Special Sessions

Number Theory and Algebra

Ting-Yu Lee, National Taiwan University
Embeddings of Maximal Tori in Classical Groups, Odd Degree Descent and Totaro's question

B. Totaro proposed the following question in his paper in 2004 : A homogeneous space has a zero-cycle of degree one. Does it also have a rational point?
In this talk, I will briefly review how to reformulate the question of embeddings of maximal tori in classical groups into a question of rational points of homogeneous spaces under classical groups. Then we will discuss the odd degree descent of the embedding problem. At the end of the talk, we give an affirmative answer to Totaro's question in this case. This is a joint work with E. Bayer and Parimala.

Changningphaabi Namoijam, National Tsing Hua University
Transcendence of Special Values of Goss $L$-functions Attached to Drinfeld Modules

Anderson introduced abelian $t$-modules, and then Goss defined $L$-function associated to abelian $t$-modules. In this talk, we discuss transcendence of special values of $L$-functions attached to Drinfeld modules (abelian t-modules of dimension one) defined over the rational function field $\mathbb{F}_q(\theta)$. We prove our result by employing a result of Anglès, Ngo Dac, and Tavares Ribeiro, and transcendence theory of Papanikolas. This is a joint work with Oğuz Gezmiş.

Yung-Ning Peng, National Central University
From nilpotent element to finite W-algebra

Finite W-algebra, which is essentially determined by a nilpotent element in a simple or reductive Lie algebra, can be viewed as a refinement of the universal enveloping algebra. It has appeared in many different fields of mathematics, possibly with different terminologies. The study of W-algebra can be traced back to Kostant’s classic works on nilpotent orbits around 1980’s. It has being studied intensively since Premet’s works on Slodowy slices around 2000’s, where the modern terminologies are introduced. In this talk, we will focus on the type A case: starting from a nilpotent matrix, we explain how the associated W-algebra is defined, and then introduce a realization of W-algebra established by Brundan-Kleshchev, in terms of some algebraic structure called shifted Yangian. Finally we will mention a recent generalization of this realization to the case of general linear Lie superalgebra.

Nobuo Sato, National Taiwan University
On alternating multiple zeta values

Alternating multiple zeta values are periods of the fundamental group of a four-point punctured projective line. I will talk about my recent work concerning alternating multiple zeta values and their linear relations arising from differential equations of hyperlogarithms. This is a joint work with Minoru Hirose at Nagoya University.

Chung-Hsuan Wang, National Cheng Kung University
Transformation formulas of $p$-adic hypergeometric functions

In this talk, we recall $p$-adic hypergeometric functions $F^{\rm Dw}_{a_1,\cdots,a_s}(t),$ $F^{(\sigma)}_{a_1,\cdots,a_s}(t)$, $\widehat{F}^{(\sigma)} _{a,\cdots,a}(t)$ and the conjecture of their transformation formulas. There are the transformation formulas of Dwork's $p$-adic hypergeometric functions $$ F^{\rm Dw}_{a,\cdots ,a}(t)=((-1)^st)^{l}F^{\rm Dw}_{a, \cdots,a}(t^{-1}) $$ and the transformation formulas of $F_{a,...,a}^{(\sigma)}(t)$ and $\widehat{F}_{a,...,a}^{\;(\sigma)}(t)$ $$ F_{a,\cdots, a}^{\;(\sigma)}(t)=-\widehat{F}_{a,\cdots, a}^{\;(\widehat{\sigma})}(t^{-1}). $$ So far, only some special cases of these transformation formulas have been proved (the case $s=1$ and the case $s=2$, $a\in \frac{1}{N}{\mathbb{Z}}$, $0\lt a\lt 1$ and $p\gt N$).

Differential and Algebraic Geometry

Sheng-Fu Chiu, Academia Sinica
From Energy-Time Uncertainty to Symplectic Excalibur

Heisenberg's Uncertainty Principle is one of the most celebrated features of quantum mechanics, which states that one cannot simultaneously obtain the precise measurements of two conjugated physical quantities such as the pair of position and momentum or the pair of electric potential and charge density. Among the different formulations of this fundamental quantum property, the uncertainty between energy and time has a special place. This is because the time is rather a variable parametrizing the system evolution than a physical quantity waiting for determination. Physicists working in quantum information theory have understood this energy-time relation by a universal bound of how fast any quantum system with given energy can evolve from one state to another in a distinguishable (orthogonal) way. In this talk, we will provide a viewpoint of this evolutional speed limit based on a persistence-like distance of the derived category of sheaves : during a fixed time period what is the minimal energy needed for a system to evolve from one sheaf to a status that is distinguishable from a given subcategory? As an application, we will discuss its relation to the dynamics of classical mechanics, namely the notion of symplectic displacement. We will show that such categorical energy gives rise to an effective lower bound of Hofer displacement energy.

You-Hung Hsu, National National Center for Theoretical Sciences
Semi-orthogonal decomposition via categorical representation theory

The derived category of coherent sheaves on an algebraic variety plays an important role in modern algebraic geometry and related areas. One fundamental way to study the structure of the derived category is via semi-orthogonal decomposition, which divides a triangulated category into simpler pieces. On the other hand, categorification has been an active research topic in representation theory and related areas. One of the essential questions is to lift the representations of Lie algebras/quantum groups from vector spaces (cohomology/K-theory) to categories (derived category), and such a notion is called categorical action. In this talk, we explain how to obtain a semi-orthogonal decomposition from a categorical action of a particular algebra, which we called it the shifted $q=0$ affine algebra. The main idea comes from the interpretation of the Kapranov exceptional collection in terms of convolutions of Fourier-Mukai kernels under the categorical action. Finally, if time permits, we provide an explicit example which shows that the projection functors to components in the semi-orthogonal decomposition are kernel functors.

