Abstract: Ramsey's theorem states that, for a fixed graph $H$, every 2-edge-colouring of $K_n$ contains a monochromatic copy of $H$ whenever $n$ is large enough. Perhaps one of the most natural questions after Ramsey's theorem is then how many copies of monochromatic $H$ can be guaranteed to exist. To formalise this question, let the \emph{Ramsey multiplicity} $M(H;n)$ be the minimum number of labelled copies of monochromatic $H$ over all 2-edge-colouring of $K_n$. We define the \emph{Ramsey multiplicity constant} $C(H)$ is defined by $C(H):=\lim_{n\rightarrow\infty}\frac{M(H,n)}{n(n-1)\cdots(n-v+1)}$. I will discuss various bounds for C(H) that are known so far.

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In this introductory talk, we will discuss how to describe and study various hydrodynamic models in Lagrangian variables, following the framework introduced by Constantin. In a unified framework and with less regularity, local well-posedness theory can be established for a range of models, including the incompressible Euler equation, the surface quasi-geostrophic equation, the Boussinesq system, the Oldroyd-B system, and more.

For a very ample line bundle L on a projective scheme X, we say that (X,L) satisfies property QR(k) if the homogeneous ideal of the linearly normal embedding can be generated by quadrics of rank <= k. Many classical varieties such as Segre-Veronese embeddings, rational normal scrolls and curves of high degree satisfy the property QR(4). In this talk, we will briefly show that the second and higher Veronese embeddings satisfies the property QR(3). And we will discuss about the asymptotic behavior of property QR(3) for any projective scheme.

In this talk, I will give some existence results on nontrivial solutions of Kirchhoff type problem by the variational methods. In this first parts, the bound state solutions of this problem with a critical exponent have been obtained by deformation results after a local compactness result has been recovered. We found a different and interesting results when this problem is considered in a high dimension N larger than 4. In the second parts, I will give some results on semi-classical solutions

In this talk, we are concerned with dynamically stability of uniformly rotating binary stars and galaxies, which are represented as stationary solutions to Euler-Poisson equations and VlasovPoisson equations respectively. These solutions were constructed as minimizers of suitable variational problems by McCann in which some kind of structural stability on them is discussed. This talk focuses on the nonlinear dynamical stability of them, based on Cazenave-Lions type arguments exploiting variational characterization of stationary solutions. We will see that the uniqueness of a minimizer, which is one of the main results of our work, plays an indispensable role in analysis. This talk is based on a joint work with Prof. Juhi Jang at USC.

Algebraic fibre spaces are relative versions of algebraic varieties. For an algebraic fibre space f:X->Y, the varieties X,Y and the general fibre F are deeply related by various formulas and conjectures on the invariants of varieties. For example, the Iitaka conjecture is still an open problem which predicts that the Kodaira dimension of X is at least the sum of kodaira dimensions of F and Y. We explain how the geometry of algebraic fibre spaces can be studied by convex bodies called Okounkov bodies.

The existence of Kaehler-Einstein metric and Kaehler-Einstein edge metric on n-dimensional complex varieties is related to the existence of Sasaki-Einstein metric on links over the varieties which is (2n+1)-dimensional real manifolds. We discuss results about existence problem of the both metrics on weighted log del Pezzo surfaces and links over the surfaces that are 5-dimensional Smale manifolds.

