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Room B232, IBS (기초과학연구원)
2019 IBS Summer Research Program on Algorithms and Complexity in Discrete Structures

### Organizers

Sang-il Oum, IBS Discrete Mathematics Group and KAIST Eunjung Kim, LAMSADE-CNRS, France## Schedule

### July 22 Monday

10:00-11:00 Introduction

11:00-12:00 Open Problems

### July 23 Tuesday

11:00-10:30 Stefan Kratsch, Humboldt-Universität zu Berlin, Germany

Elimination Distances, Blocking Sets, and Kernels for Vertex Cover

10:45-11:15 Benjamin Bergougnoux, University Clermont Auvergne, France

More applications of the d-neighbor equivalence

11:30-12:00 Yixin Cao, Hong Kong Polytechnic University, China

Enumerating Maximal Induced Subgraphs

### July 24 Wednesday

### July 25 Thursday

10:00-11:00 Nick Brettell, Eindhoven University of Technology, Netherlands

Recent work on characterising matroids representable over finite fields

11:30-12:00 Mamadou M. Kanté, University Clermont Auvergne, France

On recognising k-letter graphs

### July 26 Friday

10:00-11:00 O-joung Kwon (권오정), Incheon National University, Korea

The grid theorem for vertex-minors

11:00-12:00 Progress Report

### July 29 Monday

10:00-11:00 Archontia Giannopoulou, National and Kapodistrian University of Athens, Greece

The directed flat wall theorem

11:30-12:00 Eunjung Kim (김은정), LAMSADE-CNRS, France

Subcubic even-hole-free graphs have a constant treewidth

### July 30 Tuesday

10:00-11:00 Pierre Aboulker, ENS Ulm, France

Generalizations of the geometric de Bruijn Erdős Theorem

11:30-12:00 Michael Dobbins, Binghamton University, USA

Barycenters of points in polytope skeleta

### July 31 Wednesday

10:00-11:00 Magnus Wahlström, Royal Holloway, University of London, UK

T.B.A.

11:30-12:00 Édouard Bonnet, ENS Lyon, France

The FPT/W[1]-hard dichotomy of Max Independent Set in H-free graphs

### August 1 Thursday

10:00-11:00 Euiwoong Lee (이의웅), NYU, USA

Losing treewidth by separating subsets

11:30-12:00 Sang-il Oum (엄상일), IBS Discrete Mathematics Group and KAIST, Korea

Branch-depth: Generalizing tree-depth of graphs

### August 2 Friday

10:00-11:00 Dabeen Lee (이다빈), IBS Discrete Mathematics Group, Korea

t-perfect graphs and the stable set problem

11:00-12:00 Progress Report

August 5-9: Free Discussions / Research Collaborations / Progress Report

Tea time: Every weekday 3:30pm

## Abstracts

### July 23 Tuesday

#### Stefan Kratsch, Elimination Distances, Blocking Sets, and Kernels for Vertex Cover

The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive and negative results for kernelization for Vertex Cover subject to different parameters and graph classes, we seek to unify and generalize them using so-called blocking sets, which have played implicit and explicit roles in many results. We show that in the most-studied setting, parameterized by the size of a deletion set to a specified graph class C, bounded minimal blocking set size is necessary but not sufficient to get a polynomial kernelization. Under mild technical assumptions, bounded minimal blocking set size is showed to allow an essentially tight efficient reduction in the number of connected components. We then determine the exact maximum size of minimal blocking sets for graphs of bounded elimination distance to any hereditary class C, including the case of graphs of bounded treedepth. We get similar but not tight bounds for certain non-hereditary classes C, including the class CLP of graphs where integral and fractional vertex cover size coincide. These bounds allow us to derive polynomial kernels for Vertex Cover parameterized by the size of a deletion set to graphs of bounded elimination distance to, e.g., forest, bipartite, or CLP graphs.

