Department Seminars & Colloquia
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We consider the incompressible Navier-Stokes equations ((NS) in short) in $mathbb{R}^{3}$.
We first present basic properties of (NS):
(1) symmetry groups of (NS)
(2) vorticity formulation of (NS)
(3) some dissipative quantities
(4) notion of solutions
We then introduce the Littlewood-Paley theory and Besov spaces. We also provide some useful lemmas to deal with nonlinear terms in (NS).
We finally show how to obtain global-in-time solutions of (NS) by taking small initial data in scaling invariant spaces.
We consider the incompressible Navier-Stokes equations ((NS) in short) in $mathbb{R}^{3}$.
We first present basic properties of (NS):
(1) symmetry groups of (NS)
(2) vorticity formulation of (NS)
(3) some dissipative quantities
(4) notion of solutions
We then introduce the Littlewood-Paley theory and Besov spaces. We also provide some useful lemmas to deal with nonlinear terms in (NS).
We finally show how to obtain global-in-time solutions of (NS) by taking small initial data in scaling invariant spaces.
We consider the incompressible Navier-Stokes equations ((NS) in short) in $mathbb{R}^{3}$.
We first present basic properties of (NS):
(1) symmetry groups of (NS)
(2) vorticity formulation of (NS)
(3) some dissipative quantities
(4) notion of solutions
We then introduce the Littlewood-Paley theory and Besov spaces. We also provide some useful lemmas to deal with nonlinear terms in (NS).
We finally show how to obtain global-in-time solutions of (NS) by taking small initial data in scaling invariant spaces.
In the talk, I discuss previous works on the arithmetic of various twisted special $L$-values and dynamical phenomena behind them. Main emphasis will be put on the problem of estimating several exponential sums such as Kloosterman sums and its relation to the problem of non-vanishing of special $L$-values with cyclotomic twists. A distribution of homological cycles on the modular curves will also be discussed and as a consequence, some results on a conjecture of Mazur-Rubin-Stein about the distribution of period integrals of elliptic modular forms will be presented.
We show that the bipolar filtration of the smooth concordance group of topologically slice knots introduced by Cochran, Harvey and Horn has nontrivial graded quotients at every stage. To detect a nontrivial element in the quotient, the proof uses Cheeger-Gromov $L^2$ $rho$-invariants and infinitely many Heegaard Floer correction term invariants simultaneously. This is joint work with Jae Choon Cha.
The standard problem of optical tomography is to obtain information about the optical parameters inside of an object by making optical measurements on the boundary. Acousto-optic tomography is a variation of this problem where the object is perturbed by an acoustic field, and optical boundary measurements are taken as the parameters of the acoustic field vary. In this talk I will give a short introduction to the idea of acousto-optic tomography, and discuss some inverse problems that arise from this imaging technique. In particular, I will describe some recent results for inverse problems derived from radiative transport models of acousto-optic tomography. This is joint work with John Schotland and Guillaume Bal.
We discuss the asymptotic behavior, at a small viscosity, of solutions to some fluid equations related to the Navier-Stokes equations. The model equations, we consider in this talk, are either supplemented with the Navier-slip type boundary conditions, or simplified under some special symmetries or linearized when the no-slip boundary condition is imposed. By explicitly constructing the boundary layer correctors, which approximate the differences between the viscous and inviscid solutions, we validate the smallness of our asymptotic expansions with respect to the viscosity parameter, and prove the vanishing viscosity limit with the optimal rates of convergence.
The aim is to introduce recent progress on partial regularity problem of the Navier-Stokes equations.
They include an improved regularity criteria and the size of possible singularities in terms of the Minkowski dimension.
We also discuss about the regularity and singularity properties of the weak solutions to the Navier-Stokes equations which belong to some weak Lebesgue spaces.
The motivation for the talk is the recent result, joint with Vincent Guirardel and Camille Horbez, that Out(F_n) admits a topologically amenable action on a Cantor set. This implies the Novikov conjecture for Out(F_n) and its subgroups. Most of the talk will be an introduction to boundary amenability and ways to prove it for simpler groups.
2017 제3회 정오의 수학산책
강연자: 이수준(경희대)
일시: 2017년 5월 12일(금) 12:00 ~ 13:15
장소: 카이스트 자연과학동 E6-1 3435호
제목: 양자정보이론 소개 : 디지털정보 vs 양자정보
내용: 이 강연에서는, 디지털정보의 데이터 압축, 오류 보정에 대한 C. E. Shannon의 두 가지 중요한 정보이론의 결과를 설명하고, 이와 함께 양자정보에 대한 정보이론과의 근본적인 차이점에 대해서 소개하고자 한다.
