학과 세미나 및 콜로퀴엄
A fractional matching of a graph G is a function f giving each edge a number between 0 and 1 so that 
for each 
, where 
is the set of edges incident to v. The fractional matching number of G, written 
, is the maximum of 
over all fractional matchings f. Let G be an n-vertex graph with minimum degree d, and let 
be the largest eigenvalue of G. In this talk, we prove that if k is a positive integer and
, then 
Determination of coronary physiology is critical to the diagnosis and treatment of patients with coronary artery disease. Traditionally, assessment of coronary physiology required invasive coronary angiography. Here the non-invasive assessment of coronary physiology based on the image analysis of coronary computed tomography (CT) angiography, which might replace the invasive assessment methods, would be discussed.
The notion of a cohomological invariant of analgebraic group was introduced by J-P. Serre. Cohomological invariants of an algebraic group G relate principal homogeneous spaces of G over a field extension of the base field (G-torsors) and Galois cohomology of the field. If A is an "algebraic object", then the principal homogeneous spaces for the automorphism group G = Aut(A) are in one-to-one correspondence with the twisted forms of A. In such a way many classical algebraic objects arise: simple algebras, quadratic and hermitian forms, algebras with involutions, Cayley-Dickson algebras, etc. Thus, cohomological invariants assign to algebraic objects the cohomology classes.
We will compute cohomological invariants of small degrees. Some applications will be given. In particular, unramified invariants can be used to determine non-rationality property of classifying spaces of algebraic groups.
Abstract: This 8-hour course will cover many of the big themes in combinatorial and integer optimization introduced in the last half century.
Thurs Dec 11, 10:00-12:00
Combinatorial optimization: Shortest paths and dynamic programming. Nonbipartite matching. Matroid intersection. TSP. Submodular function maximization.
We prove that on a punctured oriented surface with Eulercharacteristic chi < 0, the maximal cardinality of a set of essential simple arcs that are pairwise non-homotopic and intersecting at most once is 2|chi|(|chi|+1). This gives a cubic estimate in |chi| for a set of curves pairwise intersecting at most once on a closed surface.
Essential dimension of an algebraic object is the smallest number of algebraically independent parameters required to define the object. This notion was introduced by Buhler, Reichstein and Serre about 20 years ago.The relations to different parts of algebra such asalgebraic geometry, Galois cohomology and representation theory will be discussed.
The notion of a cohomological invariant of analgebraic group was introduced by J-P. Serre. Cohomological invariants of an algebraic group G relate principal homogeneous spaces of G over a field extension of the base field (G-torsors) and Galois cohomology of the field. If A is an "algebraic object", then the principal homogeneous spaces for the automorphism group G = Aut(A) are in one-to-one correspondence with the twisted forms of A. In such a way many classical algebraic objects arise: simple algebras, quadratic and hermitian forms, algebras with involutions, Cayley-Dickson algebras, etc. Thus, cohomological invariants assign to algebraic objects the cohomology classes.
We will compute cohomological invariants of small degrees. Some applications will be given. In particular, unramified invariants can be used to determine non-rationality property of classifying spaces of algebraic groups.
Abstract: This 8-hour course will cover many of the big themes in combinatorial and integer optimization introduced in the last half century.
Tues Dec 9, 10:00-12:00
IP Formulation and cuts: UFL. Big M's. Generic cutting planes: Gomory for pure; BMI, MIR, GMI for mixed. Disjunctive cuts and the CGLP. Combinatorial cuts.
The notion of a cohomological invariant of analgebraic group was introduced by J-P. Serre. Cohomological invariants of an algebraic group G relate principal homogeneous spaces of G over a field extension of the base field (G-torsors) and Galois cohomology of the field. If A is an "algebraic object", then the principal homogeneous spaces for the automorphism group G = Aut(A) are in one-to-one correspondence with the twisted forms of A. In such a way many classical algebraic objects arise: simple algebras, quadratic and hermitian forms, algebras with involutions, Cayley-Dickson algebras, etc. Thus, cohomological invariants assign to algebraic objects the cohomology classes.