Shin-Yao Jow, National Tsing Hua University
Asymptotic constructions and invariants of graded linear series

Let $X$ be a complete variety of dimension $n$ over an algebraically closed field $\mathbf{K}$. Let $V_\bullet$ be a graded linear series associated to a line bundle $L$ on $X$, that is, a collection $\{V_m\}_{m\in\mathbb{N}}$ of vector subspaces $V_m\subseteq H^0(X,L^{\otimes m})$ such that $V_0=\mathbf{K}$ and $V_k\cdot V_\ell\subseteq V_{k+\ell}$ for all $k,\ell\in\mathbb{N}$. For each $m$ in the semigroup $$ \mathbf{N}(V_\bullet)=\{m\in\mathbb{N}\mid V_m\ne 0\},$$ the linear series $V_m$ defines a rational map $$ \phi_m\colon X\dashrightarrow Y_m\subseteq\mathbb{P}(V_m), $$ where $Y_m$ denotes the closure of the image $\phi_m(X)$. We show that for all sufficiently large $m\in \mathbf{N}(V_\bullet)$, these rational maps $\phi_m\colon X\dashrightarrow Y_m$ are birationally equivalent, so in particular $Y_m$ are of the same dimension $\kappa$, and if $\kappa=n$ then $\phi_m\colon X\dashrightarrow Y_m$ are generically finite of the same degree. If $\mathbf{N}(V_\bullet)\ne\{0\}$, we show that the limit $$ {\rm vol}_\kappa(V_\bullet)=\lim_{m\in \mathbf{N}(V_\bullet)}\frac{\dim_\mathbf{K} V_m}{m^\kappa/\kappa!}$$ exists, and $0\lt {\rm vol}_\kappa(V_\bullet)\lt \infty$. Moreover, if $Z\subseteq X$ is a general closed subvariety of dimension $\kappa$, then the limit $$ (V_\bullet^\kappa\cdot Z)_\mathrm{mov}=\lim_{m\in \mathbf{N}(V_\bullet)}\frac{\#\bigl((D_{m,1}\cap\cdots\cap D_{m,\kappa}\cap Z)\setminus {\rm Bs}(V_m)\bigr)}{m^\kappa}$$ exists, where $D_{m,1},\ldots,D_{m,\kappa}\in |V_m|$ are general divisors, and $$ (V_\bullet^\kappa\cdot Z)_\mathrm{mov}=\deg\bigl(\phi_m|_Z\colon Z\dashrightarrow \phi_m(Z)\bigr){\rm vol}_\kappa(V_\bullet) $$ for all sufficiently large $m\in\mathbf{N}(V_\bullet)$.

Adeel Khan, Academia Sinica
Derived specialization

I will discuss a derived version of the specialization functors of Kashiwara-Schapira and Verdier, and explain how it can be regarded as a categorification of Kontsevich's virtual fundamental class. I will also discuss some connections with topics such as Beilinson's singular support of étale sheaves, and with Joyce's categorification of Donaldson-Thomas invariants of Calabi-Yau threefolds.

Hsuan-Yi Liao, National Tsing Hua University
Formal exponential maps and Fedosov resolutions

Exponential maps arise naturally in the contexts of Lie theory and smooth manifolds. I will explain how exponential maps can be understood algebraically and how these maps can be extended to graded manifolds. As an application, I will show a new construction of Dolgushev--Fedosov resolutions in deformation quantization.

Hsueh-Yung Lin, National Taiwan University
Motivic invariants of birational maps and Cremona groups

(Joint work with E. Shinder) The study of Cremona groups has been one of the central themes in algebraic geometry since the late 19th century. We will focus on questions and results related to the non-simplicity of Cremona groups. Motivated by weak factorizations of birational maps, we construct invariants of birational maps of motivic nature. Relying on these invariants, we provide new explanations of the non-simplicity of Cremona groups, as well as negative answers to questions of Cheltsov and Dolgachev.

Ryosuke Takahashi, National Cheng Kung University
Structure of $\mathbb{Z}/2$-harmonic spinors

In this talk, I will briefly introduce the development of $\mathbb{Z}/2$-harmonic spinors and discuss the structure of it. This is a very important ingredient in the study of higher dimensional gauge theory. The main difficulty of study $\mathbb{Z}/2$-harmonic spinors is to figure out the deformation theory for the moduli space.

Kuang-Ru Wu, Academia Sinica
Positivity and negativity of vector bundles

Positivity of direct image bundles plays an increasingly important role in complex geometry since the recent works of Berndtsson, Mourougane, Takayama, and many others. In this talk, we will take a different turn and study negativity of direct image bundles. One of our results is that Kobayashi positivity implies ampleness and convex Kobayashi positivity. We will also talk about that it is possible to prove Kobayashi positivity for ample vector bundles with additional curvature assumptions. This last part is work in progress.

Analysis

Chung-Chuan Chen, National Taichung University of Education
Disjoint dynamics on weighted ORLICZ spaces

We give some sufficient and necessary conditions for translation operators on the weighted Orlicz spaces to be disjoint topologically transitive and disjoint topologically mixing. In particular, we show that in certain cases, operators are disjoint topologically transitive if, and only if, their direct sum is topologically transitive. This is a joint work with Prof. S. Öztop and Prof. S. M. Tabatabaie.
Keywords: Disjoint topological transitivity, Translation operator, Orlicz space, Locally compact group.

