Woojin Kim

Aggregation of the Alzheimer's peptide Abeta produces toxic multimeric species that play a key role in the development of Alzheimer's disease. Compounds that inhibit this aggregation may prove useful as therapeutic agents for the prevention or treatment of Alzheimer's disease. Although aggregation inhibitors may already exist in combinatorial libraries, finding these compounds in a cost-effective high-throughput manner poses an enormous challenge. To meet this challenge, we have developed a nove

A cluster tool consists of several single-wafer processing chambers and a wafer handling robot. A wafer has to wait within a chamber after being processed there until it is unloaded by the robot. Such wafer delays may cause wafer quality degradation or variability due to residual gases and heat in the chamber. The tool operation schedule has to maintain identical timing patterns or schedules for each cycle so as to keep wafer delays constant for every wafer. However, at the beginning of the tool

When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups. Motivated by this, we study the problem of obtaining persistent homology based summaries of time-dependent metric data. Given a finite dynamic metric space (DMS), we construct the zigzag simplicial filtration arising from applying the Rips simplicial complex construction (with a fixed scale par

When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups. In a similar vein, studying the dynamics of social networks leads to the problem of characterizing groups/communities as they form and disperse throughout time. Motivated by this, we study the problem of obtaining persistent homology based summaries of time-dependent data. Given a finite dyna

In topological data analysis, the shape of a dataset is often encoded into a system of vector spaces and linear maps over a partially ordered set (poset). We give an overview of how summaries of such systems can be constructed by using ideas from combinatorics.

In this paper, we study task assignment strategy for multi-agent systems in which the task space grows sequentially. Many task assignment algorithms are based on the heuristic schemes that result in optimal performance, but in time-varying conditions, real-time assignment techniques are needed. In order to maximize the real-time performance, an auction algorithm is adopted and modified with D* lite algorithm to handle various events and changes of environments. The numerical results show that th

Commutative diagrams of vector spaces and linear maps over $\mathbb{Z}^2$ are objects of interest in topological data analysis (TDA) where this type of diagrams are called 2-parameter persistence modules. Given that quiver representation theory tells us that such diagrams are of wild type, studying informative invariants of a 2-parameter persistence module $M$ is of central importance in TDA. One of such invariants is the generalized rank invariant, recently introduced by Kim and Mémoli. Via the

Abstract The notion of generalized rank in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. However, its efficient computation has not yet been studied in the literature. We show that the generalized rank over a finite interval I of a $$\textbf{Z}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math>

\textit{Formigrams} are a natural generalization of the notion of \textit{dendrograms}. This notion has recently been proposed as a signature for studying the evolution of clusters in dynamic datasets across different time scales. Although its formulation is set-theoretic, the notion of formigram is deeply related to certain algebraic-topological methods used in \textit{topological data analysis}, such as \textit{Reeb graphs} and \textit{zigzag persistence modules}. In this paper we give a self-

Identifying and representing clusters in time-varying network data is of particular importance when studying collective behaviors emerging in nature, in mobile device networks or in social networks. Based on combinatorial, categorical, and persistence theoretic viewpoints, we establish a stable functorial pipeline for the summarization of the evolution of clusters in a time-varying network. We first construct a complete summary of the evolution of clusters in a given time-varying network over a