등록: 2017년 5월 10일(수) 오후 3시까지
https://goo.gl/forms/TnDIENKzVJAnxy9v2
The motivation for the talk is the recent result, joint with Vincent Guirardel and Camille Horbez, that Out(F_n) admits a topologically amenable action on a Cantor set. This implies the Novikov conjecture for Out(F_n) and its subgroups. Most of the talk will be an introduction to boundary amenability and ways to prove it for simpler groups.
This talk will review previous work on quadrupedal gaits and recent work on a generalized model for binocular rivalry proposed by Hugh Wilson. Both applications show how rigid phase-shift synchrony in periodic solutions of coupled systems of differential equations can help understand high level collective behavior in the nervous system.
Probabilistic methods in geometric group theory have recently gained in importance. In the course I will focus on the setting where a countable group G acts on a (possibly nonproper) Gromov hyperbolic space X. Examples include a mapping class group acting on the associated curve complex, or Out(F_n) acting on the complex of free factors, or a group acting on the contact graph of a CAT(0) cube complex it acts on. Under minor assumptions, Maher-Tiozzo show that a random walk on G, projected to X, almost surely converges to a point in the Gromov boundary of X. I will discuss the proof of this theorem. As an application, we will see that "generic" elements of mapping class groups are pseudo-Anosov, and (following Horbez) we will give a random walk proof of the classical theorem of Ivanov classifying subgroups of mapping class groups.
We will give a topological generalization of the planar (p,q) theorem due to Alon and Kleitman. In particular we will show that the assertion of the (p,q) theorem holds for families of open connected sets in the plane under the hypothesis that the intersection of any subfamily is empty or connected. The proof is based on a surprising connection between nerve complexes and complete minors in graphs. This is join work with Minki Kim and Seunghun Lee.
Probabilistic methods in geometric group theory have recently gained in importance. In the course I will focus on the setting where a countable group G acts on a (possibly nonproper) Gromov hyperbolic space X. Examples include a mapping class group acting on the associated curve complex, or Out(F_n) acting on the complex of free factors, or a group acting on the contact graph of a CAT(0) cube complex it acts on. Under minor assumptions, Maher-Tiozzo show that a random walk on G, projected to X, almost surely converges to a point in the Gromov boundary of X. I will discuss the proof of this theorem. As an application, we will see that "generic" elements of mapping class groups are pseudo-Anosov, and (following Horbez) we will give a random walk proof of the classical theorem of Ivanov classifying subgroups of mapping class groups.
Probabilistic methods in geometric group theory have recently gained in importance. In the course I will focus on the setting where a countable group G acts on a (possibly nonproper) Gromov hyperbolic space X. Examples include a mapping class group acting on the associated curve complex, or Out(F_n) acting on the complex of free factors, or a group acting on the contact graph of a CAT(0) cube complex it acts on. Under minor assumptions, Maher-Tiozzo show that a random walk on G, projected to X, almost surely converges to a point in the Gromov boundary of X. I will discuss the proof of this theorem. As an application, we will see that "generic" elements of mapping class groups are pseudo-Anosov, and (following Horbez) we will give a random walk proof of the classical theorem of Ivanov classifying subgroups of mapping class groups.
We consider non-topological solutions of a nonlinear elliptic system problem derived from the SU(3) Chern-Simons models.
The existence of non-topological solutions even for radial symmetric case has been a long standing open problem.
Recently, Choe, Kim, and Lin showed the existence of radial symmetric non-topological solution when the vortex points collapse.
However, the arguments in that paper cannot work for an arbitrary configuration of vortex points.
In this talk, I introduce a new approach by using different scalings for different components of the system to construct a family of partial blowing up non-topological solutions.
This talk is based on the joint work with Prof. Chang-Shou Lin and Prof. Ting-Jung Kuo.
In this talk we discuss the generation of interface property for solutions of the nonlocal Allen-Cahn equation which was proposed by Rubinstein and Sternberg as a model for phase separation in a binary mixture. More precisely,
we show that given an arbitrarily initial condition, the solution approaches a step function and hence develops a steep transition layer (interface) within a very short time. Because of the nonlocal term, some PDE tools such as comparison principle cannot be applied so that we have to introduce new method to overcome these diculties. Furthermore, in some cases, we obtain a sharp estimate for the thickness of interface.
This is joint work with Danielle Hilhorst, Hiroshi Matano and Hendrik Weber.