Yao-Te Huang, National Sun Yat-sen University
Linear Angle Preservers of Hilbert Bundles

Let $x,y$ be two vectors in a (real or complex) Hilbert $C^*$-module $\mathcal{H}$ over a $C^*$-algebra $\mathcal{A}$. The angle $\angle(x,y)$ between $x$ and $y$ can be defined in several ways. When $\mathcal{A}=C_0(X)$ is a commutative $C^*$-algebra, in other words, $\mathcal{H}$ is a continuous field of Hilbert spaces over a locally compact space $X$, we define the cosine of the angle, $u=\cos \angle(x,y)\in C(X)$, by the equation $$ |\langle x, y\rangle| = |x||y|u. $$ We show that if $T: \mathcal{H}\to \mathcal{K}$ is a linear module map between two Hilbert $C_0(X)$-modules preserving (cosines of) non-flat angles, then $T=\alpha J$ for a bounded, strictly positive and continuous scalar function $\alpha$ on $X$ and a module into isometry $J:\mathcal{H}\to\mathcal{K}$.

Daniel Spector, National Taiwan Normal University
An Atomic Decomposition for Divergence Free Measures

In this talk we describe a recent result obtained in collaboration with Felipe Hernandez where we give an atomic decomposition for the space of divergence free measures. The atoms in this setting are piecewise $C^1$ closed curves which satisfy a ball growth condition, while our result can be used to deduce certain "forbidden" Sobolev inequalities which arise in the study of electricity and magnetism.

Ming-Cheng Tsai, National Taipei University
Additive Hermitian idempotent preservers

Let $L$ be an additive map between (real or complex) matrix algebras sending $n\times n$ Hermitian idempotent matrices to $m\times m$ Hermitian idempotent matrices. We show that there are nonnegative integers $p, q$ with $n(p+q) = r \le m$ and an $m\times m$ unitary matrix $U$ such that $$ L(A) = U[(I_{p} \otimes A) \oplus (I_{q} \otimes A^t) \oplus 0_{m-r}]U^*,$$ for any $n\times n$ Hermitian $A$ with rational trace. We also extend this result to the (complex) von Neumann algebra setting. This is a joint work with Chi-Kwong Li, Ya-Shu Wang and Ngai-Ching Wong.

Kuo-Zhong Wang, National Yang Ming Chiao Tung University
Numerical Ranges of Foguel Operators

In this talk, we will present some properties of the numerical ranges of Foguel operators $\displaystyle{F_T=\left[{\textstyle S^*\atop \textstyle 0} \ {\textstyle T \atop \textstyle S}\right]}$, where $S$ is the simple unilateral shift and $T$ an operator, both on $\ell^2$. Among other things, we show that if $T$ is nonzero compact, then the numerical radius $w(F_T)$ is strictly less than $1+(\|T\|/2)$, and if $T$ is a scalar operator $aI$, then $W(F_T)$, the numerical range of $F_T$, is open and is not a circular disc unless $a=0$. For the Foguel--Halmos operator $F_T$, where $T=\mbox{diag}(a_1, a_2, \ldots)$ with $a_n=1$ if $n=3^k$ for some $k\ge 1$ and $a_n=0$ otherwise, we show that $W(F_T)$ is neither open nor closed and give lower and upper bounds for $w(F_T)$.
Co-author: Hwa-Long Gau, Pei Yuan Wu

Ya-Shu Wang, National Chung Hsing University
2-local isometries on differentiable function spaces

Let $E$ and $F$ be Banach spaces. Let $\mathcal{S}$ be a subset of the space $L(E,F)$ of all continuous linear maps from $E$ to $F$. A (non-necessarily linear nor continuous) mapping $\Delta: E \to F$ is a 2-local $\mathcal{S}$-map if for any $x, y \in E$, there exists $T_{x,y} \in \mathcal{S}$, depending on $x$ and $y$, such that $$\Delta(x)=T_{x,y}(x) \ \hbox{and} \ \Delta(y)=T_{x,y}(y).$$ In this talk, I will present a description of the 2-local isometries between continuously differentiable function spaces $C^{(1)}[0,1]$, equipped with the norm $\displaystyle{\|f\|=\sup_{x\in [0,1]} \{|f(x)|+|f’(x)|\}}$.

Partial Differential Equations

Junsik Bae, National Center for Theoretical Sciences
Linear stability of solitary waves for the isothermal Euler-Poisson system

The Euler-Poisson system with the Boltzmann relation is a fluid model which describes the dynamics of ions in electrostatic plasmas. We introduce the linear stability result of small amplitude solitary waves of the isothermal Euler-Poisson system as well as the linear instability criterion for the large amplitude solitary waves. In order to study the eigenvalue problem, we show that the Evans function for the Euler-Poisson system converges to that for the $KdV$ equation as the small amplitude parameter tends to zero. If time is allowed, we briefly present a finite time blow-up scenario for the pressureless Euler-Poisson system. Our result particularly implies that smooth solutions to the pressureless Euler-Poisson system can break down even if the gradient of initial velocity is identically zero. These are joint works with J. Choi and B. Kwon.

Chih-Chiang Huang, National Chung Cheng University
Traveling Waves for a nonlocal equation

In this talk, we consider a nonlocal equation induced by a free energy. Based on Turing's instability, we can observe pattern formation and wave propagation. By applying a variational method, we construct the existence of traveling waves. Moreover, we pose a sufficient condition for wave speeds of two traveling waves, which ensures the existence of the third wave.
This is a joint work with Prof. Chao-Nien Chen, Prof. Yung-Sze Choi and Prof. Shyuh-yaur Tzeng.
Keywords:Traveling Wave, Variational Method, Non-local Equation

Jin-Cheng Jiang, National Tsing Hua University
On the monotonic property of the spatially homogeneous Landau equation

In this talk, I will describe a time-dependent functional involving the relative entropy and the $\dot{H}^{1}$ seminorm, which decreases along solutions to the spatially homogeneous Landau equation with Coulomb potential. The study of this monotone functionial sheds light on the competition between the dissipation and the nonlinearity for this equation. It enables to obtain new results concerning regularity/blowup issues for the Landau equation with Coulomb potential. This talk is based on the joint work with L. Desvillettes and L.-B He.

