔고, 따라서 발전 과정에 대한 역사적 고찰이 현대수학을 이해하는데 큰 도움이 된다. 과학이라는 말이 만들어진 지, 그리고 과학이 사람들의 삶을 바꾸기 시작한 지 불과 200여년 밖에 되지 않았다. 앞으로 과학은 수만 년, 수백만 년 발전을 지속할 가능성이 크다. 지금의 수학은 미래에 우주의 언어로서 필수적으로 과학의 발전에 이바지할 것이다.
콜로퀴엄 후에는 피자가 제공되며 연사님과 편하게 대화할 수 있는 자리가 있습니다.
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.
차별이나 사회적 배제와 같은 부정적 경험이 어떻게 인간의 건강을 해치는지를 규명하는 것은 보건산업, 보건정책의 측면에서 매우 중요한 문제이다. 그러한 연구는 공동체의 건강을 증진시키기 위해, 어떠한 제도적 변화가 필요한지에 대해 말해주기 때문이다. 차별경험과 건강의 관계를 고찰하는 연구에서는 주로 설문조사를 통해 차별의 경험을 측정하고 그 경험이 건강과 어떠한 연관성을 보이는지에 대해 통계적으로 규명한다. 그런데, 차별경험을 보고하는 과정은 연구 대상자의 내적 검열을 포함해 다양한 사회적 요인들에 의해 영향을 받고, 그로 인한 차별경험 측정의 불확실성을 통계적으로 어떻게 다룰 것인가는 논의가 계속되는 이슈이다. 본 발표에서는 차별경험과 건강에 대한 기존 연구를 소개하고, 차별경험을 정확히 측정하는 문제와 관련하여 역학(Epidemiologist)자로서의 고민을 공유하고자 한다.
초록: 계산수학이란 수학적 문제에 대하여 효율적이며 믿을 수 있는 컴퓨터 해를 구하고자 하는 연구분야이다. 계산수학은 엄밀한 수학적 이론을 이용하여 오차를 분석하고 대용량 계산과 병렬계산 알고리즘을 만들오 내며 계산 모델링과 이 모델에 대한 모의실험을 한다. 이러한 계산수학은 컴퓨터가 생겨나기 이전 16세기 경부터 계산을 보다 편하게 하기 위하여 발달하였다. 네이피어의 로그의 발견, 뉴턴의 비선형방정식 해법과 보간법 등이 그 예이다. 20세기 들어서 컴퓨터가 생겨나고 선형방정식계와 상미분방정식, 편미분방정식 등을 푸는 방법들이 폭발적으로 개발되었다. 몬테 카를로 방법, 대형 선형방정식계를 푸는 반복해법, fast Fourier Transform, 유한요소법 등이 개발되었다. 또한 병렬계산기가 만들어 지면서 이를 활용하는 수치방법들이 개발되고 있다. 본 강연에서는 계산수학의 태동과 발전을 살펴보고 앞으로 어떻게 발전해 나아가게 될 지를 살펴본다.
참석하고자 하시는 분은 아래 링크를 통해 사전등록을 해주시면 감사하겠습니다^^