Department Seminars & Colloquia
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E6-1, ROOM 1409
Discrete Math
Paul Wollan (University of Rome, Rome, Italy)
Solving the half-integral disjoint paths problem in highly connected digraphs
The k disjoint paths problem, which was shown to be efficiently solvable for fixed k in undirected graphs in a breakthrough result by Robertson and Seymour, is notoriously difficult in directed graphs. In directed graphs, he problem is NP-complete even in the case when k=2. In an attempt to make the problem more tractable, Thomassen conjectured that if a digraph G were sufficiently strongly connected, then every k disjoint paths problem in G would be feasible. He later answered his conjecture in the negative, showing that the problem remains NP-complete when k=2, even when we assume that the graph is arbitrarily highly connected.
We consider the following further relaxation of the problem: a half-integral solution to a k disjoint paths problem is a set of paths linking the desired vertices such that each vertex of the graph is in at most two of the paths. We will present the new result that the half-integral k disjoint paths problem can be efficiently solved (even when the parameter k is included as part of the input!) if we assume the graph is highly connected.
This is joint work with Irene Muzi and Katherine Edwards.
E6-1, ROOM 1409
Discrete Math
O-joung Kwon (Technische Universitat Berlin, Berin, Germany)
Generalized feedback vertex set problems on bounded-treewidth graphs
It has long been known that Feedback Vertex Set can be solved in time 2^O(w log w) n^O(1) on graphs of treewidth w, but it was only recently that this running time was improved to 2^O(w) n^O(1), that is, to single-exponential parameterized by treewidth. We consider a natural generalization of this problem, which asks given a graph G on n vertices and positive integers k and d, whether there is a set S of at most k vertices of G such that each block of G-S has at most d vertices. The central question of this talk is: “Can we obtain an algorithm that runs in single-exponential time parameterized by treewidth, for every fixed d?” The answer is negative. But then, one may be curious which properties of Feedback Vertex Set that make it allow to have a single-exponential algorithm. To answer this question, we add an additional condition in the general problem, and provide a dichotomy result.
Formally, for a class
A quantum walk is a (rather imperfect analog) of a random walk on a graph.
They can be viewed as gadgets that might play a role in quantum computers, and have been
used to produce algorithms that outperform corresponding classical procedures. Physical
questions about these walks lead to problems in spectral graph theory, and they also provide
interesting new graph invariants. In my talk I will present some of the background,
and some of the many open problems that they have given rise to.
BRST method was introduced in physics to quantize classical field theories with gauge symmetries, to make sure these symmetries are preserved even after quantization procedures. This requires some preparation at the classical level, such as introducing ghosts and antifields. These can be formulated in precise mathematical terms, using Koszul-Tate chain complex that incorporates antifields, and BRST cochain complex that introduces ghosts. If BRST complex can be constructed, we can find a solution to the classical master equation that extends the original Lagrangian with antifields and ghosts.
E6-1, ROOM 1409
Discrete Math
David Roberson (University College London)
Homomorphisms of Strongly Regular Graphs
A homomorphism is an adjacency preserving map between the vertex sets of two graphs. A n-vertex, k-regular graph is strongly regular, with parameters (n,k,λ, μ), if there exist numbers λ and μ such that every pair of adjacent vertices share λ common neighbors and every pair of non-adjacent vertices share μ common neighbors. A strongly regular graph is primitive if neither it nor its complement is a disjoint union of complete graphs. We prove that if G and H are primitive strongly regular graphs with the same parameters and φ is a homomorphism from G to H, then φ is either an isomorphism or a coloring (homomorphism to a complete subgraph). Moreover, any such coloring is optimal for G and its image is a maximum clique of H. Therefore, the only endomorphisms of a primitive strongly regular graph are automorphisms or colorings. This confirms and strengthens a conjecture of Peter Cameron and Priscila Kazanidis that all strongly regular graphs are cores or have complete cores. The proof of the result is elementary, mainly relying on linear algebraic techniques.