Ying-Chieh Lin, National University of Kaohsiung
Global entropy solution and relaxation limit for Greenberg-Klar-Rascle multi-lane traffic flow model

In this talk, we consider the Greenberg-Klar-Rascle multi-lane traffic flow model. This model is a relaxation system with the equilibrium state that is a discontinuous function of the car density. We study the existence of global entropy solutions and the relaxation limit for the GKR model. To construct the approximate solutions, we find two sequences of invariant regions under some suitable condition of initial data. As the relaxation time approaches 0, we prove that the limit of the entropy solutions for the GKR model is a weak solution of its equilibrium equation. It is interesting that the equilibrium equation is a scalar conservation law with discontinuous flux.

Kuan-Hsiang Wang, National Center for Theoretical Sciences
Ground states for a linearly coupled indefinite Schrödinger system with steep potential well

In this talk, we are concerned with the investigation of a class of linearly coupled Schrödinger systems with steep potential well, which arises in nonlinear optics. The existence of positive ground states is investigated by exploiting the relation between the Nehari manifold and fibering maps. Some interesting phenomena are that we do not need the weight functions in the nonlinear terms are integrable or bounded and we can relax the upper control condition of the coupling function. This is a joint work with Prof. Tsung-fang Wu and Prof. Ying-Chieh Lin.

Optimization

Jan Harold Alcantara, Academia Sinica
Proximal algorithms for a class of nonconvex nonsmooth minimization problems involving piecewise smooth and min-weakly-convex functions

We establish global convergence of a class of proximal algorithms for minimizing the sum of two functions, where one is piecewise smooth and the other is the difference between the pointwise minimum of finitely many weakly-convex functions and a piecewise smooth convex function. For minimization of a piecewise smooth function over a nonconvex set that is expressible as a finite union of convex sets, we establish the local linear convergence of a projected subgradient algorithm. We also propose two globally convergent acceleration schemes, one by extrapolation and another by component identification. Our framework is then applied to a feasibility reformulation of the linear complementarity problem. Numerical results exemplify that the proposed acceleration and identification procedure significantly improve the efficiency of the non-accelerated counterparts. This is a joint work with Ching-pei Lee.

Yu-Lin Chang, National Taiwan Normal University
Characterizations of Boundary Conditions on Some Non-Symmetric Cones

In contrast to symmetric cone optimization, there has no unified framework for non-symmetric cone optimization. One main reason is that the structure of various non-symmetric cone differs case by case. Especially, their boundary conditions are usually mysterious. In this paper, we provide characterizations of boundary conditions on some non-symmetric cones, including $p$-order cone, ellipsoidal cone, power cone and general closed convex cone. These results will be key bricks for further investigations on non-symmetric cone optimization accordingly.

Chih-Sheng Chuang, National Chiayi University
A unified Douglas-Rachford algorithm for generalized DC programming

We consider a class of generalized DC (difference-of-convex functions) programming, which refers to the problem of minimizing the sum of two convex (possibly nonsmooth) functions minus one smooth convex part. To efficiently exploit the structure of the problem under consideration, in this paper, we shall introduce a unified Douglas–Rachford method in Hilbert space. As an interesting byproduct of the unified framework, we can easily show that our proposed algorithm is able to deal with convex composite optimization models. Due to the nonconvexity of DC programming, we prove that the proposed method is convergent to a critical point of the problem under some assumptions.

Tone-Yau Huang, Feng Chia University
Optimality and Duality for Multi-objective Fractional Programming in Complex Spaces.

In this paper, we consider a complex multi-objective fractional programming problem(CMFP). We establish the necessary optimality conditions of problem (CMFP) in the sense of Pareto optimality, and derive its sufficient optimality conditions using generalized convexity. Finally, we construct the parametric dual problem to the primal problem (CMFP) and their duality theorems.
Keywords: multi-objective fractional programming, generalized convexity, duality theorems

Yu-Ching Lee, National Tsing Hua University
Competitive Demand Learning: an Equilibrium Pricing Algorithm

We consider a periodical equilibrium pricing problem for multiple firms over a planning horizon of $T$ periods. At each period, firms set their selling prices and receive stochastic demand from consumers. Firms do not know their underlying demand curve, but they wish to determine the selling prices to maximize total revenue under competition. Hence, they have to do some price experiments such that the observed demand data are informative to make price decisions. However, uncoordinated price updating can render the demand information gathered by price experimentation less informative or inaccurate. We design a nonparametric learning algorithm to facilitate coordinated dynamic pricing, in which competitive firms estimate their demand functions based on observations and adjust their pricing strategies in a prescribed manner. We show that the pricing decisions, determined by estimated demand functions, converge to underlying equilibrium as time progresses. {We obtain a bound of the revenue difference that is related to the squared number of the competitive firms and is of an order $\mathcal{O}(T^{\frac{3}{4}})$ related to $T$, and a regret bound that is related to the number of competitive firms and is of a $\mathcal{O}(\sqrt{T})$ rate related to $T$. }We also develop a modified algorithm to handle the situation where some firms may have the knowledge of the demand curve.