2020년 현재 우리가 사는 사회는 인공지능(AI), 사물인터넷(IoT), 빅데이터(Big Data) 등의 첨단 정보기술이 사회와 경제 전반에 융합되어 혁신적인 변화가 일어나고 있다. 이미 스마트폰, 데이터, 정보, 인공지능, 포노사피엔스, 디지털 전환(Digital Transformation) 등의 키워드는 현 시대의 흐름을 잘 나타내주고 있으며, 헬스케어, 지능형 로봇, 가정용 인공지능 시스템, 공유자동차 등이 우리 생활에 깊이 영향을 미치고 있다. 이런 변화의 중심에 수학이 위치하고 있으며, 사회는 적절한 수학적 지식을 갖춘 인재를 필요로 하고 있다. 따라서 수학자와 수학교육자는 이런 변화를 가능케 하는 수학에 대하여 다양한 전공의 대학생들과 일반인들에게 이해시키는 것이 꼭 필요하다. 본 발표에서는 ”AI 시대의 대학수학교육 "이란 제목으로, 행렬론을 전공한 발표자가 그간의 경험을 활용하여 ‘대학생을 위한 Math for AI’ 와 '(K-MOOC) 일반인을 위한 인공지능을 의한 기초수학 입문' 과목을 개발하고 교재와 실습실을 만들어, 지난 4학기 동안 강의하면서 경험한 내용을 공유하고자 한다. (Zoom link: https://kaist.zoom.us/j/89057059601?pwd=dGI4NHRDUGNQMVQrVXg3YWxvRXA3dz09)

The mean curvature flow is an evolution of hypersurfaces satisfying a geometric heat equation. The flow changes its topology through singularities, and there are infinitely many singularity models. However, the flow is a gradient flow of an entropy functional, and there are only a few linearly stable singularities. Hence, one can conjecture that some perturbations of initial hypersurfaces would make the perturbed flows avoid linearly unstable singularities. In this talk, we will discuss how to use ancient flows for the avoidance of unstable singularities, and its application to the knot theory.

The idea of using homogeneous dynamics to Diophantine approximation has grown to an active subfield of mathematics, with numerous results on Hausdorff dimension of sets of vectors with certain Diophantine properties. In this talk, we will start from scratch, from the Gauss map of the usual continued fraction expansion for real numbers and give a "dynamical interpretation" of Diophantine properties of continued fractions in terms of the orbits of the geodesic flow on the hyperbolic plane. We will then present a series of results of the speaker with coauthors on inhomogeneous Diophantine approximation and give ideas of proofs, especially the idea related to the partial proof of Littlewood conjecture of Einsiedler-Katok-Lindenstrauss. (The latter part of the talk is based on joint works with U. Shapira-N. de Saxce, Y. Bugeaud-Donghan Kim-M. Rams, and Wooyeon Kim-Taehyung Kim.)

The Lamb dipole is a traveling wave solution to the two-dimensional Euler equations introduced by S. A. Chaplygin (1903) and H. Lamb (1906) at the early 20th century. We prove orbital stability of this solution based on a vorticity method initiated by V. I. Arnold. Our method is a minimization of a penalized energy with multiple constraints that deduces existence and orbital stability for a family of traveling waves. As a typical case, orbital stability of the Lamb dipole is deduced by characterizing a set of minimizers as an orbit of the dipole by a uniqueness theorem in the variational setting. This is a joint work with K. Abe (Osaka City Univ).

In this talk, I will present a recent work in collaboration with physicists on the analysis of real time Transmission Electron Microscopy (TEM) images to understand molecular transition from crystal solid state to liquid state. Molecules are deposited on graphene with multilayer structures, which are projected and overlaid in noisy 2d TEM images. The problem is to find all the molecular centers in the extremely noisy 2d images where projected molecules are overlaid and to track the centers across the image frames. Before discussing the method that we considered, I will give a brief history in the development of image segmentation techniques with some theoretical and numerical details of old fashioned methods. Then, our method of image segmentation for molecular center identification follows.

