Talks & Slides

5. The 17th East Asia Section of SIAM Conference

University of Macau, China, Jun.28 (Fri.)-Jul.1(Mon.), 2024 


Title: Vector Field Reconstruction on Convex Polygonal Domains based on Hyperplane Arrangement

Abstract: We consider the problem of reconstructing a vector field from sparsely observed data in scenarios like fluidic materials or particles within a convex polygonal domain $P$. We demand the vector field to be tangent to the polygonal boundary $\partial P$. In this work, we introduce a novel scheme for reconstructing a vector field on a convex polygonal domain by a polynomial vector field, leveraging the ideas from the theory of hyperplane arrangement. Given a degree upper bound $k$ and observed vectors at finite points in the domain, our algorithm computes a degree $k$ polynomial vector field that interpolates the observations in a least squares sense while satisfying the boundary condition exactly. Additionally, the algorithm can determine a minimal degree polynomial vector field within a specified error threshold.


4. The 4th POSTECH MINDS Workshop on Topological Data Analysis and Machine Learning

POSTECH, Korea, Jan. 30 (Tue.) ~ Feb. 2 (Fri.), 2024


Title: Polynomial Interpolation of a Vector Field on a Convex Polygon

Abstract: Interpolation of sparsely observed data is a fundamental aspect of data science.

We consider fluidic materials or particles confined within a convex polygonal domain $P$.

The dynamics of these entities is modelled by a vector field which is tangent to the polygonal boundary $\partial P$.

In this work, we introduce a novel scheme for interpolating a vector field on a convex polygonal domain by a polynomial vector field, leveraging the ideas from the theory of hyperplane arrangement. Given a degree upper bound $k$ and a set of observations $\{ \left( (x_i,y_i),u(x_i,y_i) \right)\mid (x_i,y_i)\in P, u(x_i,y_i)\in R^2, i=1,\ldots, n\}$, our algorithm computes a degree $k$ polynomial vector field $u: P \to R^2$ that interpolates the observations in a least squares sense while satisfying the boundary condition exactly. The algorithm can also calculate a minimal degree polynomial vector field that meets a given error bound. We showcase the effectiveness of our algorithm through applications in vector field design.

3. Hyperplane Arrangements 2023

Rikkyo University, Japan, Dec. 11 (Mon) ~ Dec. 15 (Fri), 2023 

Title: Minimal Free Resolution of Close-to-free Arrangements

Abstract: We study the algebraic structure of a new class of hyperplane ar- rangements obtained by deleting two hyperplanes from a free arrangement. Our primary focus is on computing the minimal free resolution of the logarithmic derivation module of such a hyperplane arrangement. In particular, combined with the theory of multiarrangement, we show that the minimal free resolution is determined solely combinatorially for three-dimensional central arrangements. Our result highlights the relationship between algebraic and combinatorial prop- erties for close-to-free arrangements.

Kyushu University, Japan, Sep. 7, 2023


Title: Degrees of the logarithmic vector fields for close-to-free hyperplane arrangements

Abstract: A hyperplane arrangement A is a finite set of linear hyperplanes in a vector space K^l, where K is a field. The graded module D(A) of the logarithmic vector fields consists of polynomial vector fields over K^l tangent to A. An arrangement A is said to be free if D(A) is a free module. The degrees of a homogeneous basis of the free module D(A) are called the exponents of A. The famous factorization theorem asserts that the characteristic polynomial of the intersection lattice of a free arrangement A completely factors into linear polynomials over the integers, and the roots of the polynomial are the exponents of A. This motivates us to study the degrees of minimal homogeneous generators of the module D(A) and their connections with combinatorics. In the first part of the talk, we will discuss a complete classification of the free arrangements in the three-dimensional real vector space with exponents of the form (1,3,d) for some d≥3. To analyze the relationship between algebraic and combinatorial structures, we often consider the interplay between adding and deleting hyperplanes in an arrangement. The arrangement in which one hyperplane is deleted from a free arrangement has been extensively studied by Takuro Abe. In the second part of the talk, we will turn our attention to the algebraic structure of a new class of hyperplane arrangements obtained by deleting two hyperplanes from a free arrangement. We compute the degrees of minimal homogeneous generators of D(A). In particular, the degrees are determined solely combinatorially for three-dimensional arrangements.

We present illustrative examples that show our result’s strength and provide insights into the relation between algebraic and combinatorial properties for close-to-free arrangements.