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.

In this talk, we derive second-order expressions for both the one- and two-particle reduced density matrices of the Gibbs state at fixed positive temperatures. We consider a translation-invariant system of N bosons in a three-dimensional torus. These bosons interact through a repulsive two-body potential with a scattering length of order 1/N in the large N limit. This analysis provides a justification of Bogoliubov's prediction regarding the fluctuations around the condensate. The talk will primarily introduce basic concepts and settings, ensuring accessibility for all attendees. This work is a joint effort with Christian Brennecke and Phan Thành Nam.

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.

A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that \[r_{K_3}(K_t)=\binom{r(K_t)}3\] for all sufficiently large $t$.  Our proof involves a combination of results on the chromatic number of triangle-sparse graphs. Joint work with Jacob Fox and Jonathan Tidor.

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심사위원장: 박철우, 심사위원: 정연승, 전현호, 안정연(산업및시스템공학과), 전용호(연세대학교)

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).

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ID: 853 0775 9189, PW: 342420

In this talk, I will present the recent progress of understanding adversarial multiclass classification problems, motivated by the empirical observation of the sensitivity of neural networks by small adversarial attacks. Based on 'distributional robust optimization' framework, we obtain reformulations of adversarial training problem: 'generalized barycenter problem' and a family of multimarginal optimal transport problems. These new theoretical results reveal a rich geometric structure of adversarial training problems in multiclass classification and extend recent results restricted to the binary classification setting. From this optimal transport perspective understanding, we prove the existence of robust classifiers by using the duality of the reformulations without so-called 'universal sigma algebra'. Furthermore, based on these optimal transport reformulations, we provide two efficient approximate methods which provide a lower bound of the optimal adversarial risk. The basic idea is the truncation of effective interactions between classes: with small adversarial budget, high-order interactions(high-order barycenters) disappear, which helps consider only lower order tensor computations.

Given a set of lines in $\mathbb R^d$, a joint is a point contained in d linearly independent lines. Guth and Katz showed that N lines can determine at most $O(N^{3/2})$ joints in $\mathbb R^3$ via the polynomial method. Yu and I proved a tight bound on this problem, which also solves a conjecture proposed by Bollobás and Eccles on the partial shadow problem. It is surprising to us that the only known proof of this purely extremal graph theoretic problem uses incidence geometry and the polynomial method.

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.

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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.

TBD

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234) + Google Map

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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심사위원장: 김동수, 심사위원:안드레아스 홈슨, 엄상일, 이주영(전산학부), 서승현(강원대학교)

We investigate the global existence and optimal time decay rate of solution to the one dimensional (1D) two-phase flow described by compressible Euler equations coupled with compressible Navier-Stokes equations through the relaxation drag force on the momentum equations (Euler-Navier-Stokes system). First, we prove the global existence of strong solution and the stability of the constant equilibrium state to 1D Cauchy problem of compressible Euler-Navier-Stokes system by using the standard continuity argument for small $H^{1}$ data while its second order derivative can be large. Then we derive the optimal time decay rate to the constant equilibrium state. Compared with multi-dimensional case, it is much harder to get optimal time decay rate by direct spectrum method due to a slower convergence rate of the fundamental solution in 1D case. To overcome this main difficulty, we need to first carry out time-weighted energy estimates (not optimal) for higher order derivatives, and based on these time-weighted estimates, we can close a priori assumptions and get the optimal time decay rate by spectrum analysis method. Moreover, due to non-conserved form and insufficient decay rate of the coupled drag force terms between the two-phase flows, we essentially need to use momentum variables $(m= \rho u, M=n\omega)$, rather than velocity variables $(u, \omega)$ in the spectrum analysis, to fully cancel out those non-conserved and insufficiently time-decay drag force terms. Finally, we study the singularity formation of the two-phase flow. We consider the blow-up of Euler equations in Euler-Navier-Stokes system. For Euler equations, we use Riemann invariants to construct decoupled Riccati type ordinary differential equations for smooth solutions and provide some sufficient conditions under which the classical solutions must break down in finite time.

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

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Lecture to be recorded by KAI-X

In this talk, I will discuss how the fundamental concepts in probability theory—the law of large numbers, the central limit theorem, and the large deviation principle—are developed in the study of real eigenvalues of asymmetric random matrices.

The Nagata Conjecture governs the minimal degree required for a plane algebraic curve to pass through a collection of $r$ general points in the projective plane $P^2$ with prescribed multiplicities. The "SHGH" Conjecture governs the dimension of the linear space of these polynomials. We formulate transcendental versions of these conjectures in term of pluripotential theory and we're making some progress.

For $d\ge 2$ and an odd prime power $q$, let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb{F}_q$. The distance between two points $(x_1,\ldots,x_d)$ and $(y_1,\ldots,y_d)$ is defined to be $\sum_{i=1}^d (x_i-y_i)^2$. An influential result of Iosevich and Rudnev is: if $E \subset \mathbb{F}_q^d$ is sufficiently large and $t \in \mathbb{F}_q$, then there are a pair of points $x,y \in E$ such that the distance between $x$ and $y$ is $t$. Subsequent works considered embedding graphs of distances, rather than a single distance. I will discuss joint work with Debsoumya Chakraborti, in which we show that every sufficiently large subset of $\mathbb{F}_q^d$ contains every nearly spanning tree of distances with bounded degree in each distance. The main novelty in this result is that the distance graphs we find are nearly as large as the set $S$ itself, but even for smaller distance trees our work leads to quantitative improvements to previously known bounds. A key ingredient in our proof is a new colorful generalization of a classical result of Haxell on finding nearly spanning bounded-degree trees in expander graphs. This is joint work with Debsoumya Chakraborti.

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.