YaLun
Tsai ½²¨ÈÛ 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,
Computerassisted 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,
SturmHabicht 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 Nbody 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 nvortex
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 nvortex 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 fourvortex 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
TaiwanJapan Symposium on Celestial Mechanics and Nbody 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.