Monday, September 2, 2024

2024-09-04 / 16:00 ~ 17:00

by ()

The realization that microbiomes, associated with virtually all multicellular organisms, have tremendous impact on their host health is considered as one of the most important scientific discoveries in the last decade. The host associated microbiomes, composed of tens to hundreds of co-existing microbial species, are highly heterogenous at multiple scales (e.g. between different hosts and within a host). In this talk, I will share our recent works on understanding the heterogeneity of complex microbial communities, and how these conceptual and technological advances in microbial ecology pave the way for precision microbiome engineering to prevent and treat diseases.

2024-09-06 / 11:00 ~ 12:00

학과 세미나/콜로퀴엄 - 응용 및 계산수학 세미나:

by 송창훈()

Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. However, a critical question remains: How do we validate that PINNs accurately solve these PDEs? This talk explores the types of mathematical validation required to ensure that PINNs can reliably approximate solutions to PDEs. We will discuss the conditions under which PINNs can converge to the correct solution, the relationship between minimizing residuals and achieving accurate results, and the role of optimization algorithms in this process. Our goal is to provide a clear understanding of the theoretical foundations needed to trust PINNs in practical applications while addressing the challenges in this emerging field.

2024-09-05 / 10:30 ~ 11:30

학과 세미나/콜로퀴엄 - 대수기하학:

by ()

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.

2024-09-06 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Edge-colored Extremal Problems

by Neal Bushaw(Virginia Commonwealth University)

An edge-colored graph $H$ is called rainbow if all of its edges are given distinct colors. An edge-colored graph $G$ is then called rainbow $H$-free when no copy of $H$ in $G$ is rainbow. With this, we define a graph $G$ to be rainbow $H$-saturated when there is some proper edge-coloring of $G$ which is rainbow $H$-free, but for every pair of non-adjacent vertices $x,y\in V(G)$, the graph $G+xy$ formed by adding the edge $xy$ to $G$ cannot be given a rainbow $H$-free coloring. We think of these graphs as edge-maximal rainbow $H$-free graphs. (We note that here we make no restrictions on the colorings of $G+xy$ whatsoever, except that they are proper colorings. They may use any number of colors, and need not be extensions of the original rainbow $H$-free coloring of $G$.) With this framework in place, we define the rainbow saturation number and rainbow extremal number to be the largest and smallest number of edges, respectively, among all $n$ vertex rainbow $H$-saturated graphs. The latter of these was defined by Keevash, Mubayi, Sudakov, and Verstraëte in 2007; the former was introduced by B., Johnston, and Rombach in 2019. In this talk, we discuss recent progress on both the rainbow saturation numbers and rainbow extremal numbers. We also give several broad generalizations of these concepts and discuss many open problems. This talk contains joint work with Vic Bednar (Furman), Dan Johnston (Trinity College, CT), and Puck Rombach (Vermont).

2024-09-03 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Oriented trees in $O(k \sqrt{k})$-chromatic digraphs, a subquadratic bound for Burr’s conjecture

by Amadeus Reinald(LIRMM, Université de Montpellier, CNRS)

In 1980, Burr conjectured that every directed graph with chromatic number $2k-2$ contains any oriented tree of order $k$ as a subdigraph. Burr showed that chromatic number $(k-1)^2$ suffices, which was improved in 2013 to $\frac{k^2}{2} – \frac{k}{2} + 1$ by Addario-Berry et al. In this talk, we give the first subquadratic bound for Burr’s conjecture, by showing that every directed graph with chromatic number $8\sqrt{\frac{2}{15}} k \sqrt{k} + O(k)$ contains any oriented tree of order $k$. Moreover, we provide improved bounds of $\sqrt{\frac{4}{3}} k \sqrt{k}+O(k)$ for arborescences, and $(b-1)(k-3)+3$ for paths on $b$ blocks, with $b\ge 2$.

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