# Seminars & Colloquia

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E6-1, ROOM 1401
Discrete Math
Ilkyoo Choi (Hankuk University of Foreign Studies, Yongin-si)
Largest 2-regular subgraphs in 3-regular graphs

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.

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.

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.

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.

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.

Stochastic models of chemical reactions are of interest when considering chemical systems in which the stochastic effects are important. Examples of such systems in mathematical biology are frequent both in molecular- and population-level models (e.g., gene transcription or onset of an epidemic). In this talk I will give a brief introduction to the theory of Markovian stochastic reaction networks and describe some of the examples of its application. This talk will be understandable by undergraduate students who has a basic probability background.

It has been established now that long range dependence of stationary infinitely processes is strongly related to ergodic-theoretical properties of the shift operatior acting on its L'evy measure. We discuss one case in which these ideas can can be extended to stationary infinitely divisible random flelds.

In this talk, I will present a recent joint work with I.-J. Jeong on the issue of ill- and wellposedness for the incompressible Hall magnetohydrodynamic (MHD) equation without resistivity. This PDE is a fluid description of plasmas, where the effect of collisions is neglected (no resistivity), while the motion of electrons relative to ions (Hall current term) is taken into account. The Hall current term endows the magnetic field equation with a quasilinear dispersive character, which is key to our mechanism for both illposedness and wellposedness.

Financial markets are often driven by latent factors which traders cannot observe. Here, we address an algorithmic trading problem with collections of heterogeneous agents who aim to perform statistical arbitrage, where all agents filter the latent states of the world, and their trading actions have permanent and temporary price impact. This leads to a large stochastic game with heterogeneous agents. We solve the stochastic game by investigating its mean-field game (MFG) limit, with sub-populations of heterogenous agents, and, using a convex analysis approach, we show that the solution is characterized by a vector-valued forward-backward stochastic differential equation (FBSDE). We demonstrate that the FBSDE admits a unique solution, obtain it in closed-form, and characterize the optimal behaviour of the agents in the MFG equilibrium. Moreover, we prove the MFG equilibrium provides an -Nash equilibrium for the finite player game. We conclude by illustrating the behaviour of agents using the optimal MFG strategy through simulated examples.

Symplectic geometry has one of its origins in Hamiltonian dynamics. In the late 60s Arnold made a fundamental conjecture about the minimal number of periodic orbits of Hamiltonian vector fields. This is a far-reaching generalization of Poincaré's last geometric theorem and completely changed the field of symplectic geometry. In the last 30 years symplectic geometry has been tremendously developed due to the theory of holomorphic curves by Gromov and Floer homology theory by Floer. I will give a gentle introduction to the field of symplectic geometry and explain how modern methods give rise to existence results for periodic orbits and discover rigidity phenomena in symplectic geometry.

One of the most fundamental questions in representation theory asks for a description of the simple representations. I will give an introduction to this problem with an emphasis

on the representation theory of Lie groups and algebras.

A recurring theme is the appearance of geometric techniques in seemingly algebraic problems. I will also emphasise the important role played by invariant forms and signature.

One of the most fundamental questions in representation theory asks for a description of the simple representations. I will give an introduction to this problem with an emphasis

on the representation theory of Lie groups and algebras.

A recurring theme is the appearance of geometric techniques in seemingly algebraic problems. I will also emphasise the important role played by invariant forms and signature.

Recently, the study on particle-

uid system is gathering a lot of attentions due

to their applications, for example, in the study of sedimentation phenomena,

fuel injector in engines, and compressibility of droplets of the spray, etc. In this

talk, we discuss the global existence of weak solutions for the system consisting

of the BGK model of Boltzmann equation and incompressible Navier-Stokes

equations coupled through a drag forcing term.

We are interested in the asymptotic behavior of a ground state vector solution

for the following coupled nonlinear Schrodinger system

8<

:

u1 ? 1u1 + 11(u1)3 + 12u1(u2)2 ? 13u1(u3)2 = 0

u2 ? 2u2 + 21(u1)2u2 + 22(u2)3 ? 23u2(u3)2 = 0

u3 ? 3u3 ? 31(u1)2u3 ? 32(u2)2u3 + 33(u3)3 = 0

in

and @ui

@n = 0 on @

when ; > 0 are very large. We will see the existence of a ground

state vector solution for the system when ; are large and satisfy some relation. Then we

also see the asymptotic behavior of the solution as ; ! 1 under an additional condition.

In this talk, we are going to discuss about the well-posedness theory of dispersive equations (KdV- and

NLS-type equations) posed on T, via analytic methods. I am going to brie

y explain some notions and

methodologies required to study the (low regularity) well-posedness problems. And then, I will show the

main dierence of analysis on between non-periodic and periodic boundary conditions, and will explain

how to deal some typical, but signicant, phenomena arising in the periodic problems. We are, precisely,

going to consider fth-order (modied) KdV, higher-order Kawahara, fourth-order NLS and modied

Kawahara equations, and mainly show the Global well-posedness of the modied Kawahara in L2(T).