Ching-Pei Lee, Academia Sinica
Accelerating Inexact Successive Quadratic Approximation for Regularized Optimization Through Manifold Identification

For regularized optimization that minimizes the sum of a smooth term and a regularizer that promotes structured solutions, inexact proximal-Newton-type methods, or successive quadratic approximation (SQA) methods, are widely used for their superlinear convergence in terms of iterations. However, unlike the counter parts in smooth optimization, they suffer from lengthy running time in solving regularized subproblems because even approximate solutions cannot be computed easily, so their empirical time cost is not as impressive. In this work, we first show that for partly smooth regularizers, although general inexact solutions cannot identify the active manifold that makes the objective function smooth, approximate solutions generated by commonly-used subproblem solvers will identify this manifold, even with arbitrarily low solution precision. We then utilize this property to propose an improved SQA method, ISQA+, that switches to efficient smooth optimization methods after this manifold is identified. We show that for a wide class of degenerate solutions, ISQA+ possesses superlinear convergence not just only in iterations, but also in running time because the cost per iteration is bounded. In particular, our superlinear convergence result holds on problems satisfying a sharpness condition more general than that in existing literature. Experiments on real-world problems also confirm that ISQA+ greatly improves the state of the art for regularized optimization.

Jen-Yen Lin, National Chiayi University
Joint Replenishment Problem with Permissible Delay in Payments and the Resource Constraints under Power-of-Two Policy

When there are multiple items in an inventory system, the manufacturing process or replenishment of different goods often encounters the same elements, such as common suppliers, common personnel costs, and common transportation costs. In order to achieve Economies of Scale, the joint cost can be combined and considered in the manufacturing or replenishment process to achieve the goal of reducing the average total cost. Besides the common costs, permissible delay in period and resource constraints also have a huge impact on the supply chain. In this paper, we combine permissible delay in payments with resource constraints and add them into the joint replenishment problem. We provide some theoretical analysis for the optimal solutions, propose an efficient algorithm, discuss the complexity of the proposed algorithm, and conduct some numerical experiments.
Keywords: Joint Replenishment Problem, Permissible Delay in Payments, Resource Constraints

Li-Gang Lin, National Central University
Exact Optimization: A Status Report with Future Promises

The nonlinear programming, from a bottom-up manner, is being explicitly analyzed via a novel perspective/method. More specifically, the up-to-date optimization literature can be classified by three levels: (1) equality-constrained quadratic programming (QP); (2) linear equality-constrained optimization problem with twice-differentiable objective, as solved using Newton's method by reducing it to a sequence of equality-constrained QPs; and, after further imposing inequality constraints, (3) interior-point methods, which reduce the problem to a sequence of (2). For the first time from the proposed viewpoint toward exact optimization, (1) together with the QPs subject to inequality, equality-and-inequality, and extended constraints, respectively, can be algebraically solved in derivative-free closed formulae. All the results are derived without knowing a feasible point, a priori and any time during the process. Moreover, a variety of practical validations, evaluations, and comparisons with benchmark literature (such as MATLAB$^®$) are provided to demonstrate the superiority of the proposed method, notably the enhanced computational efficiency. Meanwhile, much more comparisons/interactions with extensive (numerical) solvers can be more efficiently obtained by virtue of collaborations. Remarkably, the very first idea along this research direction was originated in Taiwan, the progress so far is obtained by Taiwanese only, and thus it is expected that all the main observations will be Taiwan-marked before reaching out to diverse applications in the world.

Discrete Mathematics

Shagnik Das, National Taiwan University
Tight bounds for divisible subdivisions

Alon and Krivelevich proved that for every $n$-vertex subcubic graph $H$ and every integer $q\geq 2$ there exists a (smallest) integer $f=f(H, q)$ such that every $K_f$-minor contains a subdivision of $H$ in which the length of every subdivision-path is divisible by $q$. Improving their superexponential bound, we show that $f(H, q) \leq 10.5qn + 8n + 14q$, which is optimal up to a constant multiplicative factor. This is a joint work with Nemanja Draganić and Raphael Steiner.

Sen-Peng Eu, National Taiwan Normal University
Hankel Determinants from Lattice Paths

In this talk we survey results on Hankel determinants from the counting of lattice paths. Some new results and conjectures are also given.

Wei-Tian Li, National Chung Hsing University
Shifted-Antimagic Labelings for Graphs

The concept of antimagic labelings of a graph is to produce distinct vertex sums by labeling edges through consecutive integers starting from one. A long-standing conjecture proposed by Hartsfield and Ringel is that every connected graph, except a single edge, is antimagic. Many graphs are known to be antimagic, but little is known about sparse graphs as well as trees.
In this talk, we will study the $k$-shifted-antimagic labeling which uses the consecutive integers starting from $k+1$, instead of starting from 1, where $k$ is a given integer. We establish connections among various concepts proposed in the literature of antimagic labelings and extend previous results in three aspects:

  • Some classes of graphs, including trees and graphs whose vertices are of odd degrees, which have not been verified to be antimagic are shown to be $k$-shifted-antimagic for sufficiently large $k$.
  • Some graphs are proved to be $k$-shifted-antimagic for all $k$, while some are proved not for some particular $k$. In particlar, we determine the values of $k$ for which trees of diamter at most 4 are $k$-shifted antimagic.
  • Disconnected graphs are also considered. We characterize the linear forests and star forests that are $k$-shifted antimagic for every integer $k$.

This talk contains the joint work with Fei-Huang Chang, Hong-Bin Chen, and Zhishi Pan and some work with my students, Eranda Dhananjaya and Yi-Shun Wang.