Describing the eigenvalue distribution of the sum of two general
Hermitian matrices is basic question going back to Weyl. If the matrices
have high dimensionality and are in general position in the sense that one
of them is conjugated by a random Haar unitary matrix, the eigenvalue
distribution of the sum is given by the free additive convolution of the
respective spectral distributions. This result was obtained by Voiculescu on
the macroscopic scale. In this talk, I show that it holds on the microscopic
scale all the way down to the eigenvalue spacing with an optimal error bound.
Joint work with Zhigang Bao and Laszlo Erdos.
Describing the eigenvalue distribution of the sum of two general
Hermitian matrices is basic question going back to Weyl. If the matrices
have high dimensionality and are in general position in the sense that one
of them is conjugated by a random Haar unitary matrix, the eigenvalue
distribution of the sum is given by the free additive convolution of the
respective spectral distributions. This result was obtained by Voiculescu on
the macroscopic scale. In this talk, I show that it holds on the microscopic
scale all the way down to the eigenvalue spacing with an optimal error bound.
Joint work with Zhigang Bao and Laszlo Erdos.
In this talk I will start by describing a certain subset of real numbers which contain all the numbers which are of interest to arithmeticians. These numbers are called periods, and they form a countable set and in principle are much more easy to handle than a general real number. Then I will specialize to a certain subset of periods which are of mixed Tate type. The ones which have good reduction for all primes have periods called multi-zeta values, which were first defined by Euler. The remainder of the talk will be about a p-adic version of these numbers called p-adic multi-zeta values.
자연과학동 1409호
PDE Seminar
문병수 (인천대학교)
Traveling Wave Solutions to the Burgers-\\alpha \\beta equations
In this talk, the Burgers-alphabeta equation, which was first introduced by Holm and Staley, is considered in the special case where nu= 0 and b = 3. Traveling wave solutions are classified to the Burgers-alphabeta equation containing
four parameters b, alpha, nu, and beta, which is a nonintegrable nonlinear partial differential equation that
coincides with the usual Burgers equation and viscous b-family of peakon equation, respectively, for two specific
choices of the parameter beta = 0 and beta = 1. Under the decay condition, it is shown that there are smooth,
peaked and cusped traveling wave solutions of the Burgers-alphabeta equation with nu= 0 and b = 3 depending on
the parameter beta. Moreover, all traveling wave solutions without the decay condition are parametrized by the
integration constant k1 in R. In an appropriate limit beta= 1, the previously known traveling wave solutions of
the Degasperis–Procesi equation are recovered.
A reaction network is the system in which various reactions occur between several physical, chemical or biological species. To investigate the dynamics of the reaction networks such as genetic networks, metabolic networks and epidemic compartment systems, researchers use mathematical models based on the principle of mass-action kinetics or other kinetics. In this talk, we present the ways of mathematical modeling and computational methods for describing the time-evolution of the reaction networks. We show some simulation results obtained by applying the methods to motivating examples including epidemic models such as H1N1 influenza and foot-and-mouse disease spread.
Helly's theorem, a classical result in discrete geometry, asserts that if n>d convex subsets of R^d have empty intersection, some d+1 of them must already have empty intersection. I will discuss some topological generalizations of Helly's theorem, where convexity is replaced by connectivity assumptions on the nonempty intersections, that lead to non-embeddability results of Borsuk-Ulam type and to variations on Leray's acyclic cover theorem (or the Nerve theorem).
Part 2 (Monday): The Nerve theorem.
MUltiple SIgnal Classification (MUSIC) is a famous non-iterative algorithm for detecting various kind of anomaly in inverse scattering problem. Recently, it was applied to the various problems, for example, detection of antipersonnel mines buried in the ground, location search for small inclusions, internal corrosion in pipelines, and shape reconstruction of arbitrary shaped thin inclusions, cracks, and extended targets, etc. Throughout these results, it has been confirmed that MUSIC is fast, stable, and effective algorithm but some phenomena cannot be explained in the traditional approach. For example, the unexpected appearance of some artifacts or two curves along the boundary of targets instead of the true shape, an image with poor resolution with small number of incident/observation directions, and so on. In this presentation, basic concept of MUSIC algorithm is surveyed and a relationship between Bessel function of integer order of the first kind is introduced. This relationship will explain various phenomena that mentioned previously. Time permitting, a condition of incident/observation direction will be considered for a successful shape detection of small or arc-like cracks in limited-view inverse scattering problem.
