Department Seminars & Colloquia
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The phase transition deals with sudden global changes and is observed in many fundamental problems from statistical physics, mathematics and theoretical computer science, for example, Potts models, graph colourings and random k-SAT. The phase transition in random graphs refers to a phenomenon that there is a critical edge density, to which adding a small amount a drastic change in the size of the largest component occurs. In Erdös-Renyi random graph, which begins with an empty graph on n vertices and edges are added randomly one at a time to a graph, a phase transition takes place when the number of edges reaches n/2 and a giant component emerges. Since this seminal work of Erdös and Renyi, various random graph models have been introduced and studied. In this talk we discuss phase transitions in several random graph models, including Erdös-Renyi random graph, random graphs with a given degree sequence, random graph processes and random planar graphs.
지금은 융합과학기술의 시대라고 일컫는다. 하지만 아무도 융합이 무엇을 연구하는 분야인지그 정의를 명확하게 내리지는 못한다. 그 이유는 아마도 융합은 학문이나 기술의 분야가 아니라 그들을 하고자하는 방법론이어서 그럴 것이다. 수학을 바탕으로 자연과학이 나타났고 또 이들을 바탕으로 공학분야가 탄생하였다. 수 많은 학문이나 기술 분야가 각각 많은 발전을 하였고 산업에 이바지하였다. 이제 산업이 고도화되고 사회의 요구성이 매우 다양해져서 이러한 니즈를 맞추기에 단독적인 학문이나 기술로는 턱없이 부족하기에 이르렀다. 학문적으로도 새로운 생각이나 방법이 단독 분야에서 나타나는 것이 아니라 여러 분야의 화학적인 결합으로 태어난다. 따라서 어쩌면 융합이라는 분야는 무엇이 될지 모르는 정의될 수 없는 것이 당연할지 모른다.
현재 비교적 융합이 나타나는 분야는 우리의 가장 큰 관심사인 건강한 삶을 주제로 한 분야이다. 암 치료를 위한 첨단 의료기기로부터 비만 관리에 이르는 ‘라이프사이언스’에 이제까지의 모든 학문적 기술적 지식이 요구된다고 할 수 있다. 본 콜로퀴엄에서는 필자가 수행하였던 ‘라이프사이언스’ 연구 관련 주제를 소개하고 이들의 방향을 토의해보고자 한다. 여기에는 유전자나 단백질 등 유전정보의 기능을 예측하거나 분석하고 관리해주는 바이오인포매틱스, 정보통신 기술을 바탕으로 건강 관리를 해주는 u-헬스케어, 의사에게 객관적인 정보를 주어 진단의 정확도를 높일 수 있는 컴퓨터도움진단, 그리고 양성자치료기와 같은 첨단 의료기기 등의 분야가 포함된다. 덧붙여서 필자가 근무하였던 한국연구재단에서 보아왔던 연구자들의 융합분야도 소개하고자 한다.
수학은 모든 학문과 방법론의 기본이 된다. 수학적 지식과 마음가짐으로 타 분야와의 융합을 통하여 이제까지 볼 수 없었던 새로운 학문의 지평선을 열기를 기대해본다. 본 콜로퀴엄은 이러한 취지에서 융합과학기술의 일부분의 예를 소개하는 것이고 더욱 창의적인 융합의 지평선을 카이스트에서 열어주길 기대한다.
In the first half of this talk, we will review the notion of affine processes on positive semidefinite matrices, which were introduced by Cuchiero et al (2011). These processes have been applied to various modeling problems in the finance literatures. In the latter half, I will introduce our recent research result on affine processes. We establish the relationship between the Laplace transforms for linear functionals of affine processes on positive semidefinite matrices and the generalized Riccati integral equations. The result generalizes and unifies the affine transform formula and the systematic exploitation of Laplace transforms of squared Bessel functionals in a natural way. As consequences of the result, we found some formulae such as Laplace transforms of the marginal distributions, transition densities, and Pitman-Yor's formula for Wishart processes. Moreover, we also present the conditional Laplace transforms of Wishart functionals.
In one of their joint papers, Victor Reiner and Dennis Stanton introduced a (q,t)-generalization of the binomial coefficient. There was an interesting conjecture for the cases when q≤-2 is a negative integer. In this talk, I will prove this conjecture and try to give some combinatorial sense using integer partitions.
In this lecture series, we will cover basic notions of a Hitchin
system and related topics with some variations and cadenzas. . The
lectures will be generically in a expository nature: The main article
to study in the lecture series is N. Hitchin’s paper “Stable bundle
and Integrable systems”. If time permits, we will also give some
remarks about other papers “Lie groups and Teichmuller space” and
“Langlands duality and $G_2$ spectral curves”.
Many combinatorial problems and arguments concerning finite point sets in the Euclidean plane (or higher dimensions) often do not use the linear structure. A more general concept is that of an Oriented Matroid (OM). It is well-known that every OM can realized by pseudolines, and in fact most oriented matroids can not be realized by straight lines.
