Thursday, September 7, 2023

2023-09-14 / 11:00 ~ 12:00

by 이명수()

In this talk, we discuss the Neural Tangent Kernel. The NTK is closely related to the dynamics of the neural network during training via the Gradient Flow(or Gradient Descent). But, since the NTK is random at initialization and varies during training, it is quite delicate to understand the dynamics of the neural network. In relation to this issue, we introduce an interesting result: in the infinite-width limit, the NTK converge to a deterministic kernel at initialization and remains constant during training. We provide a brief proof of the result for the simplest case.

2023-09-14 / 14:30 ~ 15:45

학과 세미나/콜로퀴엄 - 기타:

by ()

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

2023-09-11 / 16:00 ~ 17:00

학과 세미나/콜로퀴엄 - 위상수학 세미나:

by 최인혁(고등과학원)

In the past decades, there has been considerable progress in the theory of random walks on groups acting on hyperbolic spaces. Despite the abundance of such groups, this theory is inherently not preserved under quasi-isometry. In this talk, I will present our study of random walks on groups that satisfy a certain QI-invariant property that does not refer to an action on hyperbolic spaces. Joint work with Kunal Chawla, Kasra Rafi, and Vivian He.

2023-09-08 / 14:00 ~ 16:00

IBS-KAIST 세미나 - 수리생물학:

by ()

Background Microbial community simulations using genome scale metabolic networks (GSMs) are relevant for many application areas, such as the analysis of the human microbiome. Such simulations rely on assumptions about the culturing environment, affecting if the culture may reach a metabolically stationary state with constant microbial concentrations. They also require assumptions on decision making by the microbes: metabolic strategies can be in the interest of individual community members or of the whole community. However, the impact of such common assumptions on community simulation results has not been investigated systematically. Results Here, we investigate four combinations of assumptions, elucidate how they are applied in literature, provide novel mathematical formulations for their simulation, and show how the resulting predictions differ qualitatively. Our results stress that different assumption combinations give qualitatively different predictions on microbial coexistence by differential substrate utilization. This fundamental mechanism is critically under explored in the steady state GSM literature with its strong focus on coexistence states due to crossfeeding (division of labor). Furthermore, investigating a realistic synthetic community, where the two involved strains exhibit no growth in isolation, but grow as a community, we predict multiple modes of cooperation, even without an explicit cooperation mechanism. Conclusions Steady state GSM modelling of microbial communities relies both on assumed decision making principles and environmental assumptions. In principle, dynamic flux balance analysis addresses both. In practice, our methods that address the steady state directly may be preferable, especially if the community is expected to display multiple steady states.

2023-09-07 / 11:50 ~ 12:30

대학원생 세미나 - 대학원생 세미나: Calculus of Variations in Materials Sciences

by 서준영(KAIST)

This talk aims to explore the application of calculus of variations in materials sciences. We will discuss the physics behind solid-solid phase transitions and elastic energy. Then the Allen-Cahn model in studying interfaces and their energy will be introduced. Finally, we will examine the Ohta-Kawasaki model and its role in understanding self-assembly in block copolymers. Recent research advancements in the Ohta-Kawasaki problem will also be presented.

2023-09-14 / 16:15 ~ 17:15

학과 세미나/콜로퀴엄 - 콜로퀴엄:

by ()

Questions of parameter estimation – that is, finding the parameter values that allow a model to best fit some data – and parameter identifiability – that is, the uniqueness of such parameter values – are often considered in settings where experiments can be repeated to gain more certainty about the data. In this talk, however, I will consider parameter estimation and parameter identifiability in situations where data can only be collected from a single experiment or trajectory. Our motivation comes from medical settings, where data comes from a patient; such limitations in data also arise in finance, ecology, and climate, for example. In this setting, we can try to find the best parameters to fit our limited data. In this talk, I will introduce a novel, alternative goal, which we refer to as a qualitative inverse problem. The aim here is to analyze what information we can gain about a system from the available data even if we cannot estimate its parameter values precisely. I will discuss results that allow us to determine whether a given model has the ability to fit the data, whether its parameters are identifiable, the signs of model parameters, and/or the local dynamics around system fixed points, as well as how much measurement error can be tolerated without changing the conclusions of our analysis. I will consider various classes of model systems and will illustrate our latest results with the classic Lotka-Volterra system.

2023-09-07 / 16:30 ~ 17:30

학과 세미나/콜로퀴엄 - 콜로퀴엄:

by ()

Hénon maps were introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model. They are among the most studied discrete-time dynamical systems that exhibit chaotic behavior. Complex Hénon maps in any dimension have been extensively studied over the last three decades, in parallel with the development of pluripotential theory. We will present the dynamical properties of these maps such as the behavior of point orbits, variety orbits, equidistribution of periodic points and fine ergodic properties of the systems. This talk is based on the work of Bedford, Fornaess, Lyubich, Sibony, Smillie, and on recent work of the speaker in collaboration with Bianchi and Sibony.

2023-09-12 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: The square of every subcubic planar graph of girth at least 6 is 7-choosable

by 김석진(건국대)

The square of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Wegner's conjecture (1977) states that for a planar graph $G$, the chromatic number $\chi(G^2)$ of $G^2$ is at most 7 if $\Delta(G) = 3$, at most $\Delta(G)+5$ if $4 \leq \Delta(G) \leq 7$, and at most $\lfloor \frac{3 \Delta(G)}{2} \rfloor$ if $\Delta(G) \geq 8$. Wegner's conjecture is still wide open. The only case for which we know tight bound is when $\Delta(G) = 3$. Thomassen (2018) showed that $\chi(G^2) \leq 7$ if $G$ is a planar graph with $\Delta(G) = 3$, which implies that Wegner's conjecture is true for $\Delta(G) = 3$. A natural question is whether $\chi_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph, where $\chi_{\ell}(G^2)$ is the list chromatic number of $G^2$. Cranston and Kim (2008) showed that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 7. We prove that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. This is joint work with Xiaopan Lian (Nankai University).

2023-09-08 / 16:00 ~ 17:00

학과 세미나/콜로퀴엄 - 대수기하학: Kummer Rigidity on K3 surfaces and its Hyperkähler Generalization

by 장승욱(Université de Rennes 1)

In this talk, we discuss some concepts that are used to study (hyperbolic) holomorphic dynamics on K3 surfaces. These concepts include Green currents, their laminations, and Green measures, which emerge as the natural choice for measuring maximal entropy. These tools effectively establish Kummer rigidity – that is, when the Green measure is absolutely continuous to the volume measure, the surface is Kummer, and the dynamics is linear. We provide an overview of the techniques employed to establish this principle and provide a glimpse into their extension within the hyperkähler context – one of the higher-dimensional analogues of K3 surfaces.

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