#### Benjamin Bergougnoux, More applications of the d-neighbor equivalence

In this talk, I will present a framework which gives efficient algorithms for several connectivity problems such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. For all these problems, we obtain efficient algorithms parameterized respectively by tree-width, clique-width, Q-rank-width, rank-width and maximum induced matching width. Our approach simplifies and unifies the known algorithms for each of the parameters and match asymptotically also the running time of the best algorithms for Vertex Cover and Dominating Set. Our framework is based on the d-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance and the generalizing power of this equivalence relation on width measures. We also prove that this equivalence relation could be useful for Max Cut: a W[1]-hard problem parameterized by clique-width.

Paper available on HAL : https://hal.archives-ouvertes.fr/hal-01799573

#### Yixin Cao, Enumerating Maximal Induced Subgraphs

We study efficient algorithms for the maximal induced subgraphs problem: Given a graph on nvertices, enumerate all its maximal induced subgraphs that belong to a particular hereditary graph class. While its optimization version, known as the vertex deletion problem in literature, has been intensively studied, algorithms for the enumeration version exist only for few simple graph classes, e.g., independent sets and cliques. Enumeration algorithms are mainly categorized in three complexity classes, polynomial total time (polynomial on n and the total number of solutions), incremental polynomial time (for all s, the time to output the first s solutions is polynomial on n and s), and polynomial delay (for all s, the time to output the first s solutions is polynomial on n and linear on s). We first show that the maximal induced subgraphs problem can be solved in polynomial total time if and only if it can be solved in incremental polynomial time. We then propose general approaches to develop polynomial-delay algorithms and incremental-polynomial-time algorithms for the problem. They enable us to develop simple algorithms to solve the problem with polynomial delay for a large number of graph classes, and in incremental polynomial time for interval graphs, chordal graphs as well as two of its well-studied subclasses, and all graph classes with finite forbidden induced subgraphs.

### July 25 Thursday

#### Nick Brettell, Recent work on characterising matroids representable over finite fields

Characterising when a matroid is representable over a certain field or set of fields is one of the oldest problems in matroid theory, dating back to Whitney’s 1935 paper. In this talk, I will discuss some of the more recent work in this area, with a focus on work I have been involved with towards characterising matroids representable over all fields of size at least four.

#### Mamadou M. Kanté, On recognising k-letter graphs

k-letter graphs are a subclass of graphs of linear clique-width k, but are well-quasi-ordered by the induced subgraph ordering. I present a simple FPT algorithm for recognising k-letter graphs and also an upper bound on the size of minimal obstructions. The characterisation is based on a simple observation allowing an MSOL-definability. (joint work with V. Lozin)

### July 26 Friday

#### O-joung Kwon (권오정), The grid theorem for vertex-minors

We prove that, for a circle graph H, every graph with sufficiently large rank-width contains a vertex-minor isomorphic to H. This is joint work with Jim Geelen, Rose McCarty, and Paul Wollan.

### July 29 Monday

#### Archontia Giannopoulou, The directed flat wall theorem

At the core of the Robertson-Seymour theory of Graph Minors lies a powerful structure theorem which captures, for any fixed graph H, the common structural features of all the graphs not containing Has a minor [Neil Robertson, Paul D. Seymour: Graph Minors. XVI. Excluding a non-planar graph. J. Comb. Theory, Ser. B 89(1): 43-76 (2003)]. An important step towards this structure theorem is the Flat Wall Theorem [Neil Robertson, Paul D. Seymour: Graph Minors .XIII. The Disjoint Paths Problem. J. Comb. Theory, Ser. B 63(1): 65-110 (1995)], which has a lot of algorithmic applications (for example, the minor-testing and the disjoint paths problem with fixed number terminals).

In this paper, we prove the directed analogue of this Flat Wall Theorem. Our result builds on the recent Directed Grid Theorem by two of the authors (Kawarabayashi and Kreutzer), and we hope that this is an important and significant step toward the directed structure theorem, as with the case for the undirected graph for the graph minor project.

Joint work with Ken-ichi Kawarabayashi, Stephan Kreutzer, and O-joung Kwon.

