A set F of graphs has the Erdős-Posa property if there exists a function f such that every graph either contains k disjoint subgraphs each isomorphic to a member in F or contains a set of at most f(k) vertices intersecting all such subgraphs. In this talk I will address the Erdős-Posa property with respect to three closely related graph containment relations: minor, topological minor, and immersion. We denote the set of graphs containing H as a minor, topological minor and immersion by M(H),T(H) and I(H), respectively. Robertson and Seymour in 1980’s proved that M(H) has the Erdős-Posa property if and only if H is planar. And they left the question for characterizing H in which T(H) has the Erdős-Posa property in the same paper. This characterization is expected to be complicated as T(H) has no Erdős-Posa property even for some tree H. In this talk, I will present joint work with Postle and Wollan for providing such a characterization. For immersions, it is more reasonable to consider an edge-variant of the Erdős-Posa property: packing edge-disjoint subgraphs and covering them by edges. I(H) has no this edge-variant of the Erdős-Posa property even for some tree H. However, I will prove that I(H) has the edge-variant of the Erdős-Posa property for every graph H if the host graphs are restricted to be 4-edge-connected. The 4-edge-connectivity cannot be replaced by the 3-edge-connectivity.

In this talk, we will survey the book "Arithmeticity in the theory of
automorphic forms - G.Shimura (2000)".

A vertex algebra is an algebraic structure which is constructed to explain
conformal field theory (CFT) rigorously. So a vertex algebra is endowed
with a non-associative product called normally ordered product which shows
operator product expansions in CFT. On the other hand, purely
mathematically, it is closely related to the theory of affine Lie algebras.
In this talk, I will introduce the most well-known definition of vertex
algebras and show simple examples.

Type 2 diabetes (T2D) is generally thought to result from the combination of two metabolic defects, insulin resistance, which increases the level of insulin required to maintain glucose within the normal range, and failure of insulin-secreting pancreatic beta cells to compensate for the increased demand. We have built up a comprehensive mathematical model of progression to T2D. The dynamics of failure and compensation can be described on two-dimensional slow manifold to investigate the mechanisms of progression to diabetes. In addition, we have extended our mathematical model by adding daily meals and hepatic glucose production, which is helpful for studying the clinical implications. These enhancements allow us to look at the mechanistic defects that underlie observed pathologies such as impaired fasting glucose (IFG) and impaired glucose tolerance (IGT). The model supports associations found in experiments between IFG and excess HGP and between IGT and peripheral insulin resistance. The model suggests how to personalize therapeutic approaches for subjects based on their underlying metabolic abnormalities.

This talk deals with some questions concerning the convergence of the solu-tions to a dynamic equation on time scale. It is also concerned with the stability domains, the spectrum of matrix pairs, the exponential stability and its robustness measure for linear implicit dynamic equations of arbitrary index.

In this talk I will discuss research with the goal of building models of brain anatomy. The neuronanatomical structures of interest can be broadly subdivided into two categories - cortical and non-cortical. Cortical structures (particularly the cerbral cortex) are typically highly folded, thin sheets of gray matter. Functionally, the cerebral cortex has been shown to have a "columnar" architecture. For this reason, we construct surface-based models for analysis of cortical properties. The construction of such models is a difficult task due to the high degree of folding of the cortical manifold in conjunction with the limited (~ 1 mm) resolution of current neuroimaging technologies. Once constructed, the cortical models can be deformed for morphometry, visualization and registration purposes. I will show some results of this type of analysis, including the morphometric changes that the cortex undergoes in disorders such as schizophrenia, Alzheimer's disease, and Huntington's disease, as well as healthy aging.
A different set of techniques have been developed for the construction of models of subcortical structures. Here, we model the segmentation as an anisotropic nonstationary Markov Random Field. The anisotropy lets us model the local spatial relationships that exist between neuroanatomical structures (e.g. hippocampus is anterior and inferior to amygdala), while the nonstationarity facilitates the encoding of inhomogeneous properties of the tissue within a structure. This approach is based on extracting the relevant model parameters from a manually labeled training set, and has been shown to be comparable in accuracy to the manual labeling.

