Ya-Lun Tsai 蔡亞倫

Assistant Professor

Department of Applied Mathematics

National Chung Hsing University

Room 614, Information Science Building,

145 Xingda Rd., South Dist., Taichung City 402,

Taiwan

 

Email: yltsai@nchu.edu.tw

http://web.nchu.edu.tw/~yltsai/

 

 

Education:

Ph.D. in Mathematics, April 2011, University of Minnesota.

Advisor: Richard Moeckel

Dissertation: Real root counting for parametric polynomial systems and Applications


Research interests: 

Polynomials appear in a wide variety of areas of mathematics. Solving systems of polynomial equations is among the oldest problems in mathematics. Algebraic geometry is the study of polynomial equations and their solutions. Computational algebraic geometry studies these mathematical objects in an algorithmic approach. Real algebraic geometry focus on the study of real solutions. One direction of my research is to study computational and real algebraic geometry in cooperating with tools of computer technique.  

 

Automated theorem proving, Computer-assisted proof, and Symbolic computation may not refer exactly the same notion, but they share one common feature. That is to prove mathematical theorems with the help of computer. Specifically, at least partially based on constructive, algorithmic, and exact computations, mathematical truths are derived. One interesting example is to output the number of real zeros for a polynomial system with rational coefficients. Using Groebner bases and Hermite quadratic forms, Professor Moeckel implemented an algorithm in Mathematica to perform such tasks. Improving the algorithm by largely reducing the running time by myself, we have a fast algorithm implementation written in Mathematica. Tools from Sturm's sequences, Sturm-Habicht sequences, Hermite quadratic forms can be used in real root counting for polynomials or polynomial systems in real coefficients. One of the main contributions of my Ph.D. thesis is to extend some usages of those theorems to parametric polynomials or systems. Specifically, my goal is to count real zeros of the system for all parameter values of our interests. The chapter 3 in my thesis covers some of the results.

 

Polynomials appear in a wide variety of areas in social science or science. According to Professor Sturmfels, "Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, game theory, statistics, machine learning, control theory, and numerous other areas." It is my personal belief that mathematics should makes impacts to the world by solving problems from the real world. Therefore, I am very interested in all kinds of problems that leads to a polynomial systems.

l   Central configurations from the study of celestial mechanics are described as roots of a parametric polynomial system. Studying its zeros can help to solve the Newtonian N-body problems.

1.      Dziobek configurations of the restricted (N+1)-body problem with equal masses,

2.      Counting central configurations at the bifurcation points.

3.  Mathematica Notebook: Some enumeration problems on central configurations at the bifurcation points.

l   Given some point charges in the space, they generate an electrostatic field. The numbers of equivalent points of the electrostatic field is conjectured by Maxwell. This problem of Maxwell’s conjecture can also be reduced to solving a parametric polynomial system.

1.      Special cases of three point charges,

2.      Maxwell's conjecture on three point charges with equal magnitudes.

l   The n-vortex problem of fluid mechanics is the study of the dynamics of n point vortices moving in an inviscid and incompressible fluid. The notion of relative equilibria in the n-vortex problem is the same as that in the n -body problem. For such solutions, n point vortices rotate around the center of vorticity uniformly.

1. Numbers of relative equilibria in the planar four-vortex problems

l   A problem about the tetrahedron is to enumerate the numbers of tetrahedron given some metric invariants. This is a polynomial system problem.

1.   Estimating the number of tetrahedra determined by volume, circumradius and four face areas using Groebner basis

 

Meetings:

l   IMA PI Summer Program for Graduate Students, July 23~August 10, 2007, USA.

l   The symposium on celestial mechanics. Oct. 26~27, 2012, Japan.

l   Groebner Bases, Resultants and Linear Algebra. Sep 3~6, 2013, Austria.(Link)

l   2013 Taiwan-Japan Symposium on Celestial Mechanics and N-body Dynamics, Dec. 6~7, 2013, Taiwan.(Link)

l   The 8th Mathematical Society of Japan Seasonal Institute Current Trends on Gröbner Bases: The 50th Anniversary of Gröbner Bases

    July 1 ~ 10, 2015, Osaka, Japan (Link)

l   Special workshop: Dynamics and Differential Equations

June 22 ~ 25, 2016, Minneapolis, USA (Link)

 

Last update on 2017/02/13.