The MathNet Korea
Information Center for Mathematical Science

### 세미나

Information Center for Mathematical Science

#### 세미나

##### 한국고등과학원 세미나

We will talk about the structure of Mordell-Weil, Selmer, and Tate-Shafarevich groups over \mathbb{Z}_p extensions as \Lambda-modules. If time permits, we will talk about the similar results for towers of modular curves in the setting of Hida's recent paper "Analytic variation of Tate-Shafarevich groups".

##### 한국고등과학원 세미나

We are reading two papers establishing the existence of a volume preserving map; On the volume elements on a manifold (J. Moser) and On a partial differential equation involving the jacobian determinant (B. Dacorogna and J. Moser). The regularity threshold on volume elements turns out C^{alpha} ( 0< alpha <1), which makes it hard to apply the PDE result directly in proving C^{1} volume preserving maps approximated by C^{infty} ones. This issue is settled by A. Avila in On the regularization of conservative maps, which will be mentioned only briefly.

##### 2018 아주대학교 수학과 colloquium

In cluster randomized designs (CRDs) structured with balanced ranked set sampling (RSS), we propose a pooled pivotal test to improve the power of detecting any treatment effect by stablizing the estimation of variance components when groups are unevenly sized. We numerically compare the power of the pooled pivotal test with the original pivotal test based on the CRD using balanced RSS and the F-test based on the CRD using simple random sampling (SRS) via simulation. Further, we provide an example using educational data.

##### 아주대학교 수학과 세미나

(Joint work with Oleg Pikhurko and Maryam Sharifzadeh) For a sequence $(H_i)_{i=1}^k$ of graphs, let $\rm{nim}(n;H_1,\ldots, H_k)$ denote the maximum number of edges not contained in any monochromatic copy of $H_i$ in colour $i$, for any colour $i$, over all $k$-edge-colourings of~$K_n$. When each $H_i$ is connected and non-bipartite, we introduce a variant of Ramsey number that determines the limit of $\rm{nim}(n;H_1,\ldots, H_k)/{n\choose 2}$ as $n\to\infty$ and prove the corresponding stability result. Furthermore, if each $H_i$ is what we call \emph{homomorphism-critical} (in particular if each $H_i$ is a clique), then we determine $\rm{nim}(n;H_1,\ldots, H_k)$ exactly for all sufficiently large~$n$. The special case $\rm{nim}(n;K_3,K_3,K_3)$ of our result answers a question of Ma. For bipartite graphs, we mainly concentrate on the two-colour symmetric case (i.e., when $k=2$ and $H_1=H_2$). It is trivial to see that $\rm{nim}(n;H,H)$ is at least $\rm{ex}(n,H)$, the maximum size of an $H$-free graph on $n$ vertices. Keevash and Sudakov showed that equality holds if $H$ is the $4$-cycle and $n$ is large; recently Ma extended their result to an infinite family of bipartite graphs. We provide a larger family of bipartite graphs for which $\rm{nim}(n;H,H)=\ex(n,H)$. For a general bipartite graph $H$, we show that $\rm{nim}(n;H,H)$ is always within a constant additive error from $\rm{ex}(n,H)$, i.e.,~$\rm{nim}(n;H,H)= \rm{ex}(n,H)+O_H(1)$.

##### 2018 아주대학교 수학과 colloquium

In this talk, an accurate and efficient finite difference method for solving the Black–Scholes (BS) equation without boundary conditions is presented. The BS equation is a backward parabolic partial differential equation for financial option pricing and hedging. When we solve the BS equation numerically, we typically need an artificial far-field boundary condition such as the Dirichlet, Neumann, linearity, or partial differential equation boundary condition. However, in this work, we propose an explicit finite difference scheme which does not use a far-field boundary condition to solve the BS equation numerically. The main idea of the proposed method is that we reduce one or two computational grid points and only compute the updated numerical solution on that new grid points at each time step. By using this approach, we do not need a boundary condition. This procedure works because option pricing and computation of the Greeks use the values at a couple of grid points neighboring an interesting spot. To demonstrate the efficiency and accuracy of the new algorithm, we perform the numerical experiments such as pricing and computation of the Greeks of the vanilla call, cash-or-nothing, power, and powered options. The computational results show excellent agreement with analytical solutions.

##### 아주대학교 수학과 세미나

근래 가장 큰 트렌드인 빅데이터 분석과 AI가 실제로는 기업들이 도입하기에 어디까지 준비되어 있는지 좀더 실체를 들여다보고, 그것이 기업의 대 고객 비지니스에 미치는 영향이 무엇인지, 마지막으로 향후 산업 지형에 끼칠 영향에 대해 짚어 보고자 합니다.

##### 2018 아주대학교 수학과 colloquium
• Title : Eigenvalues of graph
• Speaker : 오수일
• Data : 2018-05-11 17:00:00
• Host : 아주대학교
• Place :

A (Laplacian) eigenvalue of a graph G is an eigenvalue of its (Laplacian) adjacency matrix. Many relations between (Laplacian) eigenvalues and graph parameters were investigated for a long time. In 1972, Fiedler proved 2(G)  (G) for a non-complete simple graph G, where 2(G) and (G) are the second smallest Laplacian eigenvalue and vertex-connectivity of a graph G, respectively. A lot of research was stimulated by his research, and now we call 2(G) the algebraic connectivity. For a d-regular graph G, we have 2(G) = d

##### 2018 아주대학교 수학과 colloquium

Exploring electron’s degree of freedom in solids has enabled the developments of new electronics with novel functionalities. For example, electron’s charge degree of freedom has led to the development of a variety of electronics around us. Electron’s spin degree of freedom has provided us methods to store information in magnetic materials. More recently, electrons in a 2D periodic crystal are found to possess a new type of degree of freedom that may be useful beyond charge and spin, which is called valley. In this talk, I will introduce electron’s valley degree of freedom in solids and discuss topological phenomena arising from valley in 2D crystals.

##### 아주대학교 수학과 세미나

In this talk I will introduce the concepts of higher equivariant and invariant topological complexity and discuss their properties. Then I will compare them with equivariant LS-category. We give lower and upper bounds for these new invariant. This is a joint work with Marzieh Bayeh.

##### 고려대학교 수학과 세미나
• Title : Conservation laws
• Speaker : 한종규
• Data : 2018-03-15 15:30:00
• Host : KOREA University
• Place :