기나긴 수학의 역사에서 관찰되는 큰 흐름은 우아함과 완전함에 대한 열망이다. 물리적 세계의 불완전함 이면에 있는 질서를 찾아내고, 수학적 단순화 과정을 거쳐 대칭과 조화를 표현하는 일에 매료된 수학자들은 역사의 도처에서 관찰된다.하지만 세상은 여전히 완전하지 않다. 우주로 간 화성탐험선은 끊임없이 지구로 영상신호를 보내지만, 태양의 자기장과 지구의 오존층에 이리 치이고 저리 치여서 엉뚱하게 변질된 신호가 지구에 도착하지 않는가? 원자의 세계는 뉴턴역학의 결정론이 아니라 양자역학의 확률적 개념으로 접근해야 한다는 사실에 아인슈타인은 “신은 주사위 놀음을 하지 않는다”고 절망하지 않았던가? 프랙탈의 개념을 처음 발견한 망델브로는 불규칙과 무질서가 자연의 본질에 더 가깝다고 결론내리지 않았는가? 수학은 이 혼란에 어떻게 대응하고 있는지 들여다 보고자 한다.

This presentation is about recently introduced immersed finite element (IFE) methods that can solve interface problems with structured or even Cartesian meshes advantageously in some applications. After a brief survey on FE/IFE methods based on unstructured meshes for interface problems, this presentation will explain the need to construct IFE functions as macro piecewise polynomials defined with subelements formed by partitioning each interface element with the actual material interface instead of its linear approximation. We present a unified framework for developing and analyzing immersed finite element (IFE) spaces with either linear, or bilinear, or the rotated-$Q_1$ polynomials. Functions in these IFE spaces are locally piecewise polynomials defined according to the sub-elements formed by the interface itself instead of its line approximation. We show that the unisolvence for these IFE spaces follows from the invertibility of the Sherman-Morrison matrix. A group of estimates and identities are established for the interface geometry and shape functions that are applicable to all of these IFE spaces. Most importantly, these fundamental preparations enable us to develop a unified multipoint Taylor expansion procedure for proving that these IFE spaces have the expected optimal approximation capability according to the involved polynomials.

Recently, Marcus, Spielman, and Srivastava proved the  existence of infinite families of bipartite Ramanujan graphs of every  degree at least 3 by using the method of interlacing families of polynomials. In this talk, we apply their method to prove that for any  connected graph G, there exists an orientation of G such that the  spectral radius of the corresponding Hermitian adjacency matrix is at  most that of the universal cover of G.

In this talk, Bayesian semiparametric methods for function estimation and model selection problems are presented. This talk is designed to provide graduate students and researchers with an introduction to Bayesian semiparametric inference. The orientation is methodological rather than theoretical, but such asymptotic theory as is necessary for a proper understanding and validating specific Bayesian methods will be also covered in detail. The materials will include three aspects of Bayesian inference, (i) Fundamentals (ii) Asymptotics and (iii) Advanced models, focusing on nonparametric and semiparametric methods. For function estimation, the Bayesian semiparametric models using Gaussian process priors are discussed. For model selection, Bayes factors are explained for dealing with lack of fit in regression and goodness of fit in density estimation problems.

    We show that any supersingular K3 surface of Artin invariant 1 in characteristic p=11, or >13 has an automorphism the entropy of which is the natural logarithm of a Salem number of the maximal degree 22, in particular nonliftable ones to characteristic zero. This is a joint work with Professors H. Esnault and K. Oguiso.

In this talk, Bayesian semiparametric methods for function estimation and model selection problems are presented. This talk is designed to provide graduate students and researchers with an introduction to Bayesian semiparametric inference. The orientation is methodological rather than theoretical, but such asymptotic theory as is necessary for a proper understanding and validating specific Bayesian methods will be also covered in detail. The materials will include three aspects of Bayesian inference, (i) Fundamentals (ii) Asymptotics and (iii) Advanced models, focusing on nonparametric and semiparametric methods. For function estimation, the Bayesian semiparametric models using Gaussian process priors are discussed. For model selection, Bayes factors are explained for dealing with lack of fit in regression and goodness of fit in density estimation problems.