2017 제2회 정오의 수학산책
강연자: 이윤원 (인하대)
일시: 2017년 4월 28일(금) 12:00 ~ 13:15
장소: 카이스트 자연과학동 E6-1 3435호
제목: Atiyah-Singer index theorem
내용: The Atiyah-Singer Index Theorem appeared in 1960’s, which is one of great mathematical achievements in 20th century. It is a far reaching generalization of the Gauss-Bonnet-Chern Theorem for the Euler characteristic, the Hirzebruch Signature Theorem for the Signature of 4k-dimensional compact manifold, the Riemann-Roch-Hirzebruch Theorem for the Arithmetic Genus. Hence to understand this magnificent theorem, we need to investigate how the Euler characteristic, Signature and Arithmetic Genus are expressed by the Fredholm Indices for some appropriate geometric operators. In this talk, I will explain briefly the historical background of the Index Theorem, the Fredholm Indices of elliptic operators and discuss how the Index Theorem was motivated from the above classical celebrated theorems. And then, I will go through very briefly the proof of the Index Theorem by using the heat kernel method. If time permits, I will explain the Index Theorem on a compact manifold with boundary, where the eta-invariant appears as a boundary correction term.
등록: 2017년 4월 26일(수) 오후 3시까지
https://goo.gl/forms/0lqSboiW5A9vv9rd2
Let X be a smooth projective rational surface over an algebraically closed field. One version of a long standing conjecture asks whether the set of self-intersections of reduced curves is bounded below. This question can be reduced to studying singular plane curves. One approach is to measure how singular the curve is using a recently introduced quantity called an H-constant. This has raised some new open problems I will discuss.
Waldschmidt constants and Seshadri constants are asymptotic measure of effectivity and nefness related to plane curves. Evaluating them is related to the occurrence of curves of negative self-intersection on blow ups of the plane. I will discuss recent work on computing and bounding these quantities.
E6-1, ROOM 1409
Discrete Math
Brendan Rooney (KAIST)
Eigenpolytopes, Equitable Partitions, and EKR-type Theorems
E6-1, ROOM 1409
Discrete Math
The Erdos-Ko-Rado Theorem is a classic result about intersecting families of sets. More recently, analogous “EKR-type” type theorems have been developed for other types of objects. For example, non-trivially intersecting vector spaces, and overlapping strings. In this seminar we will give a proof of the EKR Theorem for permutations in Sn due to Godsil and Meagher. Along the way we will see some useful tools from algebraic graph theory. Namely, a bound on the maximum size of an independent set in a graph, equitable partitions, and eigenpolytopes.
In this talk, I will introduce the distribution of eigenvalues of random normal matrix ensembles. Specifically, I will discuss the existence and universality of scaling limits for the eigenvalues at a bulk singularity, an isolated point in the interior at which the equilibrium density vanishes. I will describe how to find a suitable scale and how the rescaled ward’s identity can be used to prove the universality. This is joint work with Yacin Ameur.
E6-1, ROOM 1409
Discrete Math
Dieter Spreen (Universität Siegen, Siegen, Germany)
Bi-Topological Spaces and the Continuity Problem
E6-1, ROOM 1409
Discrete Math
The continuity problem is the question when effective (or Markov computable) maps between effectively given topological spaces are effectively continuous. It will be shown that this is always the case if the the range of the map is effectively bi-regular. As will be shown, such spaces appear quite naturally in the context of the problem.
Evolutionary games of cyclic competitions have been extensively studied
to gain insights into one of the most fundamental phenomena in nature:
biodiversity that seems to be excluded by the principle of natural selection.
The Rock-Paper-Scissors (RPS) game of three species and
its extensions [e.g., the Rock-Paper-Scissors-Lizard-Spock (RPSLS) game] are
paradigmatic models in this field.
In all previous studies, the intrinsic symmetry associated with the cyclic competitions imposes
a limitation on the resulting coexistence states, leading to only selective types of such states.
We investigate the effect of nonuniform intraspecific competitions on coexistence and
find that a wider spectrum of coexistence states can emerge and persist.
This surprising finding is substantiated using three classes of cyclic game models through stability analysis,
Monte Carlo simulations and patterns of continuous spatiotemporal dynamical evolution.
Our finding indicates that intraspecific competitions or alternative symmetry-breaking mechanisms
can promote biodiversity to a broader extent than previously thought.
