# 세미나 및 콜로퀴엄

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The circle is the only connected closed 1-dimensional manifold, and maybe that's why it has so many interesting features. In this talk, we would like to emphasize that there are many things we still do not know about this one of the simplest manifolds. We will survey many interesting recent results around the circle in the context of low-dimensional topology.

2017 제4회 정오의 수학산책

강연자: 양성덕 (고려대)

일시: 2017년 6월 2일(금) 12:00 ~ 13:15

장소: 카이스트 자연과학동 E6-1 3435호

제목: 고차원 공간의 계량과 곡률

내용: TBA

등록: 2017년 5월 31일(수) 오후 3시까지

https://goo.gl/forms/ol819KdGNA5SdGSr2

문의: hskim@kias.re.kr / 내선:8545

The interactions between particles and

uid have received a bulk of attention

due to a number of their applications in the eld of, for example, biotechnol-

ogy, medicine, and in the study of sedimentation phenomenon, compressibility

of droplets of the spray, cooling tower plumes, and diesel engines, etc. In this

talk, we present coupled kinetic-

uid equations. The proposed equations consist

of Vlasov-Fokker-Planck equation with local alignment forces and the incom-

pressible Navier-Stokes equations. For the equations, we establish the global

existence of weak solutions, hydrodynamic limit, and large-time behavior of

solutions. We also remark on blow-up of classical solutions in the whole space.

Hyperrings and hyperfields are algebraic structures which generalize commutative rings and fields. In this talk, we aim to introduce these exotic structures and also provide examples which illustrate how hyperrings and hyperfields show up in algebraic geometry and combinatorics following the idea of Baker and Bowler on `matroids over hyperfields'.

The discrepancy of a sequence f(1), f(2), ... of numbers is defined to be the largest value of |f(d) + f(2d) + ... + f(nd)| as n,d range over the natural numbers. In the 1930s, Erdos posed the question of whether any sequence consisting only of +1 and -1 could have bounded discrepancy. In 2010, the collaborative Polymath5 project showed (among other things) that the problem could be effectively reduced to a problem involving completely multiplicative sequences. Finally, using recent breakthroughs in the asymptotics of completely multiplicative sequences by Matomaki and Radziwill, as well as a surprising application of the Shannon entropy inequalities, the Erdos discrepancy problem was solved in 2015. In this talk I will discuss this solution and its connection to the Chowla and Elliott conjectures in number theory.

2017 CMC Distinguished Lecture Series by Terence Tao

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Fusion Hall(1F), KI Bldg.(#E4), KAIST, Daejeon
해외 석학 특별 강연 시리즈
Terrence Tao (UCLA)
Finite time blowup constructions for supercritical equations

Many basic PDE of physical interest, such as the three-dimensional Navier-Stokes equations, are "supercritical" in that the known conserved or bounded quantities for these equations allow the nonlinear components of the PDE to dominate the linear ones at fine scales. Because of this, almost none of the known methods for establishing global regularity for such equations can work, and global regularity for Navier-Stokes in particular is a notorious open problem. We present here some ways to show that if one allows some modifications to these supercritical PDE, one can in fact construct solutions that blow up in finite time (while still obeying conservation laws such as conservation of energy). This does not directly impact the global regularity question for the unmodified equations, but it does rule out some potential approaches to establish such regularity.

2017 CMC Distinguished Lecture Series by Terence Tao

We consider the incompressible Navier-Stokes equations ((NS) in short) in $mathbb{R}^{3}$.

We consider the incompressible Navier-Stokes equations ((NS) in short) in $mathbb{R}^{3}$.

We consider the incompressible Navier-Stokes equations ((NS) in short) in $mathbb{R}^{3}$.

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

문의: hskim@kias.re.kr / 내선:8545

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.

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.