We will compute cohomological invariants of small degrees. Some applications will be given. In particular, unramified invariants can be used to determine non-rationality property of classifying spaces of algebraic groups.
It is known that every knot bounds a singular disk whose singular set consists of only clasp singularities. Such a singular disk is called a clasp disk. The clasp number of a knot is the minimum number of clasp singularities among all clasp disks of the knot. The $Gamma$-polynomial is the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials. I will talk about a characterization of the $Gamma$-polynomials of knots with the clasp numbers at most two.
Abstract: This 8-hour course will cover many of the big themes in combinatorial and integer optimization introduced in the last half century.
Fri Dec 5, 10:00-12:00
Convexity-based algorithms: Submodular function minimization. Subgradient optimization. Ellipsoid method and its implications for combinatorial optimization.
자연과학동 E6-1 Room 1409
Discrete Math
Jon Lee (University of Michigan, Ann Arbor, USA.)
Matroid Optimization
자연과학동 E6-1 Room 1409
Discrete Math
Matroids can be seen an an abstraction for understanding the essence of algorithms for some classical graph problems: (i) optimum-weight spanning tree, and (ii) optimum-weight assignment. We will see some recent results showing how matroids have further natural roles in various combinatorial-optimization problems where some local-search and linear-programming-based algorithms find provably-good approximations. In particular, we will look at: (iii) constrained submodular-function maximization, (iv) the well-known matroid matching problem, and (v) various nonlinear-objective matroid-intersection problems.
In this talk, I will present recent progress on the following subjects: (1) Smooth transonic flow of Euler-Poisson system; (2) Transonic shock of Euler-Poisson system. This talk is based on collaboration with Ben Duan(Dalian Univ. of Technology), Chujing Xie(SJTU) and Jingjing Xiao(Chinese Univ of Hong Kong).
자연과학동 E6-1 room 1409
Discrete Math
권오정
Upper bounds on the size of obstructions for graphs of linear rank-width at most k
자연과학동 E6-1 room 1409
Discrete Math
Graph layout problems are a class of optimization problems whose goal is to find a linear ordering of an input graph in such a way that a certain objective function is optimized. The matrix rank function has been studied as an objective function. The linear rank-width of a graph G is the minimum integer k such that G admits a linear ordering $v_1, v_2, ldots , v_n$ satisfying that the maximum over all values [operatorname{rank}A_G[{v_1, v_2, ldots, v_t}, {v_{t+1}, ldots, v_n}]] is k, where $A_G$ is the adjacency matrix of $G$ and the rank is computed over the binary field.
In this talk, we present a result that for every graph G that is vertex-minor minimal with the property having linear rank-width larger than p, the number of vertices in G is at most doubly exponential in $mathcal{O}(p)$. The number of vertex-minor obstructions for linear rank-width at most p is of interest because the only known fixed parameter tractable algorithm to test whether linear rank-width is at most p is using the finiteness of the number of forbidden vertex-minor obstructions. Our result gives an upper bound of the complexity on this algorithm. Our basic tools are the algebraic operations on labelled graphs introduced by Kante and Rao, and we extend the notion of vertex-minors in our purpose. This is joint work with Mamadou Moustapha Kante.
The warping degree of an oriented knot diagram with a base point is the number of crossing points such that we meet the point as an under-crossing first when we travel the diagram with the orientation from the base point.
In this talk, we introduce a matrix of an oriented knot diagram which is defined from a viewpoint of the warping degree, and consider its properties.
In particular, we show that the matrix represents an oriented knot diagram on the sphere uniquely.
As an application, we will consider a sudoku-like puzzle.