Tao-Ming Wang, Tunghai University
Local Antimagic Vertex Coloring of Trees

For a finite simple graph $G=(V,E)$, an antimagic labeling is an bijection from $E$ to $\{1,\cdots,|E|\}$ such that the induced vertex sums are pairwise distinct, where the induced vertex sum is the sum of all incident edge labels at such vertex. A well-known conjecture of Hartsfield and Ringel is that every connected graph other than $K_2$ admits an antimagic labelling, which is still unsettled till present. In past decades various labeling problems of antimagic types have been studied extensively.
In 2017 Arumugam et al. and Bensmail et al. independently introduced and initiated the study of the weaker notion of a local antimagic labeling, where only adjacent vertices must be distinguished by vertex sums. Thus any local antimagic labeling induces a proper vertex coloring of a graph $G$, where a vertex is assigned the color with its vertex sum. The local antimagic chromatic number $\chi_{la}(G)$ is the minimum number of colors taken over all colorings induced by local antimagic labelings of $G$. A conjecture raised by Arumugam et al. claims that for a tree $T$ with at least three vertices, the local antimagic chromatic number $\chi_{la}(T)=k+1$ or $k+2$, where $k$ is the number of leaves of the tree $T$.
In this talk, we will survey more recent progress of local antimagic labeling and local antimagic vertex coloring of graphs. In particular, to support the above mentioned conjecture, we will present the local antimagic chromatic number $\chi_{la}(T)$ for a complete full $t$-ary tree $T$, $t\ge 2$. More open problems will be posted.

Guan-Ru Yu, National Kaohsiung Normal University
Enumeration of $d$-combining Tree-Child Networks

Phylogenetic networks have replaced phylogenetic trees in many applications in biology within the past few decades. Among the many different subclasses of phylogenetic networks, the one that has attracted the most attention is the class of tree-child networks. Several recent studies have addressed counting questions for bicombining tree-child networks which are tree-child networks with every reticulation node having exactly two parents. In this talk, we extend these studies to d-combining tree-child networks where every reticulation node has now $d \geq 2$ parents. So far, we can completely solve the one-component situation. As for the general situation, we can solve the case of networks with a maximal and a fixed number of reticulation nodes. Moreover, we also tried to extend a recent conjecture by M. Pons and J. Batle from $d=2$ to $d\gt 2$. In this talk, we will explain the above results and outline some open problems which we plan to tackle in the near future. This is a joint work with Michael Fuchs, Hexuan Liu, Michael Wallner, and Yu-Sheng Zhang.

Dynamical Systems and Biomathematics

Jung-Chao Ban, National Chengchi University
Stem and topological entropy on Cayley trees

In this talk, we consider the existence of the topological entropy of shift spaces on a finitely generated semigroup whose Cayley graph is a tree. The considered semigroups include free groups. On the other hand, the notion of stem entropy is introduced. For shift spaces on a strict free semigroup, the stem entropy coincides with the topological entropy. We reveal a sufficient condition for the existence of the stem entropy of shift spaces on a semigroup. Furthermore, we demonstrate that the topological entropy exists in many cases and is identical to the stem entropy.

Yan-Yu Chen, National Taiwan University
Weak entire solution with finite excited intervals for a reaction-interface system

In this talk, we study a reaction-interface system which is a singular limit problem of a FitzHugh-Nagumo type system. To understand the global dynamics of it, we consider the entire solution. Due to the annihilation of the interface may occur, we introduce the definition of weak entire solution with finite excited intervals. Then, we show that any weak entire solution with one excited interval must be a traveling pulse solution. Also, we obtain that any weak entire solution with two excited intervals must be a two-facing traveling pulse solution annihilating at some time. Finally, we prove there are no weak entire solutions with $m$ excited intervals for $m\ge 3$
This is a joint work with Professor Hirokazu Ninomiya and Professor Chang-Hong Wu.

Shih-Hsin Chen, National Center for Theoretical Sciences
On Mathematical Analysis of Synchronization for Bidirectionally Coupled Kuramoto Oscillators

Synchronization is regarded as an universal feature in real world. There are a variety of models used to demonstrate such interesting phenomena. Among these models, Kuramoto model has attracted extensive attention and been applied in various subjects. In this talk, I will briefly introduce some literature works related to the synchronization problem for Kuramoto model. Then, I will show our main results of phase synchronization and frequency synchronization for the first order bidirectionally coupled Kuramoto model. If time permits, I will provide numerical simulations as well.

Jia-Yuan Dai, National Chung Hsing University
Selective Feedback Stabilization of Ginzburg-Landau Spiral Waves in Circular and Spherical Geometries

The complex Ginzburg-Landau equation serves as a paradigm of pattern formation. Within circular and spherical geometries, the existence and stability property of Ginzburg-Landau spiral waves have been proved. However, many spiral waves are unstable and thereby rarely visible in experiments and numerical simulation. In this talk we selectively stabilize certain significant classes of unstable spiral waves. Our tool for stabilization is the control triple method, which generalizes the celebrated Pyragas control to the setting of PDEs. This is a joint work with I. Schneider and B. de Wolff.

Chi-Jen Wang, National Chung Cheng University
Schloegl’s second model on a Bethe lattice

Schloegl's second model involves spontaneous particle annihilation at rate $p$ and autocatalytic particle creation rate. We analyze this model on a Bethe lattice when the coordination number $z=3$. Precise behavior for stochastic models on regular periodic infinite lattices is usually surmised from kinetic Monte Carlo simulation on a finite lattice with periodic boundary conditions. However, the persistence of boundary effects for a Bethe lattice complicates this process. We explored various boundary conditions and unconventional simulation ensembles on the Bethe lattice to predict behavior for infinite size. A discontinuous transition to the vacuum state on the infinite lattice occurs when the annihilation rate $p$ around 0.053.