Recently, Alfreod Hubard (Courant Institute) and myself have found a new way to represent an OM by convex sets which retains much more of the “straightness” of the Euclidean plane. Interestingly, in our model the isotopy conjecture holds in a very strong sense, and it unifies several aspects of pseudoline arrangements.
Deligne observed a long time ago that the category of split real Hodge structures is equivalent to the category of representations of a certain rank 2 torus. Later he described the Tanakian Galois group of the category of all real mixed Hodge structures as a certain extension of this torus. I'll discuss this and joint work with Greg Pearlstein where we spell out a similar picture for degenerations of real mixed Hodge structure.
자연과학동(E6) Room 3435
Face the World with Mathematical Mind
Jon McCammond (University of California, Santa Barbara)
Symmetry and Abstraction (Why do mathematicians see only 17 types of wall paper?)
자연과학동(E6) Room 3435
Face the World with Mathematical Mind
There's something about a symmetric shape that really grabs our attention, particularly if you're a mathematician. In this talk I will discuss what a mathematician sees when they look at a symmetric object, how they abstract out the essence of the situation and what one could possibly mean by the statement that "there are only 17 types of wallpaper." Applications of these abstract structures to physics, chemistry and computer science will also be discussed.
대칭적인 모양에는 사람들의 시선을 끄는 특별한 힘이 있습니다. 이 강연에서는 수학자들이 대칭적인 물체를 바라보는 방법에 대해 얘기해 보고자 합니다.
보통 사람들이 느끼는 대칭성을 수학자들이 어떻게 추상화하고, 그 추상화의 능력을 통해 이 세상의 모든 벽지를 오직 17개의 종류 만으로 분류할 수 있었는지 소개합니다. 이러한 추상적인 구조는 물리학, 화학, 전산학 등에서도 이용됩니다.
We propose a computational framework to incorporate regularization terms used in regularity based variational methods into least squares based methods. In the regularity based variational approach, the resulted image is obtained as a result of the competition between the fidelity term and a certain regularity term, while in the least squares based approach the image is computed as a minimizer to a constrained least squares problem.
The total variation minimizing denoising scheme is an exemplary scheme of the former approach with the total variation minimizing term as the regularity term, while the moving least square method is an exemplary scheme of the latter approach.
Both approaches have appeared in the literature of image processing independently.
By putting schemes from both approaches into a single framework, the resulted scheme benefits from the advantageous properties of both parties. As an example, in this talk, we propose a new denoising scheme, where the total variation minimizing term is adopted by the moving least squares method. The resulted denoising scheme overcomes the drawbacks of both schemes, i.e., the staircasing artifact in the total variation minimizing based denoising and the local boiling artifact in the moving least square based denoising method.
The proposed computational framework can be utilized to put various combinations of both approaches with different properties together.
I will discuss the existence of injective maps between right-angled Artin groups. In the case of two-dimensional right-angled Artin groups, I will give a complete classification of right-angled Artin subgroups of right-angled Artin groups. I will then derive some new rigidity results. This is joint work with Sang-hyun Kim.
I will talk about a degenerate diffusion Patlak-Keller-Segel system in $n\geq 3$ dimension. The main difference between the current work and many recent works on the same model is that we study the diffusion exponent $m=2n/(n+2)$ which is smaller than the exponent $m^{*}=2-2/n$ used in those recent works. With the exponent $m=2n/(n+2)$, the associated free energy is conformal invariant and there is a family of stationary solution $U_{\lambda,x_0}(|x|) =C(n)(\frac{\lambda}{\lambda^2+|x-x_0|^2})^{\frac{n+2}{2}}$, $\forall \lambda>0$. For radially symmetric solutions, we prove that if the initial data is strictly below $U_{\lambda,0},(|x|)$ for some $\lambda$ then the solution vanishes in $L^1_{loc}$ as $t\to\infty$; if the initial data is strictly above $U_{\lambda,0} (|x|)$ for some $\lambda$ then the solution concentrates at $r=0$ as $t\to\infty$. We then prove that there is a global weak solution provided that the $L^m$ norm of initial density is less than a universal constant, and the weak solution vanishes as time goes to infinity. We also prove a finite time blow up of the solution if the $L^m$ norm for initial data is large then that of $U_\lambda(|x|)$ and the free energy of initial data smaller than that of $U_\lambda(|x|)$.
Optimal transportation theory studies the phenomena when two mass distributions are matched in a most efficient way with respect to a transportation cost, usually a function of the distance between two points. The existence, uniqueness and characterization on such optimal transport maps is now well established and there has been a remarkable recent progress on the study of regularity of such optimal maps, which is closely related to the study of Monge-Amp\`ere type equations.