Recent development of high throughput sequencing and CRISPR genome editing technology have brought great advances in the understanding of molecular mechanisms underlying diseases. We studied infertility in Drosophila model: deep sequencing analyses of small RNA, mRNA, and RNA immunoprecipitation sequencing (RIP-seq) showed that a conserved mRNA export factor, Thoc5, represses the expression of transposable elements, which are deleterious mobile genetic elements that cause genome instability and germ cell death. We also found that Thoc5 binds nascent RNAs and facilitates biogenesis of small RNAs, which in turn regulated transposable elements. Furthermore, we used CRISPR-Cpf1, a recently found class 2/type V CRISPR RNA-guided endonuclease that is distinct from the common CRISPR-Cas9, to generate mutant mice. Whole genome analyses of the mutant mice showed precise alteration of DNA sequence at the target genomic locus with no detectable off-target mutation. We generated a cell line with a single base change in Isocitrate Dehydrogenase I (IDH1), a gain-of-function mutation widely found in glioma patients. We further applied CRISPR genome editing to pathogenic bacteria to generate bacterial strains with mutations in genes required for quorum sensing, which is an intercellular communication system for detection cell density. The quorum sensing mutant strains showed reduced biofilm formation, and changes in expression profile of genes in the metabolic pathway.
Among the most well-known examples of L-functions are the Riemann zeta
function and the L-functions associated to classical modular forms.
Less well known, but equally important, are the L-functions associated
to Maass forms, which are eigenfunctions of the Laplace-Beltrami
operator on a hyperbolic surface. Named after H. Maass, who discovered
some examples in the 1940s, Maass forms remain largely mysterious.
Fortunately, there are concrete tools to study Maass forms: trace
formulas, which relate the spectrum of the Laplace operator on a
hyperbolic surface to its geometry. After Selberg introduced his
famous trace formula in 1956, his ideas were generalised, and various
trace formulas have been constructed and studied. However, there are
few numerical results from trace formulas, the main obstacle being
their complexity. Various types of trace formulas are investigated,
constructed and used to understand automorphic representations and
their L-functions from a theoretical point of view, but most are not
explicit enough to implement in computer code.
Having explicit computations of trace formulas makes many potential
applications accessible. In this talk, I will explain the
computational aspects of the Selberg trace formula for GL(2) for
general levels and applications towards the Selberg eigenvalue
conjecture and classification of 2-dimensional Artin representations
of small conductor.
This is a joint work with Andrew Booker and Andreas Strömbergsson.
E6-1, ROOM 1409
Discrete Math
Otfried Cheong (School of Computing, KAIST)
Putting your coin collection on a shelf
E6-1, ROOM 1409
Discrete Math
Imagine you want to present your collection of n coins on a shelf, taking as little space as possible – how should you arrange the coins?
More precisely, we are given n circular disks of different radii, and we want to place them in the plane so that they touch the x-axis from above, such that no two disks overlap. The goal is to minimize the length of the range from the leftmost point on a disk to the rightmost point on a disk.
On this seemingly innocent problem we will meet a wide range of algorithmic concepts: An efficient algorithm for a special case, an NP-hardness proof, an approximation algorithm with a guaranteed approximation factor, APX-hardness, and a quasi-polynomial time approximation scheme.
2017 제1회 정오의 수학산책
강연자: 한종규 (서울대)
일시: 2017년 3월 31일(금) 12:00 ~ 13:15
장소: 카이스트 자연과학동 E6-1 3435호
제목: Symmetry, invariants and conservation laws
내용: The notion of symmetry plays a central role in understanding natural laws and in solving equations. To be symmetric means to be invariant under a group action. In this lecture we are mainly concerned with continuous groups of the symmetries of differential equations. I will explain Sophus Lie's ideas on solvability of an ordinary differential equation in terms of its symmetry group and Emmy Noether's theorem on conservation laws for variational problems. As time permits I will present other viewpoints on the conservation laws.
등록: 2017년 3월 29일(수) 오후 3시까지
https://goo.gl/forms/KllQvcZnFjp57sOk1
재미로 풀어보는 퀴즈에나 등장할 법한 추상적인 수학적 개념이 기계공학(예, 응용역학) 연구에 도움을 줄 수 있을까? 수학과 역학 사이의 간극이 가장 좁았던 때는 언제였고, 수학과 역학이 만나는 지점에서 두 학문을 두루 섭렵했던 수리과학자는 누구였을까? 이와 같은 질문에 대한 답변의 일환으로, 본 발표의 전반부에서는 수학과 역학(유체역학, 고체역학, 열역학, 파동학)의 역사가 공존했던 시절을 인물 중심으로 살펴보고자 한다. 본 발표의 후반부에서는, 역학적 파동과 메타물질에 관한 발표자의 연구주제(음향 투명망토, 음향 블랙홀, 생물음향학 등)를 간략하게 소개한다.