  Classification of topological types of Calabi-Yau threefolds is still an open problem. In this talk, we explain how to construct some new Calabi-Yau threefolds via McKay correspondence. Such construction is based on classification of automorphism groups of smooth quintic threefolds.

  The F-conjecture is a long-standing conjecture on the structure of the nef cone of the moduli space of (pointed) curves. In this talk, I will explain how one can translate the $S_n$-invariant version of the conjecture into a feasibility problem in polyhedral geometry, which is purely computational. The main ingredients are the graphical algebra in classical invariant theory, embeddings into toric varieties, and the tropical compactification. Also I will propose a stronger statement which might be true for the genus zero case in general, and give some evidences. This is joint work with David Swinarski.

  In this talk we will introduce the varieties of minimal rational tangents (VMRT), and show how the projective geometry of VMRT plays a key role in understanding the geometry of projective manifolds.

  The F-conjecture is a long-standing conjecture on the structure of the nef cone of the moduli space of (pointed) curves. In this talk, I will explain how one can translate the $S_n$-invariant version of the conjecture into a feasibility problem in polyhedral geometry, which is purely computational. The main ingredients are the graphical algebra in classical invariant theory, embeddings into toric varieties, and the tropical compactification. Also I will propose a stronger statement which might be true for the genus zero case in general, and give some evidences. This is joint work with David Swinarski.

In this talk, aimed at a general mathematical audience, I will revisit the classical problem of simplifying polynomials in one variable by Tschirnhaus transformations. Surprisingly, many of the old questions are still open. I will restate them in geometric terms and discuss recent progress in this area.

Let $overline{M_{g, n}}$ be the moduli space of
stable curves of genus $g$ with $n$ marked points.
It is a classical problem in algebraic geometry
to determine which of these spaces are rational over $mathbb C$. 
In this talk, based on joint work with Matthieu Florence,
I will address the rationality problem for twisted forms
of $overline{M_{g, n}}$.  Twisted forms of $overline{M_{g, n}}$
are of interest because they shed light on the arithmetic
geometry of $overline{M_{g, n}}$,
and because they are coarse moduli spaces for natural moduli problems
in their own right. 

A classical result of Yu.~I.~Manin and P.~Swinnerton-Dyer
asserts that every form of $overline{M_{0, 5}}$ is rational.
(Recall that the $F$-forms $overline{M_{0, 5}}$ are precisely
the del Pezzo surfaces of degree $5$ defined over $F$.) 
Mathieu Florence and I recently showed that this theorem
continues to hold for forms of  $overline{M_{0, n}}$ if
$n geq 5$ is odd but fails if $n$ is even.  We also have 
similar results for forms of $overline{M_{g, n}}$, 
where $g leq 5$ (and $n$ is small). Our approach is based on reducing
the rationality problem for twisted forms of $overline{M_{g, n}}$
to certain special cases of the Noether problem. In the talk I plan
to survey some background material on $overline{M_{g, n}}$ and 
twisted forms, and explain how the Noether problem 
arises in this context. 

일시: 5월 20일(금) 12:00 ~ 13:15

장소: 자연과학동 E6-1 3435호

주제: 기하학자의 꿈 - 공간의 분류와 대칭성

강연자: 김영훈 교수 (서울대)

Abstract:  유클리드, 데카르트, 리만, 클라인, 그로덴딕을 통해 기하학의 역사를 조망해보고 공간의 분류와 대칭성의 탐구라는 기하학자의 꿈이 어떤 것인지 얘기하고자 한다. 구체적인 예로서 곡선의 경우에 대해 생각해 본다.

