정보 이론의 주요 관심사 중 하나는 통신 과정에서 오류가 발생할 확률을 최소화하는 것이다. 예를 들어 PCR 검사 결과 음성일 경우 0으로 코드화하고 양성일 경우 1로 코드화한다고 하였을 때, 이 중요한 정보가 통신 상황에서 오류가 발생하여 0이 1로 잘못 전달되거나 1이 0으로 잘못 전달되는 경우가 발생할 수 있다. 만약 오류 발생 확률이 10%라면 적절한 방법을 동원하여 오류 발생 확률을 3% 혹은 1% 등으로 줄이기 위해 노력하는 것이 자연스럽다. 강연 전반부의 목표는 주어진 자원의 어느 정도를 오류 정정에 사용하는 것이 가장 효율적일지를 다루는 샤논 채널 코딩 정리의 의미를 이해하는 것이다. 그리고 강연 후반부의 목표는 최근 큰 주목을 받고 있는 양자 정보 이론 분야에서 2000년대 초반 확립된 코딩 정리의 의미를 파악하고, 이와 관련한 수학적 난제를 소개하는 것이다.

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.

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+.