Thursday, October 17, 2024

2024-10-23 / 16:30 ~ 18:00

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Resonance varieties are algebro-geometric objects that emerged from the geometric group theory. They are naturally associated to vector subspaces in second exterior powers and carry natural scheme structures that can be non-reduced. In algebraic geometry, they made an unexpected appearance in connection with syzygies of canonical curves. In this talk, based on works with G. Farkas, Y. Kim, C. Raicu, A. Suciu, and J. Weyman I report on some recent results concerning the geometry of resonance schemes in the vector bundle setup.

2024-10-18 / 10:30 ~ 12:00

학과 세미나/콜로퀴엄 - 대수기하학:

by 조용화(경상대)

In the 19th century, Kummer extensively studied quartic surfaces in the complex projective 3-space containing 16 nodes(=ordinary double points). One of his notable results states that a quartic surface cannot contain more than 16 nodes. This leads to a classic question: how many nodes may a surface of degree d contain? The answer to this question is known only for a very low degrees, namely, degrees 5 and 6. To find the optimal answer(31) for quintics, Beauville introduced the concept of "even sets of nodes," which turned out to be highly influential in the study of nodal surfaces. Based on the structure theorem of even sets by Casnati and Catanese, we will discuss some structure theorems of nodal quintics and sextics with maximal number of nodes. This talk is based on joint works with Fabrizio Catanese, Stephen Coughlan, Davide Frapporti, Michael Kiermaier, and Sascha Kurz.

2024-10-22 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Permutations, patterns, and twin-width

by Colin Geniet(IBS 이산수학 그룹)

This talk will first introduce combinatorics on permutations and patterns, presenting the basic notions and some fundamental results: the Marcus-Tardos theorem which bounds the density of matrices avoiding a given pattern, and the Guillemot-Marx algorithm for pattern detection using the notion now known as twin-width. I will then present a decomposition result: permutations avoiding a pattern factor into bounded products of separable permutations. This can be rephrased in terms of twin-width: permutation with bounded twin-width are build from a bounded product of permutations of twin-width 0. Comparable results on graph encodings follow from this factorisation. This is joint work with Édouard Bonnet, Romain Bourneuf, and Stéphan Thomassé.

2024-10-17 / 16:15 ~ 17:15

학과 세미나/콜로퀴엄 - 콜로퀴엄:

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2-linear varieties are a rich topic. Sijong Kwak initiated the study of 3-regular varieties. In this talk I report on joined work Haoang Le Truong on the classification of smooth 3-regular varieties of small codimension 3. Some of these varieties are analogously to the 2-regular case determinantal. This first non-determinantal cases occurs in codimension 3. In this talk I report on the classification of varieties with Betti table $$ \begin{matrix} & 0 & 1 & 2 & 3\\ \hline 0: & 1 & . & . & .\\ 1: & . & . & . & .\\ 2: & . & 10 & 15 & 6 \end{matrix} $$ Our approach consist of studying extension starting from curves. Let $X \subset \mathbb P^n$ be a variety. An e-extension $Y \subset \mathbb P^{n+e}$ of $X$ is a variety, which is not a cone, such that there exists a regular sequence $y_1,\ldots,y_e$ of linear forms for the homogeneous coordinate ring $S_Y$ of $Y$ such that $S_Y/(y_1,\ldots,y_e) = S_X$ is the coordinate ring of $X$. Using a computationally easy deformation theoretic method to compute extensions, we classify the extensions of 3-regular curves in $\mathbb P^4$ to surfaces in $\mathhbb P^5$ completely.

2024-10-18 / 11:00 ~ 12:00

IBS-KAIST 세미나 - IBS-KAIST 세미나:

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There has been an increased use of scoring systems in clinical settings for the purpose of assessing risks in a convenient manner that provides important evidence for decision making. Machine learning-based methods may be useful for identifying important predictors and building models; however, their ‘black box’ nature limits their interpretability as well as clinical acceptability. This talk aims to introduce and demonstrate how interpretable machine learning can be used to create scoring systems for clinical decision making.

2024-10-24 / 10:30 ~ 11:30

학과 세미나/콜로퀴엄 - 대수기하학:

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The lecture series gives a view on computational methods and their some applications to existence and classfication problems. In the first lectures I will introduce Groebner basis and their basic applications in commutative algebra such as computing kernel and images of morphism between finitely presented modules over polynomial rings. As a theoretical application of Groebner basis I will give Petri's analysis of the equations of a canonical curve. The second topic will be Computer aided existence and unirationality proofs of algebraic varieties and their moduli spaces. In case of curves liaison theory is needed, which will be developed. For existence proofs random searches over finite fields is a technique that has not been exploited very much. I will illustrate this technique in a number of examples, in particular for the construction of certain surfaces. Classification of non-minimal surfaces uses adjunction theory. We will discuss this from a computational point of view.

2024-10-22 / 10:30 ~ 11:30

학과 세미나/콜로퀴엄 - 대수기하학:

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2024-10-17 / 10:30 ~ 11:30

학과 세미나/콜로퀴엄 - 대수기하학:

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