#### Eunjung Kim (김은정), Subcubic even-hole-free graphs have a constant treewidth

It is known that even-hole-free graphs can have arbitrarily large rankwidth, but all known constructions have many high-degree vertices. It has been believed in the structural graph theory community that the rankwidth shall be bounded if the maximum degree is bounded. We prove that there exists a universal constant c such that every even-hole-free graph of degree at most three has treewidth at most c. As a by-product of the proof, we also show that there exists a function f such that for any r, Kr-minor-free and even-hole-free graph has treewidth at most f(r); this extends the result of Silva et. al. (Discrete Applied Mathematics 2010) addressing the case of planar graphs.

This is a joint work with Pierre Aboulker and Isolde Adler.

### July 30 Tuesday

#### Pierre Aboulker, Generalizations of the geometric de Bruijn Erdős Theorem

A classic Theorem of de Bruijn and Erdős states that every noncollinear set of n points in the plane determines at least n distinct lines. The line L(u,v)determined by two points u, v in the plane consists of all points p such that

dist(p,u)+dist(u,v)=dist(p,v) (i.e. u is betweenp and v) or dist(u,p)+dist(p,v)=dist(u,v) (i.e. p is between u and v) or dist(u,v)+dist(v,p)=dist(u,p) (i.e. v is between u and p).With this definition of line L(uv) in an arbitrary metric space (V,dist), Chen and Chvátal conjectured that every metric space on n points, where n is at least 2, has at least n distinct lines or a line that consists of all n points. The talk will survey results on and around this conjecture.

#### Michael Dobbins, Barycenters of points in polytope skeleta

In this talk I will classify the n-tuples of dimensions such that any target point in any d-polytope is the barycenter of points in faces of the prescribed dimensions. Specifically, for a (nk+r)-polytope and target point, we can always find r points in faces of dimension k+1, and n−r points in faces of dimension k, that have their barycenter at the given target. This is tight in the sense that we cannot require even one of the points to be in a face of dimension less than k, and we cannot require more than n−r of the points to be in faces of dimension k. This is joint work with Florian Frick.

### July 31 Wednesday

#### Magnus Wahlström, T.B.A.

#### Édouard Bonnet, The FPT/W[1]-hard dichotomy of Max Independent Set in H-free graphs

Maximum Independent Set (MIS) is in general graphs the paradigmatic W[1]-hard problem. In stark contrast, polynomial-time algorithms are known when the inputs are restricted to structured graph classes such as, for instance, perfect graphs (which includes bipartite graphs, chordal graphs, co-graphs, etc.) or claw-free graphs. In this talk, we introduce some variants of co-graphs with “parameterized noise”, that is, graphs that can be turned into disjoint unions or complete sums by the removal of a certain number of vertices and the edition of a certain number of edges per incident vertex, both controlled by the parameter. We give a series of FPT Turing-reductions on these classes and use them to make some progress on the parameterized complexity of MIS in H-free graphs. We present the current state of the (randomized) FPT vs W[1]-hard dichotomy of MIS in H-free graphs. We then show that for every fixed t≥1, MIS is FPT in P(1,t,t,t)-free graphs, where P(1,t,t,t) is the graph obtained by substituting all the vertices of a four-vertex path but one end of the path by cliques of size t. In particular, this helped finishing the classification when H has at most five vertices.

This follows joint works with Nicolas Bousquet, Pierre Charbit, Stéphan Thomassé, and Rémi Watrigant.

### August 1 Thursday

#### Euiwoong Lee (이의웅), Losing treewidth by separating subsets

We study the problem of deleting the smallest set S of vertices (resp. edges) from a given graph G such that the induced subgraph (resp. subgraph) G∖Sbelongs to some class of graphs H. When graphs in H have treewidth bounded by t, we construct a framework for both vertex and edge deletion versions where approximation algorithms for these problems can be obtained from approximation algorithms for natural graph partitioning problems called k-Subset Vertex Separator and k-Subset Edge Separator.

