# 세미나 및 콜로퀴엄

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We discuss a sensitivity analysis of long-term cash flows. The price of the cash flow at time zero is given by the pricing operator of a Markov diffusion acting on the cash flow function. We study the extent to which the price of the cash flow is affected by small perturbations of the underlying Markov diffusion. The main tool is the Hansen--Scheinkman decomposition, which is a method to express the cash flow in terms of eigenvalues and eigenfunctions of the pricing operator. By incorporating techniques of Malliavin calculus, the sensitivities of long-term cash flows can be represented via simple expressions in terms of the eigenvalue and the eigenfunction.

This talk introduces one of surprising empirical regularities observed in economics: Pareto distributions are everywhere. Wealth and income, the size of cities and firms, stock market returns, to list but a few, are all known to follow a Pareto distribution. I first highlight key empirical facts and describe some economic theories that have been proposed to explain the regularity. To be more specific, I will put emphasis on inequality in income distributions. A simple mechanism as well as more complex random growth models that give rise to Pareto distributions will be discussed to explore dynamics of income inequality.

Direct sampling method (DSM) is a well-known, non-iterative imaging technique in inverse scattering problem. Throughout various researches, DSM has been applied various research area e.g., diffusion tomography, electrical impedance tomography, source detection in stratified ocean waveguide, etc.; however, due to the small number of incident fields or sources, further improvements are still required. In this presentation, we carefully identify mathematical structure of indicator function of DSM to show the feasibilities and limitations, design a method of improvement, and apply in real-world microwave imaging. Simulations results with synthetic and experimental data are shown for supporting identified structure.

For a graph G, let f_{2}(G) denote the largest number of vertices in a 2-regular subgraph of G. We determine the minimum of f_{2}(G) over 3-regular n-vertex simple graphs G.

More generally, every n-vertex multigraph with maximum degree 3 and m edges has a 2-regular subgraph that omits at most max{0, ⎣(3n-2m+c-1)/2⎦} vertices.

These bounds are sharp; we describe the extremal multigraphs.

Given a group G and a manifold M, can one describe all the actions of G on M? This is a basic, natural question in geometric topology, but also a very difficult one -- even in the case where M is 1-dimensional, and G is a familiar, finitely generated group.

This talk will introduce the theory of groups acting on 1-manifolds, through the study of orderable groups. I will describe some connections between this theory and themes in topology and dynamics (like rigidity and foliation theory ), some current open problems, and indicate new approaches coming from recent joint work with C. Rivas.

One fairly standard version of the Riemann Hypothesis (RH) is that a specific probability density on the real line has a moment generating function (Laplace transform) that, as an analytic function on the complex plane, has all its zeros pure imaginary. We'll review a series of results that span the period from the 1920's to 2018 concerning a perturbed version of the RH. In that perturbed version, due to Polya, the log of the probability density is modified by a quadratic term.

This gives rise to an implicitly defined real constant known as the de Bruijn-Newman Constant, Lambda. The conjecture and now theorem (Newman 1976, Rodgers and Tao 2018) that Lambda is greater than or equal to zero is complementary to the RH which is equivalent to Lambda less than or equal to zero; The conjecture/theorem is a version of the dictum that the RH, if true, is only barely so. Until very recently, the best upper bound, was a 2009 result of Ki, Kim and Lee that Lambda is strictly less than 1/2.

A wallspace, which is named by Haglund-Paulin, has been used as a powerful tool for geometric group theory. The dual cube complex of a wallspace is the CAT(0) cube complex whose pocset structure is identical to the wallspace. In this talk, we will focus on dual cube complexes from the hyperbolic plane with finitely many simple closed geodesics in a finite-area hyperbolic surface, construct a Dehn-twist-like quasi-isometry, and give an answer to the problem suggested by Koberda.

In the k-cut problem, we are given an edge-weighted graph G and an integer k, and have to remove a set of edges with minimum total weight so that G has at least k connected components. This problem has been studied various algorithmic perspectives including randomized algorithms, fixed-parameter tractable algorithms, and approximation algorithms. Their proofs of performance guarantees often reveal elegant structures for cuts in graphs.