Helly's theorem, a classical result in discrete geometry, asserts that if n>d convex subsets of R^d have empty intersection, some d+1 of them must already have empty intersection. I will discuss some topological generalizations of Helly's theorem, where convexity is replaced by connectivity assumptions on the nonempty intersections, that lead to non-embeddability results of Borsuk-Ulam type and to variations on Leray's acyclic cover theorem (or the Nerve theorem).
Synchronization of oscillators denotes a phenomenon for the adjustment of rhythms among weakly coupled oscillators, andis one of the collective modes appearing in oscillatory complex systems such as ensembles of Josepson junctions, pacemaker cells and fireflies etc. In this talk, I will briefly discuss some challenging mathematical problems for synchronization via the Kuramoto and Lohe models, andpresent a survey of mathematical developments for these models in recent years.
The fact that the complete graph K5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that Kn embeds in a closed surface M if and only if (n − 3)(n − 4) ≤ 6b1(M), where b1(M) is the first Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of Kn+1 embeds in R2k if and only if n ≤ 2k + 1.
I will discuss a conjecture of Kuhnel that generalizes both the Heawood inequality and the van Kampen-Flores theorem, and present some partial results toward this conjecture.
Agenda
17:00~17:20 Mathematica Campus License 사용방법 및 Tip
강사 : 황지원 차장 (다한테크&Wolfram Certified Instructor)
17:20~17:50 Mathematica New Feature
강사 : Farid Pasha (International Business Manager, Wolfram Research Inc.)
17:50~18:00 Break
18:00~18:50 Mathematica Technology – Graphics & Machine Learning
강사 : 정재범 (Kernel개발자, Wolfram Research Inc.)
Mathematica의 개발사인 Wolfram Research Inc에서 Graphics Visualization 관련 Kernel 개발자인
한국인 박사를 모시고, 현재 최신 기술의 동향과 이와 관련된 Mathematica 기능을 볼 수 있는 기회입니다.
특히 Geometry와 3D Printing에 대해 자세한 설명을 들어볼 수 있으며, 최근 제일 관심을 끌고 있는
Machine Learning에 대한 Mathematica의 예제들을 살펴볼 수 있습니다.
또한 이번 세미나에서는 KAIST에서 보유하고 있는 Mathematica의 사용방법과 안내를 받아볼 수 있으니,
학생들을 비롯하여 많은 교수님들의 관심 부탁 드립니다.
The transition density (if it exists) of Markov process is the heat kernel of the generator of the process. The transition density of a general Markov process rarely admits an explicit expression. Thus obtaining sharp estimates on transition density is a fundamental problem both in probability theory and in analysis. In this talk, we discuss the behavior of transition density (Dirichlet heat kernel) for subordinate Brownian motions in $C^{1,1}$-open subsets whose scaling orders are not necessarily strictly less than $2$. Our estimate is sharp and explicitly expressed in terms of the distance to the boundary, Laplace exponent of subordinator and its derivative. This is a joint work with Ante Mimica.
We consider a parabolic-parabolic equations which describes some chemotactic model with advection and absorbing reaction. We establish the local and global well-posedness of regular solutions for the model. We also prove that the total mass of the cell density asymptotically approaches a strictly positive constant, provided that efficiency of absorbing reaction is strong enough. This talk is based on the joint work with Jaewook Ahn(Kyushu U.), Kyunkeun Kang(Yonsei U.) and Junha Kim(Chung-Ang U.).
This talk is concerned with solving some structured multi-linear systems, especially focusing on the equations whose coefficient tensors are M-tensors, or called M-equations for short. We prove that a nonsingular M-equation with a positive right-hand side always has a unique positive solution. Several iterative algorithms are proposed for solving multi-linear nonsingular M-equations, generalizing the classical iterative methods and the Newton method for linear systems. Furthermore, we apply the M-equations to some nonlinear differential equations and the inverse iteration for spectral radii of nonnegative tensors.