Computational Mathematics

Meng-Huo Chen, National Chung Cheng University
Fluid-structure interactions: one-field monolithic fictitious domain method and its parallelization

In this research we implement the parallelization of the method: one-field monolithic fictitious domain (MFD), an algorithm for simulation of general fluid-structure interactions (FSI). In this algorithm only one velocity field is solved in the whole domain (one-field) based upon the use of an appropriate $L^2$ projection. ”Monolithic” means the fluid and solid equations are solved synchronously (rather than sequentially). For simulation of fluid-structure interactions on 3D domain the algorithm and the solving of the linear systems arising from the discretization need to be parallelized in order to reduce the simulation time from several months to few days. At the initial stage of the research we focus on parallelizing the algorithm on uniform meshes. The implemented parallel algorithm is then extended to the simulations on nonuniform meshes, where an adaptive mesh refinement scheme is used to improve the accuracy and robustness. Our goal is to provide an efficient, robust algorithm which can handle the difficult fluid-structure interactions such as the collision of multiple immersed solids in fluid where the high resolution mesh is necessary for resolving the phenomena near the collision and fluid-structure interfaces.

Chun-Yueh Chiang, National Formosa University
An efficient iteration for solving the extremal solutions of discrete-time algebraic Riccati equations

Algebraic Riccati equations (AREs) have been extensively applicable in linear optimal control problems and many efficient numerical methods were developed. The most attention of numerical solutions is the (almost) stabilizing solution in the past works. Nevertheless, it is an interesting and challenging issue in finding the extremal solutions of AREs which play a vital role in the applications. In this talk, based on the semigroup property, an accelerated fixed-point iteration (AFPI) is developed for solving the extremal solutions of the discrete-time algebraic Riccati equation. In addition, we prove that the convergence of the AFPI is at least R-suplinear with order $r\gt 1$ under some mild assumptions. Numerical examples are shown to illustrate the feasibility and efficiency of the proposed algorithm.

Wei-Fan Hu, National Central University
A Discontinuity Capturing Shallow Neural Network for Anisotropic Elliptic Interface Problems

In this talk, a new Discontinuity Capturing Shallow Neural Network (DCSNN) for approximating $d$-dimensional piecewise continuous functions and for solving anisotropic elliptic interface problems is developed. There are three novel features in the present network; namely, (i) jump discontinuity is captured sharply, (ii) it is completely shallow consisting of only one hidden layer, (iii) it is completely mesh-free for solving partial differential equations (PDEs). We first continuously extend the d-dimensional piecewise continuous function in $d+1$-dimensional space by augmenting one coordinate variable to label the pieces of discontinuous function, and then construct a shallow neural network to express this new augmented function. Since only one hidden layer is employed, the number of training parameters (weights and biases) scales linearly with the dimension and the neurons used in the hidden layer. For solving elliptic interface equations, the network is trained by minimizing the mean squared error loss that consists of the residual of governing equation, boundary condition, and the interface jump conditions. We compare the results obtained by the traditional grid-based immersed interface method (IIM) which is designed particularly for interface problems. The present results show better accuracy than the ones obtained by IIM.

Yueh-Cheng Kuo, National University of Kaohsiung
The nonnegative least squares problem

We propose an index search method (ISM) for solving nonnegative least squares problems (NNLS). This method uses inner and outer schemes to find the index set corresponding to the nonzero component of the optimal solution. The outer iteration updates the approximate index set such that the objective values of the sequence generated by ISM are monotonically decreasing. Hence, the index set generated by ISM does not repeat, and the optimal solution can be achieved with finite iteration steps. Some normal equations need to be solved in the inner iteration, which is the dominant computational complexity in ISM. Numerical experiments are provided to support the theoretical results.

Te-Sheng Lin, National Yang Ming Chiao Tung University
Machine Learning to solve elliptic interface problem

We propose a shallow neural network for solving elliptic problems with delta function singular sources on an interface. In addition, we introduce the level set function as a feature input. The loss function can be formulated using either the problem residual or the corresponding energy of the problem. We perform a series of numerical tests to demonstrate the accuracy of the present network as well as its capability for problems in irregular domains and in higher dimensions.

Jephian C.-H. Lin, National Sun Yat-sen University
On the inverse eigenvalue problem for block graphs

The inverse eigenvalue problem of a graph is asking whether a given spectrum can be realized by some matrix whose zero-nonzero pattern is prescribed by the adjacency of the graph. The strong spectral property is a matrix condition that allows one to perturb the matrix without changing its spectrum, and it is powerful in creating matrices with the same spectrum and more nonzero entries. In this talk, we will introduce some techniques where the matrix does not have the strong spectral property and provide partial solutions to the inverse eigenvalue problem for block graphs.

Probability

Yuki Chino, National Yang Ming Chiao Tung University
A crossover on SAW in random environment

Self-avoiding walk is one of statistical-mechanical models which describes behaviour of polymers in homogeneous medium. The model shows the phase transition and the asymptotic behaviour around the critical point has been studied actively. From physics point of view, by introducing a random environment, we expect that the behaviour change affected from the environment. Our interest is how much the random environment affects the original system. In this talk we consider self-avoiding walk in random environment. This model shows us a crossover between weak and strong disorder depending on the structure of underlying graph. This crossover gives a part of the answer for our question. I will show how much we have known and what we want to know.

Jyy-I Hong, National Chengchi University
An $m$-spread model using branching process

When an infectious disease spreads among a population, it may cause different reactions in individuals. To understand and study the spread patterns is always one of the key things to do for decision making during an epidemic. In this talk, we will introduce a model to describe a spread pattern which depends on the spreading history within certain time period and construct an induced branching process to study the long term behavior of the spread pattern and the spread rate. This is a joint work with Jung-Chao Ban and Yu-Liang Wu.