It has still remained an open problem to (a) improve the runtime of exact algorithms, and (b) to get better approximation algorithms. In this talk, I will give an overview on recent progresses on both exact and approximation algorithms. Our algorithms are inspired by structural similarities between k-cut and the k-clique problem.

In this talk, I'll generalize the proper base change theorem in étale cohomology to space-valued sheaves, and provide two applications to the étale homotopy theory: the profinite étale homotopy type functor commutes with finite products and the symmetric powers of proper schemes over a separably closed field, respectively. In particular, the commutativity of the étale fundamental groups with finite products will be extended to all higher homotopy groups. In the applications, we'll see the advantage of the infinity categorical approach in étale homotopy theory over the model categorical one.

1. 학과 및 학생명 : 수리과학과 장부식

2. 심 사 위 원 장 : 한상근

3. 심 사 위 원 : 황강욱, 채수찬(기술경영학부), 엄재용(기술경영학부),

이주형(가천대 소프트웨어학과)

4. 논 문 명 : 세 계층으로 나뉘어진 빅데이터 마켓 모델

Three Hierarchical Levels of Big-data Market Model

over Multiple Data Sources for the Internet

of Things

5. 심 사 일 시 : 2018.11.12(월), 16:00

6. 심 사 장 소 : KAIST 산업경영학동 3221호. 끝.

In this talk we outline the construction of certain higher Chow cycles on Abelian surfaces. The existence of these cycles is predicted by certain conjectures on special values of L-functions in the local case and by the existence of certain modular forms in the case of the universal family over a Shimura curve - providing evidence for the conjecture described in the first talk. The construction uses beautiful 19th century constructions of Kummer and Humbert.

Gross and Zagier made a conjecture on the algebraicity of values of certain `higher' Greens functions at special points. Mellit proved a few cases by linking it to the existence of certain higher Chow cycles. Viazovska proved a few cases by linking it to Borcherds lifts of modular forms. We formulate a conjecture linking modular forms and higher Chow cycles which relates the two approaches and also describe a construction of higher Chow cycles which allows us to prove special cases of the Gross-Zagier conjecture as well as provide evidence for our conjecture.

Some reaction-diffusion systems appearing in chemistry have a natural entropy structure. In a series of works with K. Fellner and B. Q. Tang, we studied the effect of this structure on the large time behavior of the solutions of those systems. In some cases it is possible to obtain an explicit and quantitative estimate of convergence towards the equilibrium. We propose to explain the estimates starting from explicit examples of chemical networks.

The extremal number ex(n,F) of a graph F is the maximum number of edges in an n-vertex graph not containing F as a subgraph. A real number r∈[1,2] is realisable if there exists a graph F with ex(n , F) = Θ(n^{r}). Several decades ago, Erdős and Simonovits conjectured that every rational number in [1,2] is realisable. Despite decades of effort, the only known realisable numbers are 1,7/5,2, and the numbers of the form 1+(1/m), 2-(1/m), 2-(2/m) for integers m≥1. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than two numbers 1 and 2.

We discuss some recent progress on the conjecture of Erdős and Simonovits. First, we show that 2-(a/b) is realisable for any integers a,b≥1 with b>a and b≡±1 (mod a). This includes all previously known ones, and gives infinitely many limit points 2-(1/m) in the set of all realisable numbers as a consequence.

Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.

This is joint work with Jaehoon Kim and Hong Liu.

Recently, there has been considerable interest in both inference and predictions for compartmental epidemic models on multiple physical scales. For instance, one could be interested in analyzing response of immune system to infection within a single host or in describing infectuous interactions in a population of hosts. Both viral invasions and global pandemics are often described by similar mathematical constructs known as SIR models. In this talk I will review some basic concepts related to such models across scales and present a simple unifying framework that allows to conceptually connect both deterministic (e.g., population level) and stochastic (e.g., molecular level) SIR models with the help of tools of statistical theory of survival analysis.