There is a classical result first due to Keen known as the collar lemma for hyperbolic surfaces. A consequence of the collar lemma is that if two closed curves A and B on a closed orientable hyperbolizable surface have non-zero geometric intersection number, then there is an explicit lower bound for the length of A in terms of the length of B, which holds for any hyperbolic structure on the surface. By slightly weakening this lower bound, we generalize this statement to hold for all Hitchin representations. This is a joint work with Tengren Zhang.
The Siegel series is the local factor of the Fourier coefficient of the Siegel-Eisenstein series. It is also a crucial ingredient in Kudla's program to compare it with intersection numbers.
In this talk, I will explain a conceptual reformulation of the Siegel series. As the first application, I will explain a conceptual (and simple) proof of the equality between intersection number and the (derivative of) Siegel series. As the second application, I will explain a newly discovered identity between them. This is a joint work with T. Yamauchi.
Many problems in control and optimization require the treatment
of systems in which continuous dynamics and discrete events
coexist. This talk presents a survey on some of our recent work on such
systems. In the setup, the discrete event is given by a random
process with a finite state space, and the continuous component is the
solution of a stochastic differential equation. Seemingly similar
to diffusions, the processes have a number of salient features
distinctly different from diffusion processes. After providing
motivational examples arising from wireless communications,
identification, finance, singular perturbed Markovian systems,
manufacturing, and consensus controls, we present necessary and
sufficient conditions for the existence of unique invariant
measure, stability, stabilization, and numerical solutions of
control and game problems.
Originated from applications in signal processing, random evolution,
telecommunications, risk management, financial engineering,
and manufacturing systems, two-time-scale Markovian systems have
drawn much attention. This talk discusses asymptotic
expansions of solutions to the forward equations, scaled and unscaled
occupation measures, approximation error bounds, and associated
switching diffusion processes. Controlled dynamic systems will also
be mentioned.
1952년 영국의 수학자 A. Turing은 그 당시 생물학자들 조차 전혀 상상 할수 없었던, 다 같은 종류의 세포들이 각자 다른 세포로 분화할수 있는 메커니즘을 Reaction-Diffusion System(RD system)을 이용하여 수학적으로 제시했습니다. 그 이후로, RD system은 수리해석학적으로도 많은 발전을 거듭해왔으며, 수리모델링을 통해 생명과학 분야에 있어서도 생명의 메커니즘을 밝히는 도구로서 발전을 거듭해 오고 있습니다.
이 강연에서는 제가 최근에 연구를 진행하고 있는 다양한 생명현상을 예로 그 메커니즘을 밝히기 위해 개발한 수리모델 및 수리모델링 수법을 간단하게 소개하겠습니다. 여기에는 수학적으로 재미있는 구조를 가지고 있을 지 모르는 문제들이 숨어 있을 수 있습니다. 그런 문제들을 여러분들께서 직접 찾아 보시길 바랍니다.
Keywords: Mathematical modeling, PDE, Phase-field method
Liquid crystal is a state of matter between isotropic fluid and crystalline solid, which has properties of both liquid and solid. In a liquid crystal phase, molecules tend to align a preferred direction and molecules are described by a symmetric traceless 3x3 matrix which is often called a second order tensor. Equilibrium states are corresponding to minimizers of the governing Landau-de Gennes energy which plays an important role in mathematical theory of liquid crystals. In this talk, I will present a brief introduction to Landau-de Gennes theory and recent development of mathematical theory together with interesting mathematical questions.
자연과학동 E6-1, ROOM 1409
Discrete Math
Bernard Lidický (Iowa State University)
3-coloring triangle-free planar graphs
자연과학동 E6-1, ROOM 1409
Discrete Math
A well known theorem of Grötzsch states that every planar graph is 3-colorable. We will show a simple proof based on a recent result of Kostochka and Yancey on the number of edges in 4-critical graphs. Then we show a strengthening of the Grötzsch’s theorem in several different directions. Based on joint works with Ilkyoo Choi, Jan Ekstein, Zdeněk Dvořák, Přemek Holub, Alexandr Kostochka, and Matthew Yancey.
Consider a simple symmetric random walk $S$ and another random walk $S'$ whose $k$th increments are the $k$-fold product of the first $k$ increments of $S$.
The random walks $S$ and $S'$ are strongly dependent. Still the 2-dimensional walk $(S, S')$, properly rescaled, converges to a two dimensional Brownian motion. The goal of this talk is to present the proof of this fact, and its generalizations. Based on joint works with K. Hamza and S. Meng.