There has been a raising interest on the study of perfect matchings in uniform hypergraphs in the past two decades, including extremal problems and their algorithmic versions. I will introduce the problems and some recent developments.

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.

We prove the existence of a computable function $f\colon\mathbb{N}\to\mathbb{N}$ such that for every integer $k$ and every digraph $D$ either contains a collection $\mathcal{C}$ of $k$ directed cycles of even length such that no vertex of $D$ belongs to more than four cycles in $\mathcal{C}$, or there exists a set $S\subseteq V(D)$ of size at most $f(k)$ such that $D-S$ has no directed cycle of even length. This is joint work with Maximilian Gorsky, Ken-ichi Kawarabayashi, and Stephan Kreutzer.

Dyson Brownian motion, the eigenvalues of matrix-valued Brownian motion, has become the most standard and well-established approach to universalities for local (i.e. microscopic) eigenvalue statistics of Hermitian random matrices. When combined with a noble characteristic flow method, it can also help study the eigenvalue statistics on a mesoscopic scale. In this talk, we demonstrate this mechanism via yet another simplification of the proof of local laws for Wigner matrices and discuss some generalities.

The uniform Turán density $\pi_u(F)$ of a hypergraph $F$, introduced by Erdős and Sós, is the smallest value of $d$ such that any hypergraph $H$ where all linear-sized subsets of vertices of $H$ have density greater than $d$ contains $F$ as a subgraph. Over the past few years the value of $\pi_u(F)$ was determined for several classes of 3-graphs, but no nonzero value of $\pi_u(F)$ has been found for $r$-graphs with $r>3$. In this talk we show the existence of $r$-graphs $F$ with $\pi_u(F)={r \choose 2}^{-{r \choose 2}}$, which we conjecture is minimum possible. Joint work with Frederik Garbe, Daniel Il’kovic, Dan Král’ and Filip Kučerák.

Cell-to-cell variability in gene expression exists even in a homogeneous population of cells. Dissecting such cellular heterogeneity within a biological system is a prerequisite for understanding how a biological system is developed, homeostatically regulated, and responds to external perturbations. Single-cell RNA sequencing (scRNA-seq) allows the quantitative and unbiased characterization of cellular heterogeneity by providing genome-wide molecular profiles from tens of thousands of individual cells. Single-cell sequencing is expanding to combine genomic, epigenomic, and transcriptomic features with environmental cues from the same single cell. In this talk, I demonstrate how scRNA-seq can be applied to dissect cellular heterogeneity and plasticity of adipose tissue, and discuss related computational challenges.

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.

For any finite point set $P \subset \mathbb{R}^d$, we denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, \alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P \subset \mathbb{R}^d$ in general position, satisfying $\text{diam}(P) < \alpha\sqrt[d]{n}$ (informally speaking, `non-elongated'), contains a convex $c$-polytope. Valtr proved that $c_{2, \alpha}(n) \approx \sqrt[3]{n}$, which is asymptotically tight in the plane. We generalize the results by establishing $c_{d, \alpha}(n) \approx n^{\frac{d-1}{d+1}}$. Along the way we generalize the definitions and analysis of convex cups and caps to higher dimensions, which may be of independent interest. Joint work with Boris Bukh.

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+.

Melchior’s Inequality (1941) implies that, in a rank-3 real-representable matroid, the average number of points in a line is less than three. This was extended to the complex-representable matroids by Hirzebruch in 1983 with the slightly weaker bound of four. In this talk, we discuss and sketch the proof of the recent result that, in a rank-4 complex-representable matroid which is not the direct-sum of two lines, the average number of points in a plane is bounded above by an absolute constant. Consequently, the average number of points in a flat in a rank-4 complex-representable matroid is bounded above by an absolute constant. Extensions of these results to higher dimensions will also be discussed. In particular, the following quantities are bounded in terms of k and r respectively: the average number of points in a rank-k flat in a complex-representable matroid of rank at least 2k-1, and the average number of points in a flat in a rank-r complex-representable matroid. Our techniques rely on a theorem of Ben Lund which approximates the number of flats of a given rank. This talk is based on joint work with Rutger Campbell and Jim Geelen.

The Dedekind's Problem asks the number of monotone Boolean functions, a(n), on n variables. Equivalently, a(n) is the number of antichains in the n-dimensional Boolean lattice $[2]^n$. While the exact formula for the Dedekind number a(n) is still unknown, its asymptotic formula has been well-studied. Since any subsets of a middle layer of the Boolean lattice is an antichain, the logarithm of a(n) is trivially bounded below by the size of the middle layer. In the 1960's, Kleitman proved that this trivial lower bound is optimal in the logarithmic scale, and the actual asymptotics was also proved by Korshunov in 1980’s. In this talk, we will discuss recent developments on some variants of Dedekind's Problem. Based on joint works with Matthew Jenssen, Alex Malekshahian, Michail Sarantis, and Prasad Tetali.

We present the KKM theorem and a recent proof method utilizing it that has proven to be very useful for problems in discrete geometry. For example, the method was used to show that for a planar family of convex sets with the property that every three sets are pierced by a line, there are three lines whose union intersects each set in the family. This was previously a long-unsolved problem posed by Eckhoff. We go over a couple of examples demonstrating the method and propose a potential future research direction to push the method even further.