We say a subgraph $H$ of an edge-colored graph is rainbow if all edges in $H$ has distinct colors. The concept of rainbow subgraphs generalizes the concept of transversals in latin squares. In this talk, we discuss how these concepts are related and we introduce a result regarding approximate decompositions of graphs into rainbow subgraphs. This has implications on transversals in latin square. It is based on a joint work with K"uhn, Kupavskii and Osthus.

Graphs are mathematical structures used to model pairwise relations between objects. Graph decomposition problems ask to partition the edges of large/dense graphs into small/sparse graphs. In this talk, we introduce several famous graph decomposition problems, related puzzles and known results on the problems.

Given an action of a finite group G on an affine space A^n, the quotient variety A^n/G have been a subject of interest for over a century, partially motivated by Noether's problem concerning its rationality. Weaker properties of this variety were extensively studied, relating it to the Inverse Galois problem, parametrization of Galois extensions and essential dimension. One of its weakest possible properties, (very) weak approximation, is the subject of a more recent conjecture by Colliot-Th\'el\`ene. We shall discuss the above rationality properties, the conjecture, and the (recent) progress towards it.

Modern high-throughput technologies provide information about high dimensional features in biomedical research. These biological entities are often related to each other. When characterized well, an inferred network can lead to useful insights to researchers. However, a graph/network estimation problem becomes challenging due to the nonlinearity and outliers. In this talk, I will discuss recent network estimation approaches that address these challenges by modeling conditional medians with non-parametric methods.

High-dimensional count datasets often occur in biomedical studies where gene expression levels are measured through the RNA-sequencing technology. Although the sequencing is more promising than microarrays, the data analysis becomes more involved because the Gaussian assumption cannot be made. Besides, more refined modeling of the count data often involves many approximations, making it less appropriate for small sample problems. In this talk, I will discuss the challenges and opportunities in analyzing high-dimensional count data with various examples.

Spatial sampling is particularly important for environmental statistics. A simple reasoning developed in Grafström and Tillé (2013) shows that under a self-correlated linear model, it is more efficient to select a well-spread sample in space. If we select two neighbouring units in a sample, we will tend to collect partially redundant information. Grafström and Lundström (2013) discuss at length the concept of spreading, also known as spatial balancing and its implication on estimation. The Generalized Random Tesselation Sampling GRTS design has been proposed by Stevens Jr. and Olsen (1999, 2004, 2003) to select spread samples. The pivotal method has been proposed by Deville and Tillé (2000). Grafström et al. (2012) proposed to use the pivotal method for spatial sampling. This method, called the local pivotal method, consists, at each step, in comparing two neighboring units. If the probability of one of these two units is increased, the probability of the other is decreased, which induces a repulsion between the units. The natural extension of this idea is to confront a group of units. The local pivot method was generalized by Grafström and Tillé (2013) to provide samples that are both well-balanced in space and balanced on the totals of the auxiliary variables. This method is called the local cube method. We also propose a new method that enables us to select spreader samples that all existing methods and allows the construction of periodic sampling plans when these plans exist.

M. Krasner introduced a notion of valued hyperfield analogous to a valued field with a multivalued addition operation, and used it to do a theory of limits of local fields. P. Deligne did the theory of limits of local fields in a different way by defining a notion of triple, which consists of truncated discrete valuation rings and some additional data. Typical examples of a valued hyperfields and truncated discrete valuation rings are the $n$-th valued hyper field, which is quotient of a valued field by a multiplicative subgroup of the form +m^n$, where $m$ is the maximal ideal of a valuation ring, and the $n$-th residue ring, which is a quotient of a valuation ring by the $n$-th power of the maximal ideal.J. Tolliver showed that discrete valued hyperfields and triples are essentially same, stated by P. Deligne without a proof. W. Lee and the author showed that given complete discrete valued fields of mixed characteristic with perfect residue fields, any homomorphism between the $n$-th residue rings of the valued fields is lifted to a homomorphism between the valued fields for large enough $n$. This lifting process is functorial.

Motivated by above results, we show that given complete discrete valued fields of mixed characteristic with perfect residue fields, any homomorphism between the $n$-th valued hyperfields of the valued fields can be lifted to a homomorphism between the valued fields for large enough $n$, which is functorial. We also compute an upper bound of such a minimal $n$ effectively depending only on the ramification index. Most of all, any homomorphism between the first valued hyperfields of valued fields is uniquely lifted to a homomorphism between the valued fields in the case of tamely ramified valued fields. From this lifting result, we prove a relative completeness AKE-theorem via valued hyperfields for finitely ramified valued fields with perfect residue fields.

Any structure whose language is finite has a model of graph theory which is bi-interpretable with it. From this idea, Mekler further developed a way of interpreting a model into a group. This Mekler's construction preserves various model-theoretic properties such as stability, simplicity, and NTP2, thus helps us find new group examples in model theory. In this talk, I will introduce to you what Mekler's construction is and briefly show that this preserves NTP1.