In this talk, we consider the question of how to construct optimal maps on compact Riemannian manifolds and suggest to use a Monge-Amp\`ere type parabolic equation.
Under certain geometric conditions, we show the solution to the parabolic flow with any appropriate initial condition exists for all time and it converges exponentially to the solution to the optimal transportation problem. Such results hold in particular, on the sphere for the distance squared cost of the round metric and for the far-field reflector antenna cost, among others. This is joint work with Jeff Streets and Micah Warren.
Let $K$ be a number field. A local global principle says that if a
unit of $K$ is locally a square in all places $v$ of $K$, then it a
global square in $K$.
In certain cases, if one require the places $v$ of $K$ lying a
single prime $p$ or infinity place of $\\Q$, then the above local
global principle for unit squares still holds. The case of infinity
place is classical and dates back to Weber.
It is interesting to decide for which $K$, there exists a prime $p$
such that the above phenomena holds. For simplicity, we assume $K$
is Galois and $p$ is odd.
In this talk, I will survey our joint work on this topic.
In this talk, I will report our recent work on additive
representation functions [arXiv:1108.1920].
Precisely,
for $m\geq 1$, let $0e_0,e_1,\cdots，e_m>0$ be fixed positive integers. Assume there
exists a prime $p$ and an integer $t>0$ such that $p^t\mid b_0$, but
$p^t\nmid b_{i}\ {\rm for}\ 1\leq i\leq m$.
Then, we prove that there is no
infinite subset $\mathcal A$ of positive integers, such that the
number of solutions of the following equation
$$n=b_0(a_{0,1}+\cdot +a_{0,e_0})+\cdots +b_m(a_{m,1}+\cdots +a_{m,r_m}),\ a_{i,j}\in \mathcal A$$
is constant for $n$ large enough. This generalizes the result of
Cilleruelo and Ru\'{e} for the case of bilinear forms [On a Question
of S\'{a}rkozy and S\'{o}s for Bilinear forms, Bulletin of the
London Mathematical Society 4, 2(2009), 274-280.],
and answers a question posed by S\'{a}rkozy and S\'{o}s.
This a joint work with professor Lianrong Ma of Tsinghua University.
In this talk, we review a novel numerical method that can handle the rate-type non-Newtonian equations in a unified fashion and validate the methods in terms of various benchmark solutions as well as theoretical results. We then apply it to the real physical problems. In particular, we present our investigations and attempts to identify a mathematical model for the unusual phenomenon observed in motion of the sphere falling through the wormlike micellar fluids by Jayaraman and Belmonte; a sphere falling in a wormlike micellar fluids undergoes nontransient and continual oscillations. We tackle the Johnson-Segalman models in the parameter regimes that have been unexplored previously for the flow past a sphere and reproduce the self-sustaining, continual, (ir)regular and periodic oscillations. Our results show that the flow instability can be correlated with the critical value of the velocity gradient, as observed in experiments by Jayaraman and Belmonte in 2003. If time permits, we also present recent works on the boundary conditions for the diffusive complex fluids models as well as the fast stokes solvers implemented in a full parallel fashion.
In these lectures, we will first discuss the construction of Gibbs measure
associated to certain Hamiltonian PDEs. Then we will show their invariance
under (i) when we have an a priori (deterministic) global well-posedness, and
(ii) when we only have an a priori local well-posedness. In the second case, the
crucial point is how to construct global solutions almost surely on the support
of the Gibbs measures (due to Bourgain ’94).
In these lectures, we will first discuss the construction of Gibbs measure
associated to certain Hamiltonian PDEs. Then we will show their invariance
under (i) when we have an a priori (deterministic) global well-posedness, and
(ii) when we only have an a priori local well-posedness. In the second case, the
crucial point is how to construct global solutions almost surely on the support
of the Gibbs measures (due to Bourgain ’94).
E6-1 #1409
Discrete Math
Maryam Verdian-Rizi (SFU)
Toroidal triangulations with few odd degree vertices
E6-1 #1409
Discrete Math
We present a simple geometric description for the set of toroidal triangulations with two odd vertices where each vertex has degree five or more. Each such triangulation is described by a cut-and-glue construction starting from an infinite triangular grid. To achieve that, we define some invariants to study the cycles of toroidal triangulations. The motivation for studying such family comes from Fisk triangulations and Grünbaum coloring, which will be discussed as well.
In 1950, Northcott proved the height inequality for morphisms on projective spaces. Let f:Pn→Pn be a morphism. Then, there is a constant Cdepending on the given morphism such that the height function h satisfies h(f(P))+C>degfh(P)>h(f(P))-C for all points P∈Pn. Unfortunately, the upper bound of the inequality does not hold for rational maps. However, we can find weaker inequality by 1)
restricting points on an affine open set and 2) enlarging the upper bound a little bit. In this talk, I will introduce the D-ratio for a rational map on projective spaces and prove the Weak Northcott property for rational maps.