For the vertex deletion, our framework, combined with the previous best result for k-Subset Vertex Separator, yields improved approximation ratios for fundamental problems such as k-Treewidth Vertex Deletion and Planar-F Vertex Deletion. For the edge deletion, we give improved approximation algorithms for k-Subset Edge Separator and their applications to various problems in bounded-degree graphs.

Joint work with Anupam Gupta, Jason Li, Pasin Manurangsi, and Michal Wlodarczyk.

#### Sang-il Oum (엄상일), Branch-depth: Generalizing tree-depth of graphs

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph G=(V,E) and a subset A of E we let λG(A)be the number of vertices incident with an edge in Aand an edge in E∖A. For a subset X of V, let ρG(X)be the rank of the adjacency matrix between * https://dimag.ibs.re.kr/event/2019-ibs-summer-research-program/ *

####
최석정 강의실 (자연과학동 1401호)
수리과학과 KAIX 테마틱 프로그램 "Algebraic and Topological Methods in Geometry"

### Organizers

이용남, 백형렬

(http://www.hbaik.org/gt-fair-2019)

* 8월 5일 - 8월 14일: Summer school on Geometric Topology and Algebraic Geometry(http://www.hbaik.org/kaix-summer-school-2019)

* 8월 19일 - 8월 23일: Intensive lecture series on Algebraic Geometry첨부된 파일 참고 바랍니다. (Please refer to the file attached.)

### Organizers

Younjin Kim, Sang-il Oum

## Invited Speakers

Jaehoon Kim (김재훈), KAIST Hong Liu (刘鸿), University of Warwick Abhishek Methuku, IBS Discrete Mathematics Group Péter Pál Pach, Budapest University of Technology and Economics## Schedule

### August 12, Monday

11:00am-12:00pm Jaehoon Kim (김재훈): Tree decompositions of graphs without large bipartite holes

12:00pm-1:30pm Lunch

1:30pm-2:30pm Abhishek Methuku: Bipartite Turán problems for ordered graphs

3:00pm-4:00pm Péter Pál Pach: On some applications of graph theory to number theoretic problems

4:30pm-5:30pm Hong Liu: Recent advance in Ramsey-Turán theory

6:00pm-8:00pm Banquet

*https://dimag.ibs.re.kr/event/2019-08-12/*

### Organizers

## Scientific Organizers

BAE, Sunghan, KAIST, Daejeon TAGUCHI, Yuichiro, Tokyo Institute of Technology, Tokyo XU, Fei, Capital Normal University, Beijing YU, Jing, National Taiwan University, Taipei## Local Organizers at KAIST

BAE, Sunghan, KAIST, Daejeon IM, Bo-Hae, KAIST, Daejeon PARK, Jinhyun, KAIST, Daejeon

## Invited Speakers

ARAI, Keisuke, Tokyo Denki University, Tokyo / CHOI, Junhwa, Korea Institute for Advanced Study, Seoul / HARA, Takashi, Tsuda University, Kodaira / HSIA, Liang-Chung, National Taiwan Normal University, Taipei / HUNG, Pin-Chi, Soochow University, Taipei / HWANG, WonTae, Korea Institute for Advanced Study, Seoul / JU, Jangwon, Ulsan University, Ulsan / KIM, Wansu, KAIST, Daejeon / LIM, Subong, Sungkyunkwan University, Suwon / NG, Ming Ho, The Chinese University of Hong Kong, Hong Kong / OZEKI, Yoshiyasu, Kanagawa University, Yokohama / SAKUGAWA, Kenji, Kyoto University, Kyoto / SUN, Chia-Liang, Academia Sinica, Taipei / TONG, Jilong, Capital Normal University, Beijing / WAKABAYASHI, Yasuhiro, Tokyo Institute of Technology, Tokyo / WEI, Fu-Tsun, National Tsing Hua University, Hsinchu / XIE, Bingyong, East China Normal University, Shanghai / XU, Zhefeng, Northwest University, Xi'an / YANG, Enlin, Beijing University, Beijing / ZHAO, Yongqiang, Westlake Institute of Advanced Study, Westlake University, Hangzhou