In statistical methods for language and document modeling, there are
two major perspectives: representation at the document level, and
representation at the word level. At the document level, topic models
such as latent Dirichlet allocation (LDA) and hierarchical Dirichlet
process (HDP), based on the word-document matrix, aim to discover
topics whose dimensionality is much lower than the size of the
vocabulary. At the word-level, language models such as n-grams and
neural word embedding, based on the word co-occurrence matrix, aim to
represent each word in a high-dimensional vector space. In this work,
we develop Dual Representation Topic Model (DRTM), a novel topic model
which combines the advantages of the two approaches. DRTM models
documents and words based on the locations of the individual words
within documents, as well as the local contexts of the words. DRTM
transforms each document into a network of words by generating edges
when words of near proximity have high semantic similarity. Then it
infers the topic for each edge - a pair of words - rather than
assigning topics for individual words as in traditional topic models.
This enables the model to learn a better document representation by
inferring the global topics while considering the local contexts of
individual words.
- Speaker: Professor Lawrence Ein
(LAS Distinguished Professor, University of Illinois at Chicago)
- Title: Secant varieties and asymptotic syzygies
- Abstract: We would a give a simple proof for the theorems of Ran and Behesthi - Eisenbud on giving an upper bound of on the dimensions of the higher secant varieties, which will generalize to curves that are not just lines.
In the second half, I would like to discuss joint work with Erman and Lazarsfeld on giving a simple proof for a non-vanishing theorem for the syzygies of the Veronese embedding of the projective space.
E6-1, ROOM 1409
Discrete Math
Cyril Nicaud (Université Paris-Est, France)
Introduction to analytic combinatorics
In classical combinatorics, sequences of positive integers are usually studied through their generating series. These formal power series can be used to classify the sequences, to obtain closed formulas for the number of object of a given size, …
Seeing the generating series as analytic functions, we can use tools of complex analysis (such as the residue theorem) to obtain, typically, an asymptotic equivalent to the sequence under consideration.
In this talk I will give a quick overview of the main results obtained in this field, from the automatic construction of generating series to some theorems coming from the theory of functions of a complex variable.
The talk will not assume any specific knowledge in combinatorics or complex analysis.
E6-1, 1409
Discrete Math
Doowon Koh (Chungbuk National University)
Introduction of the finite field Erdős distance problem
The purpose of this talk is to study the finite field analog of the Erdős distance problem. First, the conjecture and known results on the problem are reviewed. Second, we introduce basic skills to deduce results on the problem. Finally, we address how to improve currently known results on the Erdős distance problem.
Sponsored by KAIST BK21 Plus.
2000년 초부터 저금리기조가 유지되면서, 주가연계증권(ELS), 파생결합증권(DLS), 구조화 채권(Structured Notes) 등의 구조화상품으로 대변되는 금융혁신은 급격한 양적 성장을 이루었다. 그러나 이러한 금융의 혁신이 금융기관의 성과 및 금융안정에 미치는 영향에 대해서는 거의 분석이 이루어지지 못하였다. 물론, 구본일, 엄영호, 지현준(2007), 엄영호(2015), 강병진(2016) 등의 몇몇 연구에서 구조화 파생상품의 가격적정성 및 기초자산에 미치는 영향, 그리고 구조화상품 투자에 따른 투자자의 효용에 대한 분석은 이루어졌으나, 구조화 상품 발행에 따라 금융기관의 경영 안정성에 미친 영향에 대해 국내의 연구는 전무한 현실이다. 해외의 사례에서도 Breuer and Perst(2007), Branger and Breuer(2008), Henderson and Pearson(2011) 등의 연구들이 구조화상품의 발행가격의 적정성 및 가격결정에 미치는 영향 등에 대한 분석은 수행하였으나 소비자를 대상으로 하는 금융혁신(파생결합증권)의 금융기관 안정성에 미치는 영향에 대한 분석은 매우 제한되어 있다. 다만, 2007-2008년 글로벌 금융위기를 겪은 후, 금융기관 사이에서 이루어진 복잡한 금융혁신(파생결합계약)이 위기를 증폭시켰다는 이론적 및 실증적 분석을 제시하였으나, 국내의 상황과는 큰 차이가 있다. 해외 시장에서 이루어지는 금융혁신이 대부분 금융기관 간 거래에서 발생하는 반면, 국내에서는 대부분의 파생결합증권이 소비자 대상 판매상품인 경우가 많기 때문에, 해외시장의 사례가 국내 금융환경에 그대로 적용하는데 무리가 따른다. 본 연구는 이러한 이유로 파생결합증권의 발행이 증권회사의 건전성에 미치는 영향에 대한 실증분석을 수행하는 것을 목적으로 한다. 파생결합증권을 발행한 증권회사의 손실 메카니즘을 분석하고 2015년 및 2016년 헤지손실을 기록했던 증권회사의 사례를 소개한다. 이를 바탕으로 ELS의 만기가 도래하는 2018년 증권사의 유동성 위기에 대한 가능성을 예상한다. 다음으로 증권회사의 건전성 규제에 대해 소개하고 파생결합증권 발행에 따른 건전성 규제에 미치는 영향을 살펴본다.