제목: 소수의 역사- 유클리드에서 리만까지
초록: 소수가 무수히 많이 있다는 유클리드의 증명에서 시작하여, 소수의 역수의 합이 무한대로 발산한다는 오일러의 결과, 가우스가 예상한 소수정리 및 관련된 내용, 1859년에 제출된 리만의 논문, 소수정리의 증명, 소수정리와 리만가설의 관계 등을 소개
참석하고자 하시는 분은 사전 등록을 해주시면 감사하겠습니다^^
Cluster algebras are fundamental objects in mathematics and physics. All important algebraic / combinatorial / geometric / topological / physical objects are conjectured to have cluster algebra structures.
We introduce cluster algebras and their remarkable properties. Positivity is a central theme in this field. In joint work with Schiffler, we prove the positivity conjecture. We also explain a very recent work of Gross, Hacking, Keel and Kontsevich along this line.
We present a complete combinatorial classification of the associated primes of every fixed power of the edge ideal of a graph. Earlier results have been obtained only for the second and third powers. The classification is achieved by using matching theory. It turns out that these associated primes are characterized by certain kind of subgraphs which can be easily detected.
우리가 TV나 서커스에서 종종 볼 수 있는 공을 사용하는 저글링은 공을 공중에 던지고 받는 행위라고 할 수 있다. 이 강연에서는 던지는 공의 높이를 순차적으로 기록해 얻어진 저글링 수열에 대해 알아본다. 가장 흔히 볼 수 있는 "3볼 캐스캐이드"는 모든 공을 같은 높이 3으로 던져서 이것으로 수열을 만들면 33...=3이 된다. 그 밖에도 441, 51, 531 등은 저글링 수열이 되지만 443은 저글링 수열이 될 수 없다. 우리는 주어진 수열이 언제 저글링 수열이 되는지, 그리고 저글링 수열이 될 때 필요한 공의 갯수에 대해 알아볼 것이다. 또한 공의 갯수와 던지는 높이가 제한된 경우 가능한 저글링 수열이 얼마나 있는지 알아본다.
The theory of minimal surbmanifolds has its origin in the theory of calculus of variations developed by Euler and Lagrange in the 18th century and in later investigations by Schwarz, Riemann and Weierstrass in the 19th century, but it has very recently seen remarkable advances that have solved lots of long standing open problems. In the first talk, we illustrate various proofs of Bernstein’s Theorem that the only entire solutions of minimal surface equation are affine functions.
The functions of living cells are regulated by the complex biochemical network, which consists of stochastic interactions among genes and proteins. However, due to the complexity of biochemical networks and the limit of experimental techniques, identifying entire biochemical interaction network is still far from complete. On the other hand, output of the networks, timecourses of genes and proteins can be easily acquired with advances in technology. I will describe how to reveal the biochemical network architecture with oscillating timecourse data. Next, I will discuss how to reduce or simplify the stochastic biochemical networks while preserving the slow timescale dynamics. Specifically, I will show when macroscopic rate functions (e.g. Michaelis-Menten equation)describing the slow timescale dynamics of deterministic systems can be used for stochastic simulations. Finally, I will discuss how the network topology affects the functions and dynamics of biochemical networks with an example of circadian (~24hr) clock.
This talk is about the extension of the previous presentation titled "Hawkes process and high-frequency financial data".
In the previous presentation, I explained the basic properties of the Hawkes process and its use in modeling price dynamics and volatility estimation.
I show the empirical studies based on the symmetric Hawkes process, a simple model to take into account for both clustering property and market micro structure noise, using S&P 500 stock price data.
In addition, I introduce the recent developments of the marked point process approach to describe not only the location of random events but also additional information, called mark, attached to each event.
The original Hawkes process method might be limited to the unit size jump movements but by the marked version, we model the random size of jump in the price dynamics.
