Schedule & Abstracts

Schedule(tentative)

Invited talks

김형준 (경북대학교)
Introduction to the Intrinsic Properties of Spatial Graphs
  Spatial graph theory is an area of knot theory which has a close connection with molecular biology and chemistry. Spatial graph theory is the study of graphs embedded in S^3. Most of the work in this area is rooted in the result of Conway and Gordon’s intrinsic properties of graphs. In this talk, I introduce intrinsic properties and their result.

정재우 (DGIST)
Castelnuovo–Mumford Regularity of Coordinate Subspace Arrangements
  Castelnuovo–Mumford regularity is a central invariant that measures the algebraic complexity of projective varieties. While it has been widely studied for irreducible varieties, the situation is more subtle for general algebraic sets since syzygies are sensitive to how components intersect.
  In this talk, we will focus on coordinate subspace arrangements which form one of the simplest and most visualizable classes of algebraic sets. Their defining ideals are square-free monomial ideals and naturally understood through the Stanley–Reisner correspondence. We will first discuss bounds on the regularity of quadratic square-free monomial ideals obtained via graph decomposition methods. Then I will turn to an Eisenbud–Goto type inequality for Stanley–Reisner ideals, highlighting how combinatorial and topological data provide effective control over syzygies.

조예원 (경상국립대학교)
Gromov Kähler hyperbolicity and eigenvalue estimates on bounded symmetric domains
  In 1991, Gromov introduced the notion of Kähler hyperbolic manifolds which in particular generalizes Kähler manifolds of Riemannian sectional curvature bounded from above by a negative constant. Gromov's basic estimate on such manifolds yields a vanishing theorem for harmonic forms and also a lower bound for the eigenvalues of the Laplacian of the given Kähler metric. The bound is determined by a uniform constant and the so-called `Kähler hyperbolicity length' of the metric.
  In this talk, I shall explain a method to obtain lower bounds for the eigenvalues of the Laplacian of the complete Kähler-Einstein metrics of Ricci curvature -1 on bounded symmetric domains, using the aforementioned estimate. The method in particular provides the optimal lower bound on the complex hyperbolic space (and polydiscs) which is sharper than McKean's estimate (1970). This is joint work with Young-Jun Choi and Kang-Hyurk Lee.

최선미 (서원대학교)
On the relationship between the triple point number and the length of a 3-cocycle of the dihedral quandle
  The triple point number, a fundamental invariant for surface-knots analogous to the crossing number in classical knot theory, is defined as the minimum number of triple points across all possible diagrams of a surface-knot. Upper and lower bounds have been explored to estimate the triple point number of surface-knots. Satoh proposed a lower bound based on the length of a 3-cocycle of the 5-dihedral quandle. In this talk, we review the established relationship between the triple point number and the length of a 3-cocycle of a quandle. We also extend this discussion to the dihedral quandle of prime order. As an application, we illustrate the computation of upper and lower bounds for the triple point number using specific examples.

표준철 (부산대학교)
Solitons for the Mean Curvature Flow
  Translating solitons and self-shrinkers are solitons for the mean curvature flow (MCF). They serve not only as blow-up models of singularities of the MCF, but also as minimal surfaces in certain Riemannian manifolds. In this talk, we compare properties of minimal surfaces and MCF solitons, particularly with respect to Bernstein-type theorems and properness. More precisely, we present rigidity results for graphical translators that move in non-vertical directions. In addition, we introduce sufficient conditions for the properness of translating solitons.This is based on joint work with Daehwan Kim, Yuan Shyong Ooi, and John Ma.

한지영 (부산대학교)
Distribution of images of a system of a quadratic form and a linear form at integer vectors
  In late 1980s, Margulis completely solved the Oppenheim conjecture, which clarifies the condition of a non-degenerate indefinite quadratic form for having the dense image set of integer vectors in the real field. He used the dynamical property of the unipotent orbit relative to the quadratic form on the homogeneous space of the special linear group divided by its arithmetic lattice subgroup. Since then, solving Oppenheim-conjecture-typed problems using homogeneous dynamical tools is one of the main topics in homogeneous dynamics. In this talk, I will present the quantitative results in the Oppenheim-typed problems and introduce dynamical ideas that are used in this subject. This is a joint work with Seonhee Lim and Keivan Mallahi-Karai.

Contributed Talks

오진석 (경북대학교)
Third and Fourth Homology of the Yang-Baxter Operators Yielding the HOMFLYPT Polynomial
  Yang–Baxter operators are bilinear endomorphisms from the tensor product of a vector space to itself that satisfy the Yang–Baxter equation. They can be viewed as generalizations of biquandles and are therefore of broad interest to knot theorists. In particular, the class of Yang–Baxter operators considered here, denoted $R_{(m)}$, was of special interest to Vaughan Jones and is closely related to the celebrated Jones polynomial. Using a standard construction, these Yang–Baxter operators give rise to a homology theory, $H_n(R_{(m)})$, modeled on the homology of pre-cubic modules. In this talk, we present a general result for $H_n(R_{(m)})$ that depends only on the explicit computation of a finite number of initial conditions. We then produce explicit formulas for the third and fourth homology by computing the requisite initial conditions.

정지원 (부산대학교)
Variation of the Solutions to the Dirichlet Problems for the Complex Monge–Ampère Equation on Bounded Strongly Pseudoconvex Domains
  Let π : D → ∆ be the projection from a bounded (strongly) pseudoconvex domain D ⊆ Cn+1 onto the unit disc ∆. For each fiber Ds = π-1(s), we solve the Dirichlet problem for complex Monge–Ampère equation obtaining a smooth family of fiberwise solutions u(s,z). It is well known that each u(s,·) is strictly plurisubharmonic along its fiber, but the total function u(s,z) need not be (strictly) plurisubharmonic in base directions. In this talk, I will introduce the following positivity theorem: if f is plurisubharmonic and smooth up to the boundary of D, then the global form ddc u(s,z) on D is (strictly) positive also in the base directions.

티엔 쉐치안 (부산대학교)
Classification of soliton solutions to the r-curvature flow on the sphere and the hyperbolic plane
  We establish a complete calssification of soliton solutions to the r-curvature flow on the sphere (or the hyperbolic plane, respectively). We prove the existence of these solitons for any given nonzero vector v in Euclidean 3-space (or Minkowski 3-space, respectively) and further analyze their qualitative behavior across different cases of the constant exponent r of the geodesic curvature, supported by illustrative examples.