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.
The r-th cactus variety of a subvariety X in a projective space generalises secant variety of X and it is defined using linear spans of finite schemes of degree r. It's original purpose was to study the vanishing sets of catalecticant minors. We propose adding a scheme structure to the cactus variety and we define it via relative linear spans of families of finite schemes over a potentially non-reduced base. In this way we are able to study the vanishing scheme of the catalecticant minors. For X which is a sufficiently large Veronese reembedding of projective variety, we show that r-th cactus scheme and the zero scheme of appropriate catalecticant minors agree on an open and dense subset which is the complement of the (r-1)-st cactus variety/scheme. As an application, we can describe the singular locus of (in particular) secant varieties to high degree Veronese varieties.
Based on a joint work with Hanieh Keneshlou.
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.
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.
"Phenotypic switching in gene regulatory networks", PNAS. (2014) will be discussed in this Journal Club. Noise in gene expression can lead to reversible phenotypic switching. Several experimental studies have shown that the abundance distributions of proteins in a population of isogenic cells may display multiple distinct maxima. Each of these maxima may be associated with a subpopulation of a particular phenotype, the quantification of which is important for understanding cellular decision-making. Here, we devise a methodology which allows us to quantify multimodal gene expression distributions and single-cell power spectra in gene regulatory networks. Extending the commonly used linear noise approximation, we rigorously show that, in the limit of slow promoter dynamics, these distributions can be systematically approximated as a mixture of Gaussian components in a wide class of networks. The resulting closed-form approximation provides a practical tool for studying complex nonlinear gene regulatory networks that have thus far been amenable only to stochastic simulation. We demonstrate the applicability of our approach in a number of genetic networks, uncovering previously unidentified dynamical characteristics associated with phenotypic switching. Specifically, we elucidate how the interplay of transcriptional and translational regulation can be exploited to control the multimodality of gene expression distributions in two-promoter networks. We demonstrate how phenotypic switching leads to birhythmical expression in a genetic oscillator, and to hysteresis in phenotypic induction, thus highlighting the ability of regulatory networks to retain memory. 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.