Abstract: We say an integer is y -smooth if all of its prime factors are less than or equal to y . We consider the Diophantine equation a+b=c where all variables are y -smooth and ( a,b,c )=1 . A recent work of Lagarias and Soundararajan showed that this equation has at least exp ( y 1/ κ ) solutions for κ >8 when y is large. In this talk, I will describe some recent progress in this problem and an analogous theorem for the polynomial rings over finite fields.
자연과학동(E6-1) Room 1409
Discrete Math
김하나 (국가수리과학연구소)
Enumeration of symmetric hex trees and the related polynomials
자연과학동(E6-1) Room 1409
Discrete Math
A hex tree is an ordered tree of which each vertex has updegree 0, 1, or 2, and an edge from a vertex of updegree 1 is either left, median, or right. In this talk, we introduce an enumeration problem of symmetric hex trees and describe a bijection between symmetric hex trees and a certain class of supertrees. Some algebraic properties of the polynomials arising in this procedure also will be discussed.
Since the revolution of molecular biology and quantitative biology in the early 1980s, mathematical modeling has been widely used to understand complex biological systems, which typically consist of non-linear and stochastic biochemical interactions. Typical process of applying mathematical models to biological systems includes mathematical representation of biological systems, the fitting of the models to experimental data, the predictions with the analysis and simulations and the validation of the prediction with experiments. In this talk, I will describe mathematical tools used in each step of this process. I will also discuss which toolboxes of mathematical biology still lack of solid mathematical foundation due to the relatively short history of mathematical biology.
Abstract. Let K be a number eld and X = Spec OK. Then we can write
the etale fundamental group et 1 (X) ' Gal(Kf ur=K), where Kf ur is the maximal extension of K which is unrami ed over all nite places. This fact is one of the motivations of the study of unrami ed extensions of number elds and their Galois groups. Under the assumption of the GRH(Generalized Riemann
Hypothesis), we show that there are number elds K such that Gal(Kf ur=K) is a nite nonabelian simple group.
We present a generalized immersed boundary (IB) method combined with the unconstrained Kirchhoff rod theory which has been developed to study the biological fluid mechanics in filamentous structures such as bacterial flagella and DNA sequences.
A thin elastic filament (rod) in the Kirchhoff model that resists bending and twisting can be modeled as a ``three-dimensional space curve'' together with an orthonormal triad (material frame) at each point of the rod. The space curve represents the centerline of the rod and the triad indicates the amount of bend and twist of the elastic rod. This is a well-established theory in the statics and dynamics of thin elastic filaments without fluid. Combining Kirchhoff rod theory with the standard models of viscous incompressible fluids will allow us to study the complicated hydrodynamics of bacterial swimming, DNA supercoiling, and more.
In this talk we will discuss structures of derived categories of surfaces with $p_g=q=0$. To be more precise we will discuss the notion of semiorthogonal decomposition and how it helps to understand the structures of derived categories of surfaces with $p_g=q=0$. Finally we will give some examples of semiorthogonal decompostions of derived categories of surfaces with $p_g=q=0$.
A hedging is an action to protect oneself from losing or failing by a counter-balancing action. Several hedging strategies will be presented to help understand the concept of hedging. In reality, complexity arising from multi-dimensional risk factors will make hedging more challenging. The Black-Scholes partial differential equation will be briefly introduced as a primary example to perfectly hedge European options. And more Greeks will be discussed for a complete hedge.
자연과학동 E6-1, ROOM 1409
Discrete Math
Bryan Wilkinson (Aarhus University, Danmark)
Generalized Davenport-Schinzel Sequences: Regaining Linearity
자연과학동 E6-1, ROOM 1409
Discrete Math
We prove the linearity (of the lengths) of some generalized Davenport-Schinzel sequences. Standard Davenport-Schinzel sequences oforder 2 (avoiding abab) are linear, while those of order 3 (avoiding ababa) and higher can be superlinear. Our goal is to determine what pattern(s), in addition to ababa, must be forbidden to regain linearity. This work is motivated by an intriguing open problem: does the lower envelope of any set of degree 3 polynomials have linear complexity?