참가 링크: http://goo.gl/forms/mqr7RHvXl7

In 2009, US National Research Council of the National Academies published a report, called “A New Biology for the 21st Century”. One of the major emphases in the New Biology is the integration between biology and mathematics. As explosion of biological measurements takes place in biology due to rapid technology development in recent years, the challenge lies in how to connect and make sense of the massive experimental data collected in various forms at different spatial and temporal scales. Mathematical modeling is becoming an increasingly important tool that enables better understanding of the complex data in biology. In this talk, I will use research carried out in my group as examples to illustrate how mathematics can help discover new knowledge in biology as well as how biological models introduce new challenges and opportunities in mathematics.

The classic Plateau's problem asks if given a smooth simple closed curve in space, does there exist a smooth orientable (embedded) surface-with-boundary spanning that curve, and having least area among all surfaces spanning the given curve? While the answer is in the affirmative, to solve Plateau's problem requires developing the theory of currents; these are generalized orientable surfaces-with-boundary. In this talk we study Plateau's problem for $C^{1,alpha}$ tangentially immersed boundaries.

 Harmonic classes for a simplicial complex are the kernel of its combinatorial Laplacians. They contain more information than homology classes for various data analytic purposes. 

In this talk, we will discuss harmonic classes as a new method for analyzing simplicial complexes associated with a given data set. As an application, we will present an intriguing connection between harmonic classes and effective conductance for a network.

  2016년 3월, 세계를 떠들썩하게 만든 세기의 대결이 펼쳐졌다. 구글 딥마인드(Google DeepMind)가 선보인 인공지능 알파고(AlphaGo)와 바둑기사 이세돌 9단의 5번기였다. "구글 딥마인드 챌린지 매치”의 뚜껑을 열어보니 결과는 알파고의 4대 1 승리, 이세돌 9단의 완패로 대결의 막을 내리게 된다.

 본 강연에서는, 알파고의 기보를 통해 알파고가 세기의 대결에서 보여준 여러 능력들을 ‘바둑이론’을 통해 점검하고, 알파고를 통해 구글이 궁극적으로 성취하고 도달하고자 하는 지향점을 검토하며, 구글 인공지능 프로젝트에 바둑적 관점이 제공할 수 있는 시사점은 무엇인지 말하고자 한다.

A new algorithm will be presented for the path tsp, with an improved analysis and ratio. After the starting idea of deleting some edges of Christofides’ trees, we do parity correction and eventual reconnection, taking the salesman to travel through a linear program determining the conditional probabilities for some of his choices; through matroid partition of a set of different matroids for a better choice of his initial spanning trees; and through some other adventures and misadventures.
The proofs proceed by global and intuitively justified steps, where the trees do not hide the forests.

One more pleasant piece of news is that we get closer to the conjectured approximation ratio of 3/2, and a hopefully last misadventure before finishing up this problem is that we still have to add 1/34 to this ratio, and also for the integrality gap. (The previous result was 8/5 with slight improvements.)

This is joint work with Anke van Zuylen.

현재 의료와 헬스케어 산업은 변혁의 시기를 지나고 있다. 의료와 헬스케어 분야를 근본적으로 변화시키고 있는 이 변혁의 시발점은 기하급수적인 디지털 기술의 발전이다. '디지털 헬스케어’ 라고 불리는 이 새로운 분야는 기존의 헬스케어 및 의료 기술이 디지털 기술과 융합되면서 태동되었다. 스마트폰, 웨어러블 디바이스, 사물인터넷, 인공지능, 3D 프린터, 클라우드 컴퓨팅, 증강 현실 등의 디지털 기술 혁신들은 헬스케어에 접목되어 기존의 건강 관리와 질병 진단 및 치료의 판도를 뒤집어 놓고 있다. 디지털 헬스케어 분야에서 어떠한 변화가 일어나고 있으며, 이 변화를 우리는 어떠한 시각으로 바라봐야 할지, 또한 우리가 맞이할 새로운 숙제와 이슈들은 무엇인지를 포괄적으로 살펴본다.