Titles and Abstracts

## Sunday, August 25, 2019

Arrival and Registration## Monday, August 26, 2019

Registration (8:30-)Wansu Kim (09:20-10:10)

Title: Geometry of Newton Stratification

Abstract: In this talk, I'd like to motivate and explain the geometry of the Newton strata in Hodge-type Shimura varieties obtain by me and Paul Hamacher. We'll focus on a few motivating examples of the moduli of polarised abelian varieties or K3 surfaces.

Keisuke ARAI (10:30-11:20)

Title: Non-existence of rational points on Shimura curves and its function field analogue

Abstract: The moduli space of QM-abelian surfaces (i.e., two-dimensional abelian varieties with an action of a maximal order of a quaternion algebra) is called a Shimura curve. I will talk about non-existence of rational points on Shimura curves as well as Hasse principle violations. The moduli space of D-elliptic sheaves (or Drinfeld-Stuhler modules) is a function field analogue of a Shimura curve. I will also talk about analogous results for function fields. The latter is a joint work with Satoshi Kondo and Mihran Papikian.

Jilong TONG (11:40-12:30)

Title: Crystalline comparison theorem in p-adic Hodge theory

Abstract: Crystalline comparison theorems in p-adic Hodge theory relate the p-adic etale cohomology with the crystalline cohomology of varieties with good reduction defined over p-adic fields. In this talk, I shall discuss some results in this direction base on my joint works with Fucheng Tan and Yichao Tian.

WonTae HWANG (14:00-14:50)

Title: Automorphism groups of simple polarized abelian varieties of (odd) prime dimension over finite fields

Abstract: Let X be an abelian variety of dimension g over a field k. In gen- eral, the group Aut_k(X) of automorphisms of X over k is not finite. But if we fix a polarization L on X, then the group Aut_k(X,L) of automorphisms of the polarized abelian variety (X, L) over k is known to be finite. Then it is natural to ask which finite groups can be realized as the full automorphism group of a polarized abelian variety over some field k.

In this talk, we give a classification of such finite groups for the case when k is a finite field, g >=3 is a prime number, and X is simple. Somewhat surprising thing is the fact that all such groups are cyclic. If time permits, we briefly introduce a similar classification for the case when g = 2.

Yongqiang ZHAO (15:10-16:00)

Title: A Galois theoretic perspective of scrollar syzygy theory

Abstract: The theory of scrollar syzygy resolution of algebraic curves was introduced by Schreyer in his work for Green's conjecture. In this talk, we will give a Galois theoretic perspective of this theory. We will explain the number theory motivations for this project and will discuss, if time permits, its number fields analogues. This is a joint project with Wouter Castryck.

Chia-Liang SUN (16:20-17:10)

Title: A new Mordell-Lang type problem motivated from a switch of roles in the unit equations

Abstract: We propose a new Mordell-Lang type problem which can be viewed as a dual to the problem generalized from that of unit equations. In positive characteristic, we will prove the following result. Let $K$ be a function field over a finite field $k$ of characteristic $p$, and $M$ be a finitely generated $k[phi]$-submodule of K with infinitely many elements, where $phi$ is the $p$-Frobenius map. Under the assumption that there is some natural number $n_0$ such that $M cap K^{p^{n_0}}subset phi ( M )$ , we fully characterize those positive-dimensional subgroups $G$ of a split algebraic torus naturally embedded into the affine space of the same dimension with the property that the set of points on $G$ with coordinates in $M$ is equal to the orbit under $phi$ of the set of points on a proper subvariety of $G$ with coordinates in $M$.

## Tuesday, August 27, 2019

Pin-Chi HUNG (09:20-10:10)Title: On the growth of Mordell-Weil ranks in $p$-adic Lie extensions

Abstract: Let $A$ be an abelian variety over a number field $F$ with ordinary reduction at every primes above $p$. Under various assumptions, we establish asymptotic upper bounds for the growth of Mordell-Weil rank of $A$ in some $p$-adic Lie extension. This is a joint work with Feng Fai Lim.