An important combinatorial result in equivariant cohomology and K-theorySchubert calculus is represented by the formulas of Billey, Graham and Willems for the localization of Schubert classes at torus fixed points. These formulas work uniformly in all Lie types, and are based on the concept of a root polynomial. In this talk we define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory). We focus on the case of the hyperbolic formal group law (corresponding to elliptic cohomology). We discuss its properties and applications. This is a report on the joint work with C. Lenart.
<9회> 11월 14일 김판기교수님 (서울대학교)
제목: Random walks and their limits
내용: Random walks 와 이의 극한으로 표현되는 브라운 운동에 대해 소개합니다. 브라운운동의 몇가지 성질과 브라운운동을 이용한 적분, 그리고 편미분 방정식과의 관계도 설명하고자 합니다.
장소: KAIST 수리과학과 E-6 3435호
시간: 12:00 ~ 13:15
점심식사: 김밥과 음료를 11시 45분부터 1000원에 제공
참석하고자 하시는 분은 사전 등록을 해주시면 감사하겠습니다^^
Oriented equivariant cohomology theories and the associated formal groups laws have been a subject of intensive investigations since 60's, mostly inspired by the theory of complex cobordism in topology. In the present talk we discuss several recent developments in the study of algebraic analogues of such theories, e.g. algebraic cobordism of Levine-Morel or algebraic elliptic cohomology, of projective homogeneous varieties. In particular, we address the problem of constructing the Schubert and the Bott-Samelson classes for such theories.
<8회> 11월 7일 오용근교수님 (IBS)
Title: Uncertainty principle, Hamiltonian dynamics and symplectic geometry
Abstract: One of the basic principle in the quantum mechanics is the Heisenberg uncertainty principle. This can be roughly stated as “one can not simultaneously locate the position of a particle and measure its momentum in the precise manner”. In the middle of 1960’s, two Russian mathematicians, V.I. Arnold and M. Gromov posed a question, called the non-squeezing question, which would be the classical analogue of the Heisenberg uncertainty principle. Formulation of the classical analogue involves the Hamiltonian formalism of the classcial mechanics, its solution requires deep knowledge of analysis of holomorphic curves, and its outcome gives rise to relatively new mathematics of symplectic topology. In this lecture, I will attempt to deliver a glimpse of the current status of symplectic topology.
장소: KAIST 수리과학과 E-6 3435호
시간: 12:00 ~ 13:15
점심식사: 샌드위치와 음료를 11시 45분부터 1000원에 제공
참석하고자 하시는 분은 아래 링크를 통해, 사전 등록을 해주시면 감사하겠습니다^^
http://goo.gl/wN3xU5
http://goo.gl/wN3xU5
오용근교수님 (IBS)
Title: Hamilton-Jacobi equation and continuous Hamiltonian dynamics
Abstract: In this talk, we will explain how a natural continuous solution to Hamilton-Jacobi equation (HJE) can be constructed by the Floer homology theory in the framework of continuous Hamiltonian dynamics. We will relate its initial value problem and boundary value problem to various constructions arising from Floer homology theory.
We will discuss filling and divergence functions. We will describe their behaviors for mapping class groups of surfaces and show that these functions exhibit phase transitions at the rank, in analogy to the corresponding result for symmetric spaces. This work is joint with Cornelia Drutu.
자연과학동 E6-1 Room 1409
Discrete Math
정지윤 (NIMS, 국가수리과학연구소)
The topology of restricted partition posets
자연과학동 E6-1 Room 1409
Discrete Math
For each composition c⃗ we show that the order complex of the poset of pointed set partitions Π∙c⃗ is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition c⃗ . Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module SB where B is a border strip associated to the composition. We also study the filter of pointed set partitions generated by a knapsack integer partition and show the analogous results on homotopy type and action on the top homology.