# 학과 세미나 및 콜로퀴엄

Stagnation of flows is an interesting phenomenon in fluid mechanics. It induces many challenging problems in analysis. We first derive a Liouville type theorem for Poiseuille flows in the class of incompressible steady inviscid flows in an infinitely long strip, where the flows can have stagnation points. With the aid of this Liouville type theorem, we show the uniqueness of solutions with positive horizontal velocity for steady Euler system in a general nozzle when the flows tend to the horizontal velocity of Poiseuille flows at the upstream. Furthermore, this kind of flows are proved to exist in a large class of nozzles and we also prove the optimal regularity of boundary for the set of stagnation points. Finally, we give a classification of incompressible Euler flows via the set of flow angles.

In the past couple of decades, mathematical fluid dynamics has been highlighted by numerous constructions of solutions to fluid equations that exhibit pathological or wild behavior. These include the loss of the energy balance, non-uniqueness, singularity formation, and dissipation anomaly. Interesting from the mathematical point of view, providing counterexamples to various well-posedness results in supercritical spaces, such constructions are becoming more and more relevant from the physical point of view as well. Indeed, a fundamental physical property of turbulent flows is the existence of the energy cascade. Conjectured by Kolmogorov, it has been observed both experimentally and numerically, but had been difficult to produce analytically. In this talk I will overview new developments in discovering not only pathological mathematically, but also physically realistic solutions of fluid equations.

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기초과학연구원 세미나실(B378)
수리생물학
김준일 (숭실대학교)
TENET+: a tool for reconstructing gene networks by integrating single cell expression and chromatin accessibility data

기초과학연구원 세미나실(B378)

수리생물학

Reconstruction of gene regulatory networks (GRNs) is a powerful approach to capture a prioritized gene set controlling cellular processes. In our previous study, we developed TENET a GRN reconstructor from single cell RNA sequencing (scRNAseq). TENET has a superior capability to identify key regulators compared with other algorithms. However, accurate inference of gene regulation is still challenging. Here, we suggest an integrative strategy called TENET+ by combining single cell transcriptome and chromatin accessibility data. TENET+ predicts target genes and open chromatin regions associated with transcription factors (TFs) and links the target regions to their corresponding target gene. As a result, TENET+ can infer regulatory triplets of TF, target gene, and enhancer. By applying TENET+ to a paired scRNAseq and scATACseq dataset of human peripheral blood mononuclear cells, we found critical regulators and their epigenetic regulations for the differentiations of CD4 T cells, CD8 T cells, B cells and monocytes. Interestingly, not only did TENET+ predict several top regulators of each cell type which were not predicted by the motif-based tool SCENIC, but we also found that TENET+ outperformed SCENIC in prioritizing critical regulators by using a cell type associated gene list. Furthermore, utilizing and modeling regulatory triplets, we can infer a comprehensive epigenetic GRN. In sum, TENET+ is a tool predicting epigenetic gene regulatory programs for various types of datasets in an unbiased way, suggesting that novel epigenetic regulations can be identified by TENET+.

In this talk, I will describe a new approach to general relativistic initial data gluing based on explicit solution operators for the linearized constraint equation with prescribed support properties. In particular, we retrieve and optimize -- in terms of positivity, regularity, size and/or spatial decay requirements -- obstruction-free gluing originally put forth by Czimek-Rodnianski. Notably, our proof of the strengthened obstruction-free gluing theorem relies on purely spacelike techniques, rather than null gluing as in the original approach.

In this talk, we will discuss nonlocal elliptic and parabolic equations on C^{1,τ} open sets in weighted Sobolev spaces, where τ ∈ (0, 1). The operators we consider are infinitesimal generators of symmetric stable Levy processes, whose Levy measures are allowed to be very singular. Additionally, for parabolic equations, the measures are assumed to be merely measurable in the time variable. This talk is based on a joint work with Hongjie Dong (Brown University).

ID: 853 0775 9189, PW: 342420

ID: 853 0775 9189, PW: 342420

Abstract: In 1993, Demeyer and Ford computed the Brauer group of a smooth toric variety over an algebraically closed field of
characteristic zero. One may pose the same question to the toric varieties over any field of positive characteristic. Another
interesting question is what will happen if we replace the base field by a discrete valuation ring, thereby replacing smooth toric varieties by smooth toric schemes over a discrete valuation ring in the sense of Kempf-Knudsen-Mumford-Saint-Donat. In this talk. I am going to discuss the answers to these questions. This is joint work with Roy Joshua.

Zoom info: meeting ID is 352 730 6970 with the password 1778. It will be open about 10-15 minutes before the scheduled talk. The talk time is in Korean Standard Time.

Zoom info: meeting ID is 352 730 6970 with the password 1778. It will be open about 10-15 minutes before the scheduled talk. The talk time is in Korean Standard Time.

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산업경영학동(E2-1) 세미나실 (2216)
응용 및 계산수학 세미나
최우철 (성균관대학교 수학과)
Distributed optimization: Theory of algorithms and applications

산업경영학동(E2-1) 세미나실 (2216)

응용 및 계산수학 세미나

Distributed optimization is a concept that multi-agent systems find a minimal point of a global cost functions which is a sum of local cost functions known to the agents. It appears in diverse fields of applications such as federated learning for machine learning problems and the multi-robotics systems. In this talk, I will introduce motivations for distributed optimization and related algorithms with their theoretical issues for developing efficient and robust algorithms.

We prove that the zero function is the only solution to a certain degenerate PDE defined in the upper half-plane under some geometric assumptions. This result implies that the Euclidean metric is the only adapted compactification of the standard half-plane model of hyperbolic space when the scalar curvature of the compactified metric has a certain sign. These Liouville-type theorems are expected to handle the boundary curvature blow-up to prove compactness results of CCE(conformally compact Einstein) manifolds with positive scalar curvature on the conformal infinity.

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산업경영학동(E2-1) 세미나실 (2216)
응용 및 계산수학 세미나
권도현 (서울시립대학교 수학과)
Applications of De Giorgi\'s Minimizing Movements and Optimal Transport

산업경영학동(E2-1) 세미나실 (2216)

응용 및 계산수학 세미나

The study of gradient flows has been extensive in the fields of partial differential equations, optimization, and machine learning. In this talk, we aim to explore the relationship between gradient flows and their discretized formulations, known as De Giorgi's minimizing movements, in various spaces. Our discussion begins with examining the backward Euler method in Euclidean space, and mean curvature flow in the space of sets. Then, we investigate gradient flows in the space of probability measures equipped with the distance arising in the Monge-Kantorovich optimal transport problem. Subsequently, we provide a theoretical understanding of score-based generative models, demonstrating their convergence in the Wasserstein distance.