강연자
[남하얀 (덕성여대)]
On the Davenport Constant and Atomic Density of Numerical Semigroups
A numerical semigroup S is a subset of the set of nonnegative integers that includes 0, is closed under addition, and has a finite complement. The Davenport constant D(G) of a finite group G is the smallest number d such that any sequence of length \ge d contains a non-empty zero-sum subsequence. In other words, D(G) - 1 is the maximum length of a zero-sum-free sequence. We define a quotient ring \mathcal{Q}_q(S) of a numerical semigroup S and derive a formula for D(\mathcal{Q}_q(S)). Furthermore, we compute the range of the atomic density of S in terms of D(\mathcal{Q}_q(S)).
[유명준 (연세대)]
The distribution of the cokernel of a random p-adic matrix
The cokernel of a random p-adic matrix can be used to study the distribution of objects that arise naturally in number theory. For example, Cohen and Lenstra suggested a conjectural distribution of the p-parts of the ideal class groups of imaginary quadratic fields. Friedman and Washington proved that the distribution of the cokernel of a random p-adic matrix is the same as the Cohen–Lenstra distribution. Recently, Wood [1] generalized the result of Friedman–Washington by considering a far more general class of measure on p-adic matrices. In this talk, we explain a further generalization of Wood’s work. This is joint work with Gilyoung Cheong, Dong Yeap Kang and Jungin Lee.
[1] M. M. Wood, Random integral matrices and the Cohen–Lenstra heuristics, Amer. J. Math. 141 (2019), no. 2, 383–398.
[정재우 (IBS-CCG)]
Secant Monomial Ideals
The secant variety of an (irreducible) variety X is the Zariski closure of the union of linear spaces spanned by closed points in X. There are active studies on secant varieties regarding several aspects, such as degree, dimension, and singular locus. In this project, we focus on the secant varieties of unions of coordinate linear spaces, which is a first step toward extending the literature on secant varieties to those of algebraic sets with multiple (connected) components.
In this talk, we introduce the correspondence between the union of coordinate linear spaces and simplicial complexes, and we present some results on the secant varieties of the union of coordinate linear spaces through analysis of the associated simplicial complexes. This is joint work with Junho Choe.
[김민규 (KIAS)]
Eigen-ring construction for linear categories
[안병희 (경북대)]
TBA
[임근호 (Einstein Institute of Mathematics)]
Quantitative bordism over acyclic groups and Cheeger-Gromov rho-invariants
In his 1999 article [1], Gromov introduced a quantitative viewpoint of the study of topology with several fundamental questions. The underlying motivation is that even when algebraic topology successfully determines the existence of a solution to a given problem, if one attempts to understand the quantitative nature of a solution, interesting new questions often arise. In the viewpoint, Gromov asked the existence of a linear complexity cobordism. In this talk, we prove a bordism version of Gromov's linearity conjecture over a large family of acyclic groups. Since all groups functorially embed into these acyclic groups [2], it follows that the linear bordism conjecture is true if one allows to enlarge a given group. Our result holds in both PL and smooth categories, and for both oriented and unoriented cases. The method of the proof hinges on quantitative algebraic and geometric techniques over infinite complexes with unbounded local (combinatorial) geometry. As an application, we prove that there is a universal linear bound for the Cheeger-Gromov rho-invariants of PL (4k-1)-manifolds associated with arbitrary regular covers.
[1] Mikhael Gromov, Quantitative homotopy theory, Prospects in mathematics (Princeton, NJ, 1996), Amer. Math. Soc., Providence, RI, 1999, pp. 45–49.
[2] Gilbert Baumslag, Eldon Dyer, and Alex Heller, The topology of discrete groups, J. Pure Appl. Algebra 16 (1980), no. 1, 1–47.
[이현석 (광운대)]
On the numbers e^{e}, e^{e^2} and e^{\pi^{2}}
In 1949, A. O. Gelfond proved that at least one of the three numbers e ^{e }, e ^{e ^2}, e ^{e ^3} is transcendental. In 1973, it was proved that at least one of the two numbers e ^{e }, e ^{e ^2} is transcendental by W. D. Brownawell [1] and M. Waldschmidt [2], independently and simultaneously. In this talk, I will present new results on this numbers. From Lindemann-Weierstrass Theorem, we can observe that e has more superior property than the usual definition of transcendental number. At this point, we can define the new type of transcendental number, that is, superior transcendental number. Lastly, I will introduce generalization of Lindemann-Weiestrass theorem with its corollaries.
[1] Brownawell, W. Dale. The algebraic independence of certain numbers related to the exponential function. J. Number Th. 6 (1974), 22-31.
[2] Waldschmidt, Michel Solution du huiti\'eme probl\'eme de Schneider. J. Number Theory, 5 (1973), 191-202.
[김민수 (경남대)]
TBA
[박지혜 (영남대)]
On Cayley maps and skew-morphisms
This presentation provides an introduction to Cayley maps and skew-morphisms, along with a brief overview of the classification problem of Cayley maps. Firstly, a Cayley map is an object generated by a given group. It can be used to visually comprehend the substructure of the group and to develop and analyze various algorithms related to the group and its generating set. Additionally, concepts and techniques of Cayley maps find applications in cryptography, graph theory, network analysis, and other fields. Skew-morphism originates from research into mappings between algebraic structures. Skew-morphism enables the transformation of a structure while preserving some of its properties. This characteristic allows skew-morphisms to be used in solving various problems. In this presentation, we utilize skew-morphism to classify Cayley maps for a given group.
[이동협 (고려대)]
Hilbert Function and Castelnuovo-Mumford Regularity of a Finite Set of Points
Castelnuovo proved that maximal genus curves lie on minimal degree surfaces. In his proof, he used the fact that a set of points in linearly general position with "the smallest" Hilbert function lies on a minimal degree curve. Within this context, there is a conjecture that any set of points in symmetric position with "small" Hilbert function lies on curves of some small degree.
In the zero-dimensional case, it is known that Castelnuovo-Mumford regularity is closely related to the Hilbert function. In this talk, we will describe the geometric configuration of points in linearly general position with large regularity, and also more general cases besides linearly general positions. This is joint work with Prof. Euisung Park.