Subong LIM (10:30-11:20)

Title: Pairs of eta-quotients with dual weights

Abstract: Let $D$ be the differential operator defined by $D = frac{1}{2pi i}frac{d}{dz}$. This induces a map $D^{k+1} : M^!_{-k}(Gamma_0(N)) to M^!_{k+2}(Gamma_0(N))$, where $M^!_{k}(Gamma_0(N))$ is the space of weakly holomorphic modular forms of weight $k$ on $Gamma_0(N)$. In this talk, we show that the structure of eta-quotients is very rarely preserved under the map $D^{k+1}$ between dual spaces $M^!_{-k}(Gamma_0(N))$ and $M^!_{k+2}(Gamma_0(N))$.

Yoshiyasu OZEKI (11:40-12:30)

Title: Torsion of abelian varieties and Lubin-Tate extensions

Abstract: In this talk, we study some finiteness properties for torsion points of abelian varieties over a p-adic field with values in a finite extension of the Lubin-Tate extension of a p-adic field. The result here is a straightforward generalization of Imai's theorem shown in 1975; his theorem describes the same finiteness result for the case of cyclotomic Z_p-extensions.

Bingyong XIE (14:00-14:50)

Title : A generalization of Colmez-Greenberg-Stevens formula

Abstract: In their proof of exceptional zero conjecture, Greenberg and Stevens showed a formula for ordinary families of 2-dinmensional Galois representations. Later Colmez generalized this formula to non-ordinary families setting. In this talk, we will give a generalization of this formula to trianguline families of higher dimensional Galois representations.

Fu-Tsun WEI (15:10-16:00)

Title: Eisenstein series on Goldman-Iwahori spaces in positive characteristic

Abstract: The Goldman-Iwahori space of rank r over a non-archimedean local field F consists of all the norms on the vector space F^r. In this talk, we shall introduce an Eisenstein series on Goldman-Iwahori spaces over local fields with positive characteristic, and present an analogue of the Kronecker limit formula. One application is to express the "Kronecker term"of a partial Dedekind-Weil zeta function of a global function field in terms of an average of certain discriminant quantities along the corresponding "Heegner cycles"on the moduli space of Drinfeld modules.

Kenji SAKUGAWA (16:20-17:10)

Title: On Jannsen's conjecture for modular forms

Abstract: Let M be a pure motive over Q and let M_p denote its p-adic etale realization for each prime number p. In 1987's paper, Jannsen proposed a conjecture about a range of integers r such that the second Galois cohomology of M(r)_p vanishes for any prime number p. Here, M(r) denotes the rth Tate twist of M. When M is an Artin motive, Jannsen's conjecture had already essentially proven a half by Soule aroud early 1980's. In this talk, we consider the case when M is a motive associated to elliptic modular forms. I will explain my ongoing work about an approach to the conjecture used a weighted completion of the arithmetic fundamental groups of modular curves.

## Wednesday, August 28, 2019

Takashi HARA (09:20-10:10)Title: On equivariant Iwasawa theory for CM number field

Abstract: I discuss the equivariant version of the Iwasawa main conjecture for CM number fields over certain noncommutative p-adic Lie extensions.

Junhwa CHOI (10:30-11:20)

Title: Iwasawa theory and CM elliptic curves

Abstract: In this talk, we will discuss some of the recent progress on the one- and two-variable main conjecture of Iwasawa theory for a finite extension of an imaginary quadratic field. We will also discuss the applications to CM elliptic curves.