Let $E$ be an elliptic curve over the rational number field $\mathbb{Q}$. Selmer Groups and Ideal Class Groups are important and widely-studied objects in number theory. Brumer and Kramer studied relations between these two objects in their paper in 1977. They actually found an upper bound for the $2$-Selmer rank of $E$ in terms of the ideal class group of a certain cubic field extension of $\mathbb{Q}$. As an application, they determined the Mordell-Weil ranks of (most) elliptic curves of prime conductor assuming the BSD conjecture. In this talk, we will talk about a generalization of Brumer-Kramer's work to the case of elliptic curves over an arbitary number field. We will give both upper and lower bounds for the $2$-Selmer rank in terms of a (modified) ideal class group, and the bounds turn out to be sharp in many cases. This is joint work with Hwajong Yoo. (If you would like to join the seminar, please let me(Bo-Hae Im) know so that I can send you the Zoom link.)

Swimming bacteria with helical flagella are self-propelled micro-swimmers in nature, and the swimming strategies of such bacteria vary depending on the number and the position of flagella on the cell body. In this talk, I will introduce some microorganisms such as E. coli, Vibrio A and P putida. The Kirchhoff rod theory is used to model the elastic helical flagella and the cell body is represented by a hollow ellipsoid that can translate and rotate as a neutrally buoyant rigid body interacting with a surrounding fluid. The hydrodynamic interaction between the fluid and the bacteria is described by the regularized version of Stokes flow. I will focus on how bacteria can swim and reorient swimming course for survival and how Mathematics can help to understand the swimming mechanism of such bacteria.

I will present my recent work on uniqueness of Riemann problem consisting of two shocks for 1D isentropic Euler system. The uniqueness is guaranteed in the class of vanishing viscosity limits of solutions to the associate Navier-Stokes system, as the Bianchini-Bressan conjecture. The main idea to achieve this issue is to get a uniform stability of any large perturbations from a composite wave of two viscous shocks to the Navier-Stokes. Especially, I will explain about this main idea in a simpler context, that is, in the case of a single shock. This is based on joint works with Alexis Vasseur.

The essential dimension quantifies the algebraic-geometric complexity of a class of algebraic objects (such as, but not necessarily, the class of Galois extensions with a given group): roughly speaking, it is the minimal number of parameters required to describe all objects in this class (over all fields containing a given field K). We introduce and discuss arithmetic-geometric and local analogues of this notion. These are supposed to quantify the difference in complexity between the local and global Galois theory of a given group over a given number field K. In particular, we show that the "local dimension" of a finite group is bounded by 2 - whereas arithmetic dimension remains mysterious in general. We give an application concerning solution of Grunwald problems.

이 강연에서는 4차산업혁명에 있어서 수학이 얼마나 중요한 역할을 하는 지 살펴 본다. 먼저 과학발전의 역사를 살펴보고 4차산업혁명시대의 도래에 대하여 논한다. 4차산업혁명의 기초이론을 살펴보고 4차산업혁명의 가장 핵심인 인공지능과 생명공학기술을 논한 다음 수학산업의 경쟁력 제고 방안을 논한다. (Zoom link: https://kaist.zoom.us/j/86300617369?pwd=WlkyNkJYY0RZR2lNZHRHazBZcU5sUT09)

In this talk, we introduce an idea of producing rational torsion points on J_0(N), which is well-known. Conjecturally, the points constructed in this way exhaust all the rational torsion points on J_0(N). So, we briefly explain how to compute the orders of such points, and prove the conjecture up to finitely many primes. (If you would like to join this online seminar, please contact Bo-Hae Im to get the Zoom link.)

Recently, with the enormous development of deep learning techniques, solving underdetermined linear systems (more unknowns than equations) have become one of major concerns in medical imaging. Typical examples include undersampled MRI, local tomography, and sparse view CT, where deep learning techniques have shown excellent performance. Although deep learning methods appear to overcome limitations of existing mathematical methods in handling various underdetermined problems, there is a lack of rigorous mathematical foundations which would allow us to understand reasons why deep learning methods perform that well. This talk deals with this learning causal relationship about structure of training data suitable for deep learning to solve highly underdetermined inverse problems. We examine whether or not a desired reconstruction map can be learnable from the training data and the underdetermined system. Most problems of solving underdetermined linear systems in medical imaging are highly non-linear.