The purpose of this work is to mathematically justify the phenomena of the plasma sheath formation near the surface of a ball-shaped material immersed in a bulk plasma, and to get some qualitative information of such a boundary sheath layer. To this end, we employ the Euler-Poisson equations (both the stationary and nonstationary models) in the three dimensional annular domain to investigate the existence, time-asymptotic behavior and quasi-neutral limit of the boundary layer solutions. This is a joint work with C.-Y. Jung (UNIST) and M. Suzuki (Nogoya Inst. Tech.).
Stochastic partial differential equations (SPDEs) are differential equations which include the effects of random forces and environments. The theory of SPDE was started in 1970s, and L_p-theory of SPDE was first introduced by Krylov in 1994.
Since then, L_p-theory has become one of main approaches to study the regularity of solutions to SPDEs.
In this talk, I will give a short description on SPDEs and introduce classical L_p-theory of second-order SPDEs.
I will also introduce recent results on SPDEs having non-local operators.
Abstract
A reaction network is the system in which various reactions occur between several physical, chemical or biological species. To investigate the dynamics of the reaction networks such as genetic networks, metabolic networks and epidemic compartment systems, researchers use mathematical models based on the principle of mass-action kinetics or other kinetics. In this talk, we present the ways of mathematical modeling and computational methods for describing the time-evolution of the reaction networks. We show some simulation results obtained by applying the methods to motivating examples including epidemic models such as H1N1 influenza and foot-and-mouse disease spread.
Black holes are perhaps the most celebrated predictions of general relativity. Miraculously, these complicated spacetimes arise as explicit (i.e., exact expression can be written down!) solutions to the vacuum Einstein equation. Looking these explicit black hole solutions leads to an intriguing observation: While the black hole exterior look qualitatively similar for every realistic black hole, the structure of the interior, in particular the nature of the `singularity' inside the black hole, changes drastically depending on whether the black hole is spinning (Kerr) or not (Schwarzschild).
A proposed picture for what happens in general is the so-called strong cosmic censorship conjecture of R. Penrose, which is a central conjecture in general relativity. In this colloquium, I will give a short introduction to general relativity and explain what this conjecture is. Time permitting, I will present some recent progress (joint work with J. Luk at Stanford) on related topics, using tools from nonlinear hyperbolic PDEs.
VOD 보기
We introduce graded manifold as an ordinary smooth manifold with Z/2Z - graded functions, that are nothing but the sections of the exterior bundle of a vector bundle. We extend ordinary differential calculus to this setting, and introduce a graded extended version of the variational bicomplex and Lagrangian field theory, that can incorporate the idea of fermions in physics.
In the inhomogeneous heat equation
ut(t; x) = u(t; x) + f(t; x);
The term f models the interruptions to the heat diusion along time and on space locations.
Especially, the eect of f in time direction is more troublesome and the regularity of u is subject
to the regularity of f. In this talk we discuss a type of modeling of f in the form Nt(!)g(t; x),
where N is a random noise process which can be whiter than the white noise. We also discuss
a regularity relation between Ng and u.