Zhefeng XU (11:40-12:30)

Title: D.H. Lehmer problem and its generlizations

Abstract: Let $qgeq2$ be an integer, for any $a$ and $overline{a}$ in the least positive reduced residue class modulo $q$, $overline{a}$ satisfies $aoverline{a}equiv 1pmod q$, let $r(q)$ be the number for cases in which $a$ and $overline{a}$ are of opposite parity. It is $q=p$ a prime that the problem is raised by D. H. Lehmer who asks us to say something nontrivial. In this talk, based on a series of works of Prof. Zhang Wenpeng, Prof. C. Cobeli, Prof. Emre Alkan, Prof. Igor E. Shparlinski and Prof. Jean Bourgain, I will introduce both historical and recent results on D.H. Lehmer problem and give some results and corollaries.

## Thursday, August 29, 2019

Enlin YANG (09:20-10:10)Title: Twist formula of epsilon factors of constructible etale sheaves

Abstract: We prove a twist formula for the epsilon factor of a constructible sheaf on a projective smooth variety over a finite field. This formula is a modified version of a conjecture by Kato and T. Saito. We also propose a relative version of the twist formula and discuss some applications. This is a joint work with Naoya Umezaki and Yigeng Zhao.

Yasuhiro WAKABAYASHI (10:30-11:20)

Title: Symplectic geometry of $p$-adic Teichm"{u}ller uniformization

Abstract: The $p$-adic Teichm"{u}ller theory established by S. Mochizuki describes the uniformization of $p$-adic hyperbolic curves and their moduli. It may be regarded as an analogue of the Serre-Tate theory of ordinary abelian varieties. In this talk, we would like to give a quick review of a part of the $p$-adic Teich"{u}ller theory, and explain a $p$-adic version of comparison theorems by S. Kawai, B. Loustau, et al..

Ming Ho NG (11:40-12:30)

Title: Distribution laws of Hecke eigenvalues

Abstract: The Erdos-Kac central limit theorem asserts that the numbers of prime factors of large integers (suitably normalised) tend to follow the Gaussian distribution. Central limit behaviour is also observed in certain family of automorphic forms. In GL(2), the central limit theorem for Hecke eigenvalues of holomorphic primitive cusp forms was obtained by Nagoshi while in case of Maass cusp forms it was given by Wang and Xiao. In this talk, we will discuss some extensions in GL(N) where N >=3. This is a joint work with Y.-K. Lau and Y. Wang.

Jangwon JU (14:00-14:50)

Title: Ternary quadratic forms representing same integers

Abstract: Kaplansky conjectured that if two positive definite integral ternary quadratic forms have perfectly identical representations, then they are isometric or, are both regular, or belong to either of two families. In this talk, we prove the existence of pairs of ternary quadratic forms representing same integers which shows that Kaplansky conjecture is not true. This is a joint work with Byeong-Kweon Oh.

Liang-Chung HSIA (15:10-16:00)

Title: Variations of canonical heights associated to a one-parameter families of H'enon maps

Abstract: Let ${mathbf H} = {H_{t}}$ be a one-parameter family of H'enon maps parameterized by points $tin {bar K}$, an algebraic closure of the number field $K$ and let ${mathbf P} = {P_{t}}$ be a family of initial points. In this talk, we're interested in the problem of variations of the canonical heights ${tilde h}_{mathbf H_{t}}(P_{t})$ associated to the specialized H'enon map $H_{t}$ and the point $P_{t}$ as $t$ varies over ${bar K}.$ Let ${tilde h}_{{mathbf H}}({mathbf P})$ be the function field canonical height associated to ${mathbf H}$ and ${mathbf P}$. Under a natural condition, we show that the function $h_{mathbf P}(t):= {tilde h}_{mathbf H_{t}}(P_{t})/{tilde h}_{{mathbf H}}({mathbf P})$ (provided that ${tilde h}_{{mathbf H}}({mathbf P}) ne 0$) on the parameter space is the restriction of the height function associated to a semipositive adelically metrized line bundle on projective line. We'll also discuss some application of this result. This is a joint work with Shu Kawaguchi.

## Friday, August 30, 2019

Morning Session: Discussions.Afternoon: Departure

*http://mathsci.kaist.ac.kr/~jinhyun/conferences/eantc2019/EANTC2019.html*