In recent years, ``stealthy'' particle systems have gained considerable attention in condensed matter physics. These are particle systems for which the diffraction spectrum or structure function (i.e. the Fourier transform of the truncated pair correlation function) vanishes in a neighbourhood of the origin in the wave space. These systems are believed to exhibit the phenomenon of ``cloaking'', i.e. being invisible to probes of certain frequencies. They also exhibit the phenomenon of hyperuniformity, namely suppressed fluctuations of particle counts, a property that has been shown to arise in a wide array of settings in chemistry, physics and biology. We will demonstrate that stealthy particle systems (and their natural extensions to stealthy stochastic processes) exhibit a highly rigid structure; in particular, their entropy per unit volume is degenerate, and any spatial void in such a system cannot exceed a certain size. Time permitting, we will also discuss the intriguing correlation geometry of such systems and its interplay with the analytical
properties of their diffraction spectrum. Based on joint works with Joel Lebowitz and Kartick Adhikari.
Hamiltonian time-series data are observations derived from a Hamiltonian dynamical system. Our goal is to analyze the time-series data using the topological information of Hamiltonian dynamical systems. Exact Multi-parameter Persistent Homology is one aspect of this analysis, in this case, the Hamiltonian system is composed of uncoupled one-dimensional harmonic oscillators. This is a very simple model. However, we can induce the exact persistence barcode formula from it. From this formula, we can obtain a calculable and interpretable analysis. Filtration is necessary to extract the topological information of data and to define persistent homology. However, in many cases, we use static filtrations, such as the Vietoris-Rips filtration. My ongoing research is on topological optimization, which involves finding a filtration in Exact Multi-parameter Persistent Homology that minimizes the cross-entropy loss function for the classification of time-series data.
2024-04-26 / 14:00 ~ 16:00
학과 세미나/콜로퀴엄 - 기타: Introduction to complex algebraic geometry and Hodge theory #4
by 김재홍(KAIST)
This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.
In this presentation, we discuss comprehensive frequency domain methods for estimating and inferring the second-order structure of spatial point processes. The main element here is on utilizing the discrete Fourier transform (DFT) of the point pattern and its tapered counterpart. Under second-order stationarity, we show that both the DFTs and the tapered DFTs are asymptotically jointly independent Gaussian even when the DFTs share the same limiting frequencies. Based on these results, we establish an α-mixing central limit theorem for a statistic formulated as a quadratic form of the tapered DFT. As applications, we derive the asymptotic distribution of the kernel spectral density estimator and establish a frequency domain inferential method for parametric stationary point processes. For the latter, the resulting model parameter estimator is computationally tractable and yields meaningful interpretations even in the case of model misspecification. We investigate the finite sample performance of our estimator through simulations, considering scenarios of both correctly specified and misspecified models. Joint work with Yongtao Guan @CUHK-Shenzhen.
In nonstationary bandit learning problems, the decision-maker must continually gather information and adapt their action selection as the latent state of the environment evolves. In each time period, some latent optimal action maximizes expected reward under the environment state. We view the optimal action sequence as a stochastic process, and take an information-theoretic approach to analyze attainable performance. We bound per-period regret in terms of the entropy rate of the optimal action process. The bound applies to a wide array of problems studied in the literature and reflects the problem’s information structure through its information-ratio.
We say that two functors Λ and Γ between thin categories of relational structures are adjoint if for all structures A and B, we have that Λ(A) maps homomorphically to B if and only if A maps homomorphically to Γ(B). If this is the case Λ is called the left adjoint to Γ and Γ the right adjoint to Λ. In 2015, Foniok and Tardif described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr in 1970. We shall present several recent advances in this direction including a new approach based on the notion of Datalog Program borrowed from logic.
"An improved rhythmicity analysis method using Gaussian Processes detects cell-density dependent circadian oscillations in stem cells", ArXiv. (2023) will be discussed in this Journal Club. Detecting oscillations in time series remains a challenging problem even after decades of research. In chronobiology, rhythms in time series (for instance gene expression, eclosion, egg-laying and feeding) datasets tend to be low amplitude, display large variations amongst replicates, and often exhibit varying peak-to-peak distances (non-stationarity). Most currently available rhythm detection methods are not specifically designed to handle such datasets. Here we introduce a new method, ODeGP (Oscillation Detection using Gaussian Processes), which combines Gaussian Process (GP) regression with Bayesian inference to provide a flexible approach to the problem. Besides naturally incorporating measurement errors and non-uniformly sampled data, ODeGP uses a recently developed kernel to improve detection of non-stationary waveforms. An additional advantage is that by using Bayes factors instead of p-values, ODeGP models both the null (non-rhythmic) and the alternative (rhythmic) hypotheses. Using a variety of synthetic datasets we first demonstrate that ODeGP almost always outperforms eight commonly used methods in detecting stationary as well as non-stationary oscillations. Next, on analyzing existing qPCR datasets that exhibit low amplitude and noisy oscillations, we demonstrate that our method is more sensitive compared to the existing methods at detecting weak oscillations. Finally, we generate new qPCR time-series datasets on pluripotent mouse embryonic stem cells, which are expected to exhibit no oscillations of the core circadian clock genes. Surprisingly, we discover using ODeGP that increasing cell density can result in the rapid generation of oscillations in the Bmal1 gene, thus highlighting our method’s ability to discover unexpected patterns. In its current implementation, ODeGP (available as an R package) is meant only for analyzing single or a few time-trajectories, not genome-wide datasets. If you want to participate in the seminar, you need to enter IBS builiding (https://www.ibs.re.kr/bimag/visiting/). Please contact if you first come IBS to get permission to enter IBS building.
2024-04-25 / 11:50 ~ 12:40
대학원생 세미나 - 대학원생 세미나: Long-time behavior of viscous-dispersive shock for the Navier-Stokes-Korteweg equations.