Chien-Hao Huang, National Chengchi University
One-dimensional polymers in random environments: the ballistic regime

The polymer gets rewards when it visits a new site. Therefore, it tends to stretch itself. When the external force is large, the polymer goes with a speed. This result is part of the joint work with Q. Berger, N. Torri and R. Wei.

Wai-Kit Lam, National Taiwan University
Dynamical critical first-passage percolation in two dimensions

In first-passage percolation (on the triangular lattice), one puts nonnegative i.i.d. random weights $(\tau_v)$ on the vertices, and studies the induced pseudometric $T$. The critical case is the case when $\mathbf{P}(\tau_v = 0) = 1/2$, where $1/2$ is the critical threshold for site percolation on the triangular lattice. In this case, there are lots of large finite clusters of zero weights, and the behavior of $T$ is very different from that in the "usual" case, namely $\mathbf{P}(\tau_v = 0) \lt 1/2$.
We consider a dynamical version of the model in the critical case, where weights are resampled according to Poisson processes. We study the "exceptional times", at which the model exhibits a completely different behavior. Based on joint work with M. Damron, J. Hanson and D. Harper.

Statistics

I-Chen Lee, National Cheng Kung University
Minimax Design for an Accelerated Life Test

Due to time constraint and experimental cost, how to plan an efficient accelerated life test (ALT) to obtain more accurate lifetime information of products is an important research issue. Many literatures proposed strategies to design a locally optimal planning of an ALT under the pre-specified planning values of parameters. On the other hand, the optimal design for an ALT also depends on the model assumption, such as lognormal or Weibull distribution. However, the true parameters and the underlying model are usually unknown before the experiment. To deal with the problems, this study adopts a minimax strategy to obtain a more robust design for an ALT. To find a minimax design efficiently, this study adopts particle swarm optimization (PSO) techniques. The result shows that the minimax design can be determined as long as we specify the range of sample failure probability and provide candidate models. Compared to the locally optimal design, the minimax design is more robust and more practical.

Liang-Ching Lin, National Cheng Kung University
Monitoring Photochemical Pollutants for Anomaly Detection based on Symbolic Interval-Valued Data Analysis

This study considers monitoring photochemical pollutants for anomaly detection based on symbolic interval-valued data analysis. For this task, we construct control charts based on the principal component scores of symbolic interval-valued data. Herein, the symbolic interval-valued data are assumed to follow a normal distribution, and an approximate expectation formula of order statistics from the normal distribution is used in the univariate case to estimate the mean and variance via the method of moments. Moreover, we consider the bivariate case wherein we use the maximum likelihood estimator calculated from the likelihood function derived under a bivariate copula. In addition, we establish the procedures for the statistical control chart based on the univariate and bivariate interval-valued variables, and the procedures are potentially extendable to higher dimensional cases. Monte Carlo simulations and real data analysis using photochemical pollutants confirm the validity of the proposed method, and the results particularly show the superiority over the conventional method using the averages in identifying the date on which the abnormal maximum occurred.
Keywords: Anomaly detection; Control chart; Monitoring photochemical pollutants; Principal component analysis; Symbolic data analysis.

Sheng-Hsuan Lin, National Yang Ming Chiao Tung University
From linear structural equation modeling to generalized multiple mediation formula

Causal mediation analysis is advantageous for mechanism investigation. In settings with multiple causally ordered mediators, path-specific effects (PSEs) have been introduced to specify the effects of certain combinations of mediators. However, most PSEs are unidentifiable. Interventional analogue of PSE (iPSE) is adapted to address the non-identifiability problem. Moreover, previous studies only focused on cases with two or three mediators due to the complexity of the mediation formula in large number of mediators. In this study, we provide a generalized definition of traditional PSEs and iPSEs with a recursive formula, along with the required assumptions for nonparametric identification. This work has three major contributions: First, we developed a general approach (that includes notation, definitions, and estimation methods) for causal mediation analysis with an arbitrary number of multiple ordered mediators and with time-varying confounders. Second, we demonstrate identified formula of iPSE is a general form of previous mediation analysis. It is reduced to linear structural equation model under linear or log-linear model, to causal mediation formula when only one mediator. Third, a flexible algorithm built based on g-computation algorithm is proposed along with a user-friendly software online. This approach is applied to a Taiwanese cohort study for exploring the mechanism by which hepatitis C virus infection affects mortality through hepatitis B virus infection, abnormal liver function, and hepatocellular carcinoma. All methods and software proposed in this study contribute to comprehensively decompose a causal effect confirmed by data science and help disentangling causal mechanisms when multiple ordered mediators exist, which make the natural pathways complicated.

Shao-Hsuan Wang, National Central University
Perturbation Theory for Cross Data Matrix-based PCA

Principal component analysis (PCA) has long been a useful and important tool for dimension reduction. Cross data matrix (CDM)-based PCA is another way to estimate PCA components, through splitting data into two subsets and calculating singular value decomposition for the cross product of the corresponding covariance matrices. It has been shown that CDM- based PCA has a broader region of consistency than ordinary PCA for leading eigenvalues and eigenvectors. In this talk, I will introduce the finite sample approximation results as well as the asymptotic behavior for CDM-based PCA via matrix perturbation. Moreover, I introduce a comparison measure for CDM-based PCA vs. ordinary PCA. This measure only depends on the data dimension, noise correlations, and the noise-to-signal ratio (NSR). Using this measure, we develop an algorithm, which selects good partitions and integrates results from these good partitions to form a final estimate for CDM-based PCA. Numerical and real data examples are presented for illustration. This joint work is with Prof. Su-Yun Huang.